下载文档原格式
a r X i v :h e p -t h /0005003v 3 8 D e c 2007.hep-th/0005003A Black Hole Farey Tail Robbert Dijkgraaf,1Juan Maldacena,2Gregory Moore,3and Erik Verlinde 41Departments of Physics and Mathematics University of Amsterdam,1018TV Amsterdam 2Department of Physics Harvard University,Cambridge,MA 021383Department of Physics and Astronomy,Rutgers University,Piscataway,NJ 08855-08494Joseph Henry Laboratories,Princeton University,Princeton NJ 08544AbstractWe derive an exact expression for the Fourier coefficients of elliptic genera on Calabi-Yau manifolds which is well-suited to studying the AdS/CFT correspondence on AdS 3×S 3.The expression also elucidates an SL (2,Z Z )invariant phase diagram for the D1/D5system involving deconfining transitions in the k →∞limit.1.Introduction and SummaryOne of the cornerstones of the AdS/CFT correspondence[1]is the relation between the partition function Z X of a superstring theory on AdS×X and the partition function Z C of a holographically related conformalfield theory C on the boundary∂(AdS).Roughly speaking we haveZ X∼Z C.(1.1) While the physical basis for this relationship is now well-understood,the precise mathe-matical formulation and meaning of this equation has not been very deeply explored.This relationship is hard to test since it is difficult to calculate both sides in the same region of parameter space.In this paper we consider the calculation of a protected supersymmetric partition function and we will give an example of a precise and exact version of(1.1).In particular,we will focus on the example of the duality between the IIB string theory on AdS3×S3×K3(arising,say,from the near horizon limit of(Q1,Q5)(D1,D5)branes)and the dual conformalfield theory with target space Hilb k(K3),the Hilbert scheme of k=Q1Q5points on K3.We will analyze the so called“elliptic genus”which is a super-symmetry protected quantity and can therefore be calculated at weak coupling producing a result that is independent of the coupling.We will rewrite it in a form which reflects very strongly the sum over geometries involved in the supergravity side.We will then give an application of this formula to the study of phase transitions in the D1D5system.Since the main formulae below are technically rather heavy we will,in this introduc-tion,explain the key mathematical results in a simplified setting,and then draw an analogy to the physics.The mathematical results are based on techniques from analytic number theory,and have their historical roots in the Hardy-Ramanujan formula for the partitions of an integer n[2].Let us consider a modular form forΓ:=SL(2,Z Z)of weight w<0with a q-expansion:f(τ)= n≥0F(n)q n+∆(1.2)where F(0)=0.In the physical context∆=−c/24,where c is the central charge of a conformalfield theory,w=−d/2if there are d noncompact bosons in the conformalfield theory,and F(0)is a ground-state degeneracy.It is well-known to string-theorists andnumber-theorists alike that the leading asymptotics of F(ℓ)for largeℓcan be obtained´a la Hardy-Ramanujan from a saddle-point approximation,and are given by:F(ℓ)∼12F(0)|∆|(1/4−w/2)(ℓ+∆)w/2−3/4exp 4π|n+∆|(w−1)/2F(n)··∞ c=11cc+d−1m1∂q 1−w f(1.7)We will call this the“fareytail transform.”Mathematically it is simply a special case of Serre duality,but the physical meaning should be clarified.(We comment on this below.) In any case,it is in terms of this new modular form that formulas look simple.One readily verifies that,for w integral,the fareytail transform takes a form of modular weight w to a form of modular weight2−w with Fourier coefficients˜F(n):=(n+∆)1−w F(n). Moreover,the transform takes a polar expression to a polar expression:Z−f=Z f−= n+∆<0˜F(n)q n+∆.(1.8)Notice that f and Z f contain the same information except for states with n+∆=0.Now, using a straightforward application of the Poisson summation formula(see appendix C), one can cast(1.4)into the form of an average over modular transformations:Z f(τ)= Γ∞\Γ(cτ+d)w−2Z−f(aτ+bQ∪∞:=2πi ǫ+i∞ǫ−i∞e2πβ(ℓ+∆) Z f(β)dβ(1.10) whereǫ→0+and we have introduced the“truncated sum”Z f(β):= (Γ∞\Γ/Γ∞)′(cτ+d)w−2Z−f(aτ+b2Actually,over two copies ofin (1.11)is a truncated versionofthat in (1.9).The prime in the notation (Γ∞\Γ/Γ∞)′means we omit the class of γ=1.Elements of the double-coset may be identified with the rational numbers −d/c between 0and 1.For further details see equations 2.7-2.10below.We may now describe the physical interpretation of the formulae (1.2)to (1.11).f (τ)will become a conformal field theory partition function.The fareytail transform Z f (τ)will be the dual supergravity “partition function.”The sum over the modular group in (1.9)will be a sum over solutions to supergravity.The fareytail transform is related to the truncated sum byZ f (τ)=Z −f (τ)+ℓ∈Z Z Z f (τ+ℓ).(1.12)In order to understand this relation,recall that in statistical mechanics a standard maneu-ver is to relate the canonical and microcanonical ensemble by an inverse Laplace transform:N (E )=14k (y∂y )2 3/2−w φ.(1.16)When w is integral we interpret this as a pseudodifferential operator:F T(φ)= ˜c(n,ℓ)q n yℓwith˜c(n,ℓ):=|n−ℓ2/4k|3/2−w c(n,ℓ).(1.17) Formal manipulations of pseudo-differential operators suggest that F T(φ)is a Jacobi form of weight3−w and the same index k.However,these formal manipulations lead to a false result,as pointed out to us by D.Zagier.Nevertheless,as we show in section four,it turns out that for n−ℓ2/4k>0the coefficients˜c(n,ℓ)can be obtained as Fourier coefficients of a truncated Poincar´e seriesˆZφdefined in equation(4.6)below.Our main result will be a formula for the fareytail transform of the elliptic genusχfor the Calabi-Yau manifold X=Hilb k(K3).The corresponding truncated Poincar´e series takes the form:Zχ(β,ω)=2π (Γ∞\Γ)01cτ+d Ψs(ωcτ+d)(1.18)The notation is explained in the following paragraph and a more precise version appears in equation(5.1)below.In sectionfive we interpret this formula physically.The sum (Γ∞\Γ)0is the sum over relatively prime pairs(c,d)with c>0(together with the pair (c,d)=(0,1).)We will interpret the average over(Γ∞\Γ)0as a sum over an SL(2,Z Z) family of black hole solutions of supergravity on AdS3×S3,related to the family discussed in[9].The sum over s is afinite sum over those particle states that do not cause black holes to form.They can be“added”to the black hole background,and the combined system has gravitational action D(s)exp(−2πi∆sτ)where D(s)is a degeneracy of states.Finally Ψs will be identified with a Chern-Simons wavefunction associated to AdS3supergravity.In fact,if we introduce the“reduced mass”L⊥0:=L0−14(1.19) then the polar part Z−χof the supergravity partition function can be considered as a sum over states for which L⊥0<0.The calculations of Cvetic and Larsen[10]show that the area of the horizon of the black hole and therefore its geometric entropy is precisely determined by a combination of the mass and(internal)angular momentum that is identical to L⊥0 (one has S BH=2π 4kn−ℓ2.)Therefore the truncation of the partition function to states with L⊥0<0describes a thermal gas of supersymmetric particles in an AdS background,truncated to those(ensembles of)particles that do not yet form blackholes.It is with this truncated partition function that contact has been made through supergravity computations[11].The sum over the quotient(Γ∞\Γ)0as in(1.18)adds in the black hole solutions.It is very suggestive that the full partition function can be obtained by taking the supergravity AdS thermal gas answer and making it modular invariant by explicitly averaging over the modular group.This sum has an interpretation as a sum over geometries and its seems to point to an application of a principle of spacetime modular invariance.The relation to Chern-Simons theory makes it particularly clear that in the AdS/CFT correspondence the supergravity partition function is to be regarded as a vector in a Hilbert space,rather like a conformal block.Indeed,the Chern-Simons interpretation of RCFT [12]is a precursor to the AdS/CFT correspondence.This subtlety in the interpretation of supergravity partition functions has also been noted in a different context in[13].The factor of(cτ+d)−3in(1.18)is perfect for the interpretation of Zχas a half-density with respect to the measure dz∧dτ.To be more precise,the wave-function is a section of a line-bundle L k⊗K where K is the holomorphic canonical line bundle over the moduli space,and k is the level of the CS-supergravity theory.The invariant norm on sections on the line-bundle L k⊗K is given byexp −4πk(Im z)2(3.22)that O3/2−wφis a Jacobi form of weight3−w and index k.This turns out to be false.In addition to this,it turns out that when one attempts to convert the truncated Poincar´e series Zφof section4to a full Poincar´e series one gets zero,and thus the series which one would expect to reproduce O3/2χin fact vanishes.As far as we are aware, the central formulae(4.5)and(4.6),which involve only the truncated Poincar´e series are nevertheless correct.Fortunately,the physical interpretation we subsequently explain is based on this truncated series,so our main conclusions are unchanged.A recent paper[14]has clarified somewhat the use of the Fareytail transform,and presented a regularized Poincar´e series forχrather than Zχ.2.The Generalized Rademacher ExpansionIn this section we give a rather general result on the asymptotics of vector-valued modular forms.It is a slight generalization of results of Rademacher[15].See also[7], ch.5,and[8].Let us suppose we have a“vector-valued nearly holomorphic modular form,”i.e.,a collection of functions fµ(τ)which form afinite-dimensional unitary representation of the modular group P SL(2,Z Z)of weight w.Under the standard generators we havefµ(τ+1)=e2πi∆µfµ(τ)(2.1)fµ(−1/τ)=(−iτ)w Sµνfν(τ)and in general we define:fµ(γ·τ):=(−i(cτ+d))w M(γ)µνfν(τ)γ= a b c d (2.2) where,for c>0we choose the principal branch of the logarithm.We assume the fµ(τ)have no singularities forτin the upper half plane,except at the cuspsWe will now state in several forms the convergent expansion that gives the Fourier coefficients of the modular forms in terms of data of the modular representation and the polar parts f−µ.We assume that w≤0.3Thefirst way to state the result isFν(n)= m+∆µ<0K n,ν;m,µFµ(m)(2.4)which holds for allν,n.The infinite×finite matrix K n,ν;m,µis an infinite sum over the rational numbers in lowest terms0≤−d/c<1:K n,ν;m,µ= 0≤−d/c<1K n,ν;m,µ(d,c)(2.5) and for each c,d we have:K n,ν;m,µ(d,c):=−i˜M(d,c)n,ν;m,µ1+i∞1−i∞dβ(βc)w−2exp 2π n+∆νβ−2π(m+∆µ)β (2.6)The matrix˜M(d,c)n,ν;m,µis essentially a modular transformation matrix and is defined (in equation(2.10)below)as follows.LetΓ∞be the subgroup of modular transformationsτ→τ+n.We may identify the rational numbers0≤−d/c<1with the nontrivial elements in the double-coset γ∈Γ∞\P SL(2,Z Z)/Γ∞so the sum on−d/c in(2.5)is more fundamentally the sum over nontrivial elements[γ]inΓ∞\P SL(2Z Z)/Γ∞.To be explicit,consider a matrixa b c d ∈SL(2,Z Z)(2.7)where c,d are relatively prime integers.Since1ℓ01 a b c d = a+ℓc b+ℓdc d .(2.8) the equivalence class inΓ∞\Γonly depends on c,d.When c=0we can take0≤−d/c<1 because: a b c d 1ℓ01 = a b+aℓc d+cℓ (2.9)shifts d by multiples of c.The term c=0corresponds to the class of[γ=1].It follows that˜M(d,c)n,ν;m,µ:=e2πi(n+∆ν)(d/c)M(γ)−1νµe2πi(m+∆µ)(a/c)(2.10) only depends on the class of[γ]∈Γ∞\P SL(2,Z Z)/Γ∞because of(2.8)(2.9).In such expressions where only the equivalence class matters we will sometimes writeγc,d.Our second formulation is based on the observation that the integral in(2.6)is essen-tially the standard Bessel function Iρ(z)with integral representation:Iρ(z)=(z2πi ǫ+i∞ǫ−i∞t−ρ−1e(t+z2/(4t))dt(2.11)for Re(ρ)>0,ǫ>0.This function has asymptotics:Iρ(z)∼(zΓ(ρ+1)z→0Iρ(z)∼ 2πz e z Re(z)→+∞(2.12) Thus,we can define˜I ρ(z):=(zcPutting these together we have the third formulation of the Rademacher expansion:Fν(n)=2π∞ c=1r µ=1c w−2Kℓ(n,ν,m,µ;c) m+∆µ<0Fµ(m)(2π|m+∆µ|)1−w˜I1−w 4π|m+∆µ|(n+∆ν) .(2.17) The function˜Iν(z)→1for z→0.The Kloosterman sum is trivially bounded by c,so we can immediately conclude that the sum converges for w<0.(In fact,by a deep result of A.Weil,the Kloosterman sum for the trivial representation of the modular group is bounded by c1/2.)From the proof in appendix B it follows that the series in fact converges to the value of the Fourier coefficient of the modular form.We will give two proofs of the above results in appendices B and C.Thefirst follows closely the method used by Rademacher[15][7].This proof is useful because it illustrates the role played by various modular domains in obtaining the expression and is closely related to the phase transitions discussed in section six below.The second proof,which is also rather elementary,but only applies for w integral,establishes a connection with another well-known formula in analytic number theory,namely Petersson’s formula for Fourier coefficients of Poincar´e series.3.Elliptic genera and Jacobi Forms3.1.Elliptic Genera and superconformalfield theoryThe elliptic genus for a(2,2)CFT is defined to be:χ(q,y):=Tr RR e2πiτ(L0−c/24)e2πi˜τ(˜L0−c/24)e2πizJ0(−1)F:= n≥0,r c(n,r)q n y r(3.1)The Ramond sector spectrum of J0,˜J0is integral forˆc even,and half-integral forˆc odd so(−1)F=exp[iπ(J0−˜J0)]is well-defined.In the path integral we are computing with worldsheet fermionic boundary conditions:e2πiz+⊗++.(3.2)We will encounter the elliptic genus for unitary(4,4)theories.These necessarily have ˆc=2k even integral and c=3ˆc=6k.We choose the N=2subalgebra so that J0=2J30 has integral spectrum.General properties of CFT together with representation theory of N =2supercon-formal theory show that χ(τ,z )satisfies the following identities.First,modular invariance leads to the transformation laws for γ∈SL (2,Z Z ):χ(aτ+bcτ+d )=e 2πik cz 2cτ+d ,z cτ+d φ(τ,z )a b c d ∈SL (2,Z Z )(3.5)φ(τ,z +ℓτ+m )=e −2πik (ℓ2τ+2ℓz )φ(τ,z )ℓ,m ∈Z Z (3.6)and has a Fourier expansion:φ(τ,z )=n ∈Z Z ,ℓ∈Z Z c (n,ℓ)q n y ℓ(3.7)where c (n,ℓ)=0unless 4nk −ℓ2≥0.Definition[18].A weak Jacobi form φ(τ,z )of weight w and index k satisfies the identities (3.5)(3.6)and has a Fourier expansion:φ(τ,z )=n ∈Z Z ,ℓ∈Z Zc (n,ℓ)q n y ℓ(3.8)where c (n,ℓ)=0unless n ≥0.The notion of weak Jacobi form is defined in [18],p.104.In physics we must use weak Jacobi forms and not Jacobi forms since L 0−c/24≥0in the Ramond sector of aunitary theory.In a unitary theory the U (1)charge |ℓ|≤14k ,ν)=c (n −νs 0−ks 20,ν)(3.10)Thus we obtain the key point,([18],Theorem 2.2),that the expansion coefficients of the elliptic genus as an expansion in two variables q,y are in fact given by:c (n,ℓ)=c ℓ(2n ˆc −ℓ2)(3.11)where c ℓ(j )is extended to all values ℓ=µmod ˆc by c ℓ(N )=(−1)ℓ−µc µ(N ).It follows that we can give a theta function decomposition to the function φ(τ,z )([18]Theorem 5.1):φ(τ,z )= −k +1≤ν<k n ∈Z Zc (n,ν)q n −ν2/4k θν,k (z,τ)(3.12)where the sum is over integral µfor 2k even and over half-integral µfor 2k odd,and where θµ,k (z,τ),µ=−k +1,...,k are theta functions:θµ,k (z,τ):=ℓ∈Z Z ,ℓ=µmod2k q ℓ2/(4k )y ℓ= n ∈Z Zq k (n +µ/(2k ))2y (µ+2kn )(3.13)In the case of the elliptic genus we have:χ(q,y ;Z )=ˆc /2 µ=−ˆc /2+1h µ(τ)θµ,ˆc /2(z,τ)(3.14)Physically,the decomposition (3.14)corresponds to separating out the U (1)current J and bosonizing it in the standard way J =i√ˆc (3.15)where O q is U(1)neutral and has weight h−q2/(2ˆc).This weight can be negative.The remaining“parafermion”contributions behave like:hµ(τ)= j=−µ2mod2ˆc cµ(j)q j/2ˆc1−ˆc/2≤µ≤ˆc/2(3.16)=cµ(−µ2)q−µ2/2ˆc+···where the higher terms in the expansion have higher powers of q.Equation(3.6)is physi-cally the statement of spectralflow invariance.Recall the spectralflow map[19]:G±n±a→G±n±(a+θ)L0→L0+θJ0+θ2ˆc√Substituting the above into the general Rademacher series(2.17)we getthe formula for the elliptic genus of an arbitrary Calabi-Yau manifold X of complex dimensionˆc:4c(n,ℓ;X)=√2ˆc n−ℓ2∞ c=1 4km−µ2<0ˆc/2 µ=−ˆc/2+1c−1/2Kℓ(n,ν,m,µ;c)c(2ˆc m−µ2;X) |2ˆc m−µ2|1/2T π(ˆc/2)2 2≥0(3.22) orˆc≤8.However,a priori for CY manifolds ofˆc≥9the elliptic genus will generally depend on other topological data besides the Hodge numbers.5Using dimension formulas for the space of Jacobi forms,the above bound has been sharpened in[22]where it has been shown that Hodge numbers of the Calabi-Yau manifold determine the elliptic genus only ifˆc<12orˆc=13.This paper also contains many explicit computations of the elliptic genus of Calabi-Yau hypersurfaces in toric varieties.3.4.Derivatives of Jacobi forms and fareytail transformsWe summarize here some formulae which are useful in discussing the fareytail trans-form of Jacobi forms.Denote the space of(weak)Jacobi forms of weight w and index k by J w,k.Introduce the operatorO:=∂8πik ∂4An unfortunate clash of notation leads to three different meanings for“c”in this formula!5We thank V.Gritsenko for pointing out to us that the elliptic genus in general depends on more data than just the Hodge numbers.This statement becomes manifest in view of the Rademacher expansion.See also[21].One then easily checks that ifφ∈J w,k thenO+(w−1/2)j!Γ(n+1−j)Γ(n+w−12iℑτ j O n−j(3.25) maps an element of J w,k to a function(in general,nonholomorphic)which transforms according to the Jacobi transformation rules with weight w+2n and index k,at least for n integral.On the other hand,for w integral and n=3/2−w the expression above simplifies to a single term O3/2−w,where the latter should be interpreted as a pseudodifferential operator.Definition(The fareytail transform):We define the fareytail transform F T(φ)of φ∈J w,k to be F T(φ):=|O3/2−w|φ.We also use the notation φ=F T(φ).Note that ifφis a weak Jacobi form and we define the polar part ofφto be:φ−:= 4kn−ℓ2<0c(n,ℓ)q n yℓ(3.26)then(F T(φ))−=F T(φ−).As pointed out to us by Don Zagier,it turns out that F T(φ)is not a(weak)Jacobi form.Nevertheless,it is related to a truncated Poincar´e series as we explain in the next section.4.The Rademacher expansion as a formula in statistical mechanicsAs we discussed in the introduction,in statistical mechanics the canonical and micro-canonical ensemble partition functions are related by an inverse Laplace transform:1N(E)=Consider the first formulation,(2.4)to(2.6)for (n +∆ν)>0.We can make the change of variable in (2.6)β→−n +∆ν|m +∆µ|β(4.2)which is valid as long as it does not shift the contour through singularities of the integrand.In the formula below we will find that Z (β)has a singularity at β=0,so we must have −n +∆ν2πiǫ++i ∞ǫ+−i ∞ Z ν(β)e 2πβ(n +∆ν)dβ(4.3)withZ ν(β)=2π0≤−d/c<1 m +∆µ<0cβ−id w −2M (γc,d )−1νµe 2πi (m +∆µ)(a/c )|m +∆µ|1−w F µ(m )exp (2π)|m +∆µ|2πie 2πβn Z φ(β,ω)(4.5)where ˜c are related to c by the fareytail transform (1.17).The basic idea of the derivation is to use the decomposition (3.14)and apply the generalized Rademacher series to the vector of modular forms given in (3.16).After some manipulation we find:Zφ(β,ω)=2πi (−1)w +1 0≤−d/c<1k µ=−k +1 m :4km −µ2<0˜c µ(4km −µ2) cτ+d w −3exp 4π|m −µ22iaτ+b cτ+d]θµ,k (ωcτ+d )(4.6)Recall that τ=iβin this formula.We will refer to our result (4.6)as the Jacobi-Rademacher formula.The most efficient way to proceed here is to work backwards by evaluating the right-hand-side of (4.5)and comparing to (4.4).The reader should note that the meaning of w has changed in this equation,and it now refers to the weight of the Jacobi form:w ((4.6))=w ((4.4))+12and w ((4.6))=0.Finally,we must stress that the derivation of equation(4.5)is only valid for n −ℓ2/4k >0.It is useful to rewrite (4.6)in terms of the slash operator.In general the slash operator for Jacobi forms of weight w and index k is defined to be [18]:p |w,k γ (τ,z ):=(cτ+d )−w exp −2πik cz 2cτ+d,z4k)τ Θ+µ,k (z,τ):=θµ,k (z,τ)+θ−µ,k (z,τ)1≤|µ|<k Θ+k,k (z,τ):=θk,k |µ|=k (4.10)As in equations (1.12)to (1.14)of the introduction the relation between the micro-canonical and canonical ensemble differs slightly from the relation between the Fouriercoefficients and the truncated Poincar´e series Z φ.In order to write the full canonical par-tition function we extend the sum in (4.8),interpreted as a sum over reduced fractions0≤c/d<1to a sum over all relatively prime pairs of integers(c,d)with c≥0.