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The Non-Perturbative SO(32) Heterotic String

a r X i v :h e p -t h /9812210v 1 22 D e c 1998QMW-98-43

hep-th/9812210

December,1998

The Non-Perturbative SO (32)Heterotic String C.M.Hull Physics Department,Queen Mary and West?eld College,Mile End Road,London E14NS,U.K.ABSTRACT The SO (32)heterotic string can be obtained from the type IIB string by gaug-ing a discrete symmetry that acts as (?1)F L on the perturbative string states and

reverses the parity of the D-string.Consistency requires the presence of 32NS 9-branes –the S-duals of D9-branes –which give SO (32)Chan-Paton factors to open D-strings.At ?nite string coupling,there are SO (32)charges tethered to the heterotic string world-sheet by open D-strings.At zero-coupling,the D-string tension becomes in?nite and the SO (32)charges are pulled onto the world-sheet,and give the usual SO (32)world-sheet currents of the heterotic string.

The perturbative type I superstring can be obtained from the perturbative type IIB superstring by orientifolding by the action of the world-sheet parity operator?[1,2].However,it seems that?extends to a symmetry of the full non-perturbative IIB theory;indeed,it is straightforward to extend its action to BPS states and to the dual IIB string theory that emerges in the strong coupling limit,where it acts through a perturbative symmetry??of the dual theory.If?does extend to such a non-perturbative symmetry,we can consider modding out the IIB theory by this symmetry for any value of the string coupling.In particular,in the strong coupling limit,this corresponds to modding out the dual IIB string theory by??, and this should give the SO(32)heterotic string,as this is the strong coupling limit of the type I string[3-6].This led to the conjecture of[7],that the SO(32) heterotic string can be obtained from the type IIB superstring by modding out by the action of the symmetry??.The purpose of this paper is to invesitgate this construction further,and in particular show how the gauge sector of the heterotic string emerges.

In the orientifold construction of the type I superstring,it is essential to add 32D9-branes which provide the Chan-Paton factors for the open strings.Extrap-olating to strong coupling,the D9-branes are replaced by their S-duals,which are the NS-NS9-branes proposed in[7].Then32NS-NS9-branes are required in the construction of the SO(32)heterotic string from the IIB string[7]and should play a key role in the construction of the gauge sector.We shall show that the9-branes indeed give rise to the gauge sector.Moreover,this extends to a construction of the non-perturbative heterotic string and in particular we will?nd that at?nite string coupling,the SO(32)charges are no longer con?ned to the string world-sheet but are tethered to it by strings,and as these strings can break,the charges can escape.

The type IIB string theory has an SL(2,Z)U-duality symmetry[8],and the massless bosonic?elds are g MN,B1MN,Φin the NS-NS sector and D MNP Q,B2MN,χin the RR sector.The2-form?elds B i MN transform as doublets under SL(2),while SL(2)acts onλ=χ+ie?Φthrough fractional linear transformations.There is also a non-dynamical RR10-form potential A2M...N that couples to D9-branes[6].This

must?t into an SL(2)doublet also[7],and its partner is a NS-NS10-form potential A1M...N.The theory has D p-branes for p=1,3,5,7,9[9]and it is interesting to ask how these transform under SL(2,Z).The D3-brane is invariant,but acting on the D1,D5and D9branes generates(p,q)-branes for all co-prime integers(p,q),as can be seen from the superalgebra[7].Of these1-branes,5-branes and9-branes,the (0,1)-branes are D-branes,the(1,0)-branes couple to?elds in the NS-NS sector, while the(p,q)branes are bound states of p(1,0)branes and q(0,1)branes(for co-prime integers(p,q)).The1-branes and5-branes couple to the2-form?elds B i MN and the9-branes can be thought of as coupling to10-form potentials A i M...N.

