Multiplicative Hitchin systems and supersymmetric gauge theory. (English) Zbl 1423.14081
The aim of this carefully written paper is to compare five different perspectives on a single moduli space: three coming from geometry and representation theory, and two coming from supersymmetric gauge theory. By leveraging these multiple perspectives the authors will equip their common moduli space with the structure of a completely integrable system which they call the multiplicative Hitchin system. Multiplicative Hitchin systems are analogues of Hitchin’s integrable system based on moduli spaces of \(G\)-Higgs bundles on a curve \(\mathcal{C}\) where the Higgs field is groupvalued, rather than Lie algebra valued. The authors discuss the relationship between several occurences of these moduli spaces in geometry and supersymmetric gauge theory, with a particular focus on the case where \(\mathcal{C}=\mathbb{CP}^1\) with a fixed framing at infinity. In this case they prove that the identification between multiplicative Higgs bundles and periodic monopoles proved by Charbonneau and Hurtubise can be promoted to an equivalence of hyperkähler spaces, and analyze the twistor rotation for the multiplicative Hitchin system. They also discuss quantization of these moduli spaces, yielding the modules for the Yangian \(Y(g)\) discovered by Gerasimov, Kharchev, Lebedev and Oblezin. The paper is divided into two parts. In Part A the authors will investigate the symplectic geometry of the moduli space of multiplicative Higgs bundles, and in Part B they will study the connection between multiplicative Higgs bundles and periodic monopoles, and the hyperkähler geometry of these moduli spaces.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 53D30 | Symplectic structures of moduli spaces |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
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