Lectures on Gauge theories and Many-Body systems

Abstract: These lectures discuss two correspondences between gauge theories and integrable many-body systems. The first correspondence goes back to the work of many mathematicians and physicists in the 1980-1990's. It is realized by an infinite dimensional Hamiltonian reduction and its quantum counterpart. In this approach the quantization parameters of the gauge theory coincide with the quantization parameters of the many-body system. The second correspondence emerged in the mid-1990's, it involves non-trivial dualities, relating classical problems on one side to quantum on another and vice versa. This duality has various reincarnations: Fourier and Legendre transforms, Langlands duality, etc. The quantization parameters are mapped to geometric parameters. Simple questions on one side solve complicated ones on the other and vice versa.

• The paper demonstrates the correspondence between supersymmetric gauge theories and integrable Calogero–Moser–Sutherland many-body systems.
• It employs Hamiltonian reduction and Lax pair formalism to establish Liouville integrability via spectral invariants in both classical and quantum settings.
• The study links gauge theory partition functions, Young diagrams, and qq-characters to fundamental structures in algebraic geometry and representation theory.

Gauge Theory–Integrable System Correspondence and Many-Body Dynamics

Introduction

The paper "Lectures on Gauge theories and Many-Body systems" (2512.23099) delivers a comprehensive exploration of the correspondences between supersymmetric gauge theories and quantum/classical integrable many-body systems, focusing particularly on the Calogero-Moser-Sutherland type and their generalizations. The narrative is built around two core mechanisms: infinite-dimensional Hamiltonian reduction (and its quantum analogues), and a collection of dualities—geometric and analytic—manifesting the equivalence between classical and quantum problems across the two domains.

The expository approach is both algebraic and geometric, addressing not only the integrable system structure but also confronting questions in partition combinatorics, localization in moduli space, and the statistical mechanics interpretation of gauge theory partition functions. The emergence and utility of spectral curves, Young diagrams, and $qq$-characters as organizing principles are also emphasized throughout.

Below, we elaborate on key technical points, main constructions, and implications as presented in the work.

Calogero–Moser System: Hamiltonian, Integrals of Motion, and Lax Formalism

The classical Calogero–Moser (CM) system is framed as a Hamiltonian system on the reduced symplectic variety $$\mathcal{P} = T^*(\mathbb{R}^N\setminus \Delta)/S_N$$, for which the Hamiltonian takes the form

$$H = \sum_{i=1}^N \frac{p_i^2}{2} + \nu^2 \sum_{i < j} \frac{1}{(x_i-x_j)^2}$$

where $\nu$ is the coupling constant. The pairwise interaction singularity encodes the nontrivial isospectral manifold structure arising from indistinguishability and collision avoidance.

Explicit Lax pair representations are constructed, establishing integrability: one constructs $L$ and $A$ matrices encoding the dynamics such that $\dot{L} = [A,L]$. The spectral invariants, i.e., the eigenvalues of $L$, or equivalently the set of $\mathrm{Tr}(L^k)$, are Poisson-commuting constants of motion. This Lax formalism not only confirms Liouville integrability but seeds the broader geometric correspondence to gauge theory moduli problems.

Symplectic Reduction and Hamiltonian Actions

A central technical device throughout is the use of symplectic (Hamiltonian) reduction. For finite ($U(N)$) and infinite-dimensional (gauge group) phase spaces, the action of the group by adjoint transformations is Hamiltonian, with an explicit moment map:

$$\mu(P,X,z) = [P,X] + \sqrt{-1}(z \otimes z^\dagger - \nu \mathbb{I})$$

Reduction at the zero level of the moment map, modulo the group action, produces the phase space of the many-body system precisely by quotienting out redundant gauge variables. The reduced Hamiltonians are then explicitly identified with gauge-invariant quantities.

Gauge Theory Realization: 2d Yang–Mills and Generalization

The paper demonstrates the correspondence between Calogero-Moser-type systems and lower-dimensional Yang–Mills/Chern–Simons gauge theories in various spacetime dimensions. For 2d Yang–Mills, the phase space and Hamiltonian are realized as connections over $S^1$ perturbed by delta-function sources ('impurities') reflecting point particle insertions. The passage to the classical/quantum correspondence is mediated by localization to holonomy eigenvalues.

