Duality group actions on fermions

In this short paper we look at the action of T-duality and string duality groups on fermions, in maximally-supersymmetric theories and related theories. Briefly, we argue that typical duality groups such as SL(2,Z) have sign ambiguities in their actions on fermions, and propose that pertinent duality groups be extended by Z_2, to groups such as the metaplectic group. Specifically, we look at duality groups arising from mapping class groups of tori in M theory compactifications, T-duality, ten-dimensional type IIB S-duality, and (briefly) four-dimensional N=4 super Yang-Mills, and in each case, propose that the full duality group is a nontrivial Z_2 extension of the duality group acting on bosonic degrees of freedom, to more accurately describe possible actions on fermions. We also walk through U-duality groups for toroidal compactifications to nine, eight, and seven dimensions, which enables us to perform cross-consistency tests of these proposals.


Introduction
In this paper we re-examine duality groups in high-dimensional string theories with maximal supersymmetry, taking a close look at duality group actions 1 on fermions in low-energy effective field theories. Historically, such duality group actions have been primarily discussed only on bosonic degrees of freedom. Briefly, we argue that in many cases, fermions are not uniquely defined under the duality groups as they are typically described, because of square-root-type branch cut ambiguities, and so propose that those duality groups be slightly enlarged.
As one prototypical example, in ten-dimensional type IIB string theory, we argue that under the S-duality group SL(2, Z), the transformation of the fermions is not uniquely defined due to a sign ambiguity, and so propose that SL(2, Z) should be replaced by a Z 2 extension. In particular, from the form of the duality group action, we propose that SL(2, Z) should be replaced by a particular nontrivial Z 2 central extension known as the metaplectic group and denoted Mp(2, Z).
We examine several different duality groups -mapping class groups of tori arising in M theory compactifications, T-dualities, ten-dimensional IIB S-duality -proposing such extensions in each case, as well as corresponding U-duality groups, which we use to perform cross-consistency tests of these proposed extensions.
We begin in section 2 by briefly reviewing the metaplectic group Mp(2, Z), which will be the most commonly appearing proposed duality group. We review its relation to SL(2, Z) and properties of this and related groups.
In section 3, we go on to discuss the linear diffeomorphism mapping class groups of tori, that play a crucial role in duality symmetries of toroidal compactifications of M theory. The mapping class group for an n-torus is ordinarily given as SL(n, Z), but we find that this group has an ambiguous action on fermions, and so we propose that in general it be replaced by a Z 2 extension which we denote SL(n, Z).
In section 4 we turn to perturbative T-duality groups of toroidally-compactified string theories. There, we argue that the worldsheet fermions themselves are well-defined under target-space T-dualities; the Ramond sector vacua, however, are only well-defined under Z 2 extensions of the ordinary duality groups, just as in the discussion of mapping class groups of tori. We discuss moduli spaces of SCFTs, and argue for similar reasons that, for example, 1 To be clear, although duality transformations should define a map of gauge-invariant local operators, their action on fundamental fields might not be so simply defined. For one example, in four-dimensional N = 1 Seiberg duality, mesons are composite operators on one side and fundamental fields on the other. However, in maximally-supersymmetric theories, actions on fields in low-energy effective actions are typically well-defined, and it is for this reason that one can e.g. describe the R-R and NS-NS two-form fields in tendimensional IIB supergravity as transforming in a doublet of the S-duality group. For simplicity, in this note we will largely focus on maximally-supersymmetric theories and related cases. the moduli space of elliptic curves SCFTs is best described as a quotient of the upper half plane by Mp(2, Z) instead of SL(2, Z) or P SL(2, Z).
In section 5, we turn to S-duality in ten-dimensional type IIB string theory. Here, no tori are manifest; nevertheless, we shall argue that the ten-dimensional fermions are also slightly ambiguous under the action of the S-duality group SL(2, Z), due to square-root branch cut ambiguities, and so for a well-defined action, we propose to promote the S-duality group to the Z 2 -extension Mp(2, Z).
In section 6, we briefly turn to four-dimensional N = 4 super Yang-Mills, and discuss how the Mp(2, Z) action in ten-dimensional type IIB appears to imply an analogous extension in S-duality in the four-dimensional N = 4 theory. That said, for most of this paper, in order to be able to speak meaningfully about duality group actions on fields rather than theories, we focus on maximally-supersymmetric theories in high dimensions.
Finally, in section 7, we put the ideas of the proceeding sections together to consider U-duality groups of toroidally-compactified M theory in nine, eight, and seven dimensions. We see that the various occurrences of the metaplectic group and analogous Z 2 extensions check one another. For example, in nine dimensions, in order for the U-duality group to be consistent, both the mapping class group SL(2, Z) of T 2 as well as the S-duality group of ten-dimensional type IIB must be extended to Mp(2, Z), which is what we have proposed in previous sections.
One of the motivations for this work is to understand the physical significance of a result in [1]. Specifically, it was argued there that the moduli stack of elliptic curves for which the Bagger-Witten line bundle is well-defined is the stacky quotient for Mp(2, Z) the metaplectic group extending SL(2, Z) by Z 2 , rather than a quotient by SL(2, Z) or P SL(2, Z). One could naturally ask whether the appearance of the metaplectic group was merely an obscure mathematical quirk of the Bagger-Witten line bundle in that circumstance, or reflected the true T-dualities of the theory. We propose in this paper that the latter is the case.
The proposal of section 5, that the fermions of ten-dimensional IIB supergravity transform under Mp(2, Z), has been independently obtained by D. Morrison [2].

