A New Construction of $$c=1$$ Virasoro Blocks

We introduce a nonabelianization map for conformal blocks, which relates $$c=1$$ c = 1 Virasoro blocks on a Riemann surface C to Heisenberg blocks on a branched double cover C C ~ of C . The nonabelianization map uses the datum of a spectral network on C . It gives new formulas for Virasoro blocks in terms of fermion correlation functions determined by the Heisenberg block on C C ~ . The nonabelianization map also intertwines with the action of Verlinde loop operators, and can be used to construct eigenblocks. This leads to new Kyiv-type formulas and regularized Fredholm determinant formulas for τ τ -functions.