Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 047, 141 pages      arXiv:1211.2240      https://doi.org/10.3842/SIGMA.2023.047
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories

Nikita Nekrasov a and Vasily Pestun b
a) Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794-3636, USA
b) Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France

Received December 19, 2022, in final form June 20, 2023; Published online July 16, 2023

Abstract
Seiberg-Witten geometry of mass deformed $\mathcal N=2$ superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space $\mathfrak M$ of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space ${\rm Bun}_{\mathbf G} (\mathcal E)$ of holomorphic $G^{\mathbb C}$-bundles on a (possibly degenerate) elliptic curve $\mathcal E$ defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group $G$. The integrable systems $\mathfrak P$ underlying the special geometry of $\mathfrak M$ are identified. The moduli spaces of framed $G$-instantons on ${\mathbb R}^{2} \times {\mathbb T}^{2}$, of $G$-monopoles with singularities on ${\mathbb R}^{2} \times {\mathbb S}^{1}$, the Hitchin systems on curves with punctures, as well as various spin chains play an important rôle in our story. We also comment on the higher-dimensional theories.

Key words: low-energy theory; instantons; monopoles; integrability.

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