A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula
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| Format: | Preprint |
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2018
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| author | Göttsche, Lothar Kool, Martijn |
| author_facet | Göttsche, Lothar Kool, Martijn |
| contents | We conjecture a formula for the virtual elliptic genera of moduli spaces of rank 2 sheaves on minimal surfaces $S$ of general type. We express our conjecture in terms of the Igusa cusp form $χ_{10}$ and Borcherds type lifts of three quasi-Jacobi forms which are all related to the Weierstrass elliptic function. We also conjecture that the generating function of virtual cobordism classes of these moduli spaces depends only on $χ(\mathcal{O}_S)$ and $K_S^2$ via two universal functions, one of which is determined by the cobordism classes of Hilbert schemes of points on $K3$. We present generalizations of these conjectures, e.g. to arbitrary surfaces with $p_g>0$ and $b_1=0$.
We use a result of J. Shen to express the virtual cobordism class in terms of descendent Donaldson invariants. In a prequel we used T. Mochizuki's formula, universality, and toric calculations to compute such Donaldson invariants in the setting of virtual $χ_y$-genera. Similar techniques allow us to verify our new conjectures in many cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1801_01878 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula Göttsche, Lothar Kool, Martijn Algebraic Geometry High Energy Physics - Theory Differential Geometry 14D20, 14D21, 14J60, 14J80, 14J81 We conjecture a formula for the virtual elliptic genera of moduli spaces of rank 2 sheaves on minimal surfaces $S$ of general type. We express our conjecture in terms of the Igusa cusp form $χ_{10}$ and Borcherds type lifts of three quasi-Jacobi forms which are all related to the Weierstrass elliptic function. We also conjecture that the generating function of virtual cobordism classes of these moduli spaces depends only on $χ(\mathcal{O}_S)$ and $K_S^2$ via two universal functions, one of which is determined by the cobordism classes of Hilbert schemes of points on $K3$. We present generalizations of these conjectures, e.g. to arbitrary surfaces with $p_g>0$ and $b_1=0$. We use a result of J. Shen to express the virtual cobordism class in terms of descendent Donaldson invariants. In a prequel we used T. Mochizuki's formula, universality, and toric calculations to compute such Donaldson invariants in the setting of virtual $χ_y$-genera. Similar techniques allow us to verify our new conjectures in many cases. |
| title | A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula |
| topic | Algebraic Geometry High Energy Physics - Theory Differential Geometry 14D20, 14D21, 14J60, 14J80, 14J81 |
| url | https://arxiv.org/abs/1801.01878 |