In Maulik and Thomas (in preparation) 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, extended to allow for strictly semistable sheaves. 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 (Monopole contributions to refined Vafa–Witten invariants. arXiv:1810.00385
 ) proves universality results for the invariants.

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Equivariant K-Theory and Refined Vafa-Witten Invariants

We conjecture a formula for the generating function of virtual $\chi_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C Vafa and E Witten 
These virtual $\chi_y$-genera can be written in terms of descendent Donaldson invariants Using T Mochizuki's formula, the latter can be expressed in terms of Seiberg-Witten invariants and certain explicit integrals over Hilbert schemes of points These integrals are governed by seven universal functions, which are determined by their values on $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$ Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases 
In an appendix by H Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include $\mu$-classes, thereby interpolating between Vafa-Witten's formula and Witten's conjecture for Donaldson invariants

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Virtual refinements of the Vafa-Witten formula

We conjecture a formula for the generating function of virtual $$\chi _y$$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. These virtual $$\chi _y$$-genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki’s formula, the latter can be expressed in terms of Seiberg–Witten invariants and certain explicit integrals over Hilbert schemes of points. These integrals are governed by seven universal functions, which are determined by their values on $${\mathbb {P}}^2$$ and $${\mathbb {P}}^1 \times {\mathbb {P}}^1$$. Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases. In an appendix by H. Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include $$\mu $$-classes, thereby interpolating between Vafa–Witten’s formula and Witten’s conjecture for Donaldson invariants.

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Virtual Refinements of the Vafa-Witten Formula

We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3 surfaces and surfaces of general type with connections to the Kawai-Yoshioka formula and the Seiberg-Witten invariants respectively. The dimension 1 theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss-Catalan numbers which is of independent interest.

Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics

The virtual K-theory of Quot schemes of surfaces

For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q q symmetry of the Calabi-Yau case. We prove rationality for nonsingular projective toric 3-folds by the theory of descendents.

Junliang Shen

Papers

Cobordism invariants of the moduli space of stable pairs

Cobordism invariants of the moduli space of stable pairs