Seminar on geometry, symmetry, and physics, Spring 2013


(aka mirror symmetry/related topics)

Mondays (usually) 1:30 pm - 2:30pm in Hill 425


Organized by Lev Borisov, Diuliu Diaconescu, and Chris Woodward

February 11 Benoit Charbonneau, St. Jerome's University in the University of Waterloo

Title: Fake G_2 instantons, singular monopoles and pairs SPECIAL PLACE: Serin E372. Abstract: In joint work with Jacques Hurtubise (McGill), we introduced some time ago a relation between singular monopoles on the product of a circle and a Riemann surface with stable pairs on the Riemann surface. In current joint work with Spiro Karigiannis and Aaron Smith (both uWaterloo), we exploit a similar approach to link G_2 instantons (and more) on the product of a circle and a Calabi--Yau 3-fold and stable pairs on the Calabi--Yau. This talk is an account of how these stories intertwine and how we failed to answer big questions about G_2-instantons and yet provide an interesting sandbox to test certain ideas in higher dimensional gauge theory.

February 18 Chris Woodward, Rutgers

Title: Quantum Kirwan and quantum Martin Abstract: I will discuss joint work with E. Gonzalez on generalizations of the Kirwan map and Martin formula for symplectic quotients, and applications to abelianization of solutions to quantum differential equations for e.g. quiver varieties.

February 25 Michael McBreen, Columbia

Title: Quantum cohomology and mirror symmetry for hypertoric varieties Abstract: Hypertoric varieties are holomorphic symplectic analogues of toric varieties. I will describe joint work with Daniel Shenfeld computing their quantum cohomology and producing a mirror formula for their quantum connection, and describe potential applications to the representation theory of the Yangian. March 4 No seminar, conference.

March 11 Howard Nuer, Rutgers

Abstract: We discuss the construction of families of Calabi-Yau 3-folds containing Enriques surfaces and their birational geometry. By considering certain 1-parameter families in the boundary of our moduli space, we find an example of a family of Calabi-Yau 3-folds exhibiting very surprising properties with regard to mirror symmetry. The first part of the talk is joint work with Lev Borisov, while the second is work-in-progress with Pat Devlin.

March 18 No seminar, spring break.

March 25 P. Georgieva, Princeton

Title: Orientability in real Gromov-Witten theory. Abstract: For a symplectic manifold M, equipped with an anti-symplectic involution, one can consider the moduli space of J-holomorphic maps from a symmetric Riemann surface to M commuting with the involutions on the domain and the target. These moduli spaces play an important role in real enumerative geometry and string theory, as seen in the works of J.-Y. Welschinger and J. Walcher. The goal of this talk is to describe what the orientability of the moduli spaces depends on, which is an essential ingredient in defining real Gromov-Witten type invariants. This is a joint work A. Zinger.

April 1, Alberto Garcia-Raboso, U. Penn.

Title: A nonabelian Hodge theorem for twisted vector bundles

April 8 A. Zinger, IAS/Stonybrook

Mirror Symmetry for Stable Quotients Invariants We describe a mirror formula for the direct analogue of Givental's J-function in the SQ theory. The mirror formula in the SQ theory is remarkably similar to that in the Gromov-Witten theory, but the former does not involve a change of variables. This suggests that the mirror map relating the GW-invariants to the B-model of the mirror is more reflective of the choice of curve counting theory on the A side than of mirror symmetry. The proof of the mirror formula in the Fano case is as in the GW-theory. On the other hand, the proof in the Calabi-Yau case consists of showing that it is a consequence of the Fano case. This is joint work with Y. Cooper.

April 15 No seminar.

April 22 Yi-Zhi Huang, Rutgers New Brunswick.

Meromorphic open-string vertex algebras, nonlinear sigma models and Witten's Dirac operators on loop spaces Abstract: Mathematicians have been searching for a construction of nonlinear sigma models for many years. Such a construction would allow us to turn the insights of physicists on the geometry and topology of Riemannian manifolds into a rigorous mathematical theory. There exist constructions of vertex operator (super)algebras from suitable manifolds in the literature. But unfortunately even the resulting algebras for flat manifolds in these constructions do not agree with the known sigma models for flat manifolds. In this talk, I will discuss a construction of modules of meromorphic open-string vertex algebras (suitable noncommutative generalizations of vertex algebras) generated from functions on Riemmannian manifolds. I believe that these modules are the basic ingredients needed in the future full construction of nonlinear sigma models. I will also discuss a recent construction of a Dirac-like operator on the loop space of a Riemmannian manifold proposed first by Witten. I will present some conjectures on the connections of the representation theory of these meromorphic open-string vertex algebras with elliptic genera, Calabi-Yau manifolds and the chiral de Rham complexes of Malikov, Schechtman and Vantrob. April 29 Sushmita Venugopalan (Tata Inst.), Introduction to pseudoholomorphic maps

May 6 No seminar, last day of class