Review of the metaplectic group
We will frequently encounter the metaplectic group Mp(2, Z) and its various cousins in this paper, so let us briefly review some pertinent facts. The metaplectic group Mp(2, Z) is the unique nontrivial central extensive of Z 2 , and can be described as the group with elements of the form and √ cτ + d is considered as a holomorphic function of τ in the upper half plane. The multiplication is defined as (A, f (·))(B, g(·)) = (AB, f (B(·))g(·)).
More generally, there is a metaplectic group Mp(2k, Z) which is the unique nontrivial Z 2 central extension of the symplectic group Sp(2k, Z). For k = 1, Sp(2k, Z) = SL(2, Z), and so this description reduces to the one above.
Metaplectic groups over R also define the symplectic analogue of spin structures on oriented Riemannian manifolds (see e.g. [3] and references therein, of which we shall give only a very brief account here). There is a formal definition of a metaplectic structure on a symplectic manifold, which is an equivariant lift of the symplectic frame bundle. Just as in ordinary spin structures, a metaplectic structure exists on a symplectic manifold (X, ω) if and only if the second Stiefel-Whitney class of M vanishes. Although we will not use metaplectic structures per se in this paper, we will often see metaplectic groups and related extensions arise via a need to define spinors in given situations.

Mapping class groups of tori
In this section, we will argue that under the action of the 'mapping class group' SL(n, Z) of a torus, describing the (analogues of) Dehn twists, spinors transform under the action of a Z 2 extension of SL(n, Z) (and the spin structures are permuted). We will argue this solely from the torus itself; in section 4.1, we will describe how the same result appears in the worldsheet theory of a sigma model on a torus.
To be clear, let us define more precisely what we mean by the 'mapping class group' of a torus. We follow the language and conventions of [5][section 3.4]. If we describe T n as a set of real variables y i subject to y i ≡ y i + n i for n i ∈ Z, then the 'mapping class group' to which we refer is the group of linear orientation-preserving diffeomorphisms y i → w i j y j . Clearly the (w i j ) ∈ GL(n, Z), and to preserve a choice of orientation, we must require (w i j ) ∈ SL(n, Z).
In this language, we can now begin to gain some intuition for the subtlety that arises when considering fermions. Broadly speaking, the mapping class group is acting by rotations.
However, to define an action on fermions, one must lift to a Spin cover. As a result, one should expect intuitively that the mapping class group will have to be replaced 2 by some sort of Z 2 cover. In this section, that is exactly the conclusion we shall reach. For n = 2, the Z 2 cover of SL(2, Z) will be the metaplectic group Mp(2, Z), and for n ≥ 3, it will be a Z 2 cover of SL(n, Z) which we will denote SL(n, Z).

