On F-term contribution to effective action

TL;DR

The work develops a two-dimensional N=2 SYM framework in the C-case by applying equivariant localization to compute the F-term contribution encoded in the twisted superpotential $W_0( ext{Sigma},\tau)$. It systematically separates perturbative and non-perturbative (vortex) contributions, employing Schwinger regularization for 1-loop determinants and a finite-dimensional vortex moduli space in an $\\ ext{Omega}$-background to obtain the leading term $W_0^{\text{vort}}(a)$, with residues organized by icicles. The analysis reveals both fruitful parallels with the four-dimensional Nekrasov–Okounkov construction and notable differences intrinsic to two dimensions, including the absence of a straightforward Seiberg–Witten prepotential in the C-case. The paper also outlines how an O-case in eight dimensions might be approached via generalized instantons and hints at potential connections to octonionic structures and exceptional groups, which could inform higher-dimensional instanton counting. Overall, the work provides a concrete localization framework for F-term computations in low-dimensional SUSY theories and clarifies the scope and limitations of the C-, H-, and O-case correspondences.

Abstract

We apply equivariant integration technique, developed in the context of instanton counting, to two dimensional N=2 supersymmetric Yang-Mills models. Twisted superpotential for U(N) model is computed. Connections to the four dimensional case are discussed. Also we make some comments about the eight dimensional model which manifests similar features.