(For c=0we only have d=1.)Let us call the result Zφ.In order to produce a truly modular object we must also allow for c<0,that is,we must sum overΓ∞\SL(2,Z Z).Extendingthe sum in this way produces zero,because the summand is odd under(c,d)→(−c,−d), and thus we fail to produce a Jacobi form whose Fourier coefficients match those of F T(φ), for n−ℓ2/4k>0.This is just as well,since,as pointed out to us by Don Zagier,F T(φ) is not a Jacobi form.5.Physical interpretation in terms of IIB string theory on AdS3×S3×K3Let us apply the the Jacobi-Rademacher formula to the elliptic genusχfor Sym k(K3). In this case w=0and(4.8)becomesZχ(β,ω)=−2πi (c,d)=1,c≥0k µ=1 4km−µ2<0˜cµ 4km−µ2;Sym k(K3)cτ+d −3exp 2πi m−µ2cτ+d exp[−2πik cω2cτ+d,aτ+b2τ;˜τ,˜ω+1where Z NSNS is defined as in(5.2)in the NSNS sector6.Note thatωis inserted relativeto(−1)F.Now we use the AdS/CFT correspondence[1].The conformalfield theory partition function is identified with a IIB superstring partition function.The reduction on AdS3×S3×K3leads to an infinite tower of massive propagating particles and a“topological multiplet”of extended AdS3supergravity[23][11].The latter is described by a super-Chern-Simons theory[24],in the present case based on the supergroup SU(2|1,1)L×SU(2|1,1)R[11].The modular parameterτand the twists in(5.3)are specified in the supergravity partition function through the boundary conditions on thefields in the super Chern-Simons theory.These involve the metric,the SU(2)L×SU(2)R gaugefields,and the gravitini.The boundary conditions are as follows:1.The path integral over3d metrics will involve a sum over asymptotically hyperbolicgeometries which bound the torus of conformal structureτmod SL(2,Z Z).Thus,we sum over Euclidean3-metrics with a conformal boundary at r=∞andds2∼dr22ℑτωσ3du(5.7) Note that in Chern-Simons theory we specify boundary conditions on A u,but leave A¯u undetermined.Similarly,we have:˜A ¯u d¯u→π6Note that a complexωin(5.3)is equivalent to inserting a phase e2πiω1in the vertical direction and a phase e2πiω2in the horizontal direction withω=ω1+τω2,whereω1,ω2are real.In other words e2πiω+=e2πiω1e2πiω2become the supercurrents in the boundary CFT.)Since these fermions are coupled to the SU(2)L×SU(2)R gaugefields the fermion conditions can be shifted by turningon aflat connection.If the boundary conditions on the gaugefields are given by (5.7)(5.8)and we wish to compute Z RR,then the fermion boundary conditions should be as in(5.3),but since we wish to specialize to the elliptic genus then we must put ˜ω=0in(5.3),leaving us withe2πiω+⊗++.(5.9)There are many geometries that contribute to this partition function.Let usfirst start by discussing the simplest ones.The simplest of these geometries are the ones which can be obtained as solutions of the SU(2|1,1)2Chern Simons theory.This Chern Simons theory is a consistent truncation of the six dimensional supergravity theory.So a solution of the Chern Simons theory will also be a solution of the six dimensional theory.These solutions correspond to choosing a way tofill in the torus.This corresponds to picking a primitive one cycleγr andfilling in the torus so thatγr is contractible.The U(1)L gauge connection isflat;in a suitable gauge it just a given by constant A u and A¯u.As we said above the constant value of A u corresponds to the parameterωin the partition function through(5.7).In the classical solution A¯u is determined by demanding that the full configuration is non-singular.This translates into the condition that the Wilson line for a unit charge particle around the contractible cycle is minus one,in other wordse i γr A=−1.(5.10) This ensures that we will have a non-singular solution because the particles that carry odd charge are fermions which in the absence of a Wilson line were periodic aroundγr.With this particular value of the Wilson line they are anti-periodic,but this is precisely what we need since the geometry near the region whereγr shrinks to zero size looks like the origin of the plane.All that we have said for the U(1)L gaugefield should be repeated for the U(1)R gaugefield.Since there are fermions that carry charges(1,0)or(0,1)we get the condition(5.10)for both U(1)L,R connections.In this way we resolve the paradox that the (++)spin structure in(5.9)cannot befilled in.For more details on these solutions in the Lorentzian context,see[26].Note that there is an infinite family of solutions that solves (5.10)since we can always add a suitable integer to A¯u.This corresponds to doing integralunits of spectralflow.For the purposes of this discussion we can take the k→∞limit, and in this limit one can examine the classical equations,and hence specify the values of both A u and A¯u.Note that thefinal effective boundary conditions of the supergravity fields depends on both A u and A¯u.In particular,they are not purely given in terms of thefield theory boundary conditions(which is the information contained in A u).7The final boundary conditions for the fermions in the supergravity theory depend also on the particular state that we are considering.The simple solutions that we have been discussing correspond to the m=0andµ=k term in(5.1).The sum over all possible contractible cycles corresponds to the sum over c,d in(5.1).And the sum over integer values of spectral flow corresponds to the different terms in the sum over integers in the theta function in (5.1).Note that from the point of view of an observer in the interior all these solutions are equivalent,(up to a coordinate transformation),to Euclidean(AdS3×S3)/Z Z.There is,however,nontrivial information in this sum since we saw that it is crucial for recovering the full partition function of the theory.Now that we have discussed the simpest solutions we can ask about all the other terms,i.e.about the sum over m,µin(5.1).These correspond to adding particles to the solutions described in the above paragraph.These particles are not contained in the Chern Simons theory.The six dimensional theory,reduced on S3gives a tower of KKfields that propagate on AdS3.If we compute the elliptic genus for them only a very small subset contributes.From the point of view of the Chern Simons theory adding these particles is like adding Wilson lines for the U(1)connection[27][25].8In this case we do not have the relation(5.10)near the boundary.This is not a problem since the connection is notflat any longer in the full spacetime.It is possible tofind complete non-singular six(or ten) dimensional solutions which correspond to various combinations of RR ground states[28].。
常微分方程的英文Ordinary Differential EquationsIntroductionOrdinary Differential Equations (ODEs) are mathematical equations that involve derivatives of unknown functions with respect to a single independent variable. They find application in various scientific disciplines, including physics, engineering, economics, and biology. In this article, we will explore the basics of ODEs and their importance in understanding dynamic systems.ODEs and Their TypesAn ordinary differential equation is typically represented in the form:dy/dx = f(x, y)where y represents the unknown function, x is the independent variable, and f(x, y) is a given function. Depending on the nature of f(x, y), ODEs can be classified into different types.1. Linear ODEs:Linear ODEs have the form:a_n(x) * d^n(y)/dx^n + a_(n-1)(x) * d^(n-1)(y)/dx^(n-1) + ... + a_1(x) * dy/dx + a_0(x) * y = g(x)where a_i(x) and g(x) are known functions. These equations can be solved analytically using various techniques, such as integrating factors and characteristic equations.2. Nonlinear ODEs:Nonlinear ODEs do not satisfy the linearity condition. They are generally more challenging to solve analytically and often require the use of numerical methods. Examples of nonlinear ODEs include the famous Lotka-Volterra equations used to model predator-prey interactions in ecology.3. First-order ODEs:First-order ODEs involve only the first derivative of the unknown function. They can be either linear or nonlinear. Many physical phenomena, such as exponential decay or growth, can be described by first-order ODEs.4. Second-order ODEs:Second-order ODEs involve the second derivative of the unknown function. They often arise in mechanical systems, such as oscillators or pendulums. Solving second-order ODEs requires two initial conditions.Applications of ODEsODEs have wide-ranging applications in different scientific and engineering fields. Here are a few notable examples:1. Physics:ODEs are used to describe the motion of particles, fluid flow, and the behavior of physical systems. For instance, Newton's second law of motion can be formulated as a second-order ODE.2. Engineering:ODEs are crucial in engineering disciplines, such as electrical circuits, control systems, and mechanical vibrations. They allow engineers to model and analyze complex systems and predict their behavior.3. Biology:ODEs play a crucial role in the study of biological dynamics, such as population growth, biochemical reactions, and neural networks. They help understand the behavior and interaction of different components in biological systems.4. Economics:ODEs are utilized in economic models to study issues like market equilibrium, economic growth, and resource allocation. They provide valuable insights into the dynamics of economic systems.Numerical Methods for Solving ODEsAnalytical solutions to ODEs are not always possible or practical. In such cases, numerical methods come to the rescue. Some popular numerical techniques for solving ODEs include:1. Euler's method:Euler's method is a simple numerical algorithm that approximates the solution of an ODE by using forward differencing. Although it may not provide highly accurate results, it gives a reasonable approximation when the step size is sufficiently small.2. Runge-Kutta methods:Runge-Kutta methods are higher-order numerical schemes for solving ODEs. They give more accurate results by taking into account multiple intermediate steps. The most commonly used method is the fourth-order Runge-Kutta (RK4) algorithm.ConclusionOrdinary Differential Equations are a fundamental tool for modeling and analyzing dynamic systems in various scientific and engineering disciplines. They allow us to understand the behavior and predict the evolution of complex systems based on mathematical principles. With the help of analytical and numerical techniques, we can solve and interpret different types of ODEs, contributing to advancements in science and technology.。
(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。
0+||zero-dagger; 读作零正。
1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。
AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。
BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。
B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。
C0 类函数||function of class C0; 又称“连续函数类”。
CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。
Cp统计量||Cp-statisticC。
a r X i v :m a t h /0211394v 2 [m a t h .N T ] 22 D e c 2003FINITENESS RESULTS FOR MODULAR CURVES OF GENUS ATLEAST 2MATTHEW H.BAKER,ENRIQUE GONZ ´ALEZ-JIM ´ENEZ,JOSEP GONZ ´ALEZ,AND BJORN POONEN Abstract.A curve X over Q is modular if it is dominated by X 1(N )for some N ;if in addition the image of its jacobian in J 1(N )is contained in the new subvariety of J 1(N ),then X is called a new modular curve.We prove that for each g ≥2,the set of new modular curves over Q of genus g is finite and computable.For the computability result,we prove an algorithmic version of the de Franchis-Severi Theorem.Similar finiteness results are proved for new modular curves of bounded gonality,for new modular curves whose jacobian is a quotient of J 0(N )new with N divisible by a prescribed prime,and for modular curves (new or not)with levels in a restricted set.We study new modular hyperelliptic curves in detail.In particular,we find all new modular curves of genus 2explicitly,and construct what might be the complete list of all new modular hyperelliptic curves of all genera.Finally we prove that for each field k of characteristic zero and g ≥2,the set of genus-g curves over k dominated by a Fermat curve is finite and computable.1.Introduction Let X 1(N )be the usual modular curve over Q .(See Section 3.1for a definition.)A curve 1X over Q will be called modular if there exists a nonconstant morphism π:X 1(N )→X over Q .If X is modular,then X (Q )is nonempty,since it contains the image of the cusp ∞∈X 1(N )(Q ).The converse,namely that if X (Q )is nonempty then X is modular,holds if the genus g of X satisfies g ≤1[9].In particular,there are infinitely many modular curves over Q of genus 1.On the other hand,we propose the following:Conjecture 1.1.For each g ≥2,the set of modular curves over Q of genus g is finite.Remark 1.2.(i)When we speak of the finiteness of the set of curves over Q satisfying some condition,we mean the finiteness of the set of Q -isomorphism classes of such curves.(ii)For any fixed N ,the de Franchis-Severi Theorem (see Theorem 5.5)implies thefiniteness of the set of curves over Q dominated by X 1(N ).Conjecture 1.1can be2BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENthought of as a version that is uniform as one ascends the tower of modular curves X1(N),provided that onefixes the genus of the dominated curve.(iii)Conjecture1.1is true if one restricts the statement to quotients of X1(N)by sub-groups of its group of modular automorphisms.See Remark3.16for details.(iv)If X1(N)dominates a curve X,then the jacobian Jac X is a quotient2of J1(N):= Jac X1(N).The converse,namely that if X is a curve such that X(Q)is nonempty and Jac X is a quotient of J1(N)then X is dominated by X1(N),holds if the genusg of X is≤1,but can fail for g≥2.See Section8.2for other“pathologies.”(v)In contrast with Conjecture1.1,there exist infinitely many genus-two curves over Q whose jacobians are quotients of J1(N)for some N.See Proposition8.2(5).(vi)In Section9,we use a result of Aoki[3]to prove an analogue of Conjecture1.1in which X1(N)is replaced by the Fermat curve x N+y N=z N in P2.In fact,such an analogue can be proved over arbitraryfields of characteristic zero,not just Q.We prove many results towards Conjecture1.1in this paper.Given a variety X over afield k,letΩ=Ω1X/k denote the sheaf of regular1-forms.Call a modular curve X over Q newof level N if there exists a nonconstant morphismπ:X1(N)→X(defined over Q)such that π∗H0(X,Ω)is contained in the new subspace H0(X1(N),Ω)new,or equivalently if the image of the homomorphismπ∗:Jac X→J1(N)induced by Picard functoriality is contained in the new subvariety J1(N)new of J1(N).(See Section3.1for the definitions of H0(X1(N),Ω)new, J1(N)new,J1(N)new,and so on.)For example,it is known that every elliptic curve E over Q is a new modular curve of level N,where N is the conductor of E.Here the conductor cond(A)of an abelian variety A over Q is a positive integer p p f p,where each exponent f p is defined in terms of the action of an inertia subgroup of Gal(2Quotients or subvarieties of varieties,and morphisms between varieties,are implicitly assumed to be defined over the samefield as the original varieties.If X is a curve over Q,and we wish to discuss automor-phisms over C,for example,we will write Aut(X C).Quotients of abelian varieties are assumed to be abelian variety quotients.MODULAR CURVES OF GENUS AT LEAST23 If we drop the assumption that our modular curves are new,we can still prove results,but (so far)only if we impose restrictions on the level.Given m>0,let Sparse m denote the set of positive integers N such that if1=d1<d2<···<d t=N are the positive divisors of N, then d i+1/d i>m for i=1,...,t−1.Define a function B(g)on integers g≥2by B(2)=13, B(3)=17,B(4)=21,and B(g)=6g−5for g≥5.(For the origin of this function,see the proofs of Propositions2.1and2.8.)A positive integer N is called m-smooth if all primes p dividing N satisfy p≤m.Let Smooth m denote the set of m-smooth integers.Theorem1.5.Fix g≥2,and let S be a subset of{1,2,...}.The set of modular curves over Q of genus g and of level contained in S isfinite if any of the following hold:(i)S=Sparse B(g).(ii)S=Smooth m for some m>0.(iii)S is the set of prime powers.Remark1.6.Since Sparse B(g)∪Smooth B(g)contains all prime powers,parts(i)and(ii)of Theorem1.5imply(iii).Remark1.7.In contrast with Theorem1.3,we do not know,even in theory,how to compute thefinite sets of curves in Theorem1.5.The reason for this will be explained in Remark5.10. If X is a curve over afield k,and L is afield extension of k,let X L denote X×k L.The gonality G of a curve X over Q is the smallest possible degree of a nonconstant morphism X C→P1C.(There is also the notion of Q-gonality,where one only allows morphisms over Q.By defining gonality using morphisms over C instead of Q,we make the next theorem stronger.)In Section4.3,we combine Theorem1.3with a known lower bound on the gonality of X1(N)to prove the following:Theorem1.8.For each G≥2,the set of new modular curves over Q of genus at least2 and gonality at most G isfinite and computable.