Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910108415426560 |
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| author | Huang, Junkang Nawata, Satoshi Zhang, Yutai Zhuang, Shutong |
| author_facet | Huang, Junkang Nawata, Satoshi Zhang, Yutai Zhuang, Shutong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19647 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category Huang, Junkang Nawata, Satoshi Zhang, Yutai Zhuang, Shutong High Energy Physics - Theory Mathematical Physics Algebraic Geometry Representation Theory Symplectic Geometry 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. |
| title | Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category |
| topic | High Energy Physics - Theory Mathematical Physics Algebraic Geometry Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2412.19647 |