# Five-Brane Webs and Highest Weight Representations

## TL;DR

This paper explores BPS-counting functions of M5-branes, revealing their structure as sums over highest weight representations of Lie algebras, and demonstrates dualities in specific brane configurations.

## Contribution

It shows that under certain conditions, the BPS partition functions simplify to sums over weights of Lie algebra representations, unveiling hidden symmetries and dualities.

## Key findings

- Partition functions can be expressed as sums over irreducible Lie algebra representations.
- Certain parameter choices lead to cancellations simplifying the partition functions.
- The duality N↔M becomes manifest in specific compact brane setups.

## Abstract

We consider BPS-counting functions $\mathcal{Z}_{N,M}$ of $N$ parallel M5-branes probing a transverse $\mathbb{Z}_M$ orbifold geometry. These brane web configurations can be dualised into a class of toric non-compact Calabi-Yau threefolds which have the structure of an elliptic fibration over (affine) $A_{N-1}$. We make this symmetry of $\mathcal{Z}_{N,M}$ manifest in particular regions of the parameter space of the setup: we argue that for specific choices of the deformation parameters, the supercharges of the system acquire particular holonomy charges which lead to infinitely many cancellations among states contributing to the partition function. The resulting (simplified) $\mathcal{Z}_{N,M}$ can be written as a sum over weights forming a single irreducible representation of the Lie algebra $\mathfrak{a}_{N-1}$ (or its affine counterpart). We show this behaviour explicitly for an extensive list of examples for specific values of $(N,M)$ and generalise the arising pattern for generic configurations. Finally, for a particular compact M5-brane setup we use this form of the partition function to make the duality $N\leftrightarrow M$ manifest.