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Register a new userTwisted sheaves and $mathrm{SU}(r) / mathbb{Z}_r$ Vafa-Witten theory
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The $mathrm{SU}(r)$ Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface $S$, was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $mu_r$-gerbes. In this paper, we instead use Yoshiokas moduli spaces of twisted sheaves. Using Chern character twisted by rational $B$-field, we give a new mathematical definition of the $mathrm{SU}(r) / mathbb{Z}_r$ Vafa-Witten partition function when $r$ is prime. Our definition uses the period-index theorem of de Jong. $S$-duality, a concept from physics, predicts that the $mathrm{SU}(r)$ and $mathrm{SU}(r) / mathbb{Z}_r$ partitions functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all $K3$ surfaces and prime numbers $r$.
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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 $chi_y$-genera of moduli spaces of rank 3 stable sheaves on $S$ in examples using Mochizukis 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).
This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.
In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented $mathbb C^*$-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in $t^{1/2}$ invariant under $t^{1/2}leftrightarrow t^{-1/2}$ which specialise to numerical Vafa-Witten invariants at $t=1$. On the instanton branch the invariants give the virtual $chi_{-t}^{}$-genus refinement of Gottsche-Kool. Applying modularity to their calculations gives predictions for the contribution of the monopole branch. We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of Gottsche-Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon-Northcott complexes, and show they calculate refined Vafa-Witten invariants. Using this Laarakker [Laa] proves universality results for the invariants.
We propose a definition of Vafa-Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce-Song pairs. For $K_Sle0$ we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for deg $K_S<0$ here, and it is proved for $S$ a K3 surface in cite{MT}. For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.
This article describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair $(mathcal{E}, varphi)$, where $mathcal{E}$ is a locally-free sheaf over a surface $X$ and $varphi$ is a section of $text{End} (mathcal{E}) otimes K_{X}$; and the existence of a solution to certain gauge-theoretic equations, the Vafa-Witten equations, for a Hermitian metric on $mathcal{E}$. It turns out to be a special case of results obtained by Alvarez-Consul and Garcia-Prada. In this article, we give an alternative proof which uses a Mehta-Ramanathan style argument originally developed by Donaldson for the Hermitian-Einstein problem, as it relates the subject with the Hitchin equations on Riemann surfaces, and surely indicates a similar proof of the existence of a solution under the assumption of stability for the Donaldson-Thomas instanton equations described in arXiv:0805.2192 on smooth projective threefolds; and more broadly that for the quiver vortex equation on higher dimensional smooth projective varieties.
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