(We could similarly prove an analogue of Theorem1.5for curves of bounded gonality instead offixed genus.)Recall that a curve X of genus g over afield k is called hyperelliptic if g≥2and the canonical map X→P g−1is not a closed immersion:equivalently,g≥2and there exists a degree-2morphism X→Y where Y has genus zero.If moreover X(k)=∅then Y≃P1k, and if also k is not of characteristic2,then X is birational to a curve of the form y2=f(x) where f is a separable polynomial in k[x]of degree2g+1or2g+2.Recall that X(k)=∅is automatic if X is modular,because of the cusp∞.Taking G=2in Theorem1.8,wefind that the set of new modular hyperelliptic curves over Q isfinite and computable.We can say more:Theorem1.9.Let X be a new modular hyperelliptic curve over Q of genus g≥3and level N.Then(i)g≤16.(ii)If Jac X is a quotient of J0(N),then g≤10.If moreover3|N,then X is the genus-3 curve X0(39).(iii)If Jac X is not a quotient of J0(N),then either g is even or g≤9.Further information is given in Sections6.3and6.5,and in the appendix.See Section3.1 for the definitions of X0(N)and J0(N).4BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENAs we have already remarked,if we consider all genera g≥2together,there are infinitely many new modular curves.To obtainfiniteness results,so far we have needed to restrict either the genus or the gonality.The following theorem,proved in Section7,gives a different type of restriction that again impliesfiniteness.Theorem1.10.For each prime p,the set of new modular curves over Q of genus at least2 whose jacobian is a quotient of J0(N)new for some N divisible by p isfinite and computable. Question1.11.Does Theorem1.10remain true if J0(N)new is replaced by J1(N)new? Call a curve X over afield k of characteristic zero k-modular if there exists a nonconstant morphism X1(N)k→X(over k).Question1.12.Is it true that for everyfield k of characteristic zero,and every g≥2,the set of k-modular curves over k of genus g up to k-isomorphism isfinite?Remark 1.13.If X is a k-modular curve over k,and we define k0=k∩k,then X=X0×k0k for some k0-modular curve X0.This follows from the de Franchis-Severi Theorem.Remark1.14.If k and k′arefields of characteristic zero with[k′:k]finite,then a positive answer to Question1.12for k′implies a positive answer for k,since Galois cohomology and thefiniteness of automorphism group of curves of genus at least2show that for each X′over k′,there are at mostfinitely many curves X over k with X×k k′≃X′.But it is not clear, for instance,that a positive answer forMODULAR CURVES OF GENUS AT LEAST25 Corollary 2.4.Let X be a curve of genus g≥2over afield k of characteristic zero. Then the image X′of the canonical map X→P g−1is the common zero locus of the set ofhomogeneous polynomials of degree4that vanish on X′.Proof.We may assume that k is algebraically closed.If X is hyperelliptic of genus g,saybirational to y2=f(x)where f has distinct roots,then we may choose{x i dx/y:0≤i≤g−1}as basis of H0(X,Ω),and then the image of the canonical map is the rational normalcurve cut out by{t i t j−t i′t j′:i+j=i′+j′}where t0,...,t g−1are the homogeneous coordinates on P g−1.If X is nonhyperelliptic of genus3,its canonical model is a planequartic.In all other cases,we use Petri’s Theorem.(The zero locus of a homogeneouspolynomial h of degree d<4equals the zero locus of the set of homogeneous polynomialsof degree4that are multiples of h.) Lemma2.5.Let X be a hyperelliptic curve of genus g over afield k of characteristic zero, and suppose P∈X(k).Let{ω1,...,ωg}be a basis of H0(X,Ω)such that ord P(ω1)<···< ord P(ωg).Then x:=ωg−1/ωg and y:=dx/ωg generate the functionfield k(X),and there is a unique polynomial F(x)of degree at most2g+2such that y2=F(x).Moreover,F is squarefree.If P is a Weierstrass point,then deg F=2g+1and ord P(ωi)=2i−2for all i;otherwise deg F=2g+2and ord P(ωi)=i−1for all i.Finally,it is possible to replace eachωi by a linear combination ofωi,ωi+1,...,ωg to makeωi=x g−i dx/y for1≤i≤g. Proof.This follows easily from Lemma3.6.1,Corollary3.6.3,and Theorem3.6.4of[21]. Proof of Proposition2.1.Suppose that X,P,q,and the w i are as in the statement of the proposition.Letωi be the corresponding elements of H0(X,Ω).We will show that X is determined by the w i when B=max{8g−7,6g+1}.Since B>8g−8,Lemma2.2implies that the w i determine the set of homogeneouspolynomial relations of degree4satisfied by theωi,so by Corollary2.4the w i determine theimage X′of the canonical map.In particular,the w i determine whether X is hyperelliptic,and they determine X if X is nonhyperelliptic.Therefore it remains to consider the case where X is hyperelliptic.Applying Gaussianelimination to the w i,we may assume0=ord q(w1)<···<ord q(w g)≤2g−2and that thefirst nonzero coefficient of each w i is1.We use Lemma2.5repeatedly in what follows.Thevalue of ord(w2)determines whether P is a Weierstrass point.Suppose that P is a Weierstrass point.Then w i=q2i−2(1+···+O(q B−2i+2)),where each“···”here and in the rest of this proof represents some known linear combination of positive powers of q up to but not including the power in the big-O term.(“Known”means “determined by the original w i.”)Define x=w g−1/w g=q−2(1+···+O(q B−2g+2)).Define y=dx/(w g dq)=−2q−(2g+1)(1+···+O(q B−2g+2)).Then y2=4q−(4g+2)(1+···+O(q B−2g+2)).Since B≥6g+1,we have −(4g+2)+(B−2g+2)>0,so there is a unique polynomial F(of degree2g+1)such that y2=F(x).A similar calculation shows that in the case where P is not a Weierstrass point,thenB≥3g+2is enough. Remark2.6.Let us show that if the hypotheses of Proposition2.1are satisfied except that thew i belong tok ,Ω),then the conclusion still holds.Let E be afinite Galois extension of k containing all the coefficients of the w i.The E-span of the w i must be stable under Gal(E/k)if they come6BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENfrom a curve over Q,and in this case,we can replace the w i by a k-rational basis of this span.Then Proposition2.1applies.Remark2.7.We can generalize Proposition2.1to the case where q is not a uniformizing parameter on X:Fix an integer g≥2,and let k be afield of characteristic zero.Let B>0be the integer appearing in the statement of Proposition2.1,and let e bea positive integer.Then if w1,...,w g are elements of k[[q]]/(q eB),then upto k-isomorphism,there exists at most one curve X over k such that thereexist P∈X(k),an analytic uniformizing parameter q′∈ˆO X,P and a relationq′=c e q e+c e+1q e+1+...with c e=0,such that w1dq,...,w g dq are theexpansions modulo q eB of some basis of H0(X,Ω).The proof of this statement is similar to the proof of Proposition2.1,and is left to the reader. The rest of this section is concerned with quantitative improvements to Proposition2.1, and is not needed for the generalfiniteness and computability results of Sections4and5. Proposition2.8.Proposition2.1holds with B=B(g),where B(2)=13,B(3)=17, B(4)=21,and B(g)=6g−5for g≥5.Moreover,if we are given that the curve X to be recovered is hyperelliptic,then we can use B(g)=4g+5or B(g)=2g+4,according as P is a Weierstrass point or not.Proof.For nonhyperelliptic curves of genus g≥4,we use Theorem2.3instead of Corol-lary2.4to see that B>6g−6can be used in place of B>8g−8.Now suppose that X is hyperelliptic.As before,assume ord q(w1)<···<ord q(w g)and that thefirst nonzero coefficient of each w i is1.The value of ord(w2)determines whether P is a Weierstrass point.Suppose that P is a Weierstrass point.Then w i=q2i−2(1+···+O(q B−2i+2)).(As in the proof of Proposition2.1,···means a linear combination of positive powers of q,whose coefficients are determined by the w i.)Define˜x=w g−1/w g=q−2(1+···+O(q B−2g+2)).For 1≤i≤g−2,the expression˜x g−i w g=q2i−2(1+···+O(q B−2g+2))is the initial expansion of w i+ g j=i+1c ij w j for some c ij∈k,and all the c ij are determined if2+(B−2g+2)>2g−2, that is,if B≥4g−5.Let w′i=w i+ g j=i+1c ij w j=q2i−2(1+···+O(q B−2i+2)).Define x=w′1/w′2=q−2(1+···+O(q B−2)).Define y=−2q−(2g+1)(1+···+O(q B−2))as the solution to w′1dq=x g−1dx/y.Then y2=4q−(4g+2)(1+···+O(q B−2)),and if−(4g+2)+B−2>0, we can recover the polynomial F of degree2g+1such that y2=F(x).Hence B≥4g+5 suffices.A similar proof shows that B≥2g+4suffices in the case that P is not a Weierstrass point.Hence max{6g−5,4g+5,2g+4}suffices for all types of curves,except that the6g−5 should be8g−7when g=3.This is the function B(g). Remark2.9.We show here that for each g≥2,the bound B=4g+5for the precision needed to recover a hyperelliptic curve is sharp.Let F(x)∈C[x]be a monic polynomial of degree2g+1such that X:y2=F(x)and X′:y2=F(x)+1are curves of genus g that are not birationally equivalent.Let q be the uniformizing parameter at the point at infinity on X such that x=q−2and y=q−(2g+1)+O(q−2g).Define q′similarly for X′.A calculation shows that the q-expansions of the differentials x i dx/y for0≤i≤g−1are even power series in q times dq,and modulo q4g+4dq they agree with the corresponding q′-expansionsMODULAR CURVES OF GENUS AT LEAST27 for X′except for the coefficient of q4g+2dq in x g−1dx/y.By a change of analytic parameter q=Q+αQ4g+3for someα∈C,on X only,we can make even that coefficient agree.A similar proof shows that in the case that P is not a Weierstrass point,the bound2g+4 cannot be improved.Remark2.10.When studying new modular curves of genus g,we can also use the multi-plicativity of Fourier coefficients of modular forms(see(3.7))to determine some coefficients from earlier ones.Hence we can sometimes get away with less than B(g)coefficients of each modular form.2.2.Descending morphisms.The next result will be used a number of times throughout this paper.In particular,it will be an important ingredient in the proof of Theorem1.9. Proposition2.11.Let X,Y,Z be curves over afield k of characteristic zero,and assume that the genus of Y is>1.Then:(i)Given nonconstant morphismsπ:X→Z andφ:X→Y such thatφ∗H0(Y,Ω)⊆π∗H0(Z,Ω),there exists a nonconstant morphism u′:Z→Y making the diagramXπuY u′Ycommute.Proof.The conclusion of(i)is equivalent to the inclusionφ∗k(Y)⊆π∗k(Z).It suffices to prove that every function inφ∗k(Y)is expressible as a ratio of pullbacks of meromorphic differentials on Z.If Y is nonhyperelliptic,then thefield k(Y)is generated by ratios of pairs of differentials in H0(Y,Ω),so the inclusion follows from the hypothesisφ∗H0(Y,Ω)⊆π∗H0(Z,Ω).When Y is hyperelliptic,we must modify this argument slightly.We have k(Y)=k(x,y),where y2=F(x)for some polynomial F(U)in k[U]without double roots. Thefield generated by ratios of differentials in H0(Y,Ω)is k(x),soφ∗k(x)⊆π∗k(Z).To show thatφ∗y∈π∗k(Z)too,write y=x dx/(x dx/y)and observe that x dx/y∈H0(Y,Ω). Now we prove(ii).The hypothesis on u∗lets us apply(i)withφ=π◦u to construct u′:Y→Y.Since Y has genus>1and k has characteristic zero,the Hurwitz formula implies that u′is an automorphism.Considering functionfields proves uniqueness. Remark2.12.Both parts of Proposition2.11can fail if the genus of Y is1.On the other hand,(ii)remains true under the additional assumption that X→Y is optimal in the sense that it does not factor nontrivally through any other genus1curve.Remark2.13.Proposition2.11remains true if k hasfinite characteristic,provided that one assumes that the morphisms are separable.8BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONEN3.Some facts about modular curves3.1.Basic facts about X1(N).We record facts about X1(N)that we will need for the proof offiniteness in Theorem1.3.See[55]for a detailed introduction.Let H={z∈C:Im z>0}.The group SL2(R)acts on H by a b c d z=az+bq for some g-dimensional C-subspaceS2(N)of q C[[q]].We will not define modular forms in general here,but S2(N)is known as the space of weight2cusp forms onΓ1(N).If M|N and d|NM .Similarly define the old subvariety J1(N)old of J1(N).The space S2(N)has a hermitian inner product called the Petersson inner product.Let S2(N)new denote the orthogonal complement to S2(N)old in S2(N).The identifications above also give us new and old subspaces of H0(X1(N)C,Ω)and H0(J1(N)C,Ω).Let J1(N)new=J1(N)/J1(N)old. There is also an abelian subvariety J1(N)new of J1(N)that can be characterized in two ways: either as the abelian subvariety such thatker(H0(J1(N)C,Ω)→H0((J1(N)new)C,Ω))=H0(J1(N)C,Ω)old,or as the abelian subvariety such that J1(N)=J1(N)old+J1(N)new with J1(N)old∩J1(N)new finite.(The latter description uniquely characterizes J1(N)new because of a theorem that no Q-simple quotient of J1(N)old is isogenous to a Q-simple quotient of J1(N)new;this theorem can be proved by comparing conductors,using[11].)The abelian varieties J1(N)new and J1(N)new are Q-isogenous.We define X0(N),J0(N),and J0(N)new similarly,starting withΓ0(N):= a b c d ∈SL2(Z) c≡0(mod N)instead ofΓ1(N).For n≥1,there exist well-known correspondences T n on X1(N),and they induce en-domorphisms of S2(N)and of J1(N)known as Hecke operators,also denoted T n.There is a unique basis New N of S2(N)new consisting of f=a1q+a2q2+a3q3+...such that a1=1and T n f=a n f for all n≥1.The elements of New N are called the newforms ofMODULAR CURVES OF GENUS AT LEAST29level N.(For us,newforms are always normalized:this means that a1=1.)Each a n is√an algebraic integer,bounded byσ0(n)k/k).The Galois group G Q acts on New N.For any quotient A of J1(N),let S2(A)denote the image of H0(A C,Ω)→H0(J1(N)C,Ω)≃S2(N)(the last isomorphism drops the dq/q);similarly for any nonconstant morphismπ:X1(N)→X of curves,define S2(X):=π∗H0(X C,Ω)q(3.1)Q-isogenyf→A f.Shimura proved that J1(N)is isogenous to a product of these A f,and K.Ribet[53]proved that A f is Q-simple.This explains the surjectivity of(3.1).The injectivity is well-known to experts,but we could notfind a suitable reference,so we will prove it,as part of Proposi-tion3.2.The subfield E f=Q(a2,a3,...)of C is a numberfield,and dim A f=[E f:Q].Moreover, End(A f)⊗Q can be canonically identified with E f,and under this identification the element λ∈End A f acts on f as multiplication byλ(considered as element of E f),and on each Galois conjugateσf by multiplication byσλ.(Shimura[57,Theorem1]constructed an injection End(A f)⊗Q֒→E f,and Ribet[53,Corollary4.2]proved that it was an isomorphism.)If A and B are abelian varieties over Q,let A Q∼B denote the statement that A and B are isogenous over Q.Proposition3.2.Suppose f∈New N and f′∈New N′.Then A f Q∼A f′if and only if N=N′and f=τf′for someτ∈G Q.Proof(K.Ribet).The“if”part is immediate from Shimura’s construction.Therefore it suffices to show that one can recover f,up to Galois conjugacy,from the isogeny class of A f. Letℓbe a prime.Let V be the Qℓ-Tate module Vℓ(A f)attached to A f.Let Qℓ. The proof of Proposition4.1of[53]shows thatQℓ[G Q]-module of dimension2over Qℓ.Moreover,for p∤ℓN,the trace of the p-Frobenius automorphism acting on Vσequalsσ(a p),where a p∈E f is the coefficient of q p in the Fourier expansion of f.If f′∈New′N is another weight2newform,and A f Q∼A f′,then(using′in the obvious way to denote objects associated with f′),we have isomorphisms of G Q-modules V≃V′and V′.Fixσ:E f֒→V′,whereσ′is some embedding E f′֒→3Earlier,in Theorem7.14of[55],Shimura had attached to f an abelian subvariety of J1(N).10BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENWe have parallel decompositionsS2(N)new= f∈G Q\New N τ:E f֒→C CτfJ1(N)new Q∼ f∈G Q\New N A fand parallel decompositionsS2(N)= M|N f∈G Q\New M d|Nc d f(q d)Mfor some M|N and f∈New M,where c d∈E f depends on f and d.Proof.By multiplying the quotient map J1(N)→A by a positive integer,we may assume that it factors through the isogenyJ1(N)→ M|N f∈G Q\New M A n f fof(3.4).We may also assume that A is Q-simple,and even that A is isomorphic to A fwith for some f,so that the quotient map J1(N)→A is the composition of J1(N)→A n ffa homomorphism A n ff→A.The latter is given by an n f-tuple c=(c d)of elements of End(A f),indexed by the divisors d of N/M.Underc→A,X1(N)֒→J1(N)→A n ffthe1-form on A C≃(A f)C corresponding to f pulls back to d|Nc d f(q d)Mfor some M|N and f∈New M,where c d∈E f depends on f and d.Proof.Apply Lemma3.5to the Albanese homomorphism J1(N)→Jac X.3.2.Automorphisms of X1(N).3.2.1.Diamonds.The action on H of a matrix a b c d ∈Γ0(N)induces an automorphism of X1(N)over Q depending only on(d mod N).This automorphism is called the diamond operator d .It induces an automorphism of S2(N).Letεbe a Dirichlet character modulo N,that is,a homomorphism(Z/N Z)∗→C∗.Let S2(N,ε)be the C-vector space{h∈S2(N):h| d =ε(d)h}.A form h∈S2(N,ε)is called a form of Nebentypusε.Every newform f∈New N is a form of some Nebentypus,and is therefore an eigenvector for all the diamond operators.Character theory gives a decompositionS2(N)= εS2(N,ε),whereεruns over all Dirichlet characters modulo N.Define S2(N,ε)new=S2(N,ε)∩S2(N)new.When we writeε=1,we mean thatεis the trivial Dirichlet character modulo N,that is,ε(n)= 1if(n,N)=10otherwise.A form of Nebentypusε=1is a form onΓ0(N).We recall some properties of a newform f= ∞n=1a n q n∈S2(N,ε).Let condεdenote the smallest integer M|N such thatεis a composition(Z/N Z)∗→(Z/M Z)∗→C∗. Throughout this paragraph,p denotes a prime,and(3.11)pv p(N)=1,ε=1=⇒f|W p=−a p f(3.12)(3.13)p∤N=⇒ε(p)a p.The equivalence(3.8)is trivial.For the remaining properties see[39]and[5].3.2.2.Involutions.For every integer M|N such that(M,N/M)=1,there is an automor-phism W M of X1(N)C inducing an isomorphism between S2(N,ε)new and S2(N,εM⊗f in S2(N,εM(p)a p if p∤M,a p if p|M.Then(3.14)f|W M=λM(f)(εN/M(−M)f,f|(W M′W M)=q.It is known thatS2(Γ(N,ε))=nk=1S2(N,εk),where n is the order of the Dirichlet characterε.The diamond operators and the Weil involution induce automorphisms of X(N,ε)C.If moreoverε=1,the curve X(N,1)is X0(N)and the automorphisms W M on X1(N)C induce involutions on X0(N)over Q that are usually called the Atkin-Lehner involutions.Remark 3.16.Define the modular automorphism group of X1(N)to be the subgroup of Aut(X1(N)Q ,which we continue to denote by d and W N respectively.Throughout the paper,D will denote the abelian subgroup of Aut(XQ )generated by Dand W N.If moreover X is hyperelliptic,and w is its hyperelliptic involution,then the group generated by D and W N.w will be denoted by D′N.Note that D is a subgroup of Aut(X),and the groups D N,D′N are G Q-stable by(3.15).If Jac X Q∼A f for some f with nontrivial Nebentypus,then D N is isomorphic to the dihedral group with2n elements,D2·n,where n is the order of the Nebentypus of f.For every nonconstant morphismπ:X1(N)→X of curves over Q such that S2(X)⊆⊕n i=1S2(N,εi)for some Nebentypusεof order n,there exists a nonconstant morphism π(ε):X(N,ε)→X over Q.This is clear when the genus of X is≤1,and follows from Proposition2.11(i)if the genus of X is>1.In particular,for a new modular curve X of genus>1,there exists a surjective morphism X0(N)→X if and only if D is the trivial group.More generally,we have the following result.Lemma3.17.Let X be a new modular curve of level N,and let G be a G Q-stable subgroup of Aut(XG:={φ∈Aut(X):φ∗ω=ωfor allω∈H0(X,Ω)G}.ThenG.Now supposeφ∈G⊆G as required.3.3.Supersingular points.We will use a lemma about curves with good reduction. Lemma3.19.Let R be a discrete valuation ring with fractionfield K.Suppose f:X→Y is afinite