Instanton calculus and chiral one-point functions in supersymmetric gauge theories

Abstract

We compute topological one-point functions of the chiral operator Tr ϕ k in the maximally confining phase of N = 1 U (N ) supersymmetric gauge theory, which is obtained from N = 2 theory by turning on a tree level superpotential W (Φ).Localization theorem for toric action allows us to express these one-point functions as polynomials in the equivariant parameter and the parameter of instanton expansion q = Λ 2N . The chiral one-point functions are of particular interest from gauge/string theory correspondence, since they are related to the Gromov-Witten theory of P 1 . Based on a combinatorial identity that gives summation formula over Young diagram of relevant functions, we find a relation among chiral onepoint functions, which recursively determines the expansion of the generating function of one-point functions.

Chiral trace relations in Ω-deformed N = 2 $$ \mathcal{N}=2 $$ theories

“…Formally, we shall be using the same localization expressions valid for U(N ). This is a useful check given the explicit results of [90] to be discussed in a moment.…”

“…Due to the fact that these expressions are polynomials in q, it is possible to find simple relations fully discussed in [90]. The authors of [90] considered the further special limit ε 1 = −ε 2 = , i.e.…”

“…The U(1) N = 2 has not been considered in [90], but it is a simple extension. Now, the chiral observables Tr ϕ n are not polynomials in q.…”

“…An explicit calculation gives the following exact expressions 17 17 Notice that to compare with [90] we need to send q → −q. This sign flip for U(N ) theories with odd N will be further discussed later.…”

Some combinatorial properties of hook lengths,contents, and parts of partitions

“…I am grateful to Soichi Okada for calling my attention to reference [4] and for providing conjecture (19). I also am grateful to an anonymous referee for many helpful suggestions.…”