Differential equations from null vectors of the Ramond algebra

TL;DR

This paper derives differential equations for chiral blocks of Ramond fields in the N=1 super Virasoro algebra, using null vectors, and applies the method to the Ising model to find exact solutions.

Contribution

It introduces a novel algebraic approach to determine differential equations for Ramond sector blocks, extending techniques from the Ising model to superconformal algebras.

Findings

Derived first-order differential equations for Ramond blocks

Obtained exact solutions for many cases

Validated results against existing methods

Abstract

We consider chiral blocks of four Ramond fields of the N=1 super Virasoro algebra where one of the fields is in the (1,2) representation. We show how the null vector in the (1,2) representation determines the chiral blocks as series expansions. We then turn to the Ising model to find an algebraic method to determine differential equations for the blocks of four spin fields. Extending these ideas to the super Virasoro case, we find a first order differential equation for blocks of four Ramond fields. We are able to find exact solutions in many cases. We compare our blocks with results known from other methods.