Quantum Holonomies from Spectral Networks and Framed BPS States

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 $U_q(g{l}_K)$. With any path on $\mathcal{C}$, the quantum holonomy associates a positive Laurent polynomial in the quantized Fock-Goncharov coordinates of higher Teichmüller space. This confirms various positivity conjectures in physics and mathematics.

40 pages, 34 figures

References in corpus (4)

• Loop operators and S-duality from curves on Riemann surfaces
• A duality map for quantum cluster varieties from surfaces
• Quantum Teichmüller spaces and quantum trace map
• Higher laminations, webs and N=2 line operators

Cited by in corpus (19)

• A slow review of the AGT correspondence
• Surface Defect Indices and 2d-4d BPS States
• Line defect Schur indices, Verlinde algebras and $U(1{)}_r$ fixed points
• BPS states in the Minahan-Nemeschansky E6 theory
• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• $q$-nonabelianization for line defects
• Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology
• Laurent positivity of quantized canonical bases for quantum cluster varieties from surfaces
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
• Quivers, Line Defects and Framed BPS Invariants
• Quantum Line Defects and Refined BPS Spectra
• Abelianisation of Logarithmic ${\mathfrak{sl}}_2$-Connections
• Note on character varieties and cluster algebras
• ${\mathrm{SL}}_2$ quantum trace in quantum Teichmüller theory via writhe
• A note on discrete dynamical systems in theories of class $S$
• BPS spectra from BPS graphs
• The quantum UV-IR map for line defects in $\mathfrak{gl}(3)$-type class $S$ theories
• ${\mathrm{SL}}_3$-laminations as bases for ${\mathrm{PGL}}_3$ cluster varieties for surfaces
• Quantum traces for ${\mathrm{SL}}_n(\mathbb{C})$: The case $n=3$