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Instanton calculus and chiral one-point functions in supersymmetric gauge theories. (English) Zbl 1151.81359

Summary: We compute topological one-point functions of the chiral operator \(\text{Tr\,} \varphi^k\) in the maximally confining phase of \(U(N)\) supersymmetric gauge theory. These one-point functions are polynomials in the equivariant parameter \(\hbar\) and the parameter of instanton expansion \(q=\Lambda^{2N}\) and are of particular interest from gauge/string theory correspondence, since they are related to the Gromov-Witten theory of \(\mathbb{P}^1\). Based on a combinatorial identity that gives summation formula over Young diagrams of relevant functions, we find a relation among chiral one-point functions, which recursively determines the \(\hbar\) expansion of the generating function of one-point functions. Using a result from the operator formalism of the Gromov-Witten theory, we also present a vacuum expectation value of the loop operator \(\text{Tr}\, e^{it\varphi}\).

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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