Elliptic curves
Let us begin by considering elliptic curves. Under the action of the group of Dehn twists SL(2, Z), if we describe the elliptic curve as for H the upper half plane, then at the same time that dz → (cτ + d) −1 dz. 2 We would like to thank D. Auroux for a discussion of this matter. 3 Briefly, the idea is that if we construct a family of elliptic curves with complex structure parameter τ as (C × H)/Z 2 , for H the upper half plane and Z 2 action given by making the right-hand side well-defined under the Z 2 quotient. As consistency tests, note that under the map on z, In passing, note that although the parameter τ is invariant under the center {±1} ⊂ SL(2, Z), the holomorphic top-form dz is not invariant under the center, and so is only uniquely defined on a (stacky) SL(2, Z) quotient of the upper half plane, not a P SL(2, Now, a chiral spinor on an elliptic curve is a section of a square root of the canonical bundle, and so should transform in the same fashion as √ dz. Thus, if ψ is a chiral spinor, then under SL(2, Z), However, the element of SL(2, Z) does not uniquely determine the sign: the group that is really acting is not SL(2, Z), but a Z 2 extension of SL(2, Z), and in fact it should be clear to the reader that the Z 2 extension in question is Mp(2, Z).

Higher-dimensional tori
So far we have argued that on a single T 2 , the action of the mapping class group SL(2, Z) on fermions is ambiguous up to a τ -dependent phase, and so the action on fermions is more properly described as an action of Mp(2, Z), the unique nontrivial central extension of SL(2, Z). Now, consider a higher-dimensional torus T n . Setting aside fermions, the mapping class group of such a torus is SL(n, Z). Now, For any T 2 ⊂ T n , the same considerations must apply to fermions, and so every SL(2, Z) ⊂ SL(n, Z) must be enhanced to Mp(2, Z).
Therefore, to describe the action of the mapping class group on fermions, we need a Z 2 extension of SL(n, Z), such that every SL(2, Z) is extended to Mp(2, Z).
We will construct this extension, which we shall denote SL(n, Z), for n ≥ 3 as a pullback of the universal cover SL(n, R) of SL(n, R).
First, let us consider the universal cover SL(n, R). Since the maximal compact subgroup of SL(n, R) is SO(n), thus the universal cover SL(n, R) is a Z 2 central extension of SL(n, R) for n ≥ 3, and a Z-fold central extension for n = 2. We can understand this topologically as follows. As a topological space, SL(n, R) is homeomorphic to SO(n) × R k for some k (ignoring the group structure), so its universal cover is homeomorphic to Spin(n) × R k . For n = 2, Spin(2) = R, a Z-fold cover, and for n ≥ 3, Spin(n) is a double cover of SO(n).
In any event, we can now construct our desired group, that extends the action of the mapping class group to fermions. Define SL(n, Z) by the pullback square where p : SL(n, R) → SL(n, R) is the projection map, and i : SL(n, Z) ֒→ SL(n, R) is inclusion. In other words, we define which is well-defined because both p and i are homomorphisms, hence For n ≥ 3, we claim that the group SL(n, Z) encodes the action of the mapping class group on the fermions. For n = 2, the desired group is the metaplectic group Mp(2, Z), which is quotient of SL(2, Z) by 2Z, a subgroup of the central Z ⊂ SL(2, Z).
It remains to check whether the restriction of SL(n, Z) to p −1 (SL(2, Z)) for any SL(2, Z) ⊂ SL(n, Z) is a trivial or nontrivial central extension. (Since Mp(2, Z) is the unique nontrivial central extension of SL(2, Z) by Z 2 , showing that p −1 (SL(2, Z)) is a nontrivial central extension for every SL(2, Z) would imply p −1 (SL(2, Z)) = Mp(2, Z) for every SL(2, Z).) Now, every copy of SL(2, Z) inside SL(n, Z) comes from taking the integral points of a copy of SL(2, R) inside SL(n, R), so the question reduces to whether for any copy of SO(2) inside SO(n), the Spin cover of SO(n) splits when restricted to that SO(2).
It can be shown that the Spin cover of SO(n) does not split when restricted to any SO(2). This follows 4 from the fact that the group homomorphism SO(2) → SO(n) is surjective on the fundamental group, and the correspondence between covering spaces and actions of the fundamental group.
As a result, the restriction of SL(n, Z) to p −1 (SL(2, Z)) for any SL(2, Z) ⊂ SL(n, Z) is a nontrivial central extension, hence and so SL(n, Z) for n ≥ 3 is the desired group describing the mapping class group action on fermions.

