THE EQUIVALENCE OF NSR AND GS FOUR-DIMENSIONAL TWISTED STRINGS A.H. CHAMSEDDINE* Institut fir Theoretische Physik, Universität Zurich, Schönberggasse 9, CH-8001 Zurich, Switzerland

J.-P. DERENDINGER Institut fz~rTheoretische Physik, ETH-Hönggetherg, CH-8093 Zurich, Switzerland C. KOUNNAS~K~ Laboratoire de Physique Théorique, Ecole Nonnale Supérieure, 24 rue Lhomond, F-75231 Pans Cedex 05, France Received 5 March 1990

Using a general construction of the modular invariant partition function for four-dimensional strings with twisted, orbifold-like, boundary conditions, we discuss the equivalence of their Neveu—Schwarz—Ramond and Green—Schwarz formulations. Four-dimensional GS strings are found to be equivalent to the class of NSR theories on orbifolds based on the canonical, free supercurrent, with underlying N = 2 linear world-sheet supersymmetry. This equivalence provides interesting insights on the consistency conditions for all order modular invariance to be applied on four-dimensional Green—Schwarz strings.

1. Introduction In recent years, large classes of consistent string theories with four-dimensional space-time symmetries have been constructed using various methods [1—131.Since the basic, underlying theory is either the heterotic string [141, or type II superstrings [151,they can be viewed as describing different classical vacua of these two strings. The fermionic sectors of these two theories (right-movers only for heterotic, left- and right-movers for type II strings) can be described alternatively * **

Supported by the Swiss National Foundation. Laboratoire propre du Centre National de Ia Recherche Scientifique, associé Supérieure et a l’Université Paris-Sud.

0550-3213/90/$03.50© 1990

—

Elsevier Science Publishers B.V. (North-Holland)

a

l’Ecole Normale

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using Neveu—Schwarz—Ramond (NSR) fermionic strings, or using Green—Schwarz (GS) superstrings [15]. The two formalisms are complimentary. Interactions, amplitudes and physical processes are much easier to describe using the NSR string which is a superconformal field theory. Boson—fermion equivalences simplify several problems. Most of the constructions [3—111of four-dimensional string vacua have then been performed in the NSR formalism. This is in particular essential in formalisms which explicitly use bosonization [3,81,or fermionization [4—7]. The advantages of GS strings are related to the fact that space-time and world-sheet statistics of the fields coincide in light-cone gauge. Space-time properties like gauge groups, spectra and especially space-time supersymmetiy are much easier to investigate. The powerful methods of conformal field theory are however lost, making the introduction of interactions problematic. The covariant form of ten-dimensional GS strings [16] has a rather subtle algebraic structure. All fields are two-dimensional bosons, but space-time fermions appear with quartic terms in the lagrangian. The action possesses a local non-linear fermionic symmetry (K-symmetry) which turns out to play a role similar to world-sheet supersymmetry for NSR strings. It allows to choose a light-cone gauge in which the action becomes simply quadratic, and where the space-time fermions also satisfy the two-dimensional Dirac equation. It is only very recently that a method to covariantly quantize this theory has been invented [17]. In ten-dimensional space-time, it is known that NSR and GS strings describe the same theories. The GS superstring was in fact invented with this equivalence in mind [181. For consistent heterotic ten-dimensional strings, the equivalence is easily checked at the level of the spectrum. It is however a much harder issue when interactions are considered. For the two space-time supersymmetric E8 X E8 and SO(32) heterotic theories [141, the identity of the NSR and GS spectra follows from a Riemann identity for the four Jacobi i~-functions 4 + i~ =

—

~2(OJr)

4 i~ —

3(OIr)

4(OIr)~]

(1.1)

When multiplied by a factor ‘i~~ ~ is the Dedekind function), the left-hand side of eq. (1.1) is the partition function of the GS fermions, and the right-hand side is the partition function of the NSR fermions in the Neveu—Schwarz and Ramond sectors, with the GSO projection appropriate for space-time supersymmetry. In the NSR partition function, the sign of the contribution ~ specifies the chirality of massless space-time fermions, in the Ramond sector. The Riemann identity (1.1), and hence the equivalence of NSR and GS strings, is a consequence of the triality property of the transverse Lorentz group SO(8) [15], which is relevant to enumerate physical degrees of freedom. Introducing the

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partition functions for SO(8) conjugacy classes,

& c2{c)

=

1 2??

—41 [t~

41

4

2(0IT)

—

~1(0~T)

j

(1.2)

,

where {v} is the vector class, {s} and {c} the two spinor classes, SO(8) triality tells us that =

~(s}

(1.3)

=

In the NSR string, bosons from the Neveu—Schwarz sector are in class {v}, and fermions from the Ramond sector in either (s) or {c}, depending on a (free) choice of chirality. Using triality (1.3), one can then write the partition function of the two-dimensional fermions either as

4fl~

~NSR

=

—

~{c}

=

—

~{c}

or as ~NSR~}

=

i~1(OIT)

=

(1.4a)

~GS

~

~{s}~{c}~{s)

(1.4b)

Eq. (1.4a) corresponds to Riemann identity (1.1), while the other chirality (1.4b) is obtained by reversing the sign of the i~ contributions. In eqs. (1.4), the first equality is the NSR partition function for both chiralities, and triality implies its equality (up to an irrelevant sign) with the GS partition function ~GS• In GS representation, space-time bosons and fermions are respectively in classes (SI and {c} with chirality choice (1.4a). Chirality choice (1.4b) exchanges these conjugacy classes. Notice that it is important in these manipulations to keep formally track of the ~~ 1(Or) terms, even though they vanish*. For the non-supersymmetric SO(16) x SO(16) theory [19], the NSR and GS partition functions are given by the same combination of i~-functions.There is no explicit need for the Riemann identity (1.1) to check the equivalence. Simply, since the space-time interpretations differ in each formalism, given states arise from different contributions. In dimensions lower than ten, and especially in four, the problem of the relationship between NSR and GS formulations of heterotic string theories is much more interesting. The number of consistent vacua is extremely large. More4 = ~[~ *

One could also use the Riemann identity i~(r’IT)

4+ 1(v ~)

~)4

—

~

4+ ~ 3(v r)

4(v I

(see appendix A) for an arbitrary (real) parameter o playing the role of a mass for the states. There is however no other identity with reversed signs of the ~1(vIr) contributions: a mass term has always the opposite chirality to the fermion (and ~i(uIT) does not vanish for all values of ~).

