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Quantized Coulomb branch of 4d $\mathcal{N}=2$ $Sp(N)$ gauge theory and spherical DAHA of $(C_N^{\vee}, C_N)$-type

Yutaka Yoshida

TL;DR

This work develops a bridge between the quantized Coulomb branch of 4d $\mathcal{N}=2$ $Sp(N)$ gauge theories and spherical double affine Hecke algebras (DAHA) of $(C_N^{\vee},C_N)$-type. Using SUSY localization in $S^1\times\mathbb{R}^3$, monopole bubbling, and a refined brane construction, the authors compute vevs of BPS loop operators and reinterpret them as operator products in a deformation-quantized Coulomb-branch algebra. They prove, for $N=1$, that the quantized Coulomb branch matches the polynomial representation of the spherical DAHA of $(C_1^{\vee},C_1)$-type, and conjecture a generalization to $N\ge 2$ where the algebra is isomorphic to the spherical DAHA of $(C_N^{\vee},C_N)$-type, with evidence that the minimal 't Hooft loop corresponds to the Koornwinder operator and higher loops relate to van Diejen operators. The work thus unifies gauge-theoretic loop algebras with rich DAHA structures, providing a concrete computational framework and suggesting a broad DAHA/Coulomb-branch dictionary across ranks and dimensions (including relevant 3d and 5d limits).

Abstract

We study BPS loop operators in a 4d $\mathcal{N}=2$ $Sp(N)$ gauge theory with four hypermultiplets in the fundamental representation and one hypermultiplet in the anti-symmetric representation. The algebra of BPS loop operators in the $Ω$-background provides a deformation quantization of the Coulomb branch, which is expected to coincide with the quantized K-theoretic Coulomb branch in the mathematical literature. For the rank-one case, i.e., $Sp(1) \simeq SU(2)$, we show that the quantization of the Coulomb branch, evaluated using the supersymmetric localization formula, agrees with the polynomial representation of the spherical part of the double affine Hecke algebra (spherical DAHA) of $(C_1^{\vee}, C_1)$-type. For higher-rank cases, where $N \geq 2$, we conjecture that the quantized Coulomb branch of the 4d $\mathcal{N}=2$ $Sp(N)$ gauge theory is isomorphic to the spherical DAHA of $(C_N^{\vee}, C_N)$-type . As evidence for this conjecture, we demonstrate that the quantization of an 't Hooft loop agrees with the Koornwinder operator in the polynomial representation of the spherical DAHA.

Quantized Coulomb branch of 4d $\mathcal{N}=2$ $Sp(N)$ gauge theory and spherical DAHA of $(C_N^{\vee}, C_N)$-type

TL;DR

This work develops a bridge between the quantized Coulomb branch of 4d gauge theories and spherical double affine Hecke algebras (DAHA) of -type. Using SUSY localization in , monopole bubbling, and a refined brane construction, the authors compute vevs of BPS loop operators and reinterpret them as operator products in a deformation-quantized Coulomb-branch algebra. They prove, for , that the quantized Coulomb branch matches the polynomial representation of the spherical DAHA of -type, and conjecture a generalization to where the algebra is isomorphic to the spherical DAHA of -type, with evidence that the minimal 't Hooft loop corresponds to the Koornwinder operator and higher loops relate to van Diejen operators. The work thus unifies gauge-theoretic loop algebras with rich DAHA structures, providing a concrete computational framework and suggesting a broad DAHA/Coulomb-branch dictionary across ranks and dimensions (including relevant 3d and 5d limits).

Abstract

We study BPS loop operators in a 4d gauge theory with four hypermultiplets in the fundamental representation and one hypermultiplet in the anti-symmetric representation. The algebra of BPS loop operators in the -background provides a deformation quantization of the Coulomb branch, which is expected to coincide with the quantized K-theoretic Coulomb branch in the mathematical literature. For the rank-one case, i.e., , we show that the quantization of the Coulomb branch, evaluated using the supersymmetric localization formula, agrees with the polynomial representation of the spherical part of the double affine Hecke algebra (spherical DAHA) of -type. For higher-rank cases, where , we conjecture that the quantized Coulomb branch of the 4d gauge theory is isomorphic to the spherical DAHA of -type . As evidence for this conjecture, we demonstrate that the quantization of an 't Hooft loop agrees with the Koornwinder operator in the polynomial representation of the spherical DAHA.

