Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra

TL;DR

This paper explores the relationship between the four-dimensional lens space index of certain superconformal theories and two-dimensional chiral algebra characters, revealing how discrete holonomies influence this connection.

Contribution

It demonstrates how the lens space index can be expressed via twisted modules of 2D chiral algebras, linking 4D supersymmetric indices to 2D algebraic structures.

Findings

Lens space index matches twisted module characters in specific limits

Discrete holonomies determine the type of twisted module

Connection established for free and Argyres-Douglas theories

Abstract

We study the supersymmetric partition function on $S^1 \times L(r, 1)$, or the lens space index of four-dimensional $\mathcal{N}=2$ superconformal field theories and their connection to two-dimensional chiral algebras. We primarily focus on free theories as well as Argyres-Douglas theories of type $(A_1, A_k)$ and $(A_1, D_k)$. We observe that in specific limits, the lens space index is reproduced in terms of the (refined) character of an appropriately twisted module of the associated two-dimensional chiral algebra or a generalized vertex operator algebra. The particular twisted module is determined by the choice of discrete holonomies for the flavor symmetry in four-dimensions.