Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity
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| Format: | Preprint |
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2018
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| _version_ | 1866908306146066432 |
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| author | Göttsche, Lothar Kool, Martijn |
| author_facet | Göttsche, Lothar Kool, Martijn |
| contents | We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfies a refined $S$-duality modularity transformation.
We provide evidence for our formula by calculating virtual $χ_y$-genera of moduli spaces of rank 3 stable sheaves on $S$ in examples using Mochizuki's formula. Further evidence is based on the recent definition of refined $\mathrm{SU}(r)$ Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank 3). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1808_03245 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity Göttsche, Lothar Kool, Martijn Algebraic Geometry High Energy Physics - Theory Differential Geometry 14D20, 14D21, 14J60, 14J80, 14J81 We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfies a refined $S$-duality modularity transformation. We provide evidence for our formula by calculating virtual $χ_y$-genera of moduli spaces of rank 3 stable sheaves on $S$ in examples using Mochizuki's formula. Further evidence is based on the recent definition of refined $\mathrm{SU}(r)$ Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank 3). |
| title | Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity |
| topic | Algebraic Geometry High Energy Physics - Theory Differential Geometry 14D20, 14D21, 14J60, 14J80, 14J81 |
| url | https://arxiv.org/abs/1808.03245 |