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over, they correspond to various realizations of the local two-dimensional symmetries. For ten-dimensional NSR strings, Lorentz covariance imposes a unique realization of world-sheet supersymmetry, up to boson—fermion equivalences. The supercurrent is always S i~I~XJ~. This is not any longer the case in smaller dimensions, where Lorentz scalar string coordinates are also present. One has a large freedom in choosing a (in general non-linear) realization of world-sheet supersymmetry [201. A question then arises: for which classes of NSR and GS four-dimensional consistent strings can one find a generalization of the equivalence existing in ten dimensions? To answer this question in general terms, the simplest method is to use an explicit construction of the modular invariant partition function valid for both NSR and GS formalisms. In light-cone gauge, both theories have the same world-sheet degrees of freedom, they differ only in the space-time interpretation. Since modular invariance is a world-sheet property only, it can be solved simultaneously for both cases. There is however a subtlety related to two-loop modular invariance. For NSR strings, a crucial phase arises, in the factorized limit of two-loop amplitudes, from the world-sheet gravitino [6,211. This has been explicitely obtained either by direct computation of the gravitino phase [6,21,23], or by requiring a sensible particle interpretation, or unitarity [5,221. These different approaches are in fact equivalent. The phase one obtains determines the GSO projections in the one-loop partition function, ensures the correct spin-statistics and includes automatically the graviton in the spectrum. In the light-cone gauge, an equivalent method to determine the correct phase is to reverse the argument: assuming formally factorization, the phase is obtained by requiring the presence of the graviton in the massless spectrum [131.For NSR strings, the resulting partition function is then identical to the one obtained with a fully covariant computation. This last method has the advantage that it can tentatively be applied also to light-cone GS strings [13], where no explicit computation exists. The equivalence with NSR strings which we establish in this article will in fact confirm that this approach makes sense, and provide informations on the correct consistency conditions to be applied on GS strings. A formalism for describing a very large class of NSR and GS strings, in a unified way, has been obtained in ref. [131.It provides a simple and explicit construction of the partition function for boundary conditions including abelian twists for fermions and bosons as well as shifts for bosons. This formalism is general enough to describe orbifolds [2, 9, 101 as well as most bosonic [3,81 and fermionic [4—7] constructions. It is essential to use a formalism which includes ZN orbifold-like boundary conditions (N ~ 2): four-dimensional GS theories, as discussed in ref. [121belong essentially to this class of theories. It is then the most appropriate for a discussion of the relationship between NSR and GS strings. We will summarize the construction of four-dimensional GS strings and introduce the parameters characterizing the fermionic sectors of models using GS and =

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NSR formulations in sect. 2. These parameters correspond to the boundary conditions of the fields, and, in some models, to some free discrete phases. As in spin-structure constructions, they must be chosen to preserve world-sheet supersymmetry in the NSR case, and K-symmetry for GS strings. The proof of the equivalence corresponds then to establish the mapping from GS to NSR parameters leading to identical partition functions. The proof of the equality of the partition functions requires a generalized Riemann identity and is discussed in sect. 3. The implications of these results are discussed in sect. 4. For completeness, appendix A gives a simple derivation of the Riemann identities used in the proof of the equivalence.

2. Twists and the Riemann identity The construction of consistent string theories given in ref. [131 is analogous to the spin structure method [221 also used in fermionic formalisms [5—71. We will shortly review it in sect. 3. A given theory is defined by the set of boundary conditions of all string fields. In some cases, there is an additional freedom in the partition function, which allows to introduce some discrete phases: some GSO projections are not completely fixed by the conditions of all order modular invariance applied on the boundary conditions, which are equivalent to the so-called level-matching conditions [2]. We will consider models with twisted complex world-sheet fermions: (2.1) and, as in orbifold compactifications, twisted (complex) bosons: z(u±T+~r)=e21~z(o-±T)+4.

(2.2)

As long as we are only interested in the partition function, we can work in the light-cone gauge where the NSR and GS formulations of heterotic strings have the same world-sheet degrees of freedom: four complex world-sheet fermions and three complex bosons for the right-movers, and eleven left-moving complex bosons, besides the transverse space-time coordinates X~.We will be only concerned with the right-movers, for which the parameters are four fermion twists (0, in eq. (2.1)), three boson twists (4) and six real shifts (4). The first and most important difference between the two formulations appears in the consistency conditions relating fermion and boson twists: the boundary conditions of the right-movers must be compatible with world-sheet supersymmetry in the NSR case, and with K-symmetry for the GS superstring. The admissible boundary conditions are then in general different in NSR and GS formulations. The second difference is the space-time interpretation of the partition function since the space-time quantum

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numbers of world-sheet fermions are different in both formulations. To obtain the relationship between NSR and GS strings, we first need to find the correspondence between boundary conditions satisfying either NSR or GS consistency conditions. The four-dimensional version of the Green—Schwarz superstring is simply obtained by regarding the ten-dimensional theory as a four-dimensional one [121. Instead of insisting on the full ten-dimensional Lorentz invariance SO(1, 9) (or the full transverse Lorentz group SO(8) in light-cone gauge), the internal part SO(6) is broken by the boundary conditions. Notice that this procedure is not exactly a dimensional reduction: since the action is free of space-time derivative, the ten- and four-dimensional lagrangians are algebraically equivalent, there is no truncation. The covariant GS string contains only world-sheet bosons. In four dimensions, it possesses four space-time Weyl spinors, which satisfy in the light-cone gauge only a two-dimensional Dirac equation, and then behave like world-sheet fennions in this gauge. For our purposes, each space-time Weyl fermion can be regarded as a complex world-sheet fermion: it contains two physical degrees of freedom. However, since Majorana—Weyl spinors do not exist in four dimensions, we are not allowed to consider a Weyl spinor as two real world-sheet fermions with independent boundary conditions. In addition, the four-dimensional GS string contains a periodic vector X’~and three complex scalars Zk. The twist parameters which can be introduced for GS right-movers are then the four fermion twists 0a’ a 0, 1,2,3 and the three boson twists tk, k 1,2,3. Consistency requires that these boundary conditions preserve the local invariances of the covariant Green—Schwarz action [16], rewritten as a four-dimensional theory: the fermionic K symmetry, the bosonic local symmetry similar to a non-linear Weyl symmetry, and world-sheet reparametrizations. In fact, all conditions can be obtained from the K-transformation of the four (left-handed) space-time spinors S~,which is of the form =

