Integrable Hierarchies and Contact Terms in u-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories

Abstract

The $u$-plane integrals of topologically twisted $N=2$ supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mariño and Moore's remark that the blowup formula of the $u$-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multi-time version without spoiling the modular invariance of the blowup formula. The multi-time tau function is comprised of a Gaussian factor $e^{Q(t_1,t_2,...)}$ and a theta function. The time variables $t_n$ play the role of physical coupling constants of 2-observables $I_n(B)$ carried by the exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are identified to be the contact terms of these 2-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.

latex, 17 pages, no figures; final version for publication

References in corpus (4)

• Issues in Topological Gauge Theory
• Calogero-Moser Systems in SU(N) Seiberg-Witten Theory
• The WDVV Equations in N=2 Supersymmetric Yang-Mills Theory
• Integrating over the Coulomb branch in N=2 gauge theory

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• Effective gravitational couplings of four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories
• Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups
• Remarks on twisted theories with matter
• Integrable hierarchies in Donaldson-Witten and Seiberg-Witten theories