An introduction to the theory of Higher rank stable pairs and Virtual localization

This article is an attempt to briefly introduce some of the results from arXiv:1011.6342 on development of a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for highly frozen triples given by the data O^r-->F for r>1 where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory. Instead, we build a zero-dimensional virtual fundamental class by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. We briefly include the results of calculations in this enumerative theory for local P^1 using the Graber-Pandharipande virtual localization technique. We emphasize that in this article we merely include the statement of our theorems and illustrate the final result of some of the computations. The proofs and more detailed calculations in arXiv:1011.6342 will appear elsewhere.Comment: 11 page

Weighted Euler characteristic of the moduli space of higher rank Joyce-Song pairs

The invariants of rank 2 Joyce-Song semistable pairs over a Calabi-Yau threefold were computed in arXiv:1101.2252, using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman. Such wall-crossing computations often depend on the combinatorial properties of certain elements of a Hall-algebra (these are the stack functions defined by Joyce). These combinatorial computations become immediately complicated and hard to carry out, when studying higher rank stable pairs with rank$>2$. The main purpose of this article is to introduce an independent approach to computation of rank 2 stable pair invariants, without applying the wallcrossing formula and rather by stratifying their corresponding moduli space and directly computing the weighted Euler characteristics of the strata. This approach may similarly be used to avoid complex combinatorial wallcrossing calculations in rank$>2$ cases.Comment: Published versio

Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms

Motivated by the S-duality conjecture, we study the Donaldson-Thomas invariants of the 2 dimensional Gieseker stable sheaves on a threefold. These sheaves are supported on the fibers of a nonsingular threefold X fibered over a nonsingular curve. In the case where X is a K3 fibration, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the K3 fiber and the Noether-Lefschetz numbers of the fibration. We prove that a certain generating function of these invariants is a vector modular form of weight -3/2 as predicted in S-duality.Comment: Some corrections were made and some arguments were extended. Many thanks to the referee's helpful comments. 22 pages, to Appear in Adv. Math. (2018). arXiv admin note: text overlap with arXiv:1305.133

Stable pairs on nodal K3 fibrations

We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.Comment: Published versio

Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces

We study moduli spaces of twisted quasimaps to a hypertoric variety $X$, arising as the Higgs branch of an abelian supersymmetric gauge theory in three dimensions. These parametrise general quiver representations whose building blocks are maps between rank one sheaves on $\mathbb{P}^1$, subject to a stability condition, associated to the quiver, involving both the sheaves and the maps. We show that the singular cohomology of these moduli spaces is naturally identified with the Ext group of a pair of holonomic modules over the `quantized loop space' of $X$, which we view as a Higgs branch for a related theory with infinitely many matter fields. We construct the coulomb branch of this theory, and find that it is a periodic analogue of the coulomb branch associated to $X$. Using the formalism of symplectic duality, we derive an expression for the generating function of twisted quasimap invariants in terms of the character of a certain tilting module on the periodic coulomb branch. We give a closed formula for this generating function when $X$ arises as the abelianisation of the $N$-step flag quiver.Comment: 46 pages. Minor changes, typos fixed. Comments are very welcom