We introduce a nonabelianization map for conformal blocks, which relates \(c=1\) Virasoro blocks on a Riemann surface C to Heisenberg blocks on a branched double cover \({\widetilde{C}}\) 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 \({\widetilde{C}}\) . 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 \(\tau \) -functions.
