Recursive representation of the torus 1-point conformal block
Leszek Hadasz, Zbigniew Jaskolski, Paulina Suchanek
TL;DR
The paper derives a recursive representation for the torus 1-point conformal block, enabling efficient computation and analytical control of its poles via the degenerate Verma-module structure. Using Gram-matrix inverses and fusion polynomials, it constructs an elliptic recursion for the torus block and proves two Poghossian identities relating torus and sphere blocks under parameter rescalings. These results provide a chiral framework for connecting torus and sphere conformal blocks and offer a practical numerical approach to test modular invariance in Liouville theory. The work has implications for the Liouville/AGT correspondence and for modular bootstrap analyses in two-dimensional CFTs.
Abstract
The recursive relation for the 1-point conformal block on a torus is derived and used to prove the identities between conformal blocks recently conjectured by R. Poghossian. As an illustration of the efficiency of the recurrence method the modular invariance of the 1-point Liouville correlation function is numerically analyzed.