Paper Structure

This paper contains 18 sections, 83 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. BPS loops in 4d $\mathcal{N}=2$ gauge theories on $S^1 \times \mathbb{R}^3$
  3. SUSY localization formula
  4. Deformation quantization and algebra of loop operators
  5. Monopole bubbling effect
  6. Naive brane setup for monopole bubbling in $U(N)$ and $SU(N)$ gauge theories
  7. Complete brane setup with extra D5-branes
  8. Monopole bubbling effect for dyonic loops
  9. $Z^{(\zeta)}_{\rm extra}$ as the decoupled states in $Z^{(\zeta)}_{\rm JK}$
  10. Quantized Coulomb branch and spherical DAHA of $(C^{\vee}_1, C_1)$-type
  11. Polynomial representation of DAHA of $(C^{\vee}_1, C_1)$-type
  12. Deformation quantization of loop operators and quantized Coulomb branch
  13. Quantized Coulomb branch of $Sp(N)$ gauge theory and spherical DAHA of $(C^{\vee}_N, C_N)$-type
  14. Polynomial representation of DAHA of $(C^{\vee}_N, C_N)$-type
  15. 't Hooft loop L(e1,0) and Koornwinder operator
  16. ...and 3 more sections

Figures (6)

  • Figure 1: (a): A brane configuration in the $(x^4,x^5)$-plane for an 't Hooft loop with magnetic charge ${\bm p}=(1,-1)$ (resp. ${\bm p}=1$) in $U(2)$ (resp. $SU(2)$) gauge theory with four hypermultiplets. The red and green circles represent a D7-brane and a D3-brane, respectively. The blue line represents an NS5-brane. (b): Another brane configuration for a 't Hooft loop. Figure (a) and Figure (b) are related by the Hanany-Witten effect: when a D3-brane crosses an NS5-brane, a D1-brane (denoted by a black line) is either created or annihilated.
  • Figure 2: (a): A D1-brane suspended between two D3-branes is added to Figure \ref{['fig:subbrane2']}. (b): three segments of D1-branes form a single D1-brane, which ended on the NS5-branes but not ended on the D3-branes. Then the D1-brane charge is screened, which is the D-brane realization of the monopole bubbling.
  • Figure 3: (a): The quiver diagram representing 1d supermultiplets associated with the D1-brane worldvolume theory in Figure \ref{['fig:subbrane4']}. (b): The quiver diagram representing 1d supermultiplets associated with the D1-brane worldvolume theory in Figure \ref{['fig:subbrane8']}. The circle represents the $U(1)$ vector multiplet. The solid and dotted lines represent $\mathcal{N}=(4,4)$ hypermultiplets. The dashed line represents $\mathcal{N}=(0,4)$ Fermi multiplets. The number in a box indicates the number of supermultiplets represented by the line connected to the box.
  • Figure 4: (a): When NS5-branes cross the branch cuts (denoted by black dashed lines) associated with D7-branes, $(1,\pm1)$ 5-branes (depicted as four oblique blue lines) are created. Note that the semi-infinite $(1,1)$ and $(1,-1)$ 5-branes intersect at upper and lower points in the $(x^4, x^5)$ plane. (b) The improved brane configuration for monopole bubbling. Two D5-branes (denoted by two horizontal blue lines) are introduced. Due to charge conservation at the junctions, the semi-infinite 5-branes are converted into NS5-branes.
  • Figure 5: (a): The brane configuration for a dyonic loop with ${\bm p}=1$ and ${\bm q}=1,$. (b): The brane configuration for monopole bubbling with $\tilde{\bm p}=0$ in the dyonic loop.
  • ...and 1 more figures