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 $\mathcal{N}=2$ 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.