The7-brane is more subtle,as it couples to the scalar?elds which transform non-linearly under SL(2,Z).The solutions of[10],in which the axion ansatz in-volves the modular invariant j-function,are SL(2,Z)invariant and lead to singlet 7-brane charges.Acting with SL(2,Z)on a7-brane leaves its charge Z i1...i7in-variant,but changes the SL(2,Z)monodromy and the couplings to strings and 5-branes.These branes are characterised by two integers[11]and are obtained from the(0,1)D7-brane of[10]by an SL(2,Z)transformation;we will refer here to the7-branes on which a(0,1)string can end as(1,0)7-branes.

At weak coupling,g≡≈0,the perturbative states are described by the NS-NS or(1,0)string while all the other branes are non-perturbative[12].The perturbative theory is formulated as the usual type IIB superstring theory,with a topological expansion in terms of the genus of the world-sheet of the NS-NS string.At strong coupling,however,it is the RR or(0,1)string that gives the states that are perturbative in an expansion in?g=1/g.The self-duality of the theory implies that the strong-coupling theory is again a type IIB string theory, but now the formulation should be in terms of the world-sheet of the(0,1)string. The perturbation theory in?g is a sum over(0,1)string world-sheets with the power of?g corresponding to the genus of the world-sheet.The dual perturbative string theory has left and right movers which each decompose into an NS and an R sector. For example,the fundamental string of the weakly coupled(g-perturbative)theory couples to the2-form in the NS-NS sector but becomes the D-string of the dual

theory,coupling to the the2-form in the RR sector of the?g-perturbative theory.

At weak coupling,the(1,0)string can end on the D-branes carrying RR charge, which are the3-brane,and the(0,1)p-branes with p=1,5,7,9.In the strongly coupled theory formulated in terms of the world-sheet of the(0,1)string,the(0,1) string,the(0,1)5-brane and the(0,1)9-brane carry charge that appears in the NS-NS sector of the dual(0,1)string while the new D-branes with density proportional to1/?g on which the(0,1)string can end are the3-brane,and the(1,0)p-branes with p=1,5,7,9.These dual D-branes all carry charge which occurs in the RR sector of the(0,1)string.This structure is obtained by acting with the SL(2,Z) transformation

S= 01?10 (1) which interchanges strong and weak coupling.For example,this takes a(1,0) string ending on(0,1)strings or5-branes to a(0,1)string ending on(1,0)strings or5-branes.

The perturbative type IIB theory is formulated in terms of a fundamental(1,0) string and has a symmetry?which reverses the parity of the(1,0)world-sheet. The dilaton is invariant under?,which is a perturbative symmetry(i.e.it takes perturbative states to perturbative states).Of the massless bosonic?elds,the ones that are invariant under?are g MN,B2MN,Φ,A2M...N,while the others are odd,and this tells us how it acts on BPS states:the(0,1)string,5-brane and9-brane are invariant while for the(1,0)string,the2-form B1MN to which it couples is odd, but the Wess-Zumino term in the world-sheet action is invariant as the world-sheet volume form is odd under world-sheet parity?.Similarly,the(1,0)5-brane and9-brane together with the7-brane and3-brane couple to p+1form gauge potentials that are odd under?,and so their world-volume volume-forms must also be odd under?,so that?must act as an orientation-reversing world-volume parity operator on these branes.

The branes that couple only to the invariant?elds are invariant,so that the

only(p,q)D-branes that are invariant are the D1,D5,D9branes with charges(0,1). This tells us how to extend the action of?to the BPS brane sector,and it will be assumed that the perturbative symmetry extends to a symmetry of the full non-perturbative type IIB string(which will also be denoted?).This is the key assumption;if?did not extend to a symmetry of the full non-perturbative type IIB theory,then orientifolding by?would be problematic even for small but?nite string coupling,whereas it is believed that the type I string extends to a consistent non-perturbative theory,and this assumption is implicitly made in many discus-sions of string dualities.The existence of such a non-perturbative symmetry can be‘derived’from M-theory:the non-perturbative IIB theory arises from M-theory compacti?ed on a2-torus in the limit in which both radii tend to zero[13]and the symmetry that provides the non-perturbative version of?is given by the IIB theory limit of the Z2M-theory symmetry used by Hoˇr ava and Witten[14].