morphism of smooth,projective,geometrically integral curves over K,and X extends to a smooth projective model X over R(in this case we say that X has good reduction).If Y has genus≥1,then Y extends to a smooth projective model Y over R,and f extends to afinite morphism X→Y over R.Proof.This result is Corollary4.10in[40].See the discussion there also for references to earlier weaker versions. The next two lemmas are well-known(to coding theorists,for example),but we could not find explicit references,so we supply proofs.Lemma3.20.Let p be a prime.LetΓ⊆SL2(Z)denote a congruence subgroup of level N not divisible by p.Let XΓbe the corresponding integral smooth projective curve overF p-points on the reduction mapping to supersingular points on X(1)is at least(p−1)ψ/12.Proof.By[33],the curve X(N)admits a smooth model over Z[1/N],and has a rationalpoint(the cusp∞).Since p∤N and X(N)dominates XΓ,Lemma3.19implies that XΓhas good reduction at p,at least if XΓhas genus≥1.If XΓhas genus0,then the rational point on X(N)gives a rational point on XΓ,so XΓ≃P1,so XΓhas good reduction at p in any case.ReplacingΓby the group generated byΓand−id does not change XΓ,so withoutloss of generality,we may assume that−id∈Γ.Thenψ=(SL2(Z):Γ).If E is an elliptic curve,thenΓnaturally acts on thefinite set of ordered symplectic bases of E[N].The curve YΓ:=XΓ−{cusps}classifies isomorphism classes of pairs(E,L),where E is an elliptic curve and L is aΓ-orbit of symplectic bases of E[N].Fix E.Since SL2(Z)acts transitively on the symplectic bases of E[N],the number of Γ-orbits of symplectic bases is(SL2(Z):Γ)=ψ.Two such orbits L and L′correspond to the same point of XΓif and only if L′=αL for someα∈Aut(E).Thenψis the sum of the sizes of the orbits of Aut(E)acting on theΓ-orbits,soψ= (E,L)∈XΓ#Aut(E)F p,we obtainsupersingular points(E,L)∈XΓ(#Aut(E,L)=ψ supersingular E/#Aut(E)=(p−1)ψF p).Thereforethe number of supersingular points on XΓmust be at least2(p−1)ψ/24=(p−1)ψ/12. Lemma3.21.Let p be a prime.Given a supersingular elliptic curve E overF pand the p2-power Frobenius endomorphism of E′equals−p.Proof.Honda-Tate theory supplies an elliptic curve E over F p such that the characteristicpolynomial of the p-power Frobenius endomorphism Frob p satisfies Frob2p=−p.All super-singular elliptic curves over Fp→E.The inseparable part of this isogeny is a power of Frob p,so without loss of generality,we mayassume thatφis separable.The kernel K ofφis preserved by−p=Frob2p,so K is defined over F p2.Take E′=E Fp2/K. The following is a generalization of inequalities used in[48].Lemma3.22.Let X be a new modular hyperelliptic curve of level N and genus g over Q. If p is a prime not dividing N,then(p−1)(g−1)<2(p2+1).Proof.We may assume g≥2.Since p∤N,Lemma3.19implies that X1(N)and X have good reduction at p,and the morphismπ:X1(N)→X induces a corresponding morphism of curves over F p.By Proposition2.11(ii),the diamond automorphism −p of X1(N) induces an automorphism of X,which we also call −p .These automorphisms induces automorphisms of the corresponding curves over F p.For the rest of this proof,X1(N),X,π, −p represent these objects over F p.Also denote by −p the induced morphism P1→P1。
a rXiv:h ep-th/9983v14A ug1999ANOMALOUS COUPLINGS OF TYPE 0D-BRANES Marco Bill´o Dipartimento di Fisica Teorica,Universit`a di Torino and I.N.F.N.,Sezione di Torino,via P.Giuria 1,I-10125,Torino,Italy Ben Craps and Frederik Roose Instituut voor Theoretische Fysica Katholieke Universiteit Leuven,B-3001Leuven,Belgium Abstract Closed type 0string theories and their D-branes are introduced.The full Wess-Zumino action of these D-branes is derived.The analogy with type II is emphasized throughout the argument.Type 0Strings and D-branes.Type 0string theory is a non-supersymmetric,modular invariant theory of closed strings.The pres-ence of a tachyonic state in their spectrum makes type 0strings much harder to analyse than the supersymmetric type II strings.Nevertheless,type 0and type II strings have many features in common.In this con-tribution this fact will be exploited to show that type 0D-branes have anomalous terms in their worldvolume action.The analysis is based on Ref.[1],to which we refer for a more detailed treatment and for a more complete list of references.In the Neveu-Schwarz-Ramond formulation,type II string theories are obtained by imposing independent GSO projections on the left and right moving part.This amounts to keeping the following (left,right)sectors:IIB :(NS+,NS+),(R+,R+),(R+,NS+),(NS+,R+);IIA :(NS+,NS+),(R+,R −),(R+,NS+),(NS+,R −),where for instance R ±is the Ramond sector projected with P GSO =(1±(−)F )/2,F being the world-sheet fermion number.12The type0string theories contain instead the following sectors:0B:(NS+,NS+),(NS−,NS−),(R+,R+),(R−,R−);0A:(NS+,NS+),(NS−,NS−),(R+,R−),(R−,R+). These theories do not contain bulk spacetime fermions,which would have to come from“mixed”(R,NS)sectors.The inclusion of the NS-NS sectors with odd fermion numbers means that the closed string tachyon is not projected out.The third difference with type II theories is that the R-R spectrum is doubled.For instance,in the0B case,beside the IIB R-R potentials C p+1contained in bispinors of the(R+,R+)sector, there are the potentials C′p+1from bispinors of the(R−,R−)sector.Note that the bispinors containing the primed and unprimed R-R potentials have opposite chirality.This implies a sign difference in the Poincar´e duality relations among thefield strengths.Thus,for instance,in type 0B there is an unconstrainedfive-formfield strength,whose self-dual (anti-self-dual)part is the unprimed(primed)field strength.For our purposes,convenient combinations of C p+1and C′p+1are(C p+1)±=12(C p+1±C′p+1).(1.1)For p=3these are the electric(+)and magnetic(−)potentials[2].We will adopt this terminology also for other values of p.There turn out to be four types of“elementary”D-branes for each p:an electric and a magnetic one(i.e.,charged under(C p+1)±),and the corresponding antibranes[2].Anomalous Couplings.The open strings stretching between two like branes are bosons,just like the bulkfields of type0.However,a bound-ary state computation shows that fermions appear from strings between an electric and a magnetic brane[2].Thus one could wonder whether there are chiral fermions on the intersection of an electric and a magnetic brane.Consider such an orthogonal intersection with no overall trans-verse directions.If the dimension of the intersection is two or six,the computation reveals that there are precisely enough fermionic degrees of freedom on the intersection to form one chiral fermion.In type II string theory,the analogous computation shows that chi-ral fermions are present on two or six dimensional intersections of two orthogonal branes with no overall transverse directions.That observa-tion has had far reaching ly,the presence of chiral fermions has been shown to lead to gauge and gravitational anoma-lies on those intersections of D-branes[3].In a consistent theory,suchType0D-branes3 anomalies should be cancelled by anomaly inflow.In the present case,the anomaly inflow is provided by the anomalous D-brane couplings in the Wess-Zumino part of the D-brane action[3].These anomalous cou-plings have an anomalous variation localized on the intersections withother branes.To sketch how this anomaly inflow comes about,let us focus on thecase of two type IIB D5-branes(to be denoted by D5and D5’)inter-secting on a string.The Wess-Zumino action on D5contains a term ofthe form D5C2∧Y4,where C2is the R-R two-form potential and Y4a certain four-form involving thefield strength of gaugefield on D5andthe curvature two-forms of the tangent and normal bundles of D5.To beprecise,one should replace this term by D5H3∧ω3,with Y4=dω3and H3the complete gauge-invariantfield strength of C2(which generically differs from dC2).Since the gauge variation of the“Chern-Simons”form ω3is given byδω3=dI2for some two-form I2,the anomalous term on D5have a variation localized on the intersection with D5’:δ D5H3∧ω3= D5dH3∧I2= D5d∗H7∧I2= D5δD5′∧I2,(1.2)which can thus cancel the anomaly due to the chiral fermions living on the intersection.A careful analysis of all the anomalies[3]shows that the anomalous part of the D p-brane action is given,in terms of the formal sum C of the various R-R forms,byS WZ=T pˆA(R T)/ˆA(R N).(1.3)Here T p/κdenotes the D p-brane tension,F the gaugefield on the brane and B the NS-NS two-form.Further,R T and R N are the curvatures of the tangent and normal bundles of the D-brane world-volume,andˆA denotes the A-roof genus.Let us now return to type0string theory.As stated above,here chiral fermions live on intersections of electric and magnetic type0D-branes. The associated gauge and gravitational anomalies on such intersections match the ones for type II D-branes.To cancel them,the minimal cou-pling of a D p-brane to a(p+1)-form R-R potential should be extended to the following Wess-Zumino action[1]:S WZ=T pˆA(R T)/ˆA(R N).(1.4)The±in Eq.(1.4)distinguishes between electric and magnetic branes. Note that T p/κdenotes the tension of a type II D p-brane,which can be computed to be√4The argument that the variation of this action cancels the anomaly on the intersection is a copy of the one described above in the type II case, apart from one slight subtlety.For definiteness,consider the intersection of an electric and a magnetic D5-brane on a string.Varying the elec-tric D5-brane action(exhibiting the(C2)+potential,or rather,itsfield strength(H3)+),onefinds that the variation is localized on the inter-section of the electric D5-brane with branes charged magnetically under the(H3)+field ing Eq.(1.1),the different behaviour under Poincar´e duality of the primed and unprimed R-Rfield strengths shows that these are precisely the branes carrying(electric)(H7)−charge,i.e. what we called the magnetic D5-branes.Schematically,δ D5+(H3)+∧ω3= D5+d(H3)+∧I2= D5+d∗(H7)−∧I2= D5+δD5−∧I2.(1.5)A completely analogous discussion goes through for the variation of the magnetic D5-brane action.AcknowledgmentsB.C.and F.R.would like to thank the organisers for a very nice school,and for financial support.This work was supported by the European Commission TMR programme ERBFMRX-CT96-0045.B.C.is Aspirant FWO-Vlaanderen. References[1]Bill´o,M.,Craps,B.and Roose,F.(1999)On D-branes in type0string theory,Phys.Lett.B457,pp.61-69,hep-th/9902196. [2]Bergman,O.and Gaberdiel,M.R.(1997)A non-supersymmetricopen string theory and S-duality,Nucl.Phys.B499,pp.183-204, hep-th/9701137;Klebanov,I.R.and Tseytlin, A.A.(1999)D-branes and dual gauge theories in type0strings,Nucl.Phys.B546,pp.155-181, hep-th/9811035.[3]Green,M.,Harvey,J.A.and Moore,G.(1997)I-brane inflow andanomalous couplings on D-branes,Class.Quant.Grav.14,pp.47-52, hep-th/9605033;Cheung,Y.K.and Yin,Z.(1998)Anomalies,branes,and currents, Nucl.Phys.B517,pp.69-91,hep-th/9710206.。
离散数学中英文名词对照表外文中文AAbel category Abel 范畴Abel group (commutative group) Abel 群(交换群)Abel semigroup Abel 半群accessibility relation 可达关系action 作用addition principle 加法原理adequate set of connectives 联结词的功能完备（全）集adjacent 相邻（邻接）adjacent matrix 邻接矩阵adjugate 伴随adjunction 接合affine plane 仿射平面algebraic closed field 代数闭域algebraic element 代数元素algebraic extension 代数扩域（代数扩张）almost equivalent 几乎相等的alternating group 三次交代群annihilator 零化子antecedent 前件anti symmetry 反对称性anti-isomorphism 反同构arboricity 荫度arc set 弧集arity 元数arrangement problem 布置问题associate 相伴元associative algebra 结合代数associator 结合子asymmetric 不对称的(非对称的)atom 原子atomic formula 原子公式augmenting digeon hole principle 加强的鸽子笼原理augmenting path 可增路automorphism 自同构automorphism group of graph 图的自同构群auxiliary symbol 辅助符号axiom of choice 选择公理axiom of equality 相等公理axiom of extensionality 外延公式axiom of infinity 无穷公理axiom of pairs 配对公理axiom of regularity 正则公理axiom of replacement for the formula Ф关于公式Ф的替换公式axiom of the empty set 空集存在公理axiom of union 并集公理Bbalanced imcomplete block design 平衡不完全区组设计barber paradox 理发师悖论base 基Bell number Bell 数Bernoulli number Bernoulli 数Berry paradox Berry 悖论bijective 双射bi-mdule 双模binary relation 二元关系binary symmetric channel 二进制对称信道binomial coefficient 二项式系数binomial theorem 二项式定理binomial transform 二项式变换bipartite graph 二分图block 块block 块图（区组）block code 分组码block design 区组设计Bondy theorem Bondy 定理Boole algebra Boole 代数Boole function Boole 函数Boole homomorophism Boole 同态Boole lattice Boole 格bound occurrence 约束出现bound variable 约束变量bounded lattice 有界格bridge 桥Bruijn theorem Bruijn 定理Burali-Forti paradox Burali-Forti 悖论Burnside lemma Burnside 引理Ccage 笼canonical epimorphism 标准满态射Cantor conjecture Cantor 猜想Cantor diagonal method Cantor 对角线法Cantor paradox Cantor 悖论cardinal number 基数Cartesion product of graph 图的笛卡儿积Catalan number Catalan 数category 范畴Cayley graph Cayley 图Cayley theorem Cayley 定理center 中心characteristic function 特征函数characteristic of ring 环的特征characteristic polynomial 特征多项式check digits 校验位Chinese postman problem 中国邮递员问题chromatic number 色数chromatic polynomial 色多项式circuit 回路circulant graph 循环图circumference 周长class 类classical completeness 古典完全的classical consistent 古典相容的clique 团clique number 团数closed term 闭项closure 闭包closure of graph 图的闭包code 码code element 码元code length 码长code rate 码率code word 码字coefficient 系数coimage 上象co-kernal 上核coloring 着色coloring problem 着色问题combination number 组合数combination with repetation 可重组合common factor 公因子commutative diagram 交换图commutative ring 交换环commutative seimgroup 交换半群complement 补图(子图的余) complement element 补元complemented lattice 有补格complete bipartite graph 完全二分图complete graph 完全图complete k-partite graph 完全k-分图complete lattice 完全格composite 复合composite operation 复合运算composition (molecular proposition) 复合（分子）命题composition of graph (lexicographic product)图的合成（字典积）concatenation (juxtaposition) 邻接运算concatenation graph 连通图congruence relation 同余关系conjunctive normal form 正则合取范式connected component 连通分支connective 连接的connectivity 连通度consequence 推论（后承）consistent (non-contradiction) 相容性（无矛盾性）continuum 连续统contraction of graph 图的收缩contradiction 矛盾式（永假式）contravariant functor 反变函子coproduct 上积corank 余秩correct error 纠正错误corresponding universal map 对应的通用映射countably infinite set 可列无限集（可列集）covariant functor (共变)函子covering 覆盖covering number 覆盖数Coxeter graph Coxeter 图crossing number of graph 图的叉数cuset 陪集cotree 余树cut edge 割边cut vertex 割点cycle 圈cycle basis 圈基cycle matrix 圈矩阵cycle rank 圈秩cycle space 圈空间cycle vector 圈向量cyclic group 循环群cyclic index 循环（轮转）指标cyclic monoid 循环单元半群cyclic permutation 圆圈排列cyclic semigroup 循环半群DDe Morgan law De Morgan 律decision procedure 判决过程decoding table 译码表deduction theorem 演绎定理degree 次数,次(度)degree sequence 次(度)序列derivation algebra 微分代数Descartes product Descartes 积designated truth value 特指真值detect errer 检验错误deterministic 确定的diagonal functor 对角线函子diameter 直径digraph 有向图dilemma 二难推理direct consequence 直接推论（直接后承）direct limit 正向极限direct sum 直和directed by inclution 被包含关系定向discrete Fourier transform 离散 Fourier 变换disjunctive normal form 正则析取范式disjunctive syllogism 选言三段论distance 距离distance transitive graph 距离传递图distinguished element 特异元distributive lattice 分配格divisibility 整除division subring 子除环divison ring 除环divisor (factor) 因子domain 定义域Driac condition Dirac 条件dual category 对偶范畴dual form 对偶式dual graph 对偶图dual principle 对偶原则(对偶原理) dual statement 对偶命题dummy variable 哑变量（哑变元）Eeccentricity 离心率edge chromatic number 边色数edge coloring 边着色edge connectivity 边连通度edge covering 边覆盖edge covering number 边覆盖数edge cut 边割集edge set 边集edge-independence number 边独立数eigenvalue of graph 图的特征值elementary divisor ideal 初等因子理想elementary product 初等积elementary sum 初等和empty graph 空图empty relation 空关系empty set 空集endomorphism 自同态endpoint 端点enumeration function 计数函数epimorphism 满态射equipotent 等势equivalent category 等价范畴equivalent class 等价类equivalent matrix 等价矩阵equivalent object 等价对象equivalent relation 等价关系error function 错误函数error pattern 错误模式Euclid algorithm 欧几里德算法Euclid domain 欧氏整环Euler characteristic Euler 特征Euler function Euler 函数Euler graph Euler 图Euler number Euler 数Euler polyhedron formula Euler 多面体公式Euler tour Euler 闭迹Euler trail Euler 迹existential generalization 存在推广规则existential quantifier 存在量词existential specification 存在特指规则extended Fibonacci number 广义 Fibonacci 数extended Lucas number 广义Lucas 数extension 扩充（扩张）extension field 扩域extension graph 扩图exterior algebra 外代数Fface 面factor 因子factorable 可因子化的factorization 因子分解faithful (full) functor 忠实（完满）函子Ferrers graph Ferrers 图Fibonacci number Fibonacci 数field 域filter 滤子finite extension 有限扩域finite field (Galois field ) 有限域（Galois 域）finite dimensional associative division algebra有限维结合可除代数finite set 有限（穷）集finitely generated module 有限生成模first order theory with equality 带符号的一阶系统five-color theorem 五色定理five-time-repetition 五倍重复码fixed point 不动点forest 森林forgetful functor 忘却函子four-color theorem(conjecture) 四色定理（猜想）F-reduced product F-归纳积free element 自由元free monoid 自由单元半群free occurrence 自由出现free R-module 自由R-模free variable 自由变元free-Ω-algebra 自由Ω代数function scheme 映射格式GGalileo paradox Galileo 悖论Gauss coefficient Gauss 系数GBN (Gödel-Bernays-von Neumann system)GBN系统generalized petersen graph 广义 petersen 图generating function 生成函数generating procedure 生成过程generator 生成子（生成元）generator matrix 生成矩阵genus 亏格girth （腰）围长Gödel completeness theorem Gödel 完全性定理golden section number 黄金分割数（黄金分割率）graceful graph 优美图graceful tree conjecture 优美树猜想graph 图graph of first class for edge coloring 第一类边色图graph of second class for edge coloring 第二类边色图graph rank 图秩graph sequence 图序列greatest common factor 最大公因子greatest element 最大元（素）Grelling paradox Grelling 悖论Grötzsch graph Grötzsch 图group 群group code 群码group of graph 图的群HHajós conjecture Hajós 猜想Hamilton cycle Hamilton 圈Hamilton graph Hamilton 图Hamilton path Hamilton 路Harary graph Harary 图Hasse graph Hasse 图Heawood graph Heawood 图Herschel graph Herschel 图hom functor hom 函子homemorphism 图的同胚homomorphism 同态（同态映射）homomorphism of graph 图的同态hyperoctahedron 超八面体图hypothelical syllogism 假言三段论hypothese (premise) 假设(前提)Iideal 理想identity 单位元identity natural transformation 恒等自然变换imbedding 嵌入immediate predcessor 直接先行immediate successor 直接后继incident 关联incident axiom 关联公理incident matrix 关联矩阵inclusion and exclusion principle 包含与排斥原理inclusion relation 包含关系indegree 入次（入度）independent 独立的independent number 独立数independent set 独立集independent transcendental element 独立超越元素index 指数individual variable 个体变元induced subgraph 导出子图infinite extension 无限扩域infinite group 无限群infinite set 无限（穷）集initial endpoint 始端initial object 初始对象injection 单射injection functor 单射函子injective (one to one mapping) 单射（内射）inner face 内面inner neighbour set 内（入）邻集integral domain 整环integral subdomain 子整环internal direct sum 内直和intersection 交集intersection of graph 图的交intersection operation 交运算interval 区间invariant factor 不变因子invariant factor ideal 不变因子理想inverse limit 逆向极限inverse morphism 逆态射inverse natural transformation 逆自然变换inverse operation 逆运算inverse relation 逆关系inversion 反演isomorphic category 同构范畴isomorphism 同构态射isomorphism of graph 