Localization on Hopf surfaces

Abstract

We discuss localization of the path integral for supersymmetric gauge theories with an R-symmetry on Hermitian four-manifolds. After presenting the localization locus equations for the general case, we focus on backgrounds with S^1 x S^3 topology, admitting two supercharges of opposite R-charge. These are Hopf surfaces, with two complex structure moduli p,q. We compute the localized partition function on such Hopf surfaces, allowing for a very large class of Hermitian metrics, and prove that this is proportional to the supersymmetric index with fugacities p,q. Using zeta function regularisation, we determine the exact proportionality factor, finding that it depends only on p,q, and on the anomaly coefficients a, c of the field theory. This may be interpreted as a supersymmetric Casimir energy, and provides the leading order contribution to the partition function in a large N expansion.

v2: discussion of background reality conditions modified and other minor changes, references added; v3: further minor corrections, version accepted for publication in JHEP

References in corpus (7)

• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• Calculating the Superconformal Index and Seiberg Duality
• On the Superconformal Index of N=1 IR Fixed Points: A Holographic Check
• Supersymmetric gauge theories on squashed five-spheres and their gravity duals
• The gravity dual of supersymmetric gauge theories on a squashed five-sphere
• Supersymmetric dualities beyond the conformal window
• Factorization of 4d N=1 superconformal index

Cited by in corpus (14)

• Cardy Formulae for SUSY Theories in d=4 and d=6
• From Rigid Supersymmetry to Twisted Holomorphic Theories
• Constraints on Chiral Operators in N=2 SCFTs
• Generalized Indices for $\mathcal{N}=1$ Theories in Four-Dimensions
• Comments on the Casimir energy in supersymmetric field theories
• N = 4 Super-Yang-Mills on Conic Space as Hologram of STU Topological Black Hole
• Supersymmetric counterterms from new minimal supergravity
• $c-a$ from the $N=1$ superconformal index
• Central charges from the $\mathcal{N}=1$ superconformal index
• Supersymmetric index on T^2 x S^2 and elliptic genus
• Wilson loops on three-manifolds and their M2-brane duals
• Partition functions for equivariantly twisted $\mathcal{N}=2$ gauge theories on toric Kähler manifolds
• A Localization Computation in Confining Phase
• Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension