The $\mathcal{N}=1$ Chiral Multiplet on $T^2\times S^2$ and Supersymmetric Localization

Abstract

We compute the supersymmetric partition function of an $\mathcal{N}=1$ chiral multiplet coupled to an external Abelian gauge field on complex manifolds with $T^2\times S^2$ topology. The result is locally holomorphic in the complex structure moduli of $T^2\times S^2$. This computation illustrates in a simple example some recently obtained constraints on the parameter dependence of supersymmetric partition functions. We also devise a simple method to compute the chiral multiplet partition function on any four-manifold ${\mathcal{M}}_4$ preserving two supercharges of opposite chiralities, via supersymmetric localization. In the case of ${\mathcal{M}}_4=S^3\times S^1$, we provide a path integral derivation of the previously known result, the elliptic gamma function, which emphasizes its dependence on the $S^3\times S^1$ complex structure moduli.

39 pages. v2: reference added; v3: corrected typos, JHEP version

References in corpus (6)

• Notes on SUSY Gauge Theories on Three-Sphere
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• Counting Gauge Invariants: the Plethystic Program
• Calculating the Superconformal Index and Seiberg Duality
• Dirac operator on the Riemann sphere
• Constraints on the R-charges of Free Bound States from the Römelsberger Index

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