Quantized Coulomb Branch Algebra

• Quantized Coulomb branch algebra is a deformation quantization of the supersymmetric gauge theory’s Coulomb branch using a convolution framework.
• It encodes operator product expansions of loop and monopole operators, with noncommutativity controlled by equivariant parameters like ħ.
• The algebra bridges connections to spherical DAHA, shifted Yangians, and quantum cohomology, informing studies in mirror symmetry and integrable systems.

The quantized Coulomb branch algebra is a rigorous mathematical structure capturing the deformation quantization of the Coulomb branch of supersymmetric gauge theories, particularly in three and four dimensions, via the convolution formalism of Braverman–Finkelberg–Nakajima (BFN). This algebra encodes the operator product expansions of BPS loop or monopole operators in presence of an Ω-background, realizing a flat noncommutative deformation governed by an equivariant parameter (often denoted $\hbar$ or $\epsilon$).

1. Mathematical Framework and Physical Motivation

The Coulomb branch of a supersymmetric gauge theory is an affine (holomorphic symplectic) variety whose coordinate ring is generated by local gauge-invariant operators, notably adjoint scalars and monopole operators. Braverman–Finkelberg–Nakajima introduced a geometric construction: for a complex reductive group $G$ and representation $N$, one considers the based affine Grassmannian $Gr_G = G(\mathbb{C}((z))) / G(\mathbb{C}[[z]])$ and the ind-scheme $\mathcal{R}_{G,N}$ parameterizing $(g, x)$ with $x$ regular modulo $g$. The quantized Coulomb branch algebra is defined as the equivariant Borel–Moore homology $$H^{G_{O} \times \mathbb{C}^{*}}_{*}(\mathcal{R}_{G,N})$$ endowed with the convolution product, where $\mathbb{C}^*$ (rotation of the loop parameter $z$) introduces the quantization parameter $\hbar$, which is interpreted as the Ω-background parameter physically (Yoshida, 19 Mar 2025, Shen et al., 14 Oct 2025).

2. Deformation Quantization and Noncommutative Structure

The quantization manifests as an explicit association of operator insertions to elements of the algebra, with the Ω-background parameter $\epsilon_+$ or $\hbar$ controlling noncommutativity. For example, in the 4d $\mathcal{N}=2$ $Sp(N)$ theory with suitable matter, the localization formula for loop operator vevs yields operators $X_i = e^{-a_i}$ (complexified holonomy variables) and 't Hooft loop operators $M_{\pm}^{(i)}$, with relations: $$X_i M_{\pm}^{(i)} = e^{\pm 2\epsilon_+} M_{\pm}^{(i)} X_i,$$ and the critical nontrivial commutator

$[M_+, M_-] = f(X; \epsilon_+)$

which encodes correction terms derived from monopole bubbling and one-loop determinants (Yoshida, 19 Mar 2025). The quantization parameter $\hbar = \epsilon_+$ is closely tied to the generator of the loop rotation equivariant cohomology.

3. Connections to Spherical DAHA and Representation Theory

A striking structure is revealed in the isomorphism between the quantized Coulomb branch algebra and the spherical double affine Hecke algebra (DAHA). In the rank-one case $Sp(1) \simeq SU(2)$, the quantized algebra matches the polynomial representation of the spherical part of the DAHA of $(C_1^\vee, C_1)$ type; explicitly, the operators correspond to DAHA generators with the $q$ parameter identified as $e^{2\hbar}$, and the relations mirror those in the DAHA (Yoshida, 19 Mar 2025). For higher rank $N$, it is conjectured (and supported by explicit checks) that the full quantized Coulomb branch for 4d $\mathcal{N}=2$ $Sp(N)$ theory with four fundamentals and one antisymmetric hypermultiplet is isomorphic to the spherical DAHA of $(C_N^\vee, C_N)$ type. The algebraic generators correspond to Koornwinder and van Diejen operators in the DAHA, matching precisely the quantized 't Hooft loops and their composite products (Yoshida, 19 Mar 2025).

