Localization on $AdS_2\times S^1$

TL;DR

This paper applies localization techniques to supersymmetric Chern-Simons theory on the non-compact space $AdS_2 imes S^1$, demonstrating that the partition function matches that on a $q$-covering of $S^3$, thus providing a new testing ground for localization.

Contribution

The authors determine the localizing Lagrangian and compute the partition function of ${ m extbf{N}=2}$ supersymmetric Chern-Simons theory on $AdS_2 imes S^1$, establishing a precise match with the $S^3$ covering case.

Findings

Partition function on $AdS_2 imes S^1$ matches that on the $q$-covering of $S^3$.

Supersymmetry on $AdS_2 imes S^1$ is successfully implemented with appropriate boundary conditions.

Localization techniques are extended to non-compact spaces with boundary conditions ensuring normalizability.

Abstract

Conformal symmetry relates the metric on $AdS_2 \times S^{1}$ to that of $S^3$. This implies that under a suitable choice of boundary conditions for fields on $AdS_2$ the partition function of conformal field theories on these spaces must agree which makes $AdS_2 \times S^{1}$ a good testing ground to study localization on non-compact spaces. We study supersymmetry on $AdS_2\times S^1$ and determine the localizing Lagrangian for ${\cal N}=2$ supersymmetric Chern-Simons theory on $AdS_2\times S^1$. We evaluate the partition function of ${\cal N}=2$ supersymmetric Chern-Simons theory on $AdS_2 \times S^1$ using localization, where the radius of $S^1$ is $q$ times that of $AdS_2$. With boundary conditions on $AdS_2\times S^1$ which ensure that all the physical fields are normalizable and lie in the space of square integrable wave functions in $AdS_2$, the result for the partition function precisely agrees with that of the theory on the $q$-fold covering of $S^3$.