Boundary Conditions and Localization on AdS: Part 1

TL;DR

This paper investigates how boundary conditions affect the one-loop partition function of a supersymmetric chiral multiplet on AdS2×S1, revealing conditions under which different actions yield consistent results.

Contribution

It introduces a detailed analysis of boundary conditions' impact on one-loop determinants in supersymmetric localization on AdS spaces, highlighting when different computational methods agree.

Findings

Normalizable boundary conditions may not preserve supersymmetry when certain integer conditions are met.

The one-loop determinants from different actions agree only when no integers lie in a specific interval.

The index method and Green's function method produce consistent results under particular boundary condition regimes.

Abstract

We study the role of boundary conditions on the one loop partition function of ${\cal N}=2$ chiral multiplet of R-charge $\Delta$ on $AdS_2\times S^1$. The chiral multiplet is coupled to a background vector multiplet which preserves supersymmetry. We implement normalizable boundary conditions in $AdS_2$ and develop the Green's function method to obtain the one loop determinant. We evaluate the one loop determinant for two different actions: the standard action and the $Q$-exact deformed positive definite action used for localization. We show that if there exists an integer $n$ in the interval $D: ( \frac{\Delta-1}{2L}, \frac{\Delta}{2L} )$, where $L$ being the ratio of radius of $AdS_2$ to that of $S^1$, then the one loop determinants obtained for the two actions differ. It is in this situation that fields which obey normalizable boundary conditions do not obey supersymmetric boundary conditions. However if there are no integers in $D$, then fields which obey normalizable boundary conditions also obey supersymmetric boundary conditions and the one loop determinants of the two actions precisely agree. We also show that it is only in the latter situation that the one loop determinant obtained by evaluating the index of the $D_{10}$ operator associated with the localizing action agrees with the one loop determinant obtained using Green's function method.