4. Difference Operator Realization and Minuscule Monopoles

One powerful technique is the embedding of the quantized Coulomb branch algebra into an algebra of shift (difference) operators. After abelianization (i.e., restriction to the maximal torus), the algebra acts as difference operators on equivariant parameters (Chern roots), with monopole operators realized as explicit difference-operator classes. For generic quivers (including those with involutions), generators are constructed from minuscule coweight monopoles, and their images define algebra homomorphisms from shifted twisted Yangians (for theories with symmetries) or truncated shifted Yangians (for BCFG quiver types) onto the Coulomb branch algebra (Shen et al., 14 Oct 2025, Nakajima et al., 2019, Wang, 30 Dec 2025). This realization demonstrates a flat deformation and PBW property, and can often be fully described via Drinfeld-type current relations.

5. Quantum Cohomology, Shift Operators, and Integrable Systems

Recent developments elucidate connections to quantum cohomology: the quantized Coulomb branch acts (via shift operators) on equivariant quantum cohomology of symplectic resolutions, e.g., Gromov–Witten theories of varieties with group actions. The assignment $$\Gamma \otimes \alpha \mapsto S_{G,N,X}(\Gamma, \alpha)$$ extends the Coulomb algebra to modules over big quantum cohomology, which reduces to classical structures at $\hbar = 0$ and realizes generalized Peterson isomorphisms (Chan et al., 29 May 2025). The images of Cartan subalgebra generators correspond to the Gelfand–Tsetlin subalgebra, exposing the algebra as a quantum integrable system.

6. Bubbling Effects, Trace Maps, and Mirror Symmetry

Nontrivial features such as monopole bubbling corrections are inherent in the algebraic relations. Bubbling terms enter as additional difference operator contributions, determined either by localization or by enforcing polynomiality constraints. For abelian and certain non-abelian cases, trace functionals on the quantized Coulomb branch can be constructed (often as correlation functions in a topological quantum field theory), with explicit integral formulas compatible with conformal field theory correlators (Zhang, 22 Oct 2025, Dedushenko et al., 2018, Assel et al., 2019). In the context of 3D mirror symmetry, these algebras are related via equivalences with quantum groups, e.g., folded Hecke algebras or twisted Yangians, thus providing categorical matches between modules on Higgs and Coulomb branches (Shen et al., 14 Oct 2025).

7. Structural Properties, Generalizations, and Open Problems

The quantized Coulomb branch algebra is flat over the quantization parameter and exhibits a Poisson commutative limit at $\hbar = 0$. It admits a wealth of presentations: as a convolution algebra, as a spherical DAHA, as a shifted Yangian, and as a difference operator algebra. It also supports central functors and has Frobenius-constant quantization properties over finite fields, with deep links to modular representation theory (Lonergan, 2017). Generalizations to K-theoretic, elliptic, and noncotangent-type theories extend the framework, admitting connections to skein algebras, symmetric pairs, and integrable Hamiltonian reductions (Allegretti et al., 2024, Dedushenko et al., 30 Dec 2025, Gannon et al., 2023).

Open problems include:

• Explicit combinatorial derivation of bubbling terms for higher magnetic charges, especially for Sp(N) with antisymmetric matter;
• Complete identification of composite loop operators with higher van Diejen operators, including constant shifts;
• Full realization of elliptic and rational degenerations (elliptic DAHA, Cherednik algebras);
• Representation-theoretic analysis of finite dimensional modules and factorization/co-product structures;
• Geometric explanation of dualities (e.g., S-duality permutations of Coulomb branches in the $K$-theoretic setting) (Allegretti et al., 2024).

The quantized Coulomb branch algebra is a cornerstone connecting supersymmetric gauge theory, geometric representation theory, and quantum integrable systems, supporting a variety of structural, categorical, and physical dualities and quantizations.