T-duality 4.1 Action on worldsheet theories
In this subsection, we will consider the action on worldsheet fermions from two different sources: the action on the worldsheet theory of T-duality on a target elliptic curve, and the action of Dehn twists when the worldsheet itself is an elliptic curve.
Let us first consider a (2,2) supersymmetric sigma model with target space T 2 . Now, as is well-known, the set of T-dualities is larger than merely a Z 2 for each circle in T 2 , as for example, those T-dualities can be combined with Dehn twists to form a larger symmetry group. Across both complex and Kähler structures on T 2 , the T-duality group acting on the CFT is well-known to be O(2, 2; Z). Now, the part of the T-duality group O(2, 2; Z) that preserves GSO projections is where one SL(2, Z) acts on complex structures and the other on Kähler structures. As anticipated above, the largest part of the T-duality group is therefore two SL(2, Z) factors of Dehn twists, for complex and Kähler structures separately.
Let us consider the SL(2, Z) action on the complex structure of the target-space T 2 . The worldsheet fermions transform in the pullback of the tangent bundle on the target space, hence (up to dualizations and complex conjugations), they transform as dz, not √ dz, just as the worldsheet bosons (consistent with worldsheet supersymmetry). As a result, it is SL(2, Z), not Mp(2, Z), that acts on worldsheet fermions themselves under T-duality on the target space.
The Ramond sector vacua, on the other hand, are a different matter. The Ramond sector vacua of a physical theory transform in principle as sections of the pullback of the canonical bundle on the target space, i.e. as √ dz. (When the target is Calabi-Yau, this can be subtle to see, but is much more manifest in other cases, see for example [6] for a recent discussion of Fock vacua transforming as sections of bundles.) Thus, in principle, the Ramond sector vacua should pick up factors of ± √ cτ + d −1 under the action of T-duality on the target space, and in particular transform under Mp(2, Z) rather than SL(2, Z). (Similarly, though less relevantly for our purposes, target-space spin structures are also encoded in the R sector vacua, in signs under target-space periodicities.) Since the Z 2 extension acts only on the vacua, it should commute with the GSO projection, and so our analysis should have no effect on physical states.
In passing, we should note that the transformation of the Ramond sector vacua under target-space Dehn twists is the worldsheet realization of the target-space mapping class group action on fermions discussed in section 3.
T-duality for higher-dimensional tori can be treated in a very similar fashion. Under SL(n, Z) factors of Dehn twists, the worldsheet fermions themselves will be well-defined, but the Ramond sector vacua will pick up ambiguous signs (in addition to permutations of the spin structures). For the same reasons as in our discussions of mapping class groups of tori, the Ramond sector vacua are well-defined under the action of the Z 2 extension SL(n, Z). Now, let us ask a different question: how do the worldsheet fermions transform under Dehn twists of the worldsheet itself? Of course, Dehn twists will permute the worldsheet spin structures, but let us consider possible additional phase factors picked up the fields themselves. In principle, the worldsheet fermions in a physical (untwisted) theory with worldsheet Σ are sections of √ K Σ , hence transform as √ dz for z a coordinate on the worldsheet. As a result, it is natural to propose that the worldsheet fermions transform under Mp(2, Z) under the action of worldsheet Dehn twists.

Moduli spaces of SCFTs
Let us consider first moduli spaces of SCFTs constructed as sigma models on elliptic curves. As we just argued in the preceding subsection, although the worldsheet fermions themselves are single-valued under the action of SL(2, Z) on the target space, the R sector vacua are not. In addition to the expected permutations of spin structures, the R sector vacua also necessarily pick up phases, which due to a square root branch cut are ambiguous under SL(2, Z). (We have only discussed the role of the metaplectic group on the space of complex structures, but by mirror symmetry, analogous considerations must also apply to the Kähler structures.) This sheds new light on a result in [1]. There, it was argued that the Bagger-Witten line bundle over a moduli space of SCFTs for elliptic curves was only well-defined over the stack above, an Mp(2, Z) quotient of the upper half plane rather than SL(2, Z) or P SL(2, Z). Now, the Bagger-Witten line bundle over any moduli space of SCFTs is a (possibly fractional) line bundle of spectral flow operators, that play an essential role in target-space supersymmetry. Here, we have proposed that the Mp(2, Z) quotient (as opposed to an SL(2, Z) or P SL(2, Z) quotient) is necessary in order to make the R sector vacua well-defined.
Analogous results hold in higher dimensions. For simplicity, let us consider the space of complex structures on a torus X that is a complex projective manifold, i.e. an abelian variety, and has complex dimension g, say. For a fixed choice of polarization (Kähler form) on X, the complex structures preserving the polarization are parametrized by a quotient where H g is the higher-dimensional Siegel upper half plane (meaning, symmetric g × g complex matrices with positive-definite imaginary part), and Sp(2g, Z) is the symplectic group of integral 2g × 2g matrices preserving the polarization -the ordinary symplectic group Sp(2g, Z) for nondegenerate cases (more properly, principal polarizations). (See e.g. [7][section 2.6] for more information.) Here also, for much the same reasons, the spin structures are parametrized by the stacky quotient where Mp(2g, Z) is the metaplectic group extending Sp(2g, Z) by Z 2 .
In general, for an n-dimensional torus T n , the group of all T-dualities is O(n, n; Z); see e.g. [8] for a thorough discussion, hence ordinarily the complete moduli space of tori is described as a quotient by O(n, n; Z). We will not try to give a thorough description of the precise metaplectic replacement here, but we will note that elements of O(n, n; Z) whose determinant is different from one also modify the worldsheet GSO projection (see e.g. [8][9][10], [11][section 3.2]), for example exchanging type IIA and IIB, so to give a thorough description of the moduli space of SCFTs with target an n-dimensional torus T n will presumably involve a Z 2 extension of SO(n, n; Z) (as well as a means of following spin structures).

