Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category
Abstract
We study the representation theory of the spherical double affine Hecke algebra (DAHA) of , using brane quantization. By showing a one-to-one correspondence between Lagrangian -branes with compact support and finite-dimensional representations of the spherical DAHA, we provide evidence of derived equivalence between the -brane category of -character variety of a four-punctured sphere and the representation category of DAHA of . The root system plays an essential role in understanding both the geometry and representation theory. In particular, this -model approach reveals the action of an affine braid group of type on the category. As a by-product, our geometric investigation offers detailed information about the low-energy effective dynamics of the SU(2) Seiberg-Witten theory.
Keywords
Cite
@article{arxiv.2412.19647,
title = {Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category},
author = {Junkang Huang and Satoshi Nawata and Yutai Zhang and Shutong Zhuang},
journal= {arXiv preprint arXiv:2412.19647},
year = {2026}
}
Comments
62 pages + Appendix, 30 figures, 11 tables. v2: Corrections made to the proof of Claim 4.2, typos fixed, and an additional reference included. v3: add the generalized polynomial representation, provide an explicit description of the affine braid group action incorporating the Maslov index, and include additional references