Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category

TL;DR

This work ties together the representation theory of the spherical DAHA of type $C^\vee C_1$ with brane quantization on the SL(2,ℂ) character variety of a four-punctured sphere. By establishing a correspondence between compact Lagrangian $A$-branes and finite-dimensional spherical-DAHA modules, it provides strong evidence for a derived equivalence between the $A$-brane category and DAHA representations, organized by the $D_4$ root system. An affine $D_4$ braid-group action emerges on the brane/representation categories, realized via wall-crossing and monodromy on the Hitchin moduli space. The analysis connects to Seiberg–Witten theory with $N_f=4$, Hitchin fibrations, and Kodaira singular fibers, offering a geometric and categorical framework for understanding DAHA representations and their symmetries. Overall, the paper advances a deep, structure-rich bridge between algebra, geometry, and high-energy physics, with potential implications for boundary conditions in quantum field theories and geometric representation theory.

Abstract

We study the representation theory of the spherical double affine Hecke algebra (DAHA) of $C^\vee C_1$, using brane quantization. By showing a one-to-one correspondence between Lagrangian $A$-branes with compact support and finite-dimensional representations of the spherical DAHA, we provide evidence of derived equivalence between the $A$-brane category of $\mathrm{SL}(2,\mathbb{C})$-character variety of a four-punctured sphere and the representation category of DAHA of $C^\vee C_1$. The $D_4$ root system plays an essential role in understanding both the geometry and representation theory. In particular, this $A$-model approach reveals the action of an affine braid group of type $D_4$ on the category. As a by-product, our geometric investigation offers detailed information about the low-energy effective dynamics of the SU(2) $N_f=4$ Seiberg-Witten theory.

Figures (30)

• Figure 1: (Left) Two sets of generators of DAHA of type $C^\vee C_1$ presented by the fundamental group of a fourth-punctured sphere. (Right) Braid group action of $B_3$ on the generators of DAHA.
• Figure 2: Generators of the fundamental group of a four-punctured sphere and the generators of spherical DAHA of type $C^\vee C_1$.
• Figure 3: An algebra of line operators (colored circles) in a 4d $\mathcal{N}=2$ theory becomes non-commutative in the $\Omega$-background $S^1 \times \mathbb{R} \times_{q} \mathbb{R}^2$, which provides deformation quantization of holomorphic coordinate ring of the Coulomb branch. The 4d $\mathcal{N}=2$ theory compactified on $S^1 \times S_q^1$ is described by 2d $A$-model $\Sigma \rightarrow \mathcal{M}_C$ on the Coulomb branch where the boundary condition at $\partial \Sigma$ is given by $\mathfrak{B}_\text{cc}$. Here $\mathbb{R}^2 \supset S_q^1$ is the circle generating the $\Omega$-deformation.
• Figure 4: (Left) Open strings that start and end on the same brane $\mathfrak{B}_\text{cc}$ form an algebra. (Right) Joining a $(\mathfrak{B}_\text{cc},\mathfrak{B}_\text{cc})$-string with a $(\mathfrak{B}_\text{cc},\mathfrak{B}')$-string leads to another $(\mathfrak{B}_\text{cc},\mathfrak{B}')$-string.
• Figure 5: The schematic figure of the Hitchin fibration ${\mathcal{M}}_H\to \mathcal{B}_H$ when the ramification parameters $\talpha_j$ are generic, and the others are zero $\tbeta_j=\tgamma_j=0$, $(j=1,2,3,4)$. A generic fiber $\mathbf{F}$ is topologically a two-torus, and the global nilpotent cone $\mathbf{N}$ at the origin of the Hitchin base $\mathcal{B}_H$ is a singular fiber of Kodaira type $I_0^*$, which is illustrated on the right.

Theorems & Definitions (6)

• Claim 1.1
• Claim 1.2
• Claim 1.3
• Claim 4.1
• Claim 4.2
• proof