$\mathcal{N} = (0, 2)$ higher-spin supergravity in AdS$_3$

TL;DR

This work constructs a linearized ${ m N}=(0,2)$ higher-spin gravity in AdS$_3$ by implementing a holomorphic decomposition that preserves a left ${ m shs}[(1- u)/2]_{ m L}$ sector and bosonizes the right sector, yielding an asymptotic symmetry consistent with 2d ${ m N}=(0,2)$ superconformal structure. The matter sector is organized into four ${ m N}=(0,2)$ short multiplets across the $$ and $$ projections, with masses controlled by the parameter $ u$, enabling a well-defined one-loop analysis. The paper derives the 1-loop partition function around thermal AdS$_3$ using heat-kernel techniques, obtaining functional-determinant expressions for half-integer spin fields and Dirac operators, and culminating in a compact product formula for the total partition function. These results establish a perturbative HS/CFT framework in 3d with asymmetric supersymmetry and pave the way for exploring nonlinear completions, symmetry breaking, and potential Schwarzian sectors in dual 2d theories.

Abstract

In this paper we generalize Vasiliev's higher-spin gravity theory in 3d into $\mathcal{N} = (0, 2)$ case, by which we mean that the asymptotic symmetry of such a gravity theory have the structure of 2d $\mathcal{N} = (0, 2)$ superconformal algebra. While the construction is limited to linearized level, asymptotic symmetry and possible matter content of such theories is discussed. Also, the 1-loop partition function of this theory around thermal Euclidean AdS space-time, with different matter fields, is calculated by heat-kernel method.