Abstract
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.
Highlights
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 Z2 extension
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 four-dimensional N = 4 super Yang-Mills, and in each case, propose that the full duality group is a nontrivial Z2 extension of the duality group acting on bosonic degrees of freedom, to more accurately describe possible actions on fermions
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 Z2 extension which we denote SL(n, Z)
Summary
We will frequently encounter the metaplectic group M p(2, Z) and its various cousins in this paper, so let us briefly review some pertinent facts. Is the unique nontrivial central extensive of Z2, and can be described as the group with elements of the form ab cd. There is a metaplectic group M p(2k, Z) which is the unique nontrivial Z2 central extension of the symplectic group Sp(2k, Z). Metaplectic groups over R define the symplectic analogue of spin structures on oriented Riemannian manifolds 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. 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
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