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

TL;DR

This paper explores the representation theory of the spherical DAHA of type C^, establishing a correspondence with brane categories and revealing affine braid group actions, with implications for Seiberg-Witten theory.

Contribution

It demonstrates a derived equivalence between brane categories of character varieties and DAHA representations, highlighting the role of the D4 root system and affine braid group actions.

Findings

Established a one-to-one correspondence between Lagrangian A-branes and DAHA representations.

Revealed an affine braid group action of type D4 on the category.

Provided geometric insights into the low-energy dynamics of SU(2) N_f=4 Seiberg-Witten 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.