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arXiv · 2412.19647

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

Junkang Huang·Satoshi Nawata·Yutai Zhang·Shutong Zhuang
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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.

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Junkang Huang, Satoshi Nawata, Yutai Zhang, Shutong Zhuang. 2024-12-27. Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category. https://arxiv.org/abs/2412.19647

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