Then

??=S?S?1(2)

is also a symmetry of the full IIB theory,where S is the SL(2,Z)transformation(1) interchanging weak and strong coupling.However,??leaves the dilaton invariant, and is a symmetry order by order in IIB string perturbation theory.The weakly coupled type IIB string has a perturbative symmetry(?1)F L,where F L is the left-handed fermion number of the conventional world-sheet formulation,and as??and (?1)F L act in exactly the same way on perturbative states,??can be thought of as a non-perturbative extension of(?1)F L[7].The massless bosonic?elds that are invariant under??are g MN,B1MN,Φ,A1M...N,which are precisely the NS-NS?elds, so that D-branes are odd and NS or(1,0)strings,5-branes and9-branes are even under??.In the same way that?inverts the parity of the(1,0)string world-sheet, together with the world-volume orientations of the(1,0)5-brane and9-brane and the7-brane and3-brane,??inverts the parity of the(0,1)string world-sheet,and also inverts the world-volume orientations of the(0,1)5-brane and9-brane and the 7-brane and3-brane.

The orientifold of the weakly coupled IIB theory constructed using?gives the weakly coupled type I theory[1,2].The orientifolding can be thought of as introducing an orientifold9-plane,and32D9-branes must be added to cancel the anomalies and divergencies introduced by the orientifold9-plane.The invariant sector is formulated in terms of unoriented closed type I strings,with massless bosonic?elds g MN,B2MN,Φ.In addition there is an open string sector,which can(in some ways)be thought of as a twisted sector,with SO(32)Chan-Paton factors arising from32RR9-branes;the Chan-Paton factor labels which9-brane the string ends on[6].The RR string,5-brane and9-brane are invariant under?, and so survive the projection.The closed fundamental IIB strings are BPS,and as ?acts as world-sheet parity on these,they become the nonoriented fundamental type I strings,which are no longer BPS and can break.As well as non-oriented closed(1,0)strings,there should also be non-oriented(1,0)5-branes,3-branes and7-branes(and,in principle,(1,0)9-branes)in the theory,as?acts as an orientation-reversing parity operator on their world-volumes.

An isomorphic construction emerges if we instead mod out the strongly coupled IIB string using??.Indeed,the strongly coupled IIB theory is formulated as a world-sheet theory of the(0,1)string,which is a perturbation theory in?g=1/g, and??reverses the parity of the(0,1)string world-sheet.The resulting type I theory is one in which the fundamental open and closed strings are(0,1)strings which can end on the(1,0)strings,5-branes and9-branes.Consistency requires 32(1,0)9-branes(which can be thought of as cancelling out the e?ects of a(1,0) orientifold9-plane),and these give SO(32)Chan-Paton factors to the open(0,1) strings.This is the standard type I superstring,but embedded di?erently in the type IIB theory.If this extrapolates to all values of the coupling,then continuing back to weak coupling,we?nd that modding out weakly-coupled type IIB string by??in the presence of32(1,0)9-branes should give the weakly coupled SO(32) heterotic string.This construction of the SO(32)heterotic string from the IIB string modded out by??with329-branes can be understood as a particular limit of the Hoˇr ava-Witten construction.In[14],M-theory on T2is modded out by a

Z2symmetry to give M-theory on a cylinder S1×S1/Z2.In the limit in which the radius R of the S1and the length L of the S1/Z2both tend to zero,with the limiting form of L/R small,the M-theory on T2becomes the IIB string with coupling L/R, the Z2symmetry becomes??and the resulting theory is the SO(32)heterotic string.Similarly,if L/R is large,this reduces to the orientifold construction of the type I string;see[15]for details.

This construction implies that the gauge structure of the heterotic string must arise from the32(1,0)9-branes.We will now show how this comes about,and verify that the perturbative heterotic string indeed emerges from this construction.

