A Unified Scheme for Modular Invariant Partition Functions of WZW Models

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.。