Ten-dimensional IIB S-duality
In this section, we will describe the action of S-duality on fermions in ten-dimensional IIB strings, and we will propose that SL(2, Z) should in principle be replaced by the metaplectic group Mp(2, Z). (The result of this section has been independently obtained by D. Morrison [2].) Classically, recall [12][13][14] that type IIB supergravity in ten dimensions has an SU(1, 1; C) ∼ = SL(2, R) symmetry (modulo finite factors) extending a local U(1) symmetry. In that theory, the fermions only transform under the local U(1) (see e.g. [12][equ'n (7c)]). After gaugefixing, the local transformation is determined so as to preserve the gauge, and so an SL(2, R) transformation determines an action on the fermions [15]. (See also [16][section 13.5] for a review.) Furthermore, in the quantum theory, the continuous SL(2, R) symmetry is replaced by a local SL(2, Z), and it is in this fashion that we can understand that the fermions transform under the action of S-duality.
In the conventions of [15], if we pick gauge-fixing conditionφ = 0, then under the U(1) subgroup of SL(2, R), the (complex) gravitino ψ µ transforms as [15][equ'n (2.11)] ψ µ → exp(iΣ/2)ψ µ , and the (complex) dilatino λ transforms as [15][equ'n (2.11)] the U(1) rotation is defined by Put more simply, and restricting to the local discrete gauge symmetry SL(2, Z), These transformations were used in e.g. [17] to argue that the coefficients of certain higherdimension operators should behave as nonholomorphic analogues of modular forms. Now, as written, the transformations above appear to be invariant under the center of SL(2, Z). However, there is a potential subtlety present in the one-quarter-root branch cuts. If we perform a field redefinition, we can construct new fermions with cleaner transformation laws, which is what we will describe next.
Following [18], one can define a modular form of weight (m + , m − ) to be a real analytic function F on the upper half plane H such that For example, Im τ is such a modular form, of weights (−1, −1). Given any such F , one can transform it to a modular form F ′ of weight (m + − m − , 0) defined by In the present case, ψ µ transforms like a modular form of weights (−1/4, +1/4) and λ a modular form of weights (+3/4, −3/4), so we define Then, under the action of SL(2, Z), these new fields transform as Furthermore, we are free to redefine the fermions as we wish -ψ ′ µ , λ ′ are no more or less physical than ψ µ , λ, so our field redefinition has merely made more manifest a subtle symmetry of the theory.
The transformations above are only defined up to signs -the usual ambiguity in square roots. As a result, the action of SL(2, Z) is ambiguous, as an element of SL(2, Z) does not uniquely determine a choice of sign. The group that really is acting is some two-fold cover of SL(2, Z). Given the form of the transformations above, it is natural to propose that the full duality group is the metaplectic group Mp(2, Z).