Consider?rst the D-string in the weakly-coupled type I string theory.The zero-mode structure of the supergravity solution corresponding to the D-string was studied in[4,5]and shown to give precisely the world-sheet structure of the heterotic string.The excitation spectrum of the D-string can be calculated by quantizing the open string allowing Dirichlet(D)boundary conditions with the fundamental strings ending on the D-string,as well as Neumann(N)boundary conditions[6].There are three sectors:(i)the NN sector in which both ends of the string satisfy Neumann boundary conditions(or,equivalently,end on the D9-branes)and carry SO(32)Chan-Paton factors,giving the usual open strings of type I string theory;(ii)the DD sector in which both ends of the string lie on the D-string has a massless sector which is a world-sheet theory on the D-string, consisting of8scalars and8right-handed Majorana-Weyl world-sheet fermions; (iii)the DN sector in which one end of the string lies on the D-string and the other satis?es Neumann boundary conditions and carries an SO(32)Chan-Paton factor has a massless sector which is again a theory on the D-string world-sheet,this time consisting of32left-handed Majorana-Weyl world-sheet fermions transforming as a32of SO(32).Thus the massless modes of the DD and DN sectors gives an e?ective D-string world-sheet theory which is precisely that of the heterotic string [6].

Consider now the theory obtained by modding out the weakly coupled IIB

string by??,with32NS9-branes.On the perturbative states,??acts as(?1)F L and so,of the massless?elds in the IIB supergravity multiplet,an N=1super-gravity supermultiplet survives,with bosonic sector consisting of the NS-NS?elds g MN,B1MN,Φ,A1M...N.The oriented fundamental string and NS5-brane of the IIB theory,coupling to B1MN,also survive,as do the9-branes coupling to A1M...N.

As??acts as the parity operator on the IIB D-string world-sheet,there are (0,1)strings or‘D-strings’of the orientifolded theory that are non-oriented and can be closed or open.The open D-strings can be thought of as ending on the NS9-branes,and these give them SO(32)Chan-Paton factors.They are not BPS states and have interactions through which they can break and join.In the strong coupling limit,these D-strings become the fundamental open and closed strings of the dual type I theory,while in the weakly coupled theory,they have tensions of order1/g and their e?ective dynamics is governed by a Born-Infeld action(without Wess-Zumino term).These D-strings can end on the fundamental string and the solitonic5-brane.There are then DD D-strings which have both ends lying on the fundamental string or solitonic5-brane,and DN D-strings,one end of which end of which is free and carries an SO(32)charge,and the other end of which ends on the fundamental string or solitonic5-brane.There are thus SO(32)charges tethered to the fundamental string or solitonic5-brane by open D-strings.

In the zero coupling limit,the D-string tension becomes in?nite,and these D-strings collapse to zero length.Then the SO(32)charges tethered to the fun-damental string or solitonic5-brane are pulled onto the world-volume,giving rise to an SO(32)current density on the world-volume.As the D-strings collapse,all that survives is the zero-slope limit or massless sector of the D-string spectrum, and this must be the same as the massless sector of the excitations of type I strings ending on a D1or D5-brane in the weakly-coupled type I theory,as the massless sector is in a short multiplet protected by supersymmetry that can be extrapolated from weak to strong coupling.This means that for the D-strings ending on the fundamental string,all that survives are8scalars and8right-handed Majorana-Weyl world-sheet fermions from the DD sector and32left-handed Majorana-Weyl

world-sheet fermions transforming as a32of SO(32)from the DN sector.This gives precisely the right world-sheet structure for the free fundamental heterotic string.What is novel is the picture of what happens to the heterotic string if the coupling constant is?nite.Then the SO(32)charges,which are con?ned to the world-sheet at zero coupling,can move o?the world-sheet,but are tethered to it by strings of tension1/g.However,these can also break,so that a DN string splits into a DN string and an NN string,which is an open D-string that can then escape from the fundamental string.