图的同构join of graph 图的联JJordan algebra Jordan 代数Jordan product (anti-commutator) Jordan乘积（反交换子）Jordan sieve formula Jordan 筛法公式j-skew j-斜元juxtaposition 邻接乘法Kk-chromatic graph k-色图k-connected graph k-连通图k-critical graph k-色临界图k-edge chromatic graph k-边色图k-edge-connected graph k-边连通图k-edge-critical graph k-边临界图kernel 核Kirkman schoolgirl problem Kirkman 女生问题Kuratowski theorem Kuratowski 定理Llabeled graph 有标号图Lah number Lah 数Latin rectangle Latin 矩形Latin square Latin 方lattice 格lattice homomorphism 格同态law 规律leader cuset 陪集头least element 最小元least upper bound 上确界（最小上界）left (right) identity 左（右）单位元left (right) invertible element 左（右）可逆元left (right) module 左（右）模left (right) zero 左（右）零元left (right) zero divisor 左（右）零因子left adjoint functor 左伴随函子left cancellable 左可消的left coset 左陪集length 长度Lie algebra Lie 代数line- group 图的线群logically equivanlent 逻辑等价logically implies 逻辑蕴涵logically valid 逻辑有效的（普效的）loop 环Lucas number Lucas 数Mmagic 幻方many valued proposition logic 多值命题逻辑matching 匹配mathematical structure 数学结构matrix representation 矩阵表示maximal element 极大元maximal ideal 极大理想maximal outerplanar graph 极大外平面图maximal planar graph 极大平面图maximum matching 最大匹配maxterm 极大项（基本析取式）maxterm normal form(conjunctive normal form) 极大项范式（合取范式）McGee graph McGee 图meet 交Menger theorem Menger 定理Meredith graph Meredith 图message word 信息字mini term 极小项minimal κ-connected graph 极小κ-连通图minimal polynomial 极小多项式Minimanoff paradox Minimanoff 悖论minimum distance 最小距离Minkowski sum Minkowski 和minterm (fundamental conjunctive form) 极小项（基本合取式）minterm normal form(disjunctive normal form)极小项范式（析取范式）Möbius function Möbius 函数Möbius ladder Möbius 梯Möbius transform (inversion) Möbius 变换（反演）modal logic 模态逻辑model 模型module homomorphism 模同态(R-同态)modus ponens 分离规则modus tollens 否定后件式module isomorphism 模同构monic morphism 单同态monoid 单元半群monomorphism 单态射morphism (arrow) 态射（箭）Möbius function Möbius 函数Möbius ladder Möbius 梯Möbius transform (inversion) Möbius 变换（反演）multigraph 多重图multinomial coefficient 多项式系数multinomial expansion theorem 多项式展开定理multiple-error-correcting code 纠多错码multiplication principle 乘法原理mutually orthogonal Latin square 相互正交拉丁方Nn-ary operation n-元运算n-ary product n-元积natural deduction system 自然推理系统natural isomorphism 自然同构natural transformation 自然变换neighbour set 邻集next state 下一个状态next state transition function 状态转移函数non-associative algebra 非结合代数non-standard logic 非标准逻辑Norlund formula Norlund 公式normal form 正规形normal model 标准模型normal subgroup (invariant subgroup) 正规子群（不变子群）n-relation n-元关系null object 零对象nullary operation 零元运算Oobject 对象orbit 轨道order 阶order ideal 阶理想Ore condition Ore 条件orientation 定向orthogonal Latin square 正交拉丁方orthogonal layout 正交表outarc 出弧outdegree 出次(出度)outer face 外面outer neighbour 外（出）邻集outerneighbour set 出(外)邻集outerplanar graph 外平面图Ppancycle graph 泛圈图parallelism 平行parallelism class 平行类parity-check code 奇偶校验码parity-check equation 奇偶校验方程parity-check machine 奇偶校验器parity-check matrix 奇偶校验矩阵partial function 偏函数partial ordering (partial relation) 偏序关系partial order relation 偏序关系partial order set (poset) 偏序集partition 划分,分划,分拆partition number of integer 整数的分拆数partition number of set 集合的划分数Pascal formula Pascal 公式path 路perfect code 完全码perfect t-error-correcting code 完全纠-错码perfect graph 完美图permutation 排列（置换）permutation group 置换群permutation with repetation 可重排列Petersen graph Petersen 图p-graph p-图Pierce arrow Pierce 箭pigeonhole principle 鸽子笼原理planar graph （可）平面图plane graph 平面图Pólya theorem Pólya 定理polynomail 多项式polynomial code 多项式码polynomial representation 多项式表示法polynomial ring 多项式环possible world 可能世界power functor 幂函子power of graph 图的幂power set 幂集predicate 谓词prenex normal form 前束范式pre-ordered set 拟序集primary cycle module 准素循环模prime field 素域prime to each other 互素primitive connective 初始联结词primitive element 本原元primitive polynomial 本原多项式principal ideal 主理想principal ideal domain 主理想整环principal of duality 对偶原理principal of redundancy 冗余性原则product 积product category 积范畴product-sum form 积和式proof (deduction) 证明（演绎）proper coloring 正常着色proper factor 真正因子proper filter 真滤子proper subgroup 真子群properly inclusive relation 真包含关系proposition 命题propositional constant 命题常量propositional formula(well-formed formula,wff)命题形式（合式公式）propositional function 命题函数propositional variable 命题变量pullback 拉回(回拖) pushout 推出Qquantification theory 量词理论quantifier 量词quasi order relation 拟序关系quaternion 四元数quotient (difference) algebra 商（差）代数quotient algebra 商代数quotient field (field of fraction) 商域（分式域）quotient group 商群quotient module 商模quotient ring (difference ring , residue ring) 商环（差环，同余类环）quotient set 商集RRamsey graph Ramsey 图Ramsey number Ramsey 数Ramsey theorem Ramsey 定理range 值域rank 秩reconstruction conjecture 重构猜想redundant digits 冗余位reflexive 自反的regular graph 正则图regular representation 正则表示relation matrix 关系矩阵replacement theorem 替换定理representation 表示representation functor 可表示函子restricted proposition form 受限命题形式restriction 限制retraction 收缩Richard paradox Richard 悖论right adjoint functor 右伴随函子right cancellable 右可消的right factor 右因子right zero divison 右零因子ring 环ring of endomorphism 自同态环ring with unity element 有单元的环R-linear independence R-线性无关root field 根域rule of inference 推理规则Russell paradox Russell 悖论Ssatisfiable 可满足的saturated 饱和的scope 辖域section 截口self-complement graph 自补图semantical completeness 语义完全的（弱完全的）semantical consistent 语义相容semigroup 半群separable element 可分元separable extension 可分扩域sequent 矢列式sequential 序列的Sheffer stroke Sheffer 竖（谢弗竖）simple algebraic extension 单代数扩域simple extension 单扩域simple graph 简单图simple proposition (atomic proposition) 简单（原子）命题simple transcental extension 单超越扩域simplication 简化规则slope 斜率small category 小范畴smallest element 最小元（素）Socrates argument Socrates 论断（苏格拉底论断）soundness (validity) theorem 可靠性（有效性）定理spanning subgraph 生成子图spanning tree 生成树spectra of graph 图的谱spetral radius 谱半径splitting field 分裂域standard model 标准模型standard monomil 标准单项式Steiner triple Steiner 三元系大集Stirling number Stirling 数Stirling transform Stirling 变换subalgebra 子代数subcategory 子范畴subdirect product 子直积subdivison of graph 图的细分subfield 子域subformula 子公式subdivision of graph 图的细分subgraph 子图subgroup 子群sub-module 子模subrelation 子关系subring 子环sub-semigroup 子半群subset 子集substitution theorem 代入定理substraction 差集substraction operation 差运算succedent 后件surjection (surjective) 满射switching-network 开关网络Sylvester formula Sylvester公式symmetric 对称的symmetric difference 对称差symmetric graph 对称图symmetric group 对称群syndrome 校验子syntactical completeness 语法完全的（强完全的）Syntactical consistent 语法相容system Ł3 , Łn , Łא0 , Łא系统Ł3 , Łn , Łא0 , Łאsystem L 公理系统 Lsystem Ł公理系统Łsystem L1 公理系统 L1system L2 公理系统 L2system L3 公理系统 L3system L4 公理系统 L4system L5 公理系统 L5system L6 公理系统 L6system Łn 公理系统Łnsystem of modal prepositional logic 模态命题逻辑系统system Pm 系统 Pmsystem S1 公理系统 S1system T (system M) 公理系统 T(系统M)Ttautology 重言式（永真公式）technique of truth table 真值表技术term 项terminal endpoint 终端terminal object 终结对象t-error-correcing BCH code 纠 t -错BCH码theorem (provable formal) 定理（可证公式）thickess 厚度timed sequence 时间序列torsion 扭元torsion module 扭模total chromatic number 全色数total chromatic number conjecture 全色数猜想total coloring 全着色total graph 全图total matrix ring 全方阵环total order set 全序集total permutation 全排列total relation 全关系tournament 竞赛图trace (trail) 迹tranformation group 变换群transcendental element 超越元素transitive 传递的tranverse design 横截设计traveling saleman problem 旅行商问题tree 树triple system 三元系triple-repetition code 三倍重复码trivial graph 平凡图trivial subgroup 平凡子群true in an interpretation 解释真truth table 真值表truth value function 真值函数Turán graph Turán 图Turán theorem Turán 定理Tutte graph Tutte 图Tutte theorem Tutte 定理Tutte-coxeter graph Tutte-coxeter 图UUlam conjecture Ulam 猜想ultrafilter 超滤子ultrapower 超幂ultraproduct 超积unary operation 一元运算unary relation 一元关系underlying graph 基础图undesignated truth value 非特指值undirected graph 无向图union 并(并集)union of graph 图的并union operation 并运算unique factorization 唯一分解unique factorization domain (Gauss domain) 唯一分解整域unique k-colorable graph 唯一k着色unit ideal 单位理想unity element 单元universal 全集universal algebra 泛代数（Ω代数）universal closure 全称闭包universal construction 通用结构universal enveloping algebra 通用包络代数universal generalization 全称推广规则universal quantifier 全称量词universal specification 全称特指规则universal upper bound 泛上界unlabeled graph 无标号图untorsion 无扭模upper (lower) bound 上（下）界useful equivalent 常用等值式useless code 废码字Vvalence 价valuation 赋值Vandermonde formula Vandermonde 公式variery 簇Venn graph Venn 图vertex cover 点覆盖vertex set 点割集vertex transitive graph 点传递图Vizing theorem Vizing 定理Wwalk 通道weakly antisymmetric 弱反对称的weight 重（权）weighted form for Burnside lemma 带权形式的Burnside引理well-formed formula (wff) 合式公式(wff)word 字Zzero divison 零因子zero element (universal lower bound) 零元（泛下界）ZFC (Zermelo-Fraenkel-Cohen) system ZFC系统form)normal(Skolemformnormalprenex-存在正则前束范式(Skolem 正则范式)3-value proposition logic 三值命题逻辑。
[Hoff93] A.Hoffmann, H.-J. Eikerling, W.Hardt, R.Geneveriere, and R.Camposano. “ISF-Eine Entwurfsdarstellung für High-Level-Synthese.”Workshop des SFB 358 der TU Dresden und der ITG/GI,1993.[Kern78] B.W. Kernighan and D.M. Richtie.The C Programming Language. Prentice Hall, 1978.[Leis91] C.E. Leiserson and J.B. Saxe. “Retiming synchronous circuits.”In Algorithmica 6, 1991.[Mali91]S.Malik, E.M. Sentovich, R.K. Brayton, and A.Sangiovanni-Vincentelli. “Retiming and Resynthesis: Optimizing sequential networks with combinational.”IEEE Transactions on Computer-Aided Design,1991.[McFa78]M.C. McFarland. “The Value Trace: A data base for automated digital Design.” Technical report, Design Research Center, Carnegie-Mellon University, Pittsbourg, USA, December 1978.[Orai86] A.Orailogu and D.D. Gajski. “Flow Graph Representation.”Proc. of 23rd Design Automation Conference, Las Vegas, 1986.[Rund90] E.A. Rundensteiner and D.D. Gajski. “A Design representation model for High-Level-Synthesis.”Technical report, University of California, University of Irvine, 1990.[Sent92] E.M. Sentovich. “SIS: A System for sequential circuit Synthesis.” Technical report, University of California at Berbeley, 1992. Memorandum No. UCB/ERL M92/41.[Stal92]R.M. ing and Porting GNU CC. Free Software Foundation Inc., December 1992.[Stro91] B.Stroustrup.The C++ Programming Language - Second Edition. Addison-Wesley Publishing Company, 1991.[Syno92]Synopsys, Inc. “VHDL Design Analyser Manual”, Synopsys Inc., 1992.[Univ91]Stanford University. “Interviews Reference Manual Version 3.0.” Technical report, University of California, Palo Alto, USA, 1991.[Vlis89]J.M. Vlissides and M.A. Linton. “Unidraw: A Framework for building domain-specific graphical Editors.”Proc. of the ACM SIGGRAPH/SIGCHI User Interface Software and Technologies, 1989.ularity in the synthesis process. Therefore the transformation between the two layers is possible with-out loss of information. In our opposite approach we maintain information to show the correspondence between operations in the behavioral description and functional modules in the netlist.The system provides high flexibility done by comfortable interfaces. Input to the PSF data represen-tation is possible by specifications in C/C++ and VHDL. Output of the PSF can be done in C from the behavioral level and in VHDL and BLIF from the structural level. Using this features, also redesign is possible. Several editors exist to control the modular synthesis process.The PSF and its environment, has been implemented in C++ on SUN SPARC stations under UNIX. The state of implementation allows to handle the standard HLS benchmarks. Additionally it is possi-ble to transform various hierarchical C-programs into the extended CDFG.Forthcoming research activities will apply a wide range of applicationsonto PSF from HW/SW-Code-sign [Hard93] to sequential logic synthesis [Fesk93].These work was funded by the Deutsche Forschungsgemeinschaft, under the project SFB 358.6References[Aho87] A.V. Aho, R.Sethi, and J.D. PILERS Principles, Techniques and Tools. Bell Telephone Laboratories, Inc., 1987.[Camp85]R.Camposano and R.Weber. “Compilation and Internal Representation of Digital Systems Specification in DSL.”Sixth Conference Latino-Americana in Informatics, Brazil, 1985.[Camp87]R Camposano and J.T.J. van Eijndhoven. “Partitioning a Design in structural synthesis.”ICCD, pages 564–566, 1987.[Camp89]R.Camposano and R.M. Tabet. “Design Representation for the Synthesis of Behavioral VHDL models.”CHDL’89 Eliesevier Science Publishers, 1989.[Camp92]R.Camposano and P.G.Ploeger. “Retiming and High-Level Synthesis.”Sixth ACM / IEEE Int. WS on High-Level Synthesis, 1992.[Eike93]H.-J. Eikerling and R.Camposano. “CP/DP-Partitionierung und Resynthese.”Handouts 6. EIS Workshop Entwurf Integrierter Schaltungen, 1993.[Erns92]R.Ernst and J.Henkel. “Hardware-Software Co-Design of Embeded Controllers based on Hardware Extraction.”Proceedings of the ACM Workshop on Hardware-Software Co-design, October 1992. [Fesk93]K.Feske, G.Franke, M.Koegst, and ttig. “Iterative fsm structuring by partial state assignment.”Technical report, Fraunhofer-Institut fuer Integrierte Schaltungen, Erlangen, Aussenstelle Dresden,Germany, 1993.[Gajs92] D.Gajski, N.Dutt, A.Wu, and S.Lin.HIGH-LEVEL-SYNTHESIS - Introduction to Chip and System Design. Kluwer Academic Publishers, 1992.[Gupt92]R.K. Gupta and G.DeMicheli. “System Synthesis via Hardware - Software Co - Design.”CSL TR-93-548, 1992.[Hard93]W.Hardt and R.Camposano. “Trade-Offs in HW/SW Codesign.” 1993.Fig. 5: Inspection of scheduling in of Fig. 4’s loopbodyIn the left most picture you can see the data flow graph. The editor option “show data” makes it pos-sible to prove the annotations of the selected node. In Fig. 5 you can see the value for the asap and alap values (to calculate the mobility for other scheduling algorithms), and the value for the “result-ing” control step. In addition the symbolic state of the add operation is displayed. This is used by the forthcoming creation of the FSM. In the right most display, there are more detailed informations of the add operation, i.e. bitwidth.Fig. 4: CDFG of the design flow exampleThe presentation of the data flow shows the structuring in basic blocks. The loop body is represented by one single basic block. The black colored node in the DFG is the begin of the basic block’s data flow. Hence, this shows the interdependence. Edges represent data flow. Variable nodes in the DFG reflect the variables of the input source code. Nodes which are presenting constants are presented by its value and operators are presented by its C code symbol. The “ArrRd” node in the DFG should rep-resent the reading of a value of an array element. The index of the array element is connected with this2Paderborn Synthesis Format PSFA special item of the Paderborn Synthesis Format is the partitioning into two abstract layers. The ob-jective of this concept is to supply a data representation which suits different applications in the design process. The layer with higher abstraction is called extended Control-Data-Flow-Graph, CDFG. This representation has been chosen to support software oriented applications on algorithmic level like compiler transformations or HW/SW partitioning. The second layer is represented by a finite state ma-chine and a netlist. This structural layer is called Control-Data-Path, CDP. This abstraction suits syn-thesis algorithms like retiming [Leis91] and its extensions [Mali91], [Camp92] FSM partitioning and redesign [Eike93]. Both layers are supported by using the Library of efficient Datatypes and Algo-rithms, LEDA [Naeh92]. Its graph package has been incremented by hyperedges and hierarchy. 2.1Extended Control-Data-Flow-GraphFlow graphs are widely accepted representations in the design automation community. Representa-tions like VT by McFarland [McFa78], DSL by Camposano et. al. [Camp85] do only take data flow into account. This data flow is augmented by explicit control flow. The representation of different se-mantics in one single structure makes the interpretation more difficult. A hybrid representation of con-trol and data flow provides easier access to necessary information [Orai86].The Control-Data-Flow-Graph of the PSF consists out of two hierarchical graphs, the data flow graph, DFG and the control flow graph, CFG. Both together present a behavioral description of the system. The opportunity to cover hierarchy in the CDFG’s representation is necessary to reflect the software orientated requirements. This supports exposition of information which is not done by inlining. Inlin-ing is the flattening of function calls into a non hierarchical form.There exists a correspondence between these two graphs to identify the parts of control flow with the data flow. The representation of the control flow graph uses common compiler approaches. Loops and conditional branches will be presented by special nodes i.e. LoopBegin- and LoopEnd-Nodes and edges to connect them (Fig. 4). Data flow is structured by basic blocks. Basic blocks which are code sequences which are performed in connection are defined in compiler theory [Aho87]. This structur-ing supports the easy recognition of maximal parallel degree. In contrast to other systems the data flow graph connects the basic blocks into one global data flow graph of a system. This allows global data analysis by presenting an overall data base to variables, which gives potential optimization over basic blocks boundaries.A specialty of the CDFG’s representation is the extended amount of data which is handled in its struc-tures. The CDFG is augmented by synthesis applications with data i.e. scheduling information. An important aspect is the structural information which is applied to the CDFG. For instance, the infor-mation about the functional unit to perform an operation in hardware and their multiplexer inputs are bound to the CDFG. This method allows to present interdependence between the PSF’s behavioral layer CDFG and the structural layer CDP. For example an operation in the CDFG can be identified by its functional unit in the synthesized netlist.1Motivation and IntroductionState-of-the-Art synthesis systems try to allow IC design independent of the specification language. Often it is necessary to implement a system’s specification in one language to use commercial tools. In “classical” task of design automation like processor design the usage of one specification language is no disadvantage because the processor designers do usually have high skills in hardware design lan-guages (VHDL, Verilog,..). Current trends are to synthesize parts of a design which were former im-plemented in software into hardware. This requires that programming languages which are used for software design are to be taken in consideration in hardware design approaches. Under the catchword HW/SW-Codesign there have been several publications [Hard93], [Erns92], [Gupt92].The presented Paderborn Modular Synthesis System, PMOSS for HW/SW-Codesign and High-Lev-el-Synthesis is based upon the Paderborn Synthesis Format PSF. PSF which is a further development of the Internal Synthesis Format, ISF [Hoff93]. The PSF satisfies the needs of a uniformed data format for High-Level-Synthesis, HLS. Rundensteiner & Gajski [Rund90] and Camposano & Tabet [Camp89] describe the necessity of such a data base.Gajski et. al.[Gajs92] describe distinguished classes of HLS applications like•scheduling•allocation•bindingand illustrate their exchangeability in one algorithmic class. HLS is performed in several steps which do have a massive interdependence. The algorithms which are used in design automation use different representations. As examples can be given scheduling which requires a design representation of con-trol and data flow and retiming introduced by Leiserson & Saxe [Leis91] which requires a data repre-sentation as a netlist.The PSF presents two representations in different levels of abstraction to provide these algorithms an abstraction level which satisfies their needs. The behavioral description of a system will be transferred into a Control-Data-Flow-Graph, CDFG. This representation consists out of two graphs which de-scribe control and data flow as well as their interdependence. The structural description is presented by a Control-Data-Path, CDP. This structure consists out of a finite state machine and a hierarchical netlist.PMOSS synthesis approach annotates the CDFG with several information. Like mentioned in the be-ginning the single tasks of HLS are in great interdependence. The PSF synthesis process’ major con-cept is modularity. This approach has been taken to provide optimal succession of each task and an optimal choice of synthesis. The CDFG is to be assigned with HLS informations like scheduling steps, functional units to perform an operation and its instance etc. Each algorithm can be started after the required data has been assigned to the CDFG. These additional structural information is the reason why this is called “extended CDFG”.The front ends which transfer a system specification into the behavioral layer of the PSF takes the HW/SW-Codesign into account by using a software language as input language. Systems which are specified in C or the objectoriented extension C++ can be handled with the Control-Data-Flow-Graph. This front end is based upon the GNU Compiler [Stal92]. An additional front end exists to transfer VHDL descriptions into the CDFG of the PSF. This second input path provides the combination of。
