Wall-crossing and invariants of higher rank Joyce-Song stable pairs

We introduce a higher rank analog of the Joyce-Song theory of stable pairs. Given a nonsingular projective Calabi-Yau threefold $X$, we define the higher rank Joyce-Song pairs given by ${O}^{r}_{X}(-n)\rightarrow F$ where $F$ is a pure coherent sheaf with one dimensional support, $r>1$ and $n\gg 0$ is a fixed integer. We equip the higher rank pairs with a Joyce-Song stability condition and compute their associated invariants using the wallcrossing techniques in the category of weakly semistable objects.Comment: Added "Joyce-Song" to the title. Revised version according to referee's corrections, 26 pages, Illinois. J. Math., Vol 59, 1, 55-83, (2016

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

Super Gromov-Witten Invariants via torus localization

In this article we propose a definition of super Gromov-Witten invariants by postulating a torus localization property for the odd directions of the moduli spaces of super stable maps and super stable curves of genus zero. That is, we define super Gromov-Witten invariants as the integral over the pullback of homology classes along the evaluation maps divided by the equivariant Euler class of the normal bundle of the embedding of the moduli space of stable spin maps into the moduli space of super stable maps. This definition sidesteps the difficulties of defining a supergeometric intersection theory and works with classical intersection theory only. The properties of the normal bundles, known from the differential geometric construction of the moduli space of super stable maps, imply that super Gromov-Witten invariants satisfy a generalization of Kontsevich-Manin axioms and allow for the construction of a super small quantum cohomology ring. We describe a method to calculate super Gromov-Witten invariants of $\mathbb{P}^n$ of genus zero by a further geometric torus localization and give explicit numbers in degree one when dimension and number of marked points are small