Recursion relations in CFT and N=2 SYM theory

TL;DR

This paper derives recursion relations for the generalized prepotential in ${ m N}=2$ SYM theories using conformal block techniques and the AGT conjecture, establishing links between conformal blocks and partition functions.

Contribution

It introduces new recursion relations for ${ m N}=2$ SYM prepotentials based on conformal blocks and clarifies their connection to partition functions via the AGT conjecture.

Findings

Explicit large expectation value limits derived.

Recursion relations established for various hypermultiplet configurations.

Relationship between torus and sphere conformal blocks demonstrated.

Abstract

Based on prototypical example of Al.Zamolodchikov's recursion relations for the four point conformal block and using recently proposed Alday-Gaiotto-Tachikawa (AGT) conjecture, recursion relations are derived for the generalized prepotential of ${\cal N}=2$ SYM with $f=0,1,2,3,4$ (anti) fundamental or an adjoint hypermultiplets. In all cases the large expectation value limit is derived explicitly. A precise relationship between generic 1-point conformal block on torus and specific 4-point conformal block on sphere is established. In view of AGT conjecture this translates into a relation between partition functions with an adjoint and 4 fundamental hypermultiplets.