a r X i v :m a t h /0603065v 3 [m a t h .Q A ] 16 O c t 2006Full field algebras,operads and tensor categories Liang Kong Abstract We study the operadic and categorical formulations of (conformal)full field algebras.In particular,we show that a grading-restricted R ×R -graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates.This result is general-ized to conformal full field algebras over V L ⊗V R ,where V L and V R are two vertex operator algebras satisfying certain finiteness and reductivity conditions.We also study the geometry interpretation of conformal full field algebras over V L ⊗V R equipped with a nondegenerate invariant bi-linear form.By assuming slightly stronger conditions on V L and V R ,we show that a conformal full field algebra over V L ⊗V R equipped with a non-degenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of V L ⊗V R -modules.The so-called diagonal constructions [HK2]of conformal full field algebras are given in tensor-categorical language.0Introduction In [HK2],Huang and the author introduced the notion of conformal full field algebra and some variants of this notion.We also studied their basic properties andgave constructions.We explained briefly without giving details in the introduction of [HK2]that our goal is to construct conformal field theories [BPZ][MS].It is one of the purpose of this work to explain the connection between conformal full field algebras and genus-zero conformal field theories.[HK2]and this work are actually a part of Huang’s program ([H1]-[H13])of constructing rigorously conformal field theories in the sense of Kontsevich and Segal [S1][S2].Around 1986,I.Frenkel initiated a program of using vertex operator algebras to construct,in a suitable sense,geometric conformal field theories,the precise mathematical definition of which was actually given independently by Kontsevich1and Segal[S1][S2]in1987.According to Kontsevich and Segal,a conformalfield theory is a projective tensor functor from a category consisting offinite many ordered copies of S1as objects and the equivalent classes of Riemann surfaces with parametrized boundaries as morphisms to the category of Hilbert spaces.This beautiful and compact definition of conformalfield theory encloses enormously rich structures of conformalfield theory.In order to construct such theories,it is more fruitful to look at some substructures of conformalfield theories atfirst.In the first category,the set of morphisms which have arbitrary number of copies of S1in their domains and only one copy of S1in their codomain has a structure of operad [Ma],which is induced by gluing surfaces along their parametrized boundaries.We denoted this operad as K H.Let H,a Hilbert space,be the image object of S1. The projective tensor functor in the definition of conformalfield theory endows H with a structure of algebra over an operad,which is a C-extension of the operad K H[H3].It is very difficult to study this algebra-over-operad structure on H directly.It was suggested by Huang[H12][H13]that one shouldfirst look at a dense subset of H,which carries a structure of an algebra over a partial operad˜K c⊗2-power of the determine line bundle over thepartial operad K.The restriction of the line bundle˜K c on K H,denoted as˜K c H, is also a suboperad.˜K c⊗˜Kproducing modular invariant genus-one correlation functions when c L=c R.Weshow in Section1that the grading-restricted R×R-graded fullfield algebras are exactly smooth algebras over K,which is partial suboperad of K,and confor-mal fullfield algebra over V L⊗V R are exactly smooth algebras over˜K c L⊗c R.The modular invariance property of conformal fullfield algebras will be studied in [HK3].In order to cover the entire genus-zero conformalfield theories,one still needsto consider the Riemann surfaces with more than one copy of S1in the codomain. In chiral theory,it was studied by Hubbard in the framework of so-called vertex operator coalgebras[Hub1][Hub2].In a theory including both chiral and antichiral parts,it amounts to have a conformal fullfield algebra equipped with a nondegen-erate invariant bilinear form[HK2].Conversely,a conformal fullfield algebra with a nondegenerate invariant bilinear form gives a complete set of data needed for a genus-zero conformalfield theory(nonunitary).In particular,we show in Section 2that such conformal fullfield algebras are just algebraic representations of the sewing operations among spheres with arbitary number of negatively oriented and positively oriented punctures as long as the resulting surfaces after sewing are still of genus-zero.The invariant property of bilinear form used in this work is slightly different from that in[HK2].Both definitions have clear geometric meanings.But we need the new definition for a reason which is explained later.Although the notion of conformal fullfield algebras has the advantage of being a pure algebraic formulation of a part of genus-zero conformalfield theories,its axioms are still very hard to check directly.The theory of the tensor products of the modules of vertex operator algebras is developed by Huang and Lepowsky [HL1]-[HL4][H2][H7].The following Theorem proved by Huang in[H7]is very crucial for our constructions of conformal fullfield algebras.Theorem0.1.Let V be a vertex operator algebra satisfying the following condi-tions:1.Every C-graded generalized V-module is a direct sum of C-graded irreducibleV-modules,2.There are onlyfinitely many inequivalent C-graded irreducible V-modules,3.Every R-graded irreducible V-module satisfies the C1-cofiniteness condition. Then the direct sum of all(in-equivalent)irreducible V-modules has a natural structure of intertwining operator algebra and the category of V-modules,denoted as C V has a natural structure of vertex tensor category.In particular,C V has natural structure of braided tensor category.3Assumption0.2.All the vertex operator algebras appeared in this work,V,V L and V R are all assumed to satisfy the conditions in Theorem0.1without further announcement.Sometimes we even assume stronger conditions on them,as we will do explicitly in Section4and Section5.In[HK2],we have studied in detail the properties of conformal fullfield algebras over V L⊗V R.In particular,an equivalent definition of conformal fullfield algebra over V L⊗V R is also given.We recalled this result in Theorem3.2.The axiom in this definition are much easier to verify than those in the original definition of conformal fullfield algebra.Also by Theorem0.1,the categories of V L-modules, V R-modules and V L⊗V R-modules,denoted as C V L,C V L and C V L⊗V R respectively, all have the structures of braided tensor category[H7][HK2].In this work,the braiding structure in C V L⊗V R is chosen to be different from the one obtained from ing Theorem3.2and this new braiding structure on C V L⊗V R,we can show,without much effort,that a conformal fullfield algebra over V L⊗V R is equivalent to a commutative associative algebra in C V L⊗V R with a trivial twist.We would also like to give a categorical formulation of conformal fullfield algebra over V L⊗V R equipped with a nondegenerate invariant bilinear form, where V L and V R are assumed to satisfy the conditions in Theorem4.5,which are slightly stronger than those in Theorem0.1.It turns out that the way we define the invariant property of bilinear form of conformal fullfield algebra in[HK2]is not easy to work with categorically.This is the reason why a slightly modified notion of invariant bilinear form is introduced in Section2.This modification leads us to consider on the graded dual space of a module over a vertex operator algebra a module structure which is different from(but equivalent to)the usual contragredient module structure[FHL].Because of this,we redefine the duality maps[H9][H11]in this new convention and prove the rigidity(in appendix).Then we show in detail,in Section4,that a conformal fullfield algebra over V L⊗V R equipped with a nondegenerate invariant bilinear form in the new sense exactly amounts to a commutative Frobenius algebra with a trivial twist in C V L⊗V R.Once the categorical formulation is known.We can give a categorical con-struction of conformal fullfield algebras equipped with a nondegenerate invariant bilinear form.This construction was previously given by Huang and the author in [HK2]by using intertwining operator algebras,and was also known to physicists as diagonal construction(see for example[FFFS]and references therein).Recently,Fuchs,Runkel,Schweigert and Fjelstad have proposed a very gen-eral construction of all correlation functions of boundary conformal conformalfield theories using3-dimensional topologicalfield theories in a series of papers[FRS1]-[FRS4][FjFRS][RFFS][SFR].In particular,a construction of commutative asso-ciative algebras in C V⊗V is explicitly given in[RFFS].Our approach is somewhat4complementary to their approach(see[RFFS]for comments on the relation of two approachs).We hope that two approachs can be combined to obtain a rather complete picture of conformalfield theory in the near future.The layout of this paper is as follow.In Section1,we study the operadic formulation of grading-restricted R×R-graded fullfield algebra and its variants, following the work of Huang[H3].In Section2,we give a the geometric description of a conformal fullfield algebra over V L⊗V R equipped with a nondegenerate invariant bilinear form.In Section3,we give a categorical formulation of conformal fullfield algebra over V L⊗V R.In Section4,we give a categorical formulation of conformal fullfield algebra over V L⊗V R with a nondegenerate invariant bilinear form for V L and V R satisfying the conditions in Theorem4.5.In Section5,we give a categorical construction of conformal fullfield algebras over V L⊗V R and prove that such obtained conformal fullfield algebras over V L⊗V R are naturally equipped with a nondegenerate invariant bilinear form.For the convenience of readers,the materials in Section3,4,5are completely independent of those in Section1,2.For those who is only interested in categorical formulation of conformal fullfield algebras over V L⊗V R,it is harmless to start from Section3directly.Convention of notations:N,R,R+,C,H,ˆC,ˆH denote the set of natural num-bers,real numbers,positive real numbers and complex numbers,and upper half plane,one point compactification of C and H∪R,respectively.We also use I F to denote the identity map on a vector space F.Note.After this paper appeared online in math arxiv,the author noticed that the categorical construction of commutative associative algebra in C V⊗V given in this work is nothing but a basis independent version of that in[FrFRS]which appeared earlier.There is another natural point of view of this construction in terms of adjoint functors.It will be discussed elsewhere.Acknowledgment This work grows from a chapter in author’s thesis.I want to thank my advisor Yi-Zhi Huang for his constant support and many important suggestions.I also want to thank him for spending incredible amount of time in helping me to improve the writting of my thesis and this work.I thank J.Lepowsky and C.Schweigert for many inspiring conversations related to this work.1Operadic formulations of fullfield alegbrasIn this section,we study the operadic formulation of fullfield algebra.In section 1.1,we recall the notion of sphere partial operad K and its partial suboperad K. We introduce the notion of smooth function on K.In section1.2,we recall the5notions of determinant line bundle over K and the C-extensions of K,such as˜K c and˜K c L⊗c R for c,c L,c R∈C.Section1.1and1.2are mainly taken from[H3].The readers who is interested in knowing more on this subject should consult with [H3]for details.In section1.3,we recall the notion of algebra over partial operad [H3],and explain what it means for an algebra over K and˜K c L⊗c R to be smooth. Then we give two isomorphism theorems.Thefirst one says that the category ofgrading-restricted R×R-graded fullfield algebras is isomorphic to the category of smooth K-algebras.The second one says that the category of conformal fullfield algebras over V L⊗V R is isomorphic to the category of smooth˜K c L⊗c R-algebras over V L⊗V R.We give a selfcontent proof of thefirst isomorphism theorem.The proof of the second isomorphism theorem is technical,and heavily depends on the results in[H3].1.1Sphere partial operad KA sphere with tubes of type(n−,n+)is a sphere S with n−ordered punctures p i,i=1,...,n−and n+ordered punctures q j,j=1,...,n+,together with a negatively oriented local chart(U i,ϕi)around each p i and a positively oriented local chart(V j,ψj)around each q i,where U i and V j are neighborhood of p i and q j respectively and the local coordinate mapsϕi:U i→C andψj:V j→C are conformal maps so thatϕ(p i)=ψj(q j)=0.The conformal equivalence of sphere with tubes is defined to be the conformal maps between two spheres so that the germs of local coordinate mapϕi andψj are preserved.We then obtain a moduli space of sphere with tubes of type(n−,n+). In this section,We are only interested in spheres with tubes of type(1,n).For this type of spheres with tubes,we label the only negative oriented puncture as the0-th puncture.We denote the moduli space of sphere with tubes of type(1,n) as K(n).Using automorphisms of sphere,we can select a canonical representative from each conformal equivalence class in K(n)for all n>0byfixing the n-th puncture at0∈C,the0-th puncture at∞,andψ0,the local coordinate map at∞,to be so thatwψ0(w)=−1.(1.1)limw→∞As a consequence,the moduli space K(n),n∈Z+can be identified withK(n)=M n−1×H×(C××H)nwhereM n−1={(z1,...,z n−1)|z i∈C×,z i=z j,for i=j}6andH={A=(A1,A2,...)∈∞i=1C|A i∈C,e ∞j=1A j x j+1ddx x x=1dw w,∀i=1,...,n.(1.4)Let P∈K(m)and Q∈K(n).Let¯B r be the closed ball in C centered at0 with radius r,ϕi the germs of local coordinate map at i-th puncture p i of P,and ψ0the germs of local coordinate map at0-th puncture q0of Q.Then we say that the i-th tube of P can be sewn with0-th tube of Q if there is a r∈R+such that p i and q0are the only punctures inϕ−1i(¯B r)andψ−10(¯B1/r)respectively.A new sphere with tubes in K(m+n−1),denoted as P i∞0Q,can be obtained by cutting outϕ−1i(¯B r)andψ−10(¯B1/r)from P and Q respectively,and then identifying the boundary circle via the mapψ−1◦JˆH◦ϕiwhere JˆH:w→−1wis an automorphism of upper hand plane.7Therefore,we have sewing operations:i∞0:K(m)×K(n)→K(m+n−1)partially defined on the entire K between two spheres with tubes along two oppo-sitely oriented tubes.Remark1.2.Our definition of sewing operation is defined differently from that:w→1defined in[H3],where JˆCof K (1)together with the sewing operation 1∞0is a group isomorphic to C ×.There is also an obvious action of permutation group S n on K (n ).The following result is proved in [H3].Proposition 1.5.The collection of setsK ={K (n )}n ∈Ntogether with I K ,sewing operations,the actions of S n on K (n )and the group (1.5),is a C ×-rescalable partial operad.Let A (a ;i )={A j |A i =a,A j =0,j ∈Z +,j =i }.For simplicity,we will also use A (a ;i )to denote the element in K (0)such that the local coordinate map at ∞is given by −exp a 1d 1w=−exp −aw −i +1d w .Let K(n )be the subset of K (n )consisting of elements of the form (z 1,...,z n −1;A (a ;1),(a (1)0,0),...,(a (n )0,0)).(1.6)Then K={ K (n )}n ∈N is a partial suboperad of K [H3].We use “overline”to denote complex conjugation.A function f on K (n ),n ∈N is called smooth if there is a N ∈N such that f can be written asN k =1g k (A (0),a (1)0,A (1),...,a (n )0,A (n );a (1)0,a (n )0,A (i )j ,for i =0,...,n,j ∈Z +and linear combination of n i =1(a (i )0)r i ∂ǫ ǫ=0f (P (z )1∞0A (ǫ;2)),¯L I (¯z )f :=∂Proposition 1.6.L I (z )=z −2∂∂A (i )j I ,¯L I (¯z )=¯z −2∂a (1) I +1 i =0∞ j =1¯z −(2i −1)j −2∂A (i )j I .