Lectures on Gauge theories and Many-Body systems
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| Format: | Preprint |
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2025
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| _version_ | 1866911586180923392 |
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| author | Chaban, Igor Nekrasov, Nikita |
| author_facet | Chaban, Igor Nekrasov, Nikita |
| contents | These lectures study two correspondences between gauge theories and integrable many-body systems. The first arises from infinite-dimensional Hamiltonian reduction and relates gauge-theoretic dynamics directly to Calogero--Moser-type systems and their quantum counterparts. The second emerges in supersymmetric gauge theory through instanton counting and non-perturbative dualities, linking classical problems on one side to quantum problems on the other. A central motivation comes from the observation that conjugacy classes of holonomies in gauge theory can be interpreted as configurations of indistinguishable particles on a circle. In quantum theory these particle positions become random variables, and the correspondence may be either exact or approximate depending on spacetime dimension and supersymmetry. We focus on the Calogero--Moser--Sutherland family associated with root systems of type A and with SU(N) gauge theories in dimensions from one to six. In low dimensions the correspondence is direct and involves matrix quantum mechanics, two-dimensional Yang--Mills theory, and higher-dimensional Chern--Simons-type theories. In four, five, and six dimensions with eight supercharges, the correspondence takes a more indirect form through supersymmetric gauge theory and the Omega-deformation. We introduce non-local observables whose correlation functions satisfy non-perturbative Dyson--Schwinger equations and, in four dimensions, lead to Schrodinger equations for many-body systems. The notes are divided accordingly: the first part develops symplectic-reduction constructions of Calogero--Moser systems, while the second studies localization, partition measures, and order/disorder observables in supersymmetric gauge theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_23099 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lectures on Gauge theories and Many-Body systems Chaban, Igor Nekrasov, Nikita Mathematical Physics High Energy Physics - Theory These lectures study two correspondences between gauge theories and integrable many-body systems. The first arises from infinite-dimensional Hamiltonian reduction and relates gauge-theoretic dynamics directly to Calogero--Moser-type systems and their quantum counterparts. The second emerges in supersymmetric gauge theory through instanton counting and non-perturbative dualities, linking classical problems on one side to quantum problems on the other. A central motivation comes from the observation that conjugacy classes of holonomies in gauge theory can be interpreted as configurations of indistinguishable particles on a circle. In quantum theory these particle positions become random variables, and the correspondence may be either exact or approximate depending on spacetime dimension and supersymmetry. We focus on the Calogero--Moser--Sutherland family associated with root systems of type A and with SU(N) gauge theories in dimensions from one to six. In low dimensions the correspondence is direct and involves matrix quantum mechanics, two-dimensional Yang--Mills theory, and higher-dimensional Chern--Simons-type theories. In four, five, and six dimensions with eight supercharges, the correspondence takes a more indirect form through supersymmetric gauge theory and the Omega-deformation. We introduce non-local observables whose correlation functions satisfy non-perturbative Dyson--Schwinger equations and, in four dimensions, lead to Schrodinger equations for many-body systems. The notes are divided accordingly: the first part develops symplectic-reduction constructions of Calogero--Moser systems, while the second studies localization, partition measures, and order/disorder observables in supersymmetric gauge theory. |
| title | Lectures on Gauge theories and Many-Body systems |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2512.23099 |