Four-dimensional N = 4 SYM
Four-dimensional N = 4 super-Yang-Mills can be directly connected to ten-dimensional IIB string theory by virtue of the AdS/CFT correspondence. In particular, in [19] it was observed that as the components of the N = 4 supercurrent multiplet couple to the fields of the ten-dimensional IIB supergravity, the transformation properties of the ten-dimensional fields under SL(2, Z) (or, as we have observed here, Mp(2, Z)) imply transformation laws for the four-dimensional multiplets.
Given the role of Mp(2, Z) we have discussed in this section, one might ask what happens after compactification of the four-dimensional theory on a curve. Such compactifications were discussed historically in [20,21], and more recently in e.g. [27]. For example, these papers argued that the SL(2, Z) of the four-dimensional N = 4 theory becomes T-duality in the two-dimensional theory. Briefly, if the fermions of the four-dimensional theory transform under the Z 2 extension Mp(2, Z) of SL(2, Z), then the same must be true of the fermions in the two-dimensional theory, which is consistent with observations in section 4 regarding T-duality and fermions.
In passing, for completeness we should also mention that SL(2, Z) actions on fermions in four-dimensional N = 2 U(1) gauge theory are discussed in [22]. In the conventions of that reference, SL(2, Z) acts honestly on the fermions, but the zero modes and partition function pick up factors which could have square-root sign ambiguities. Specifically, if the fermions into modes of R-charge R = 1, denoted α, corresponding to a pair of positivechirality gluinos, and conjugate fields of R = −1, denoted α and of negative chirality, then The zero modes are more nearly relevant for our purposes. The normalized integration measure for any fermi zero mode β is of the form [22][equ'n (3.15)] Im τ As a result, since the number of α zero modes minus the number of α zero modes is −(χ + σ)/2, the fermion measure picks up a factor of [22]

U-duality
In this section we will discuss U-dualities appearing in toroidally-compactified theories, which will not only allow us to display how U-duality groups are modified when one takes into account fermions, but also perform cross-checks of our proposals. Briefly, in each dimension we will propose a Z 2 extension of the ordinary U-duality group, reflecting the fact that taking fermions into account should only generate sign ambiguities and hence we expect only an overall Z 2 extension, rather than an extension by a larger finite group.

Nine dimensions
U-duality groups of nine-dimensional theories were discussed in e.g. [5,23]. Briefly, a ninedimensional theory can be obtained as either M theory on T 2 or, equivalently, type II on S 1 . As M theory on T 2 , for the reasons discussed in section 3, this theory has an Mp(2, Z) duality when one takes into account fermions. If we think about this as type IIB on S 1 , then as discussed in section 5, the ten-dimensional type IIB theory has an Mp(2, Z) symmetry, which coincides with the M theory U-duality group. In addition, there is also the ordinary T-duality on S 1 , but as this exchanges IIA and IIB (albeit shifting the dilaton in the process), it does not contribute to the duality group of IIB per se.
In any event, regardless of how we construct the nine-dimensional theory, we see that when taking into account the fermions, in our proposal it has an Mp(2, Z) symmetry, slightly enlarging what was previously described as an SL(2, Z) symmetry. We also see that the results of section 3 and section 5 constrain one another: consistency of the nine-dimensional theory requires the two duality groups obtained independently in those sections to match, as indeed they do.

Eight dimensions
In this section we shall discuss the eight-dimensional theory which can be obtained alternatively as a compactification of M theory on T 3 , or of type IIA/B on T 2 .
In the past, omitting fermions, it was said that the U-duality group is SL(3, Z)×SL(2, Z). The SL(3, Z) factor arose from the mapping class group of the T 3 in the M theory compactification, and as explained in e.g. [11][section 4.3], the other factor arises from T-duality of the type II compactification on T 2 .
Omitting fermions, the (GSO-preserving) T-duality group acting on the SCFT is However, in the spacetime theory, we cannot 5 quotient out the Z 2 , as it acts nontrivially on the RR fields. In fact, to better understand this statement, let us summarize the actions of various SL(2, Z)'s appearing. The ten-dimensional SL(2, Z) S of IIB (omitting fermions) acts as [24][25][26] τ → aτ + b cτ + d , where τ is the complexified ten-dimensional string coupling, B N is the ten-dimensional NS-NS B field, and B R is the R-R B field. Part of the T-duality group of the IIB compactification on T 2 , which we shall denote SL(2, Z) T , acts as (e.g. [24]) forT = B N + iV 2 the T 2 Kähler modulus, with B N the NS-NS two-form on T 2 and V 2 the volume of the T 2 , and ρ = −B R + iτ 1 V 2 . The other half of the T-duality group we denote SL(2, Z) U , and acts on the complex structure modulus U of T 2 as (e.g. [24]) From the transformation law of ρ under SL(2, Z) T , we see that the RR 2-form B R picks up a sign under the Z 2 center of SL(2, Z) T , and so in forming the string duality group, we must lift SO(2, 2; Z) to a double cover, namely SL(2, Z) T × SL(2, Z) U .
The SL(2, Z) S , SL(2, Z) T are combined as a pair of 2 × 2 blocks inside a 3 × 3 matrix to form the SL(3, Z) factor in the U-duality group, which altogether is SL(3, Z) × SL(2, Z) U .
We claim that, when fermions are taken into account, the U-duality group in this theory is ( SL(3, Z) × Mp(2, Z))/Z 2 . The SL(3, Z) factor arises from the mapping class group of T 3 in the M theory compactification, as in section 3.
In any event, there would appear to be two natural possibilities for the U-duality group in eight dimensions: either SL(3, Z) × Mp(2, Z), or ( SL(3, Z) × Mp(2, Z))/Z 2 , where the Z 2 quotient acts on the two central Z 2 extension factors. As we are looking for a Z 2 extension of the ordinary U-duality group rather than a Z 2 × Z 2 extension, we propose that the correct U-duality group in eight dimensions is We shall also see that this group arises in the decompactification limit from seven dimensions in the next section.