(1.9)Proof.Let P =P (z )1∞0A (ǫ,2)with its coordinates in moduli space given by (z ;A (0),(a (1)0,A (1)),(a (2)0,A (2))).It is shown in [H3]thatz ,A (0)j ,a (1)0,A (1)j ,a (2)0and A (2)j ,j ∈Z +are holomorphic functions of ǫ.Hence their complex conjugation ¯z ,a (1)0,a (2)0and1.2Determinant line bundle over KThe determinant line bundle over K and the C -extensions of K are studied in[H3].For each n ∈N ,the determinant line bundle Det(n )over K (n )is a trivial bundle over K (n ).We denote the fiber at Q ∈K (n )as Det Q .There is a canonical section of Det(n ),denoted by ψn ,for each n ∈N .For any element Q ∈K (n ),let µn (Q )be the element of the fiber over Q given byψn (Q )=(Q,µn (Q )).Then there is a λQ for each element ˜Qof Det Q such that ˜Q =(Q,λQ )and λQ =αµn (Q )for some α∈C .Consider the following two general elements in K (m )and K (n ).P =(z 1,...,z m ;A (0),(a (1)0,A (1)),...,(a (m )0,A (m )))Q =(ξ1,...,ξn ;B (0),(b (1)0,B (1)),...,(b (n )0,B (n )))(1.10)If P i ∞0Q exists,then there is a canonical isomorphism:l i P,Q :Det P ⊗Det Q →Det P i ∞0Q ,given as:l i P,Q(a 1µm (P )⊗a 2µn (Q ))=a 1a 2e 2Γ(A (i ),B (0),a (i )0)µm +n −1(P i ∞0Q ),(1.11)10whereΓis a C-valued analytic function of complex variables A(i)j,B(0)k ,a(i)0,j,k∈N.If we expandΓas formal series,we haveΓ(A(i),B(0),α)∈Q[α,α−1][[A(i),B(0)]].For a detailed discussion ofΓ,see chapter4in[H3].In particular,we haveA(i),a(i)0).(1.12) For(P,λP)∈Det(m)and(Q,λQ)∈Det(n)such that P i∞0Q exists,then we define a partially defined mapi ∞20:Det(m)×Det(n)→Det(m+n−1)by(P,λP)i ∞20(Q,λQ)=(P i∞0Q,l i P,Q(λP⊗λQ)).Using this partial operation,one obtain a C×-rescalable partial operad structure on Det={Det(n)}n∈N.For c∈C,the so-called vertex partial operad of central charge c,denoted as ˜K c,is the c˜K¯c of holomorphic line bundle˜K¯c also has a natural structure of partial operad.The section ofψon˜K¯c canonically gives a section¯ψonDet¯c P⊗Det¯c Pi∞0Q,for any pair of P,Q∈K so that P i∞0Q exists,can be written as¯l iP,Q(λ1⊗λ2)=λ1λ2eΓ(B(0),˜K¯c=In this work,we are also interested in the tensor product bundle˜K c L⊗c R for c L,c R∈C.It is clear that it is a C×-rescalable partial operad as well.The natural section induced from˜K c L and c R is simplyψ⊗¯ψ.We will denote the sewing operation on˜K c L⊗c R simply as ∞without making its dependents on c L,c R explicit.A function on˜K c or˜K c L⊗c R is called smooth if it is smooth on the base space K and linear onfiber.1.3Two isomorphism theoremsIn this subsection we discuss the operadic formulations of grading-restricted R×R-graded fullfield algebras and conformal fullfield algebras over V.We apologize for not recalling the definitions of these two notions here.They can be found in [HK2].Wefirst recall the definition of algebra over partial operad.Let G be a group and U a complete reducible G-module and W a G-submodule of U.We will useU such that image of W⊗n is in1Here,νn for n∈N is simply the restriction ofνon P(n).12If P is rescalable[H3],we call a P-pseudo-algebra a P-algebra.In this work,we are interested in studying K-algebras and˜K c L⊗c R-algebras, both of which are C×-rescalable partial operads.Definition1.8.A K-algebra(or˜K c L⊗c R-algebra)(F,W,ν)is called smooth if it satisfies the following conditions:1.F(m,n)=0if the real part of m or n is sufficiently small.2.For any n∈N,w′∈F′,w1,...,w n+1∈F,an element Q in K(n)(or˜K c L⊗c R(n)),the functionQ→ w′,ν(Q)(w1⊗···⊗w n+1)is smooth on K(n)(or on˜K c L⊗c R(n)).We choose the branch cut of logarithm as followlog z=log|z|+Arg z,0≤Arg z<2π.(1.15) We define the power functions,z m and¯z n for m,n∈R,to be e m log z and e nbe so that ˜ρ(0)=0.Therefore k =0.˜ρis actually a linear map from C →C .Namely,˜ρcan be written as˜ρ: x y→ a b c d x y (1.17)where x,y ∈R are the real part and the imaginary part of a complex number inC .Moreover,the group homomorphism preserves the identity,i.e.ρ:1→1.It implies that ˜ρ(2πi )=l 2πi for some l ∈Z .Applying this result to (1.17),we obtain that b =0and d =l ∈Z .Conversely,it is easy to see that every ˜ρof form (1.17)with b =0and d ∈Z gives arise to a group homomorphism ρ:C ×→C ×of the following form:z =e x +iy →e ax +i (cx +dy )=e (a +ic )x e diy =|z |a +ic z 2−d 2+d Now we study the basic properties of a smooth K -algebras.We fix a smooth K -algebras (F,W,ν).The set I :={(m,n )∈C ×C |m −n ∈Z }together with the usual addition operation gives an abelian group.By Lemma 1.9any K -algebras must be I ly,F = (m,n )∈I F (m,n ).Since ν1((0,(a,0)))for a ∈C ×gives the representation of C ×on F by the definition of algebra over operad,we must have ν1((0,a ))u =a −m ¯a −n u for u ∈F (m,n ).Let d L and d R be the grading operators such that d L u =mu and d R u =nu for u ∈F (m,n ).We call m the left weight of u ∈F (m,n )and n the right weight of u ,and denote them as wt L u and wt R u respectively.We also define wt u :=wt L u +wt R u which is called total weight.These two grading operators can also be obtained from ν1((0,(a,0)))as follow:d L =∂∂¯a a =1ν1((0,(a,0))).(1.19)Conversely,using d L and d R ,we can also express the action of ν1((0,(a,0)))on F as ν1((0,(a,0)))=a −d L ¯a −d R .(1.20)14By the definition of smooth function on K,the correlation functions are linear combination of i(a(i)0)r ily,1:=ν0((0)).(1.21) We call1the vacuum state.We have(0,(a,0))1∞0(0)=(0)for all a∈C×.This implies1∈F(0,0).Since K(0)={(0)},W is just C1.Let P(z)=(z;0,(1,0),(1,0))∈ K(2).We denote the linear mapν2(P(z)):F⊗F→Proof.First,for u,v∈F,we haveν2((z;0,(a1,0),(a2,0))(u⊗v)=ν2((P(z)1∞0(0,(a1,0)))2∞0(0,(a2,0)))(u⊗v)=(ν2(P(z))1∗0ν1((0,(a1,0))))2∗0ν1((0,(a2,0)))(u⊗v)=Y(a−d L1¯a−d R1u;z,¯z)a−d L2¯a−d R2v.(1.25)On the other hand,we also have(0,(a−1,0))1∞0P(z)=(az;0,(a−1,0),(a−1,0)).(1.26) Using(1.25),the image of(1.26)under the morphismνgivesa d L¯a d R Y(u;z,¯z)=Y(a d L¯a d R u;az,¯a¯z)a d L¯a d R,which implies(1.24).∂z Y(u;z,¯z)+Y(d L u;z,¯z)(1.27)d R,Y(u;z,¯z) =¯z∂∂s|s=0and∂We further define two operators D L,D R∈Hom(F,∂a a=0ν1((A(a;1),(1,0)),D R:=−∂It implies the following identity:w′,(ν1((A(a;1),(1,0)))1∗0ν1((0,(a1,0)))1∗0(ν1((A(b;1),(1,0)))1∗0ν1((0,(b1,0)))(w)= w′,ν1((A(a+b/a1;1),(1,0)))1∗0ν1((0,(a1b1,0)))(w) (1.30) for all w∈F,w′∈F′.Apply−∂∂b b=0,a=0,b1=1− −∂∂b1 b=0,a1=1,b1=1to both sides of equation(1.30).The left hand side of(1.30)gives(m,n)∈I w′,(d L P(m,n)D L−D L d L)w ,while the right hand of(1.30)gives−∂∂b b=0 w′,ν1((A(b/a1;1),(1,0)))a−d L1¯a−d R1w− −∂∂b1 b1=1 w′,ν1((A(a;1),(1,0)))b−d L1¯b−d R1w= −∂a1=1 −∂∂a a=0 −∂∂¯a1a1=1 w′,D L1∂a1Compare above two equations.It is clear that[d R,D L]=0.Combining above two results,we conclude that D L∈End V and has weight (1,0).Similarly,we can show[d L,D R]=0and[d R,D R]=D R.Therefore,D R is in End V as well and has weight(0,1).Next we show that[D L,D R]=0.The identity(A(a+b;1),(1,0))=(A(a;1),(1,0))1∞0(A(b;1),(1,0))=(A(b;1),(1,0))1∞0(A(a;1),(1,0))(1.32) in K implies that for w′,w∈F,w′,ν1((A(a;1),(1,0)))1∗0ν1((A(b;1),(1,0)))w= w′,ν1((A(b;1),(1,0)))1∗0ν1((A(a;1),(1,0)))w .(1.33) Apply −∂∂¯bb=0to both sides of(1.33).We obtain[D L,D R]=0.∂z Y(u;z,¯z),(1.34)D R,Y(u;z,¯z) =Y(D R u;z,¯z)=∂∂z z=0ν2(P(a)1∞0(A(−z;1),(1,0))),(1.37)and∂∂z z=0ν2(P(z+a)).(1.38) Hence it is clear thatY(D L u,z,¯z)=∂Similarly,we can show that the D L-bracket property follows from the following sewing identity:(A(−z;1),(1,0))1∞0(P(a)2∞0(A(z;1),(1,0)))=P(z+a).We omit the detail.The proof of D R-bracket and D R-derivative properties is similar.For the convergence property of R×R-graded fullfield algebra,we use the weaker version of convergence property discussed in the remark1.2in[HK2].Then it is clear that this weaker version of convergence property is automatically true by the definition of algebra over partial operad.The permutation axiom of full field algebra follows automatically from that of partial operad.The single-valuedness property follows from Lemma1.9.The rest of axioms follows from(1.24),(1.27),(1.28),(1.34)and(1.35).z1+a,...,z n+a,z1+a,...,z n+a,Theorem1.17.The category of grading-restricted R×R-graded fullfield algebras is isomorphic to the category of smooth K-algebras.Proof.The proof is similar to that of Theorem5.4.5in[H3].Our case is much simpler.Given a grading-restricted R×R-graded fullfield algebra(F,m,1,d L,d R,D L,D R).20We define a mapνfrom K to H C×F C1as follow.We defineν0((0)):=1.(1.48) For n>0,u i∈F,a,z i∈C,i=1,...,n and Q∈ K(n)as(1.6),we define ν(Q)(u1⊗···⊗u n):=e−aD L−¯a D R m n((a(1)0)−d L(a(n)0)−d R u n;z1,z n−1,0,0).(1.49) In particular,ν1((0,(a,0))=m1((a(1)0)−d L(a(1)0)−d R u1.(1.50) This gives the representation of rescaling group C×according to the R×R-grading of F.The R×R-grading,together with the single-valuedness property,guarantees that the grading is in the set{(m,n)|m,n∈R,m−n∈Z}as required by the ax-ioms of smooth K-algebra.The grading restriction conditions and the smoothness of all correlation functions are all automatically satisfied.Let P∈ K(m)and Q∈ K(n)given as follow:P=(z1,...,z m−1;A(a;1),(a(1)0,0),...,(a(m),0))Q=(ξ1,...,ξn−1;A(b;1),(b(1)0,0),...,(b(n)0,0)).(1.51) We assume that P i∞0Q exists.Then we haveP i∞0Q=(z1,...,z i−1,ξ1−aa(i)0+z i,z i+1,...,z n;A(a;1),(a(1)0,0),...,(a(i−1)0,0),(a(i)0b(1)0,0),...,(a(i)0b(m),0),(a(i−1),0),...,(a(n)0,0)).(1.52) 21By(1.46),we haveνm(P)i∗0νn(Q)= k,l∈R e−aD L−¯a D R m m((a(1)0)−d L(a(i)0)−d R P k,l m n((b(1)0)−d L(b(n)0)−d R v n;ξ1−b,ξm−b),...,(a(m))−d L(a(1)0)−d R u1,...,P k,l m n((a(i)0b(1)0)−d L(a(i)0b(n)0)−d R v n;(ξ1−b)/a(i)0,(ξm−b)/a(i)0),...,(a(m))−d L(a(1)0)−d R u1,...,(a(i)0b(1)0)−d L(b(n)0)−d R v n,...,(a(m))−d L(a(i)0+z i,a(i)0+z i,...,ξm−bξm−bBy construction(1.41)and(1.42),for z i=0,i=1,...,n,we have˜m n(u1,...u n;z1,¯z1,...,z n,¯z n)=νn+1((z1,...,z n;0,(1,0),...,(1,0)))(u1⊗u n⊗1)=m n+1(u1,...,u n,1;z1,¯z1,...,z n,¯z n,0,0)=m n(u1,...,u n;z1,¯z1,...,z n,¯z n).(1.53) The cases when z i=0for some i=1,...,n follows from smoothness.Therefore ˜m=m.By1.29,we also have˜D L=−∂∂a a=0e−aD L−¯a D R=D L,˜D R=−∂∂a a=0e−aD L−¯a D R=D R.(1.54) The coincidence of two gradings is also obvious.Therefore,we have proved that one way of composing two functors gives the identity functor on the category of grading restricted R×R-graded fullfield algebra.Similarly,one can show that the opposite way of composing these two functors also gives the identity functor on the category of smooth K-algebras.˜K˜K˜K˜Kto denote this function.For the simplicity of notation,we will not distinguish L L(n)with L L(n)⊗1 and L R(n)with1⊗L R(n)in this work.Proposition1.18.A conformal fullfield algebra over V L⊗V R,(F,m,ρ),has a canonical structure of smooth˜K c L⊗c R-algebras over V L⊗V R.Proof.Wefirst define a mapνn:˜K c L⊗c R(n)→Hom(F⊗n,˜K˜KA(0))mn (e−L L+(A(1))−L R+(a(1)0−L R(0)u1,...,e−L L+(A(n))−L R+(a(1)0−L R(0)u n;z1,¯z1,...,z n−1,¯z n−1,z n,¯z n),(1.57)whereL L±(A)=∞j=1A j L L(±j),L R±(A)=∞ j=1A j L R(±j)for A∈ n∈N C.24。
a r X i v :h e p -t h /9306072v 2 21 J u n 1993SUTDP/11/93/72May,1993A UNIFIED SCHEME FOR MODULAR INV ARIANTPARTITION FUNCTIONS OF WZW MODELSM.R.Abolhassan i a , F.Ardala n a,b a Department of Physics,Sharif University of Technology P.O.Box 11365-9161,Tehran,Iran b Institute for studies in Theoretical Physics and Mathematics P.O.Box 19395-1795,Tehran,Iran Abstract We introuduce a unified method which can be applied to any WZW model at arbitrary level to search systematically for modular invariant physical partition functions.Our method is based essentially on modding out a known theory on group manifold G by a discrete group Γ.We apply our method to su (n )with n =2,3,4,5,6,and to g 2models,andobtain all the known partition functions and some new ones,and give explicit expressions for all of them.1.IntroductionConformalfield theories(CFT’s)play an important role in two dimensional critical statisti-cal mechanics and string theory,and has been extensively studied in the past decade.1−4 Among these theories WZW models have attracted considerable attentions,because as two dimensional rational conformalfield theories they are exactly solvable,5,6and most known CFT’s can be obtained from them via coset construction.7Moreover these models explicitly appear in some statistical models like quantum chains,and describe their critical behavier.8WZW models in addition to conformal symmetry,have an infinite dimensional symmetry whose currents satisfy a Kac-Moody algebraˆg at some level k.The partition function of a WZW model takes the formZ(τ,¯τ)= χλ(τ)Mλ,λ′χ∗λ′(¯τ),(1.1)whereχλis the character of the affine module whose highest weigth(HW)isλ,Mλ,λ′are positive integers which determine how many times the HW representationsλ,λ′in the left and right moving sectors couple with each other,and the sum is over thefinite set of integrable representations(see Subsec. 2.1.for a precise definition).For the consistency of a physical theory it is necessary that the partition function(1.1)be invariant under the modular group of the torus.9Construction and classification of partition functions of WZW models has been the goal of a large body of work in the past few years.However,up to now only the classifications of su(2),10and su(3)11−13at arbitrary level,and of simple affine Lie algebras at level one14,15have been completed.Tofind moudular invariant partition functions of WZW models a number of methods have been used:Automorphism of Kac-Moody algebras,16,17simple currents,18,19confor-mal embedding,20−22automorphism of the fusion rules of the extended chiral algebra,23,24 lattice method,26,27andfinally direct computer calculations.28In these efforts many mod-ular invariant partition functions have been found,and may be arranged in three broad categories:i)Diagonal Series−For every WZW model with a simply connected group manifold, there exists a physical modular invariant theory with diagonal matrix Mλ,λ′=δλ,λ′.9,17 They are often designated as a member of the A series.ii)Complementary Series−There are some nondiagonal series for every WZW model whose Kac-Moody algebra has a nontrivial centre,16,17associated to subgroups of the centre.They are often designated as members of the D series.iii)Exceptional Series−In addition to the above two series,WZW models have a number of nondiagonal partition functions which occur only at certain levels.They are called E series.Some of the known E series have been found by the conformal embedding method (see e.g.Ref.13),some by utilizing the nontrivial automorphism of the fusion rules of the extended algebra,24,25and some others by computer calculations.28Alhough many exceptional partition functions have been obtained by these methods,however they don’t follow from a unified method and prove to be impractical for high rank groups and high levels.Furthermore,these methods don’t answer the question of why there are exceptional partition functions only at certain levels.It must be mentioned that corresponding to anyphysical theory,there exists a charge conjugation,c.c.,counterpart,such that M(c.c.)λ,λ′=Mλ,C(λ′),where C(λ)is the complex conjugate representation ofλ.In this paper a unified approach which we call orbifold-like method,is presented and shown to lead to all the known nondiagonal theroies.The method is easily applied to highrank groups at arbitrary level.The organization of the paper is as follows:In Sec.2,we briefly review some characteristic features of Kac-Moody algebra,such as their unitaryhighest weight representations and modular transformation properties of their characters. Then we present our approach.We start with some known theory whose partition function is Z(G)and mod it out by a discrete groupΓ.We will takeΓto be a cyclic group Z N.It is not necessarily a subgroup of the centre ofˆg.In order for the modding to gives rise to a modular invariant combination with rational coefficients,certain relations must be satisfied,which will be explicated.In Sec.3,we apply our method to su(n)models with n=2,3,4,5,6and as an example of a non-simply laced affine Lie algebra,to g2models, and generate all the known nondiagonal theories and some new exceptional ones.In Sec.4,we conclude with some remarks.In Appendix A,we gather some formulas and relations that are used in the body of the paper.2.The Orbifold-like methodObserving that all modular invariant partition functions of su(n)models at level one are obtained by modding out the group SU(n)by subgroups of its centre15,the authors in Ref. 29for thefirst time found that not only the D series of su(2)WZW models are obtained with modding out the diagonal theories by the Z2centre,but also all their exceptional partition functions can be found by modding out the A or D series by a Z3which is not obviously a subgroup of the centre.In this paper we are going to generalize that work to WZW models with any associated affine Lie algebra at arbitrary level.We have called our approach orbifold-like method, because thefinite group which one uses in modding may not be the symmetry of a target manifold.Befor going to the details of our approach,we collect in the following subsection, some facts about untwisted Kac-Moody algebras and set up our notations.2.1.Preliminaris and NotationsA WZW model is denoted byˆg k where its affine symmetry algebra is the untwisted Kac–Moody algebraˆg associated with a compact Lie algebra g,and the positive integer number k is the level of the Kac-Moody algebra(see Refs.30,31for details).The primaryfields of the model can be labelled by the highest weight representations of horizontal Lie algebra g.In the basis of fundamental weightsωi of g these HW representations are expressed byλ=Σr i=1λiωi whereλi’s are positive integers(Dynkin labels)and r is the rank of g. Imposing unitarity condition,restricts the numbre of HW representations which appear in a theory at a given level.These representations which are usually called integrable representations satisfy the relation2(see e.g.Ref.32).The character of an integrable HW representation transforms under the action of the generators of the modular group S:(τ→−1/τ)and T:(τ→τ+1)as χλ(−1/τ)=C {λ′∈B h} {ω∈W(G)}ε(ω)e(2πi(k+ˇh)r/2 voll.cell of Q∗|Γ| h1,h2∈Γ[h1,h2]=0(h1,h2),(2.4)where|Γ|is the order ofΓ.ForΓ=Z N which is the interesting group for us,eq.(2.4) reduces to:Z(G/Z N)=1NNα=1(1,hα),(2.6)and acting properly on it by the generators of the modular group,S and T,one can obtain the full partition function Z(G/Z N).9In Ref.35the following formula is drived for Z(G/Z N)when Z N is a subgroup of the centre of G and N is prime:Z(G/Z N)= N α=1 TαS+1 Z1(G/Z N) −Z(G).(2.7)Concerning the method mentioned above we make two crucial observations.First,the orbifold method can be similarly applied to the case where G is not the covering group, and even when Z N is not a subgroup of the centre of G,but is a symmetry of the classical theory.However,in those cases in order for the modding to be meaningful,i.e.,the sum of the terms in the bracket of eq.(2.7)and therefor the whole expression be modular invariant with real coefficients,the following relation must be satisfiedT N SZ1(G/Z N)=SZ1(G/Z N).(2.8) Secondly,for the case when N is not prime,the expression(2.7)does not completely describe an orbifold partition function and in order to generate the full partition function Z(G/Z N),some additional terms must be included into the bracket.But then,extra constraints beyond(2.8)have to be satisfied for modular ivariance(see eq.