# Seminar on geometry, symmetry, and physics, Spring 2019

(aka mirror symmetry/related topics)

Thursdays (usually) 1:00-2:00 pm in Serin Lab E372

Organized by Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward

On the BKMP Remodeling Conjecture for toric Calabi-Yau 3-orbifolds Zhengyu Zong, Tsinghua University

Location: Serin E372

Date & time: Thursday, 31 January 2019 at 1:30PM - 2:30PM

Abstract: The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti (BKMP) relates the all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this talk, I will talk about the proof of the Remodeling Conjecture in arXiv: 1604.07123 which is a joint work with Bohan Fang and Melissa Liu. The key idea of the proof is to realize both A-model and B-model higher genus potentials as quantizations of two isomorphic semi-simple Frobenius structures.

Thursday April 4: Florian Beck, University of Hamburg

Title: Hyperholomorphic line bundles and energy functionals

Abstract: Hyperkähler (HK) manifolds have attracted a lot of attention both by mathematicians and physicists due to their rich and intricate structure. Most prominently, they come equipped with three complex structures satisfying the quaternionic relations. Even though an HK manifold M is a differential-geometric object, the HK structure can be encoded in one complex-analytic object, the twistor space Z of M. By construction, it fibers over the complex line and the sections of this fibration provide the link to the HK structure on M. In this talk, we pursue and extend this complex-analytic approach for HK manifolds that admit an appropriate circle action. For each such HK manifold M, Haydys constructed a hyperholomorphic line bundle, i.e. it is holomorphic with respect to each of the three complex structures on M. In turn, this induces a holomorphic line bundle L on Z. By employing L, we define an energy functional on the space of sections of Z. We show that it is a natural complex-analytic extension of the moment map of the circle action on M. This gives new insights on the space of sections of Z if M is the moduli space of solutions to Hitchin's self-duality equations.

Monday, April 8, 2019

Laura Schaposnik, SUNY Stonybrook

Special Room: Physics & Astronomy - 385E

Special Time: 03:30:00PM - Monday 08 April 2019

Title: Geometric correspondances between singular fibres of the Hitchin fibration

Abstract: Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle, and their moduli spaces carry a natural hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention both within mathematics and string theory. In this talk we shall consider novel correspondences between branes lying completely within the singular fibres of the Hitchin fibration, which can be understood through group isomorphisms.

Thursday April 11: Francesco Sala, IPMU

Title: Categorification of 2d cohomological Hall algebras

Abstract: Let $\mathcal{M}$ denote the moduli stack of either coherent sheaves on a smooth projective surface or Higgs sheaves on a smooth projective curve $X$. The convolution algebra structure on the Borel-Moore homology of $\mathcal{M}$ is an instance of two-dimensional cohomological Hall algebras.

In the present talk, I will describe a full categorification of the cohomological Hall algebra of $\mathcal{M}$. This is achieved by exhibiting a derived enhancement of $\mathcal{M}$. Furthermore, this method applies also to several other moduli stacks, such as the moduli stack of vector bundles with flat connections on $X$ and the moduli stack of finite-dimensional representations of the fundamental group of $X$. In the second part of the talk, I will focus on the case of curves and discuss some relations between the Betti, de Rham, and Dolbeaut categorified cohomological Hall algebras. This is based on a paper with Mauro Porta ( arXiv:1903.07253).