A new construction of $c=1$ Virasoro blocks

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. 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 $τ$-functions.

💡 Research Summary

The authors present a novel construction of conformal blocks for the Virasoro algebra at central charge c = 1, based on a “non‑abelianization” map that translates Heisenberg (free‑boson) blocks on a branched double cover (\widetilde C) of a Riemann surface C into Virasoro blocks on C. The key new ingredient is a spectral network W on C, a collection of arcs of gl₂‑type which encodes extra data needed to lift the free‑field construction to genuine c = 1 Virasoro blocks without unwanted insertions.

In the traditional free‑field approach one writes the stress tensor as (T=\frac12 J^2) with (J) the Heisenberg current, which maps Heisenberg blocks to a very special subclass of Virasoro blocks. A more general “branched free‑field” construction uses the anti‑symmetric combination (eJ^{(-)}=\frac{1}{\sqrt2}(eJ_1-eJ_2)) on the two sheets of (\widetilde C) and defines (T=\frac12(eJ^{(-)})^2). This yields Virasoro blocks on C but introduces extra primary insertions of weight (h=1/16) at each branch point, reflecting the singularities of the covering map.

To cancel these spurious insertions the authors introduce a fermionic operator \