Branes and DAHA Representations

TL;DR

This work builds a geometric bridge between the representation theory of the spherical double affine Hecke algebra $S\!H$ of type $A_1$ and the topology of Hitchin moduli spaces via brane quantization in the $A$-model. Finite-dimensional $S\!H$-modules are realized as spaces of open strings between canonical coisotropic branes and compact Lagrangian branes, yielding explicit object–morphism correspondences and new indecomposable representations. The authors situate this 2d story inside a web of higher-dimensional physics, using M-theory branes, the 3d/3d correspondence, and class $\mathcal{S}$ to access PSL$(2,\mathbb{Z})$ actions and modular data, including modular tensor categories arising from Argyres–Douglas theories and refined Chern–Simons theory. They further connect to skein modules and line operators on Coulomb branches in 4d $\mathcal N=2^*$ theories, proposing a canonical higher-rank coisotropic brane that realizes the full DAHA and establishes Morita-type equivalences between representation categories. Overall, the paper provides a coherent geometric realization of DAHA representations across 2d–4d physics, tying Hitchin geometry, modularity, and quantum algebras to concrete brane constructions and operator algebras.

Abstract

Using brane quantization, we study the representation theory of the spherical double affine Hecke algebra of type $A_1$ in terms of the topological A-model on the moduli space of flat SL(2,C)-connections on a once-punctured torus. In particular, we provide an explicit match between finite-dimensional representations and A-branes with compact support; one consequence is the discovery of new finite-dimensional indecomposable representations. We proceed to embed the A-model story in an M-theory brane construction, closely related to the one used in the 3d/3d correspondence; as a result, we identify modular tensor categories behind particular finite-dimensional representations with PSL(2,Z) action. Using a further connection to the fivebrane system for the class S construction, we go on to study the relationship of Coulomb branch geometry and algebras of line operators in 4d N=2* theories to the double affine Hecke algebra.

Figures (22)

• Figure 1: Schematic illustration of the Hitchin fibration ${\mathcal{M}}_H(C_p,\mathop{\mathrm{SU}}\nolimits(2))\to \mathcal{B}_H$ and global nilpotent cone at $\beta_p=0=\gamma_p$ and a generic value of $\talpha_p$.
• Figure 2: The Hitchin fibration with a generic ramification contains three singular fibers of Kodaira type $I_2$ at the base points $b_i$ ($i=1,2,3$).
• Figure 3: Generators and relations in the orbifold fundamental group of the once-punctured torus. On the left, generators and relations are drawn on the double cover. The relations depicted are $TXT = X^{-1}$, $TY^{-1}T = Y$, and $Y^{-1}X^{-1}YXT^2 = 1$.
• Figure 4: (Left) Open strings that start and end on the same brane $\mathfrak{B}$ 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 action of raising and lowering operators on Macdonald polynomials

Theorems & Definitions (16)

• Claim 1.1
• Conjecture 2.1
• Conjecture 2.2
• Theorem C.1: Kleppner
• Definition C.2
• Theorem C.3
• Definition C.4: cyclic representation
• Definition C.5: polynomial representation
• Proposition C.6
• proof