Quantum Holonomies from Spectral Networks and Framed BPS States

We propose a method for determining the spins of BPS states supported on line defects in 4d 𝒩=2 theories of class S. Via the 2d–4d correspondence, this translates to the construction of quantum holonomies on a punctured Riemann surface 𝒞 . Our approach combines the technology of spectral networks, which decomposes flat GL(K,ℂ) -connections on 𝒞 in terms of flat abelian connections on a K -fold cover of 𝒞 , and the skein algebra in the 3-manifold 𝒞× [0,1] , which expresses the representation theory of the quantum group U q ( gl K ). With any path on 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.