=

=

6S~ 2i(öaX~)’j/~E~ + 2\/~7(8aZat))~b =

—

4~’~SLböaSRCE~d +

other terms, (2.3)

where a is a two-dimensional vector index, and e~ are the fermionic (in spacetime) local parameters of K-transformations. We have defined complex scalar fields Zab ~Zba such that =

k=1,2,3;

ZkZok,

Zab2EZCd.

With this definition, a,..., d 0, 1,2,3 are SU(4) indices for representation 4, and the transformation (2.3) is explicitly SU(4) covariant. To be compatible with =

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transformation (2.3), twists must then satisfy the necessary and sufficient conditions [12]: 4~k=0O~0k’

k=1,2,3,

modi,

(2.4a)

modi.

(2.4b)

3

0=00+ ~ k—i

For convenience, one fermion, with twist 0~,is singled out: the comparison with NSR strings will later on be easier. It is only an irrelevant choice of basis for the GS spinors. It is however important to notice that a choice of space-time chirality has been made when writing eqs. (2.4). The condition (2.4b) is due to the trilinear terms in eq. (2.3). These terms associate spinors of well-defined chirality and we have chosen to identify each left-handed Weyl spinor with a complex world-sheet fermion with twist 0. Its complex conjugate has a right-handed space-time chirality, and a twist —0. With this choice, the conditions (2.4) have a simple interpretation. Eq. (2.4b) indicates that the fermion twists can be viewed as an SU(4) transformation, the fermions transforming in representation 4. Then, eq. (2.4a) tells us that the bosons are twisted by the same SU(4) element, but for the representation 6. The twists are then elements of a subgroup of SU(4) SO(6): this is the same condition as for the holonomy group of orbifolds [21.It is also a consequence of the procedure providing the four-dimensional theory. One starts in ten dimensions (and in the light-cone gauge) with space-time fermions in the left-handed SO(8) spinor, 8~.Under SO(6) x SO(2) SU(4) x U(1), the fermions are in ~—1/2 + the U(1) quantum numbers being space-time helicity. The twists in eqs. (2.4) apply to the left-handed fermions. Other chirality choices would lead to replace some twists in eqs. (2.4) by their opposite value. For instance, eq. (2.4a) could as well be replaced by 0=00+01+02—03,

modi,

(2.4c)

corresponding to another choice of space-time chirality. The GS partition functions would not be affected by this change of convention. However, as already mentioned, the GSO projection for the Ramond sectors of the equivalent NSR theory would be inverted. In fact, all possible chirality choices are equivalent either to eq. (2.4b) (with fermions arising from the spinor 8), or to eq. (2.4c) (with fermions in the other spinor, 8). In the following discussion we will always use the more symmetric convention (2.4b). Thus, GS strings have three free twist parameters. Since only derivatives of bosons appear in the K-transformations, shift (lattice) parameters 4 can be omitted in our discussion. Fermion twists will have the effect of reducing the number of space-time supersymmetries. It turns out [12, 13] that models without

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any (right-moving) twisted bosons are always direct four-dimensional generalizations of the ten-dimensional theories, with N 4 or N 0 supersymmetry. For all other N 0, 1,2 cases, twisted bosons always appear: GS four-dimensional strings are mostly orbifolds. It will then be essential to use a construction of the modular invariant partition functions including orbifolds. Notice that the choice of chirality is physically irrelevant as long as heterotic theories only are considered. This would not be the case for type II strings where the relative chirality of left- and right-movers makes the difference between type ha and lIb models. Even though our discussion of the equivalence of NSR and GS strings will be explicitly given for heterotic strings, it is straightforward to apply it to type II theories, with the two relative chiralities. The analogous discussion for the NSR string is well known. The NSR fourdimensional fields are X~,i/ti’, internal complex fermions ~I’kand bosons Zk, k 1,2, 3. We will only consider linear realizations of world-sheet supersymmetry, corresponding to the “canonical” form of the supercurrent =

=

=

=

3

S

=

~ +

(~/Jkazk+ c.c.),

~

(2.5)

k=1

the space-time supercurrent being 5st ~J’~9X~.This supercurrent is equivalent to a SU(2)6 realization with free fermions. Even though this is only a very limited class of realizations of world-sheet supersymmetry, it will turn out to be sufficient for our purpose: to every GS theory as defined above, with K-symmetry, there will correspond a NSR theory where world-sheet supersymmetry is realized with the supercurrent (2.5). In the light-cone gauge, ~ reduces to two transverse components, ~fra’ a 1,2, and can be considered as a complex fermion, analogous to lfJk. Clearly, its twist 0~can only be 0 or leading to a periodic or antiperiodic space-time supercurrent ~ The global existence of the supercurrent (2.5) indicates then that NSR twists must satisfy =

=

~-,

4)k60+Ok,

modi,

k=1,2,3,

(2.6a)

0 0=0or~.