Seven dimensions
In this section, we will discuss the U-duality group of the theory which can alternatively be described as M theory compactified on T 4 , or as type II on T 3 .
In the past, omitting fermions, it was said that the U-duality group is SL(5, Z). The mapping class group of the T 4 appearing in the M theory compactification, namely SL(4, Z), appears as a subgroup of SL(5, Z) (embedded in the obvious way, as a 4 × 4 block inside 5 × 5 matrices [5][section 3.4]).
Similarly (see e.g. [11][section 4.3]), the (GSO-preserving and RR-compatible) T-duality group SL(4, Z) of the type II compactification does not commute with the mapping class group of the M theory compactification, and the two of them together generate SL(5, Z).
In the present case, taking into account fermions, our proposal is that the contribution to the U-duality group from the mapping class group arising from M theory on T 4 is the Z 2 extension SL(4, Z) of SL(4, Z), as discussed in section 3. Similarly, we expect (though have not carefully checked) that the relevant T-duality group is a (different) SL(4, Z). They should combine into a Z 2 extension of SL(5, Z), and the natural possibility is SL(5, Z), which we conjecture to be the case.
Formally repeating the arguments in [5][section 3.4], the U-duality group of the eightdimensional decompactification limit should be the subgroup a Z 2 extension of SL(2, Z) × SL(3, Z), consistent with our results in section 7.2.

Conclusions
In this short paper we have proposed that duality group actions in high-dimensional theories with maximal supersymmetry should be slightly enlarged, by nontrivial Z 2 extensions, to correctly describe duality group actions on fermions. We have argued this separately for mapping class groups of tori in M theory compactifications, T-duality groups, ten-dimensional IIB S-duality, and briefly four-dimensional N = 4 theories, and checked the proposals against one another by exploring U-duality groups in dimensions nine, eight, and seven.
We have only considered U-duality groups of high-dimensional string compactifications. It would be interesting, albeit more technically complex, to extend to lower-dimensional cases and cases with less supersymmetry.
One possible application of such an extension would be to try to identify the Bagger-Witten bundle over moduli spaces of K3 superconformal field theories. One of the original motivations for this work, after all, was to understand whether the metaplectic group appearing in the description of the stringy moduli stack of elliptic curves given in [1] was merely a formal quirk or reflected physical dualities. An understanding of U-duality groups in six-dimensional compactifications of string theory could be used to give analogous information concerning moduli stacks of K3 superconformal field theories.
Another natural question concerns spinors and Bagger-Witten line bundles on special Kähler moduli spaces. As briefly outlined in section 2, the analogue of a spin structure on a symplectic manifold is a metaplectic structure, defining a bundle whose structure group is a metaplectic group. This suggests that a detailed investigation of spinors and Bagger-Witten line bundles on special Kähler manifolds will reveal that the metaplectic group plays an important role there as well. We leave such considerations for future work.
In passing, it is also tempting to wonder whether considerations such as those in this paper for N = 4 theories in four dimensions are relevant to metaplectic geometric Langlands theory, as in [27,28].