Abstract: Computing the L2 cohomology of moduli spaces of monopoles and instantons is a challenging problem. It is significant in physics having an interpretations as counting the BPS states in quantum gauge theories, as well as in mathematics, manifesting itself in the geometric Langlands correspondence for complex surfaces. We propose a rather unconventional compactification of these moduli spaces … Read More
Abstract: The minicourse will be comprised of three talks, which can be summarized by the slogan “Donaldson-Thomas theory of Surface x Curve = Gromov-Witten theory of moduli spaces of sheaves on Surface”.
In the second talk, using quasimaps, we establish a very general wall-crossing formula relating Donaldson-Thomas theory of Surface x Curve and Gromov-Witten theory of moduli spaces of sheaves on Surface. The wall-crossing is governed by Vertex (also known as I-function). We then discuss the old and the new instances of this wall-crossing.
Abstract : I will report an on-going joint project with Hanany and Finkelberg. We identify Coulomb branches of orthosymplectic quiver gauge theories with orthosymplectic bow varieties. Then we use this identification to realize closures of nilpotent orbits for SO, and more as Coulomb branches.
In the third talk, we discuss how quasimaps can be used to translate S-duality of Vafa-Witten invariants of Curve x Curve’ to a duality of quasimap invariants of moduli spaces of Higgs bundles on Curve. This provides an enumerative realization of Kapustin-Witten’s dimensional reduction in the case of SL and PGL.
Abstract : I will report an on-going joint project with Hanany and Finkelberg. We identify Coulomb branches of orthosymplectic quiver gauge theories with orthosymplectic bow varieties. Then we use this identification to realize closures of nilpotent orbits for SO, and more as Coulomb branches. (continuation).
Abstract: Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of C^3, when the Donaldson-Thomas (DT) invariant of length d sheaves is p(d), the number of plane partitions of d. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies … Read More
Abstract: Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of C^3, when the Donaldson-Thomas (DT) invariant of length d sheaves is p(d), the number of plane partitions of d. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies … Read More
I will give an update on the dictionary between the representations of quantum toroidal algebras and branes of Type IIB string theory. I will argue that brane crossings correspond to R-matrices, of which the “degenerate resolved conifold” is one example. A more interesting example is given by the “Hanany-Witten R-matrix”, for which the brane creation effect and Hanany-Witten rules become … Read More
In a few cases, for a quasiprojective suface S it is known that there are vertex algebras which provide formulas for enumerative invariants of moduli spaces of sheaves on S. I will explain a number of examples where this is known or expected to hold and focus on extending this structure to the case of elliptic surfaces. I will describe … Read More