(2.6b)

Again, bosons only appear with derivatives in the supersymmetry transformations and there is no constraint on shifts. Eqs. (2.6) leave three independent parameters, which can be taken as the fermion twists Ok: the twist of O~,does not introduce any freedom: both values are present in any consistent theory, which must include a massless graviton. In both formalisms, the same combination of fermion twists defines the boson twists [eqs. (2.4a) and (2.6a)]. The difference is in the GS condition on the sum of fermion twists, eq. (2.4b), which is replaced by the reality constraint on the ~

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transverse part of

~

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eq. (2.6b). But consider the following transformation: 3 =

~ A~1d1, j=0

(2.7)

where the matrix A is given by 1 A=~ 1

1 1 —1

1 1 1

1 1

11

1

—1

—1

1

(2.8)

Clearly, for any set of GS fermion twists 0~ d,, i 0,. 3 satisfying eqs. (2.4), the twists O~ D1 are then admissible NSR fermion twists, verifying eqs. (2.6). And, since (2.9) 4k and 4’k do not need to be the oppositesince is also true. Also, boson twists transformed =

=

. . ,

=

D 0+D~=d0+d1, i=1,2,3.

(2.10)

As it should for relating NSR and GS theories, only fermions are affected by the transformation. In fact, A is nothing other than the matrix generating the triality transformation which exchanges the conjugacy classes {v) and (s} of SO(8). It is also the matrix responsible for the equivalence of ten-dimensional GS and NSR strings. It is well known that with any matrix satisfying eq. (2.9), one can construct Riemann identities. In ten dimensions, the relevant Riemann identity, eq. (1.1), is directly related to the matrix A. We will show in sect. 3 that the relationship between NSR and GS strings in four dimensions is also related to the transformation (2.8), with the help of a generalized Riemann identity which can be applied to arbitrary boundary conditions for the fermion string fields. Conditions (2.4) and (2.6) have been derived from the covariant GS and NSR theories. They can also be directly obtained in the light-cone formalisms, where they ensure that physical states form complete representations of the four-dimensional Lorentz group. Eq. (2.6b) is well known to imply that all states have integer or half-integer helicity. Transformation (2.7) indicates that this interpretation is also valid for eq. (2.4b). With the second chirality choice, for which the GS condition (2.4b) is replaced by eq. (2.4c), the matrix A must also be replaced by

B=

~( ~ ~

~

=B’.

(2.11)

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B generates the S0(8) triality transformation exchanging the classes (v} and {c). The Riemann identities associated to A and B are different (see appendix A) leading to NSR theories with GSO projections selecting the two chiralities in the Ramond sectors where space-time fermions arise. In sect. 3, we will only use the chirality choice corresponding to A, but the same discussion using B would only introduce some minus signs: an overall change of space-time chirality does not make physical differences. In the case where the boundary conditions are such that the resulting string theory has at least N 1 space-time supersymmetry, the relation between NSR and GS strings is especially simple. As usual, supersymmetry is easier to investigate in the GS formalism [12]. Since space-time supersymmetry is linearly realized in GS strings, the fields are classified in supermultiplets: X~and S~belong to a vector multiplet, and Zk and S~to three chiral multiplets. To obtain a supersymmetric theory, one must require that_boundary conditions preserve this supermultiplet structure. This is the case if 0~ 0, in which case bosons and fermions in supermultiplets have the same twist, according to eq. (2.4a). Each sector has then a vanishing vacuum energy and a supersymmetric spectrum. (In fact, 00 0 is sufficient but not necessary: there is one exception. It can happen that models with =

=

=

a 1,.. 4 possess a “hidden” space-time supersymmetry. Supersymmetric partners appear however in different sectors of the theory. This phenomenon is analogous to the appearance of supersymmetry in NSR strings. It is discussed in the second article of ref. [12]). If 0~ 0, the transformation (2.7) becomes almost trivial. Since =

~,

=

. ,

=

3

(2.12) k= i

transforming the GS fermion twists

ö~

(0, 0~,ö2,

ö3) with the matrix A leads to the NSR twists 0a 0a + N/2. However, if N is odd, the second term corresponds to a NSR boundary condition where all fermions are antiperiodic, and all bosons periodic according to eq. (2.6a). A NSR theory always contains a sector with Oa since it is in this sector that the graviton appears. This term can then be omitted when defining a string theory in terms of linearly independent boundary conditions. This is obviously also true for N even since boundary conditions are defined modulo 1. Then, for space-time supersymmetric string model, the same boundary conditions can be used for the GS and NSR formulations*. There is however a complication in the partition function: even though boundary conditions are only defined modulo 1 in eqs. (2.1) and (2.2), this is not true for partition functions where non-trivial phases arise. Notice also that the transformation (2.7), when applied to twists w (~,~,~,~) leads to (1,0,0,0), for which all fields are periodic. In ten dimensions, w is the =

=

=

=

*

This case has already been discussed in ref. [13].

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unique non trivial twist. It is also the twist vector giving rise to the Neveu—Schwarz sector in the NSR formalism, in which the graviton appears. In order to break supersymmetiy, an N 0 GS theory in ten dimensions must also contain a sector with fermion twist w, accompanied with non trivial boundary conditions. One can then directly see that the boundary conditions for an N 0 ten-dimensional theory, like the SO(16) X S0(16) theory [191,are necessarily identical when formulated with NSR or GS fermions. The partition function is then given by a unique combination of It-functions, even though space-time interpretations differ in the two cases. =

=

3. The comparison of the partition functions A construction of the partition function for consistent NSR and GS string theories with twisted boundary conditions has been presented in ref. [13]. Since we will use this formalism to explicitly prove the equivalence of GS and NSR formulations, at the level of the spectrum of physical states, we begin by summarizing its main aspects. As usual in spin structure constructions, a consistent twisted string model is specified by a set of boundary condition vectors, (WM). Each vector contains the parameters defining the boundary conditions (2.1) and (2.2) for all left- and right-moving fields, i.e. all twists and shifts. We only consider vectors with a finite order NM. A vector has the following form: (3.1)

WM_(O0,Ol,02,03;(4?k,Vk,vflJ(cb/,vj,v!)).

The first four angles are the twists of the complex fermions, and for each complex boson (three right-movers, k 1,2,3 and eleven left-movers, I 1,..., 11), three parameters are necessary: the twist 4, and two real shifts, vx Re 4 and v~’ Im 4 [see eq. (2.2)]. The order NM is then the smallest integer for which all components of NMWM are integers. Modular invariance implies that all linear combinations of the vectors WM will necessarily be included so that a general boundary condition is =

=

=

AW=EaMWM, M

aM=O,...,NM1.

