Supersymmetric localization: ${\cal N}=(2,2)$ theories on S$^2$ and AdS$_2$

TL;DR

This paper applies supersymmetric localization to compute exact partition functions of ${ m N}=(2,2)$ theories on Euclidean AdS$_2$ and S$^2$, revealing size-dependent contributions from anomalies and zero modes.

Contribution

It extends localization techniques to AdS$_2$, clarifies boundary conditions, and computes the 1-loop determinants using index theory, providing new insights into size dependence of partition functions.

Findings

Partition functions depend on the size of AdS$_2$ and S$^2$ backgrounds.

Both local conformal anomalies and zero modes contribute to the size dependence.

The 1-loop determinants are computed explicitly using index theory and heat kernel methods.

Abstract

Application of the supersymmetric localization method to theories on anti-de Sitter spacetime has received recent interest, yet still remains as a challenging problem. In this paper, we focus on (global) Euclidean AdS$_2$, on which we consider an Abelian ${\cal N}=(2,2)$ theory and implement localization computation to obtain the exact partition function. For comparison, we also revisit the theory on S$^2$ and perform a parallel computation. We refine the notion of equivariant supersymmetry and use appropriate functional integration measure. For AdS$_2$ we choose a supersymmetric boundary condition which is compatible with the principle of variation. To evaluate the 1-loop determinant about the localization saddle, we use index theory and fixed point formula, where we pay attention to the effect of zero modes and their superpartners. The existence of fermionic superpartner of 1-form boundary zero modes is proven. Obtaining the 1-loop determinant requires expansion of the index that presents an ambiguity, which we resolve using boundary condition. The resulting partition function reveals an overall dependence on the size of the background manifold, AdS$_2$ as well as S$^2$, as a sum of two types of contributions: a local one from local conformal anomaly through the index computation and a global one coming from zero modes. This overall size dependence is confirmed by the perturbative 1-loop evaluation using heat kernel method.