The many-body system Hamiltonian arises from the constraint (Gauss Law) in the gauge theory reduced to the Cartan (diagonal) directions, subject to residual Weyl symmetry. The quantum analog features the path integral reducing to a sum/integral over partitions or dominant weights, highlighting a deep spectral connection.

Elliptic, Trigonometric, and Rational Reductions

Through Poisson geometry and the gauge theory path integral machinery, the paper further generalizes from rational (inverse-square), to trigonometric ($1/\sin^2$), and to the elliptic Calogero–Moser system, with corresponding gauge theory incarnations in higher dimensions, including 4d/5d $\mathcal{N}=2^*$ and $\mathcal{N}=1$ supersymmetric compactifications.

Moduli and spectral parameter spaces are shown to organize as complex algebraic varieties, with the phase space compactification constructed via blowups and deformation theory, e.g., the compactification of $T^*C_\tau/\mathbb{Z}_2$ in the $N=2$ case.

Partition Functions, Localization, and Measures on Young Diagrams

The second major arc tackles the combinatorial and enumerative aspects: supersymmetric partition functions in $\Omega$-background compactifications (notably Nekrasov partition functions) are obtained via equivariant localization, reducing infinite-dimensional integrals to explicit combinatorial sums over collections of Young tableaux (multipartitions), with the measure derived from the equivariant weights and the fixed-point data in the instanton moduli space.

The structure of $qq$-characters—nontrivial combinations of observables (order operators) whose vacuum expectation values are holomorphic sections of associated vector bundles—emerges as an effective tool for organizing gauge theory correlation functions, further manifesting the gauge/integrable correspondence.

Figure 1: A Young diagram with boxes in $\partial_{+}\lambda$ (outer boundary) and $\partial_{-}\lambda$ (inner boundary) highlighted, significant in the combinatorics of instanton partition functions.

Figure 2: Illustration of the arm and leg lengths for a box $\square = (1,2)$, crucial for partition statistics and in evaluating weights for the partition measure.

Surface Defects, Disorder Operators, and Parabolic Sheaves

Order/disorder dichotomy is refined via the introduction of codimension-2 surface defects, realized as modifications of the gauge bundle along lower-dimensional subloci. The statistical model is altered (via parabolic structures on sheaves) and the corresponding partition function involves a refined coloring of the partitions. The rich structure of these defects undergirds connections to geometric representation theory and hints at the generalization to more intricate defect networks and category-theoretic structures.

Gauge Theories, Quantum versus Classical Correspondence, and Spectral Curves

The indirect correspondence: the expectation values of $qq$-characters or certain surface defect observables, in the thermodynamic ($\varepsilon_1, \varepsilon_2 \to 0$) limit, encode the quantized spectral data (e.g., Baxter TQ equations) of the associated integrable system. The derivation of matrix-valued Lax operators, construction of Baker-Akhiezer functions, and explicit spectral curve equations are demonstrated, culminating in a unified algebraic description of the integrable system's dynamics and its quantum/classical duals.

Implications and Outlook

This framework links Hamiltonian systems, representation theory, algebraic geometry, and gauge field theory, enabling computation of nonperturbative quantities (e.g., prepotentials, BPS indexes) through combinatorial and geometric means, and vice versa. It provides algorithmic handles on integrable dynamics (such as via Young diagram growth processes) and exposes deep geometric structures underlying supersymmetric quantum field theory. The described correspondences also serve as foundational input for the geometric Langlands program and the study of quantum Hitchin systems.

Anticipated future directions include a more detailed analysis of non-simply laced gauge algebras, further generalization to higher-dimensional varieties, quantization of moduli stacks beyond cohomology ($K$-theory, elliptic), and direct quantization of spectral curves, deepening connections with quantum groups, 2d CFT, and string theory backgrounds.

Conclusion

The lecture volume (2512.23099) provides a mathematically rigorous and physically comprehensive synthesis of the gauge-integrable system correspondences, one of the central pillars of modern mathematical physics. By unifying reduction procedures, partition combinatorics, and the language of spectral geometry, it supplies both powerful computational techniques and conceptual insights, with implications spanning quantum field theory, representation theory, and enumerative geometry.