  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 as regularized Fredholm determinants of integral operators, with kernel
given by an appropriate free-fermion two-point function 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 Fredholm determinant formulas for $\tau$-functions.
