The exact 8d chiral ring from 4d recursion relations

TL;DR

This paper derives exact expressions for the eight-dimensional chiral ring elements in a 4d N=2 super Yang-Mills theory with SU(2) gauge group and four flavors, using recursion relations and modular functions, extending beyond previous localization results.

Contribution

It introduces recursion techniques to compute the large-z expansion coefficients of the tau profile in terms of modular functions, providing exact results for the chiral ring elements.

Findings

Exact expressions for the eight-dimensional chiral ring elements.

Resummation of instanton expansions beyond initial orders.

Connection between 4d recursion relations and 8d chiral ring structure.

Abstract

We consider the local F-theory set-up corresponding to four D7 branes in type I' theory, in which the exact axio-dilaton background tau(z) is identified with the low-energy effective coupling of the four-dimensional N=2 super Yang-Mills theory with gauge group SU(2) and Nf=4 flavours living on a probe D3 brane placed at position z. Recently, an intriguing relation has been found between the correlators forming the chiral ring of the eight-dimensional theory on the D7 branes and the large-z expansion of the tau profile. Here we apply to the SU(2), Nf=4 theory some recursion techniques that allow to derive the coefficients of the large-z expansion of tau in terms of modular functions of the UV coupling. In this way we obtain exact expressions for the elements of the eight-dimensional chiral ring that resum their instanton expansions, previously known only up to the first few orders by means of localization techniques.