Quantum Holonomies from Spectral Networks and Framed BPS States

TL;DR

This paper introduces a novel method combining spectral networks and skein algebra to compute quantum holonomies, revealing the spins of BPS states in 4d $ ext{N}=2$ theories and confirming positivity conjectures.

Contribution

It develops a new approach to determine BPS state spins using spectral networks and quantum holonomies on punctured Riemann surfaces, linking physics and mathematics.

Findings

Constructs quantum holonomies as positive Laurent polynomials

Confirms positivity conjectures in physics and mathematics

Provides a framework for analyzing BPS states in class S theories

Abstract

We propose a method for determining the spins of BPS states supported on line defects in 4d $\mathcal{N}=2$ theories of class S. Via the 2d-4d correspondence, this translates to the construction of quantum holonomies on a punctured Riemann surface $\mathcal{C}$. Our approach combines the technology of spectral networks, which decomposes flat $GL(K,\mathbb{C})$-connections on $\mathcal{C}$ in terms of flat abelian connections on a $K$-fold cover of $\mathcal{C}$, and the skein algebra in the 3-manifold $\mathcal{C}\times [0,1]$, which expresses the representation theory of the quantum group $U_q(gl_K)$. With any path on $\mathcal{C}$, the quantum holonomy associates a positive Laurent polynomial in the quantized Fock-Goncharov coordinates of higher Teichm\"uller space. This confirms various positivity conjectures in physics and mathematics.