Abstract
We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions. This modular property is, in a sense, the generalization of the modular invariance of the supersymmetric partition function of two-dimensional supersymmetric theories on a torus i.e. of the elliptic genus. The partition functions in question are on manifolds homeomorphic to the ones obtained by gluing solid tori. Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background flavor symmetry connections, if present. Our modular property is a manifestation of the consistency of the gluing procedure. The modular property is used to rederive a supersymmetric Cardy formula for four dimensional gauge theories that has played a key role in computing the entropy of supersymmetric black holes. To be concrete, we work with four-dimensional mathcal{N} = 1 supersymmetric theories but we expect versions of our result to apply more widely to supersymmetric theories in other dimensions.
Highlights
Manifold does not have a non-trivial large diffeomorphism group
We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions
Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background flavor symmetry connections, if present
Summary
Before we get into the modular properties of the four-dimensional partition functions, let us review the modular properties of the T2 partition function of two-dimensional theories, i.e. of the elliptic genus. As the phase φg(z, τ ) encodes the anomalies under large transformations, it can not be removed by a local counter term This means there does not exist a redefinition of the index Z(z, τ ) which absorbs φg(z, τ ) for all group elements g ∈ G(2d). The presence of the phase eiφg(z,τ) ∈ H1(G(2d), M) means that the index is not a function of the parameters (z, τ ) but rather a section of a non-trivial bundle It captures the anomaly of the theory under large diffeomorphisms and large gauge transformations. With the brief discussion of the group cohomology above and illustration of its usefulness in classifying supersymmetric partition function in 2d, we can spoil the punchline of the paper for the benefit of an eager and mathematically initiated reader: “The normalized part of the supersymmetric index” of a four-dimensional N = 1 supersymmetric field theory is a non-trivial class in H1(G, N/M). What we mean by “the normalized part of the supersymmetric index” will become clear in due course
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