Abstract
We consider supersymmetric field theories on compact manifolds \( \mathcal{M} \) and obtain constraints on the parameter dependence of their partition functions \( {Z_{\mathcal{M}}} \). Our primary focus is the dependence of \( {Z_{\mathcal{M}}} \) on the geometry of \( \mathcal{M} \), as well as background gauge fields that couple to continuous flavor symmetries. For \( \mathcal{N} \) = 1 theories with a U(1) R symmetry in four dimensions, \( \mathcal{M} \) must be a complex manifold with a Hermitian metric. We find that \( {Z_{\mathcal{M}}} \) is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over \( \mathcal{M} \) and \( {Z_{\mathcal{M}}} \) is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional \( \mathcal{N} \) = 2 theories with a U(1) R symmetry, where the necessary geometric structure on \( \mathcal{M} \) is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that \( {Z_{\mathcal{M}}} \) is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S 3 × S 1 or S 2 × S 1, which are related to supersymmetric indices, and manifolds diffeomorphic to S 3 (squashed spheres). In examples where \( {Z_{\mathcal{M}}} \) has been calculated explicitly, our results explain many of its observed properties.
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Closset, C., Dumitrescu, T.T., Festuccia, G. et al. The geometry of supersymmetric partition functions. J. High Energ. Phys. 2014, 124 (2014). https://doi.org/10.1007/JHEP01(2014)124
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DOI: https://doi.org/10.1007/JHEP01(2014)124