Mayer expansion of the Nekrasov pre potential: the subleading $\epsilon_2$-order

TL;DR

This paper uses the Mayer cluster expansion to analyze the subleading $ ext{epsilon}_2$ correction to the Nekrasov prepotential in $ ext{N}=2$ super Yang-Mills, confirming previous results through detailed diagram analysis.

Contribution

It provides a detailed diagrammatic analysis of the subleading $ ext{epsilon}_2$ correction, confirming earlier field theory results with new insights into diagram cancellations.

Findings

Confirmed the field theory result for the $ ext{epsilon}_2$ correction

Identified cancellations among diagram contributions

Validated assumptions used in previous computations

Abstract

The Mayer cluster expansion technique is applied to the Nekrasov instanton partition function of $\mathcal{N}=2$ $SU(N_c)$ super Yang-Mills. The subleading small $\epsilon_2$-correction to the Nekrasov-Shatashvili limiting value of the prepotential is determined by a detailed analysis of all the one-loop diagrams. Indeed, several types of contributions can be distinguished according to their origin: long range interaction or potential expansion, clusters self-energy, internal structure, one-loop cyclic diagrams, etc.. The field theory result derived more efficiently in [1], under some minor technical assumptions, receives here definite confirmation thanks to several remarkable cancellations: in this way, we may infer the validity of these assumptions for further computations in the field theoretical approach.