A note on the Virasoro blocks at order $1/c$

TL;DR

This paper derives explicit formulas for the order $1/c$ corrections to Virasoro blocks in 2D CFT, enabling better understanding of their structure and monodromies in the large central charge limit.

Contribution

It provides a new explicit series expansion for Virasoro blocks at order $1/c$, including summation in hypergeometric functions and handling of integer weights.

Findings

Explicit $1/c$ correction formulas for Virasoro blocks.

Series expansion around $z=0$ expressed via hypergeometric functions.

Manifest monodromies of the blocks around $z=1$.

Abstract

We derive an explicit expression for the $1/c$ contribution to the Virasoro blocks in 2D CFT in the limit of large $c$ with fixed values of the operators' dimensions. We follow the direct approach of orthonormalising, at order $1/c$, the space of the Virasoro descendants to obtain the blocks as a series expansion around $z=0$. For generic conformal weights this expansion can be summed in terms of hypergeometric functions and their first derivatives with respect to one parameter. For integer conformal weights we provide an equivalent expression written in terms of a finite sum of undifferentiated hypergeometric functions. These results make the monodromies of the blocks around $z=1$ manifest.