Related papers: Refined sheaf counting on local K3 surfaces

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…

We conjecture an explicit formula for the $K$-theoretically refined Vafa-Witten invariants of the Enriques surface. By a wall-crossing argument the conjecture is equivalent to a new conjectural formula for the K-theoretically refined…

There are two natural ways to count stable pairs or Joyce-Song pairs on $X=\mathrm{K3}\times\mathbb C$; one via weighted Euler characteristic and the other by virtual localisation of the reduced virtual class. Since $X$ is noncompact these…

Using the multiple cover formula of Y. Toda for counting invariants of semistable twisted sheaves over twisted local K3 surfaces we calculate the $\SU(r)/\zz_r$-Vafa-Witten invariants for K3 surfaces for any rank $r$ for the Langlands dual…

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 $K_S\le0$ we expect our definition coincides with an…

We study the equivariant sheaf counting theory on K3 surfaces with finite group actions. Let $\sS=[S/G]$ be a global quotient stack, where $S$ is a K3 surface and $G$ is a finite group acting as symplectic homomorphisms on $S$. We show that…

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds,…

For any smooth complex projective surface $S$, we construct semistable refined Vafa-Witten invariants of $S$ which prove the main conjecture of arXiv:1810.00078. This is done by extending part of Joyce's universal wall-crossing formalism to…

We consider three different families of Vafa-Witten invariants of $K3$ surfaces. In each case, the partition function that encodes the Vafa-Witten invariants is given by combinations of twisted Dedekind $\eta$-functions. By utilising known…

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local K3 surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai-Yoshioka's formula for stable pairs with…

Supersymmetric D-branes supported on the complex two-dimensional base $S$ of the local Calabi-Yau threefold $K_S$ are described by semi-stable coherent sheaves on $S$. Under suitable conditions, the BPS indices counting these objects (known…

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…

We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces.…

We compute the sheaf cohomology with constant $\mathbb{Z}_2$ coefficients of a concrete class of locally profinite sets of independent interest. We introduce $k$-sheer partitions to aid in constructions. It is also shown that questions of…

We provide a definition of Vafa-Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the Vafa-Witten invariants for projective surfaces inspired by the S-duality conjecture. We…

We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories…

We define and compute higher rank analogs of Pandharipande-Thomas stable pair invariants in primitive classes for K3 surfaces. Higher rank stable pair invariants for Calabi-Yau threefolds have been defined by Sheshmani \cite{shesh1,shesh2}…

We study the reduced descendent Gromov-Witten theory of K3 surfaces in primitive curve classes. We present a conjectural closed formula for the stationary theory, which generalizes the Bryan-Leung formula. We also prove a new recursion that…

We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving the holomorphic volume form. We define integers $n_{g,h}(\beta) $ which give a virtual count…

By showing a structural result on how the Rozansky-Witten invariants of Hilbert schemes of points on a K3 surface and the Generalised Kummer varieties depend on the dimension, we give a method how to compute a large class of Rozansky-Witten…