=

(3.2)

It is then sufficient to specify a basis {WM} to define the theory. The vectors in the basis must satisfy two classes of consistency conditions: generalized level-matching constraints, which imply the existence of a modular invariant partition function, and the conditions for world-sheet supersymmetry or K-symmetry discussed in sect. 2. Two vectors have a special role. Firstly, the shift vector V~—(o,o,o,o;(o,~,~-)3i(o,~,fl11)

(3.3)

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is always present in the basis (the index M takes then values 0, 1,..., and a0 0, 1 since N0 2). For GS strings, V0 contains the boundary conditions for the compactification of scalar string coordinates on the torus with the largest symmetry. It also generates the sector containing the graviton. The partition function constructed with the minimal basis (V0} is then the N 4, SO(44) model. In NSR strings, it generates the Ramond sector, where gravitino states appear, or are projected out by GSO projectors generated by other boundary vectors WM. Secondly, the vector =

=

=

Wo= ~

(3.4)

appears in all NSR models: it is used to generate the Neveu—Schwarz sectors where the space-time bosons appear. In particular, the graviton arises in the sector W6 + V0. Again, the basis {l/~,W6} for a NSR model leads to the N 4, SO(44) model. We want to show the connection between any theory for which the basis of vectors has the form {V0, WM} for a GS theory, or {V0, W~,WM} for a NSR model. GS vectors Wm satisfy conditions (2.4), while NSR vectors Wm satisfy eqs. (2.6). For a given basis, the modular invariant partition function is given by =

Ee(~)~:~

~=(flNM)~ M

(3.5)

A,B

where NM is the order of the vector WM, and

~(~)

=exP[2i3r ~

aMbNMNI

(3.6)

M, N

contains the only possible free discrete parameters for a given basis of vectors, satisfying ~MN ENM and NMEMN NNEMN 0, (modulo 1), since NMWM and OWM are equivalent boundary conditions. The partition function ~ for the boundary conditions AW and BW along the cycles of the torus is a combination of functions =

—

d(0’,O;T)

=

=

=

e~000It[2~](0~r)fl(TY1

(3.7)

for shifted bosons and twisted fermions, and of d(0’,0;T)~ multiplied by the appropriate number of fixed points of the corresponding orbifold for twisted bosons. The choice of phase in the definition of the function d is such that modular transformations do not generate phases depending on 0 or 0’: this is convenient to solve modular invariance, and the coefficients (3.6) in the partition function do not explicitly depend on the vectors WM.

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To compare the two formulations, we will start from GS theories for which the partition function and the set of vectors are simpler. The partition function has been constructed [13] by demanding modular invariance at one and two loops. Then we will use transformation (2.7), (2.8) to generate the corresponding NSR vectors, and the associate Riemann identity to obtain a new form of the partition function. The comparison will be completed by showing that this new partition function characterizes a NSR theory with the correct GSO phases given by NSR two-loop modular invariance, as derived in refs. [6, 211. This last point is non-trivial: the two-loop GS condition has not been derived from explicit calculations, in ref. [13] it was simply postulated. The Riemann identity for It-functions with characteristics associated to the matrix A is very simple to establish. (For completeness, a derivation is given in appendix A). It is of the form

flIt[!t](0PT)=~ 0/

~

exp -i7rb~0~flIt[~i~](0lT)~

a,b=0

i~0

j

i=t)

i

(3.8)

/

where 0, EJA~JO and 0’ ~JA~JOJ.It is straightforward to translate eq. (3.8) in terms of the functions d(O’, 0; T). One finds =

=

3

fl

1

d(0,0,.;~)

=

—~

exp[i~r(a

~

—

1)(b— 1)]exp ~i’zr(b~0t—aE0i’)

a,b—0

i

i

3

x fld(O;+~b,O,+~a;r) 1

=

(3.9a)

0

~

e~ °o~°~d(0~+ ~b,00+ ~a;r)

a,b=0

x fl

0k + (ek~°~

~a; T)).

(3.9b)

d(O~+ ~b,

The last equality is due to the properties d(O’+ 1 O~r)=e

°~d(0’ O~T)

d(O’,O+1;r)=e’~°’~d(O’,9;r).

(3.10)

This less symmetric form of the Riemann identity will be the most useful for discussing NSR partition functions. Then, since bosons are left untouched by the transformation (2.7), one can promote eqs. (3.9) to an identity between partition

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729

Twisted strings

functions for all fields. For a GS vector W (W’), containing the fermion twists ã (0;), one defines the corresponding NSR vector W (W’), containing fermion twists 0, (0;). All entries corresponding to bosons are identical in W and W (W’ and W’). One can then write

=

~

—

(3.lla)

~

a,b=0

or, using eq. (3.9b), c~W I

i~r(W

—

W’

—

(~ 11].,

0±S)(bW—aW)~W+a(W0+S) W±b(W0+S)

e

2

a,b=0

where the vector W6 is defined in eq. (3.4) and S is the spin-statistics vector given

by S= (1,0,0,0;(0,0,0)31(0,0,0)u1),

(3.12)

which role will be clarified later on. The product of two vectors, W1 W2, has the usual lorentzian (left—right) signature. Finally, one can convert eq. (3.llb) to an identity for the complete partition function of a GS model, with basis of vectors (l/~,WM}: .

