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 × 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.

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

“…superconformal index proposed in [35], in analogy with the relation between 4d indices and partition functions considered in [5]- [9] and the present work. It is given by 16) where G(u; ω) := G(u; ω 1 , .…”

“…In this section, we discuss basic facts about Hopf surfaces as described in [5] and [36] (see also [37]). This is needed to understand the action of SL(3, Z) on the superconformal index geometrically.…”

“…This is needed to understand the action of SL(3, Z) on the superconformal index geometrically. To avoid confusion, we mostly stick to the notation of [5].…”

“…To study Killing spinors on primary Hopf surfaces explicitly, a non-singular metric compatible with integrable complex structures was introduced in [5]. This metric admits a complex Killing vector K that satisfies K µ K µ = 0 and commutes with its own complex conjugate.…”

“…It receives contributions only from short representations and has served as a powerful check for many dualities. By applying the technique of localization [5,6] (see also [7,8]), it was found that the index I SC is related to the partition function Z SUSY by Z SUSY = e −βE Casimir I SC , (1.1)…”