(A.3)).We call the appropriate moddings for a given theory which satisfy these constraints,allowed moddings.We have collected in Appendix A,a list of formulas for the cases of interest to us.Our strategy infinding the nondiagonal WZW theories with the affine symmetry algegraˆg k is as follows.First we start with a diagonal theory and mod it out by some group Z N.The untwisted part of the partition function Z1,is realized by representing the action of Z N on the characters of HW representations in the left−moving sector asp·χλ(τ)=e2πicombinations of characters that have been found in the process of the previous moddings, a new modular invariant partition function will be found.See for example,the case of modding D8by Z9in su(3)models on page15.Case III)Some of the terms in the bracket have negative coefficients but after sub-tracting from it some known physical partitions or/and some modular invariant combina-tions of characters,at most a new modular invariant combination will be obtained,which is not a partition function.See for example,the case of modding D24by Z2in su(3) models on page16.3.ApplicationsIn this section we apply our method to su(n)WZW models with n=2,3,4,5,6,and as an example of a nonsimply-laced affine Lie algebra to g2WZW models.3.1. su(2)WZW modelsFor su(2)models besides the usual diagonal series A h= {λ∈B h}|χλ|2at each level, whereB h= λ=mω 1≤m<h=k+2 (3.1)is the fundamental domain andωis the fundamental weight of su(2),and a nondiago-nal D series at even levels,three exceptional modular invariant partition functions have been found at levels k=10,16,28;and it has been shown that this set completes the classification of su(2)WZW models.10In what follows we will review the results of Ref. 29,where it was shown that the exceptional partition functions can be obtained by our orbifold method,and present some further calculations.The action of the Z N group on the characters of the left−moving HW representations of su(2)is defined due to eq.(2.9) byp·χm=e2πi|W| {λ∈W B h;m odd}|χλ|2,(3.4)using the identityχω(λ)=ǫ(ω)χλ,where W B h is the Weyl reflection of the fundamental domain B h.Then we rewrite(3.4)in the following formZ1(A h/Z2)=1h Q consists of the following HWrepresentations:Q∗2h 1h λ,ω′(λ′)−ω′′(λ′′)χλ′¯χλ′′.(3.7)The sum overλis easily done using eq.(3.6),and it appears that the sum over one of the two Weyl groups can be factored out and give an overall factor equal to the order of Weyl group.Finally the sum over the other Weyl group must be done.Substituting(3.7)in eq.(2.7),we easily do the sum andfind a nondiagonal partition function at every even level given byD h≡Z(A h/Z2)=h−1m odd=1 χm 2+h/2−2 m odd=1(χm¯χh−m+c.c.)+2 χh/2 2(3.8a)for h=2mod4,andD h≡Z(A h/Z2)=h−1m odd=1 χm 2+h/2−2 m even=2(χm¯χh−m+c.c.)+ χh/2 2(3.8b)for h=0mod4.these are exactly the D series given in Ref.10.E-SeriesOne expects tofind possibly,exceptional partition functions at levels k=4,10,28 according to the following conformal embeddings:20su(2)k=4⊂ su(3)k=1, su(2)k=10⊂ (B2)k=1, su(2)k=28⊂ (g2)k=1.(3.9)1.At level k=4(h=6),we start with A6and mod it out by Z3,and check that the eq.(2.8)is satisfied;then we do the sum in the bracket of eq.(2.7)with N=3,and finally get3α=1TαSZ1+Z1 =4A6−D6.(3.10)We continue modding by allowed Z N’s up to N=48,but nothing more than A6and D6 appears.For example,in the case N=6we obtain6α=1Tα+3α=1TαST2+2 α=1TαST3 SZ1+Z1 =4A6+D6.(3.11)Then we start with D6,mod it out by allowed moddings but also nothing more is found. For example,in modding by Z3wefind3α=1TαSZ1+Z1 =2D6.(3.12) This is not surprising,since it can easily be seen that D6given byD6= χ1+χ5 2+2 χ3 2,(3.13)exactly corresponding to conformal embedding su(2)k=4⊂ su(3)k=1.2.At level k=10(h=12),we start with A12and mod it out by Z6which is allowed. After doing the sum in the bracket of eq.(A.5),we encounter the case I of Subsec. 2.2., which after subtracting the known partition functions A12and D12each with mutiplicity 2,we obtain the exceptional partition function E6,which in our notation is described by E126α=1Tα+3α=1TαST2+2 α=1TαST3 SZ1+Z1 =2A12+2D12+2E12,(3.14)whereE12= χ1+χ7 2+ χ4+χ8 2+ χ5+χ11 2.(3.15) In Ref.29,E12was obtained by modding A12or D12by Z3,but there,we encounter the case II of Subsec. 2.2.We also get E12in modding D12by Z6;the result is exactly the same as eq.(3.14).3.At level k=16(h=18),we start with A18and mod it out by Z6which is allowed. After doing the sum in the bracket of eq.(A.5),we encounter the case I of Subsec. 2.2., which after subtraction A18of mutiplicity3,the exceptional partition function E18is obtained:6α=1Tα+3α=1TαST2+2 α=1TαST3 SZ1+Z1 =3A18+3E18,(3.16)whereE18= χ1+χ17 2+ χ5+χ13 2+ χ7+χ11 2+ χ9 2+ χ9(We alsofind E18in modding D18by the allowed modding like Z3:3α=1TαSZ1+Z1 =D18+2E18.(3.18)4.At level k=28(h=30),we start with D30and mod it out by Z3which is allowed. After doing the sum in the bracket of eq.(2.7),we encounter the case I of Subsec. 2.2, which after subtraction D12of mutiplicity2,leads to the exceptional partition function E30:3α=1TαSZ1+Z1 =2D30+E30,(3.19)whereE30= χ1+χ11+χ19+χ29 2+ χ7+χ13+χ17+χ23 2.(3.20) So in this subsection we have generated,by orbifold method,not only the partition functions which correspond to a conformal embedding like E12and E30,22but also the one which follows from a nontrivial automorphism of the fusion rules of the extended algebra i.e.E18.24It is interesting to notice that all the nondiagonal partition functions of su(2) are obtained from moddings by Z N’s,with N a divisor of2h and h=k+2.3.2. su(3)WZW modelsFor su(3)models besides the usual diagonal series A h= {λ∈B h}|χλ|2at each level,whereB h= λ 2≤2 i=1m i<h=k+3 (3.21)andλ=Σ2i=1m iωi,and a nondiagonal D h series at each level;four exceptional modular invariant partition functions have been found at levels k=5,9,21.12,13Recently,it was shown that this set completes the classification of su(3)WZW models.11In what follows we obtain all of these by the orbifold method.We define the action of the Z N group on the characters of the left−moving HW representations of su(3)byp·χ(m1,m2)=e2πiThe partition function Z(G/Z3)can be calculated using eq.(2.7)with N=3.Following the same recipe mentioned in section3.1,first we write the untwisted part(3.23)in the following form1Z1(A h/Z3)== λ=mω1+m′α1 1≤m≤3h+2;−1≤m′≤h−2 .(3.25)h Qso that,just as in eq.(3.5)we can rewrite eq.(3.24)in the form,1Z1(A h/Z3)=*There is a minus sign error in Ref.17.Definingσ(λ)the same as in our case,the two terms in the exponential of eq.(6.4)of Ref.17must both have negative signs,in order for Mλ,λ′to commute with the operator T.Then,we start with D6,mod it out by allowed moddings but also nothing more is found. For example,in modding by Z6wefind6α=1Tα+3α=1TαST2+2 α=1TαST3 SZ1+Z1 =9(χ1,3+χ4,3)+(χ2,3+χ6,1)2 5D8−A c.c.8+E8 (3.35)4α=1Tα+ST2 SZ1+Z1 =13.At level k=9(h=12),we start with D12and mod it out by Z9and do the sum according to the bracket of eq.(A.7)with N=9,finally we encounter the case II of Subsec. 2.2.,which after subtraction D12of multiplicity12,we obtain an exceptional partition function,denoted by E(1)12with an overal multiplicity−3:9α=1Tα+ST3+ST6 SZ1+Z1 =12D12−3E(1)12,(3.37) whereE(1)12= χ1,1+χ1,10+χ10,1+χ2,5+χ5,2+χ5,5 2+2 χ3,3+χ3,6+χ6,3 2.(3.38) This partition function corresponds to conformal embedding su(3)k=9⊂ (e6)k=1.13In modding D12by Z2,after doing the sum in the bracket of eq.(2.7)with N=2,one encounters case II,however in this case the trace of another modular invariant can easily be seen.Actually subtracting D12,its charge conjugation counterpart D c.c12and E(1)12with multiplicities5/2,−1/2,and1/2respectively,we obtain another exceptional partition function which we denote by E(2)122α=1TαSZ1+Z1 =1(χ2,2+χ2,8+χ8,2)+c.c. ,(3.40)which does not correspond with a conformal embedding;and was found using a nontrivial automorphism of the fusion rules of the extended algebra.24We continued modding up to Z72,but no other exceptional theory appears at this level.4.At level k=21(h=24),starting with D24and modding by Z2leads to the case III of Subsec. 2.2.,which after subtracting D24of multiplicity2,yields a modular invariant combination with some of its coefficients negative integers,which we call M242α=1TαSZ1+Z1 =2D24+M24,(3.41) withM24= χ[1,1]+χ[2,11] 2+ χ[5,5]+χ[7,7] 2+ χ[1,7]+χ[8,5] 2+ χ[7,1]+χ[5,8] 2 + χ[3,3]+χ[6,9] 2+ χ[1,4]−χ[4,7] 2+ χ[4,1]−χ[7,4] 2+2 χ[3,9] 2+2 χ[9,3] 2− χ[2,2]+χ[2,8]+χ[8,2]−χ[4,10] 2+3 χ[4,4]−χ[8,8] 2 ,(3.42)whereχ[m1,m2]≡χ(m1,m2)+χ(m2,h−m1−m2)+χ(h−m1−m2,m1).(3.43)Modding by Z3gives D12itself,and modding by Z4,Z8,Z9,Z18,Z24,Z36give rise to three extra modular invariant combinations,which we do not mention here their explicit form.Finally,modding by Z72and doing the sum in the bracket of eq.(A.16),we encounter the case II of Subsec. 2.2.,which after Subtracting D24and M24and their charge conjugation counterparts D c.c24,M c.c24with multiplicities12and3respectively,we are left with an exceptional partition function which is denoted by E24:72α=1Tα+18α=1TαST2+8 α=1TαST3+9 α=1TαST4+2 α=1TαST6+9α=1TαST8+8 α=1TαST9+ST12+8 α=1TαST15+2 α=1TαST18+ST24+2α=1TαST30+ST36+ST48+ST60 SZ1+Z1=12D24+12D c.c24+3M24+3M c.c24+9E24,(3.44) withE24= χ[1,1]+χ[5,5]+χ[2,11]+χ[7,7] 2+ χ[1,7]+χ[7,1]+χ[5,8]+χ[8,5] 2.(3.45)This theory corresponds to conformal embedding su(3)k=21⊂ (e7)k=1.13 So with our method we reproduce not only exceptional partition functions correspond-ing to a certain conformal embedding like E8,E(1)12,and E24,but also the one which can not be obtained by a conformal embedding i.e.E(2)12.Note that at each level the allowed modding Z N has N a divisor of3h,where h=k+3.3.3. su(4)WZW modelsIn addition to the diagonal series A h= {λ∈B h}|χλ|2,whereB h= λ 3≤3 i=1m i<h=k+4 ,(3.46)andλ=Σ3i=1m iωi,there exist two D series corresponding to the two subgroups of the centre of SU(4).Furthermore,up to now three exceptional partition functions have been found in levels k=4,6,8.In the following we obtain all of these by orbifold approach.We define the action of the Z N on the characters of left−moving HW representations of su(4) due to eq.(2.9)byp·χ(m1,m2,m3)=e2πiand p is the generator of Z N .Thus,the untwisted part of a partition function,consists of left −moving HW representations λwhich satisfy:ϕλ=20mod N .D -SeriesWe follow the same recipe of calculation that was described in Subsec. 3.2.,but without going into the details,and find the general form of D h series at each level.Starting with A h ,first we mod it out by a subgroup Z 2of the centre and obtain at every level anondiagonal partition function which we denote by D (2)h ,D (2)h ≡Z (A h /Z 2)= {λ|Σ3i =1im i =0mod 2}|χλ|2+ {λ|Σ3i =1im i =kmod 2}χλ¯χµ(λ),(3.48)where µ(m 1,m 2,m 3)=(m 3,h −Σ3i =1m i ,m 1).Then we mod out the A h series by Z 4according to eq.(A.4)and find at every even level a nondiagonal partition function D (4)h ,which has the formD (4)h ≡Z (A h /Z 4)= {λ|Σ3i =1im i =2mod 4}|χλ|2+{λ|Σ3i =1im i =2+k2mod 4}χλ¯χσ3(λ),(3.49)where σ(m 1,m 2,m 3)=(m 2,m 3,h −Σ3i =1m i ).These results agree with the ones obtainedin Ref.17modulo the comment in the footnote of page 13.E -SeriesIt is expected that there are nondiagonal partition functions at levels k =2,4,6,8,due to the following conformal embeddings:20su (4)k =2⊂ su (6)k =1, su (4)k =4⊂ (B 7)k =1su (4)k =6⊂ su (10)k =1,(D 3)k =8⊂ (D 10)k =1.(3.50)1.At level k =2(h =6),we have only D (2)6,and D (4)6=A c.c.6.First,we start withA 6and mod it out by all allowed Z N ’s up to N =24.Nothing other than A 6and D (2)6and their charge cojugation counterparts is found.For example,in the cases N =3,6,8we obtain3α=1T αSZ 1+Z 1=4A 6−2D (2)6(3.51) 6α=1T α+3α=1T αST 2+2α=1T αST 3 SZ 1+Z 1 =4A 6+4D (2)6(3.52) 8α=1T α+2α=1T αST 2+ST 4SZ 1+Z 1 =2A 6+2A c.c.6+D (2)6,(3.53)respectively.Then we start with D(2)6and do the allowed moddings.Again,nothing more is found.For example,in modding by Z3wefind3α=1TαSZ1+Z1 =2D(2)6.(3.54) This is not surprising,since it can easily be seen that D(2)6D(2)6= χ(1,1,1)+χ(1,3,1) 2+ χ(1,1,3)+χ(3,1,1) 2+2 χ(1,2,1) 2+2 χ(2,1,2) 2,(3.55) exactly corresponding to conformal embedding su(4)k=2⊂ su(6)k=1.2.At level k=4(h=8),there exist two D(2)8and D(4)8partition functions.We choose to start with D(2)8.Modding by Z2gives D(2)8itself,and modding by Z4and Z8 give a combination of D(2)8and D(4)8:4α=1Tα+ST2 SZ1+Z1 =2D(2)8+2D(4)8(3.56)8α=1Tα+2α=1TαST2+ST4 SZ1+Z1 =4D(2)8+4D(4)8.(3.57)The next modding which satisfies the condition(A.3)is Z16.After doing the sum in the bracket of eq.(A.10)we encounter the case I of Subsec.2.2.,which after subtracting the D(2)8and D(4)8each with multiplicity4,an exceptional partition function is found which we denote by E8:16α=1Tα+4α=1TαST2+ST4+ST8+ST12 SZ1+Z1 =4D(2)8+4D(4)8+4E8,(3.58)whereE8= χ(1,1,1)+χ(1,5,1)+χ(1,2,3)+χ(3,2,1) 2+ χ(1,1,5)+χ(5,1,1)+χ(2,1,2)+χ(2,3,2) 2+4 χ(2,2,2) 2.(3.59)It can easily be shown that this partition function corresponds to conformal embedding su(4)k=4⊂ (B7)k=1.The above exceptional partition function was obtained in the context offixed-point resolution of Ref.19.We then repeat the above moddings starting with D(4)8 andfind exactly the same results as with D(2)8.3.At level k=6(h=10),there exist D(2)10and D(4)10.Starting with D(2)10the following results are obtained.Modding by Z2and Z4gives D(2)10itself,but modding by Z8and doing the sum in the bracket of eq.(A.6)we encounter the case III of Subsec.2.2.,whichafter subtraction D(2)10of multiplicity6leads to a modular invariant combination,which we call M108α=1Tα+2α=1TαST2+ST4 SZ1+Z1 =6D10−M10(3.60)whereM10= (χ(1,1,1)+χ(1,7,1))−(χ(2,1,6)+χ(6,1,2))−(χ(3,1,3)+χ(3,3,3)) 2+ (χ(1,1,7)+χ(7,1,1))−(χ(1,2,1)+χ(1,6,1))−(χ(1,3,3)+χ(3,3,1)) 2+3 χ(2,2,4)+χ(4,2,2) 2+3 χ(2,2,4)+χ(4,2,2) 2.(3.61) In modding by Z5after doing the sum in the bracket of eq.(2.7)with N=5,we encounterthe case II of Subsec.2.2.,which by subtracting D(2)10,and its charge conjugation D(2)c.c.10,and M10by multiplicities−3/5,4/5,and8/5respectively,an exceptional partition function is found which we call E105α=1TαSZ1+Z1 =15 2D(2)10+4D(2)c.c.10+3M10+2E10(3.64)4.At level k =8(h =12),there exist D (2)12and D (4)12.We choose D (4)12and obtainthe following results.Modding by Z 2and Z 4gives D (4)12itself,but in modding by Z 8weecounter the case II of Subsec. 2.2.,which after subtraction D (4)12of multiplicity 12,anexceptional partition function is found,which we call E (1)128α=1T α+2α=1T αST 2+ST4SZ 1+Z 1=12D (4)12−2E (1)12,(3.65)where *E (1)12= χ(1,1,1)+χ(1,1,9)+χ(1,9,1)+χ(9,1,1)+χ(2,3,2)+χ(2,5,2)+χ(3,2,5)+χ(5,2,3)2+ χ(1,3,1)+χ(1,7,1)+χ(3,1,7)+χ(7,1,3)+χ(1,4,3)+χ(3,4,1)+χ(4,1,4)+χ(4,3,4) 2+2χ(2,2,4)+χ(2,4,4)+χ(4,2,2)+χ(4,4,2) 2.(3.66)It can easily be seen that E (1)12just corresponds to conformal embedding (D 3)k =8⊂ (D 10)k =1with the following branching rules:ch 1=χ(1,1,1)+χ(1,1,9)+χ(1,9,1)+χ(9,1,1)+χ(2,3,2)+χ(2,5,2)+χ(3,2,5)+χ(5,2,3)ch 2=χ(1,3,1)+χ(1,7,1)+χ(3,1,7)+χ(7,1,3)+χ(1,4,3)+χ(3,4,1)+χ(4,1,4)+χ(4,3,4)ch 3=ch 4=χ(2,2,4)+χ(2,4,4)+χ(4,2,2)+χ(4,4,2),(3.67)where chi i ’s are the characters of the integrable representations of ( (D 10)k =1andχ(m 1,m 2,m 3)’s are those of SU (4)k =8.Then,modding by Z 3and doing the sum in eq.(2.7)with N =3leads to the case I ofSubsec.2.2.,which after subtraction D 12,D c.c.12,and E (1)12each of multiplicity 1/3anotherexceptional partition function is found,which we call E (2)124α=1T α+ST 2SZ 1+Z 1 =1χ(3,3,3)+(χ(1,5,1)+χ(5,1,5))*We are not aware of the explicit form of this partition function in the literature.This exceptional partition function which doesn’t correspond to a conformal emedding, was recently found by a computational method which essentially looks for the eigenvectors of matrix S(the generator of the modular group)with eigenvalues equal to one,25,28and can be shown to be a consequence of an automorphism of the fusion rules of the extended algebra.28We continue the modding by allowed groups up to Z96,but no new partition function is found.For example in modding by Z24according to eq.(A.12),we get24α=1Tα+6α=1TαST2+8 α=1TαST3+3 α=1TαST4+2 α=1TαST6+3α=1TαST8+ST12 SZ1+Z1 =4 D12+D c.c12+E(1)12+4E(2)12 .(3.70)So again,as in the case of su(3)models,with the orbifold method not only exceptional partition functions which correspond to conformal embeddings like E8,E10,and E(1)12are found,but also the a partition function which does not corresponds to a conformal em-bedding i.e.E(2)12,is generated.In this subsection we have been able to obtain explicitly all the exceptional partition functions which correspond to a conformal embedding and moreover the one(E(2)12)which follows from an automorphism of the fusion rules of the extended algebra.Note that at each level the allowed modding Z N has N a divisor of4h,where h=k+4.3.4. su(5)WZW modelsFor su(5)models besides the usual diagonal series A h= {λ∈B h}|χλ|2,at each level, whereB h= λ 4≤4 i=1m i<h=k+5 ,(3.71)andλ=Σ4i=1m iωi,and one nondiagonal D series at each level,up to now some exceptional partition functions have been found which we are going to obtain by our method.We define the action of the Z N on the characters of HW representations of su(5)due to eq.(2.9)byp·χ(m1,m2,m3,m4)=e2πiWe start with A h series,following the same recipe mentioned in Subsec.3.2.,mod it out by Z5using the eq.(2.7)with N=5.Doing the sum in the bracket,we obtain the general form of D h series:D h≡Z(A h/Z5)= {λ|Σ4i=1im i=0mod5}|χλ|2+ {λ|Σ4i=1im i=3kmod5}χλ¯χσ(λ)+ {λ|Σ4i=1im i=kmod5}χλ¯χσ2(λ)+ {λ|Σ4i=1im i=4kmod5}χλ¯χσ3(λ)+ {λ|Σ4i=1im i=2kmod5}χλ¯χσ4(λ)(3.73)whereσ(m1,m2,m3,m4)=(m2,m3,m4,h−Σ4i=1m i).These results agree with the ones obtained in Ref.17,modulo the comment mentioned in the footnote of page13.E-SeriesOne expects tofind,possibly,the exceptional series at levels k=3,5,7according to the following conformal embeddings:20su(5)k=3⊂ su(10)k=1, su(5)k=5⊂ (D12))k=1, su(5)k=7⊂ su(15)k=1.(3.74)We will limit ourselves only to thefirst two cases in this work.1.At level k=3(h=8),we start with A8,doing the allowed moddings and obtain the following results.Modding by Z2gives A8itself,but modding by Z4we encounter the case II of section2.2.,which after subtracting A8and D c.c8with multiplicities5and−1 respectively,an exceptional partition function is found which we call E84Tα+ST2 SZ1+Z1 =5A8−D c.c.8+E8(3.75)α=1withE8= χ(1,1,1,1)+χ(1,2,2,1) 2+ χ(1,1,1,4)+χ(2,2,1,2) 2χ(1,1,2,1)+χ(1,3,1,2) 2+ χ(1,1,3,1)+χ(3,1,1,2) 2χ(1,1,4,1)+χ(2,1,2,1) 2+ χ(1,2,1,3)+χ(3,1,2,1) 2χ(1,2,1,2)+χ(1,4,1,1) 2+ χ(1,3,1,1)+χ(2,1,1,3) 2χ(1,2,1,1)+χ(2,1,3,1) 2+ χ(2,1,2,2)+χ(4,1,1,1) 2.(3.76)We have checked that E8exactly corresponds to conformal embedding su(5)k=3⊂ su(10)k=1.We carried out all the allowed moddings up to Z40and no other exceptional partition function was found.Then,we start with D8and mod it out by allowed moddings up to Z40,but again no more exceptional partition function is found.For example,modding。