~=(flNM)~ M

=

(~-~:

A,B

(2flNM)~

~

i

e’~0~

aB4

(3.13)

A,a,B,b

M

The coefficients are expressed in terms of parameters MN as in eq. (3.6). The GS theory is completely specified by the basis vectors and the values of these discrete parameters. We must now check that the last expression is a correct NSR theory. The NSR model can be defined by the set of boundary conditions {AW+ a(W0 + 5)), generated by the basis ([/~,W~+ 5, W~4.This set is however not in the standard form: Wu + S indicates that all fermions are antiperiodic and it is usually replaced by the equivalent but simpler vector W~.This replacement introduces however a phase which follows from eq. (3.lla):

=

(2UNM)’ M

(~)et~

~

~

(3.14)

A,a,B,b

The phase exp[i~(ab + a

+

b)], which is directly generated by the spin-statistics

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/ Twisted strings

vector 5, is well known to result from the contribution of the two-dimensional gravitino to the two-loop partition function. It provides the correct connection between spin and statistics of the string spectrum, or equivalently the CPT theorem of field theory [21,6, 23]. This is remarkable since in the covariant GS formalism there is no gravitino to provide this phase. To complete our discussion of the equivalence of NSR and GS theories, we still have to obtain the new NSR discrete torsion parameters ~MN• Since the basis of vectors contains now also W~,new discrete parameters can be introduced. By assigning ~MN~MN’

CMOIWMWO,

(3.15)

EMO,

~OM

we can rewrite eq. (3.14) in the form

E e(~

~=(NoflNM)~ M

where No

=

(3.16)

~

A,a,B,b

2 is the order of W0 and

(~

~)

=ex~[~i~r(~

aMEMNbN+

M,N

EaEONbN+

EaMMOb)1.

N

M

(3.17)

Eq. (3.16) is in fact the general form of a NSR partition function with correct spin and statistics. As an exercise, one can explicitly prove that the graviton state of a GS string theory remains in the spectrum when the theory is transformed to its NSR formulation. We need to examine the GSO conditions ensuring the presence of the graviton. In a GS theory with vectors {J7~,WM}, the graviton is in sector V0. In this sector, the GSO projector generated by each WM is of the form 1 NM-i

NM

(3.18)

2~M~’~o)

aM—O ~ e

where the phase ~M(Vo), obtained in ref. [13], satisfies NMIM(Vo) 0 (mod 1) and depends on the oscillator content of the state under consideration. Using its explicit form, the graviton is in the spectrum if =

0

=~I~M(V0)Igraviton

—EOM 2VOWM+WMWO,

(modi),

(3.19)

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731

Twisted strings

where W0 is an auxiliary (non physical) vector with all components equal to (3.19) determines the GS parameters EOM=(2VoWo)WM,

(modi).

~.

Eq.

(3.20)

In the equivalent NSR theory, the graviton is in the sector V0 + W0. The GSO conditions corresponding to eq. (3.19) read 1M(Vo+ WO)Igravjton O= _EOM+EOM 2WM(VO+WO)+WMWOWM~,

where ~‘V is the number operator for the graviton states, i.e. V= ±S.One must then take COM+EOM— WM(2V0

(modi), ~~~i/2~O)R

(3.21)

XaIIO)L,

w0)+ ~WM~W0+ WM~S, (modi).

(3.22)

But the NSR vectors WM are obtained by applying transformation (2.7) on the GS vectors. One has then WMWO—(SWO)WM+WMWO, WMVO—WMVO.

(3.23)

Using these last relations, we find that the GSO conditions (3.21) are automatically satisfied since the transformation of the partition function leads directly to the definition of EOM given in eqs. (3.15). This exercise can easily be generalized to arbitrary physical states, showing again the equivalence of the GS and NSR spectra. Even though we have basically reached our goal, two further technical peculiarities should be mentioned. A string model is defined in terms of vectors WM and parameters EMN, satisfying level-matching conditions for modular invariance. These conditions however depend explicitly on the order NM of the vectors WM. They ensure that the GSO projector for WM obtained when summing over the NM inequivalent boundary conditions takes values 0 or 1 on all states. The order of a vector is however not always preserved by the mapping. It can happen that starting from a vector W, with fermion twists 0 m’ m2 m3 m N’N’N’N

where ma and N are integers, and N is the order of W, the mapping with A leads to a vector WA with a different order. It is straightforward to prove that the order

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Twisted strings

N can only be reduced (or increased, by the opposite transformation) by a factor 2.

This happens if and only if (i) all integers m’~are odd and ~am’~ is a multiple of 4, and (ii) all other boundary conditions in W, not restricted by world-sheet or K-symmetry (i.e. twists of left-movers and shifts), are of order N/2. (Because of eqs. (2.4a) or (2.6a), when condition (i) is satisfied then the twists of right-moving bosons are of order N/2.) If the order of a vector is reduced, it seems that the transformed partition function [see eq. (3.16)] contains a summation over too many, redundant boundary conditions. Consider however the set of boundary conditions of a model with vectors {W, WM), assuming that the order N of W is reduced to N/2 by the mapping. According to condition (i) above, the vector (N/2)W is equal to W0, modulo an integer, irrelevant vector. The set of boundary conditions can then equivalently be written as aW+ LMaMWM, with a 0,..., N 1, or as aW+ a’W0 + ~MaMWM, with a 0, N/2 1 and a’ 0, 1. The vector W0 is then automatically included in the set, and should not be considered as independent from the basis {W, WM). One can then write a partition function of the form (3.5), but with a, b 0,.. N/2 1 only: =

=

..

=

.,

—

. ,

— —

—

~IMI NMJ\

~ ~ a,b=0 aM,bM

1N/2—i =

—

=

(NH NM) M

aa b ‘ bM M

~aW+ aMWM bW+bMWM

I

L a,b=0 a’,b’=O aM,bM

~ ‘

~

‘

aM M

~

(3.24)

Applying now the transformation by A, W6 becomes trivial, with integer components only. It is then simple to check that the Riemann identity leads to a partition function for the transformed vectors WA and W~,with orders N/2 and NM, and without any remaining contribution of the vector W6, which disappears as it should under the mapping. A second technical remark has to do with NSR vectors. A NSR model, with basis vectors ~ W0, WM} can also be described using another basis ~ W0, UM) with the additional condition that for all UM’s, the fermion is periodic: 0~ 0. This choice of vectors is much more convenient when discussing the spectrum or space-time supersymmetry for a NSR model [13]. It is then useful to obtain explicitely the mapping from a GS model into the equivalent NSR theory expressed in this specific basis. Starting from a GS vector WM, one first obtains as usual the corresponding NSR vector WM by the mapping (2.7). Clearly, UM can be defined by =

UM=WM—nM(S—Wo),

(3.25)

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733

Twisted strings

where ~ We can then express the partition function (3.16) in terms of the set of boundary conditions {aMU~+ aW6) at the expense of generating external ~M

=

phases that can be absorbed in e(~).Using extensively the periodicity of the d-functions and some algebraic manipulations, the partition function becomes

~=

(ifli’si~.4’ M

where e

~ A,a,B,b

(~ ~

(3.26)

(~~)is of the same form as in eq. (3.17) but with the identifications: w0),

EMN_EMN+(WMS)(WN

Wo)—(~’N•S)(J’f’M~

(3.27)

EMO_2(WOS)WM.