It has been realised for some time that the5-brane in the SO(32)theory should carry SO(32)currents[16-23],and in[24]a world-volume structure was proposed, which was shown to lead to cancellation of all anomalies in[25].For the D5-brane of the type I theory,the massless world-volume?elds arise from type I strings with DD and DN boundary conditions[24].For N coincident D5-branes,the following 6-dimensional(1,0)supermultiplets with SO(4)R-symmetry arise.The DD sector gives a Yang-Mills multiplet with Sp(N)gauge symmetry and a scalar multiplet (transforming as an N(2N?1)?1antisymmetric tensor representation of Sp(N), plus a singlet)with4scalars X and two spinors transforming as a vector and chiral spinor of SO(4),respectively;the4X i are the collective coordinates for the brane.The DN sector gives a pseudo-real hypermultiplet with4scalars which are a vector of SO(4)and a4-component fermion which is an SO(4)singlet.These hypermultiplets transform as a(32,d a)of SO(32)×Sp(N)where the representation of dimension d a=N(2N+1)is the adjoint of Sp(N).

The same massless modes should be present on the heterotic5-brane world-volume,as the massless sector can be extrapolated to strong type I(weak heterotic) coupling.In the construction of the heterotic string by gauging??,the world-sheet structure emerges from D-strings ending on the heterotic5-brane,and in the weak coupling limit of the heterotic string,the D-strings collapse to zero length and only the massless sector survives.The SO(32)currents on the5-brane world-sheet arise from SO(32)charges tethered to the5-brane that are‘pulled in’in the weak coupling limit,while the Sp(2)currents arise from non-oriented D-strings with

both ends attached to the5-brane.

The presence of the9-branes plays a vital role in the construction and are responsible for the gauge structure.If no9-branes are added,then the perturbative IIB string can be orbifolded by(?1)F L to give the IIA string[26],and no gauge sector emerges.It is not clear whether this can be extended to the full non-perturbative theory,and will be discussed further in[15].

One can apply similar considerations to other string theories.Consider the IIB theory.For weak coupling,the D-string dynamics is governed by fundamental strings ending on the D-string,which is useful as it allows detailed calculation. The D-string can be thought of as being surrounded by a cloud of fundamental strings ending on it,and closed fundamental strings can break o?from the teth-ered strings and propagate into the bulk,mediating the interaction with the bulk degrees of freedom.By duality,at strong coupling the(1,0)string dynamics is governed by(0,1)strings ending on the(1,0)string,and this again can be studied using(the S-dual)conformal?eld theory.Back at weak IIB coupling,D-strings can end on fundamental strings,but this is not so useful in giving the dynamics of the fundamental strings,as we only have a formulation of the D-string dynamics in terms of the fundamental strings themselves(from which an e?ective Born-Infeld action can be derived).However,the excitations of the D-string that are in short multiplets can be understood,since they can be extrapolated to strong coupling where they are the short-multiplet states of the fundamental string.In particular, the massless sector of the the D-string is the same as that of the fundamental string,and consists of8world-sheet scalars X i and16world-sheet fermions,which transform as a vector and complex chiral spinor of the transverse SO(8),respec-tively.At zero IIB string coupling,the tension of the D-strings becomes in?nite and only the massless sector survives,so that the familiar world-sheet degrees of freedom of the fundamental IIB string arise from the D-strings attached to the fundamental string that collapse to points,giving local?elds on the fundamental string world-sheet.The picture suggests that at?nite coupling,the world-sheet is replaced by a cloud of D-strings and the non-BPS states might be understood

in terms of D-string excitations;it would of course be very interesting to?nd a non-trivial check of this.

The description of the heterotic string proposed here arises from this picture of the type IIB string on gauging??.The D-strings become non-oriented and non-BPS,but the world-sheet structure of the free heterotic string again emerges from D-strings attached to the world-sheet,in the limit in which they collapse to points.

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钠与硫酸铜反应

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