Modularity of Vafa-Witten Partition Functions from SymTFT

TL;DR

This paper explores the modular properties of Vafa-Witten partition functions derived from 6d M5 brane theories on complex surfaces, using SymTFT to verify dualities and construct invariant 2d theories.

Contribution

It demonstrates the modularity and duality properties of Vafa-Witten partition functions via SymTFT, providing new insights into 2d-4d correspondences and duality defects.

Findings

Matching modular transformation properties for Hirzebruch and Del Pezzo surfaces.

Construction of modular invariant 2d partition functions.

Evidence for duality defects in the 2d SCFT.

Abstract

The 6d (2,0) theory of $N$ M5 branes compactified on the product geometry $T^2\times S$, where $S$ is a K\"ahler 4-manifold, can be studied in two different limits. In one limit, the size of $T^2$ is taken to zero and together with a topological twist one arrives at the Vafa-Witten partition function on $S$. On the other hand, taking the size of $S$ to zero leads to a 2d $\mathcal{N}=(0,4)$ theory. This gives rise to a 2d-4d correspondence where the Vafa-Witten partition functions are identified with the characters of the 2d theory. In this paper, we test this conjecture for Hirzebruch and Del Pezzo surfaces by employing the technique of SymTFT to show that the modular transformation properties of the two sides match. Moreover, we construct modular invariant 2d absolute partition functions and verify that they are invariant under gauging of a discrete symmetry at the self-dual point in coupling space. This provides further hints for the presence of duality defects in the 2d SCFT.