Using the relation between

WM

and (W0

MNEMN

MO2flM2WO

UM

in eq. (3.25), we can equivalently write

UM)No + UM.

(W0 (3.28)

Choosing to work with vectors WM or UM, and with partition functions (3.16) or (3.26) is only a matter of convenience. This last remark completes our discussion of the equivalence of the GS and NSR partition functions for the class of string theories considered in sect. 2. To perform our analysis, we have explicitely used the heterotic string, for which bosonic left-movers are passive spectators. It is simple to generalize the discussion to type II strings. The main difference appears when the Riemann identity is applied to the partition function of all fields, ~ as in eqs. (3.11). Since both leftand right-movers contain fermions, with twists O~and 0~,one must use the Riemann identity twice. As a consequence, the NSR version will contain two new vectors W1~’-and W0R, with twists for left- and right-moving fermions respectively. A new discrete parameter ELR 0 or which provides the two relative chiralities, must be introduced: distinguishes between type ha and lIb theories. In GS strings, chirality is specified by either eq. (2.4b) or eq. (2.4c). The GS—NSR mapping uses accordingly the matrix A or B, and this choice directly determines -~-

=

-~-,

~LR

LR~

4. Conclusions To summarize, we have shown that the GS superstring, when regarded as a four-dimensional theory is equivalent to the class of NSR theories with twisted,

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734

Twisted strings

orbifold-like boundary conditions based upon the canonical, free internal supercurrent 5jflt ~k( ~/‘azk + c.c.), for complex fields ~Jkand Zk. The fact that the boundary conditions preserve a complex basis is related to the space-time properties of GS fermions: each represents a two-component Weyl space-time spinor. This NSR system possesses then an underlying internal N 2 superconformal symmetry. The relationship between the symmetry structures of the two fermionic strings is then very suggestive. Both are reparametrization invariant, the Weyl symmetry of NSR strings has a corresponding local bosonic symmetry in the GS case [16], world-sheet supersymmetry and K-symmetry clearly play an analogous role. Moreover, by construction, the GS action has an explicit linear global space-time supersymmetry. This is an additional restriction on the class of models, even if boundary conditions can destroy this space-time supersymmetiy. For NSR strings, it is known in general that space-time supersymmetry implies the existence of a second (global) world-sheet supersymmetry [24]. The opposite statement is not true: N 2 superconformal symmetry is not sufficient to obtain space-time supersymmetry. One further needs that physical states have integer charge under the SO(2) transformation which rotates the two supercharges. By violating this condition, one can construct NSR theories without space-time supersymmetry based on an underlying N 2 superconformal algebra. Through the NSR—GS mapping obtained in sect. 2, one can show that this mechanism for destroying space-time supersymmetry translates into the analogous condition on boundary conditions for GS strings. The N 2 superconformal structure is then the NSR counterpart of the underlying global space-time supersymmetry of the GS superstring. Extended N 2 superconformal symmetry alone does not imply that the supercurrent is of the canonical form obtained here. However, GS theories involve naturally twisted bosons. This is a strong additional condition on the possible supercurrents, implying that these bosons must appear in S,,, only via their derivative 3Zk. From the NSR point of view, all order modular invariance is well understood, provided factorization is assumed. This is not the case for GS strings, for which explicit higher-order calculations are missing. This is particularly true for the covariant GS theory: its quantization [17]is still at a preliminary stage. In the construction of the light-cone GS partition function [13], a condition for all order modular invariance was postulated. This was done by first assuming a formal factorization, up to a phase, and then by demanding the existence of a graviton in the spectrum to determine this phase. The same physical argument gives the correct answer for NSR strings. This is in fact similar to the requirement of a sensible particle interpretation used in ref. [5]. The equivalence of GS and NSR theories shows that it makes sense to follow the same approach for GS theories. In fact, the condition equivalent to two-loop modular invariance is very simple for GS fermions. The phase due to the world-sheet gravitino is completely hidden in the Riemann identity [seeeq. (3.9a)]: there is no such phase for GS strings. In fact, =

~/~k

=

=

=

=

=

A.H.

Chamseddine eta!.

/ Twisted strings

735

fermions in ten-dimensional GS supersymmetric theories have partition function ?146i(0Ir)4, without summation over spin structures: there is no space for a non-trivial phase. The fact that the class of theories which can be described using four-dimensional GS twisted strings is a small subset of the consistent NSR models is not a surprise. The covariant GS action is based on the unique realization of its very large symmetry algebra which is known at present. In fact, classically, this realization exists in space-time dimensions D 4, 6, 10 [16]. These three possibilities, when considered as four-dimensional theories contain respectively 1, 2 and 4 space-time Weyl spinors. At the quantum level, only the D 10 action, which was used in our discussion, makes sense: in light-cone gauge, D 10 is required to close Lorentz algebra. Contrary to the superconformal case, there is no other known realization of the symmetry algebra which can be used as an internal system able to saturate this “central charge”. We then loose the large freedom in the choice of the internal system which characterizes NSR theories. Besides these formal aspects, an explicit translation from the NSR to the GS formulation of twisted theories has practical advantages for string model building. The GS formalism is simpler for computing spectra, space-time supersymmetries, gauge groups. This is true also for theories without space-time supersymmetry. The number of terms in the partition function is four times smaller than for the equivalent NSR theory. However, interactions and studies of the effective field theory require the techniques of superconformal field theory, which are only available in the NSR formulation. =

=

=

J.-P.D. would like to thank the Laboratoire de Physique Théorique, Ecole Normale Supérieure for hospitality during completion of this work. Appendix A THE RIEMANN IDENTITIES

The Riemann identity (3.8) used to prove the equality of the NSR and GS partition functions is associated to the matrix 1

1

1

1

1

—1

~

1

—1

—1

1 =A’.

