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

TL;DR

This work establishes a precise link between the four-dimensional lens space index on $S^1\times L(r,1)$ and twisted characters of two-dimensional chiral algebras associated to 4d $\mathcal{N}=2$ SCFTs. By focusing on free theories and Argyres-Douglas theories, the authors show that in the Macdonald limit the lens index equals the refined, twisted character of a tailored module of the corresponding VOA, with the twist controlled by discrete flavor holonomies. They develop explicit constructions for the twisted beta–gamma system, the twisted $(b,c)$ system, and then apply these ideas to AD theories of type $(A_1,A_N)$ and $(A_1,D_N)$, deriving matching formulae and detailed examples including $\widehat{\mathfrak{su}}(2)_{-4/3}$ and $\widehat{\mathfrak{su}}(3)_{-3/2}$ cases, as well as their minimal-model relatives. The results provide strong evidence that lens indices encode rich 2d VOA data via twisted modules, suggesting a broad, top-down framework for relating 4d protected sectors to 2d chiral algebras across general $\mathcal{N}=2$ 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.