(A.1)

1

Applied on an SO(8) weight, the transformation generated by A, 4

k;

=

~ A,~k 1, j=I

(A.2)

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Twisted strings

The basis vectors e1, I 1,. 4 of the root lattice of SO(8), and the basis vectors é1 of the dual lattice (weight lattice) can be written in the standard form: is easily seen to exchange the conjugacy classes =

and

{s)

{v).

. . ,

e1

=

e2=

(1, —1,0,0),

e1

(1,0,0,0),

(0,1, —1,0),

ë2= (1,1,0,0),

=

e3=(0,0,1, —1),

~

—fl,

e4_(0,0,1,1),

e4=(~J,~J).

Denoting by A the root lattice generated by the SO(8) are then (0) =A,

(v} =ë1+A,

e1’s,

(A.3)

the four conjugacy classes of

(s} =ë4+A,

{c} =E3+A.

(A.4)

The transformation (A.2) exchanges e1 with e4, and also é1 (vector class) with é4 (spinor s class), and leaves e2, e3, ë2 (root class) and ë3 (spinor c) invariant. To derive the Riemann identity associated to A, one considers the quantity J=

11It[0°j(c1+d~T~r) =exP[_iTT~d71~I1It[~j(0~T). (A.5)

Since the theta functions are given by 2+2(n+a)(b+v)I}. =

(A.6)

nE7L ~ exp(i~[T(n+a)

the quantity J can be written +d 2r+ 2E(n 1) 1

j= ~

+

di)cil}, 2

with an unconstrained sum on all integers n,, i variables D~=~

=

1,..

. ,

=

(A.7)

1, the new

4. Since A

C,.= EA, 1c1,

I(~=EA11n3,

(A.8)

can be directly substituted in the expression (A.7) of J. However the summation over the unconstrained integers n is replaced by a summation over constrained variables K,: the form of A indicates that for any four integers n,, the four

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737

Twisted strings

numbers K, are either all integer or all half-integer, and their sum is always even: 2n~.One can then write =

~

exp{_i1rr~Dj2}

a=0,~

x

~ exp{i~[E(mi+Di+a)2r+2~(mi+Di+a)Ci]}~(1 m 1~7Z i

+(_1)~1m1),

(A.9) with a sum over unconstrained integers m,. The projector + (_i)E1m1)

~

=

exp{2i~rb~m1}

(A.10)

retains only the contributions from integers satisfying )1m1 even: it will introduce the correct signs for the usual NSR GSO projection. One finally finds =

J=exp{_i3Tr~D7}~ ~ I

exp{2iirb~Dj} ~

a,b=0,~

i

m,~L

2(mj+Dj+a)(Cj+b)1}, (A.11)

x~exp{i~[(mj+Dj+a)2r+

which leads to the Riemann identity

=

L1[~ ~](oIr).

a,b~O{

(A.12)

Eq. (A.12) holds for arbitrary parameters d, and c,, and for D, and C. as defined in (A.8). For the particular values d1=c1=~=(ë4)1,

i=1,...,4,

(A.13)

one obtains the Riemann identity for Jacobi functions which is relevant for the equivalence of ten-dimensional GS and NSR supersymmetric strings, eq. (1.1): 4+ It =

~[It1(0~r)~

—

It2(0~r)

4 3(OIr)

—

It 4(0~T)~].

(A.14)

A.H.

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Twisted strings

In terms of partition functions for SO(8) conjugacy classes, defined as 2

e ,WTP

V’

—

PE{s,,’,c,O}

eq. (A.14) expresses the equality ~?{S} If one now considers the matrix 1 B=— 2

which follows from triality of SO(8).

=

1

1

1

i

i

I

—1

I

I

I

1—1 —1 1

1 1

1 1

=Bi

1A15 ‘

the transformation analogous to (A.2) exchanges e 1 with e3, and also é1 with ë3, and leaves e2, e4, ë2 and e4 invariant. It exchanges the conjugacy classes {v) and {c). The Riemann identity associated to B can be derived following steps analogous to eqs. (A.4)—(A.11). The only difference is in the projector (A.10) which must be replaced by ~

exp{2i~b(m1 + m2 + m3

—

m4

+

2a)}

b=0,

(A.16)

~

=

The Riemann identity associated to B reads then ‘~

d

~

~

D.+a C:+b

(0~r), (A.17)

where D, E1B1~d1,and C. the values =

=

~1B,1c~,the matrix B being given in eq. (A.15). For

d,=c,= (e~),’

i= 1,...,4,

(A.18)

one obtains

4

It1(OIr)

=

+[ItloIT~+ It

4

2(OIT)

—

It

4 + It

3(OIr)

4],

4(OIr)

(A.19)

instead of eq. (A.14). In terms of conjugacy classes, it corresponds to completing the information given by triality on partition functions for S0(8) classes. =

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739

Twisted strings

The two identities (A.14) and (A.19) imply the Jacobi identity It2(0~r)4—It3(0Ir)4+It4(0~r)4=0,

(A.20)

besides the trivial result It 1(OIT) 0. There is another particular case of our Riemann identities which is worth mentioning. Choosing =

d,=(é4),,

c~=(ë4),+2~(ë3)1,

(A.21)

and using the Riemann identity for A, eq. (A.11), leads to

4

=

4[Iti(vlr)

+

It

4 It

4+

—

2(~Ir)

3(~r)

It 4(v~T)~],

(A.22)

for arbitrary (real) values of ~, which can be regarded as a formal mass parameter for the states in the partition functions. However, using the identity (A.17) with d,

=

(e3)1 ,

c,

=

(e3),

+

2v(ë4),

(A.23)

does not give a new identity: it leads again to eq. (A.22).

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Twisted strings

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