## Geometry and Physics Seminar

### General information

The Geometry and Physics seminar series is operated jointly between UAlberta and UBC. The seminars hosted by UBC are on Mondays 15:10-16:10 with coffee and cookies beforehand. The seminar hosted by UAlberta are Wednesdays 15:10-16:10. Exceptions to these dates do occur. The seminars at UBC take place in ESB 4127 (Earth Sciences Building).

### Seminars W2013

• 9/9/2013 - Hsian Hua Tseng (Ohio State University) @ UAlberta: Counting disks in toric varieties
• For a toric manifold X and a Lagrangian torus fiber L in X, Fukaya-Oh-Ohta-Ono defined open Gromov-Witten invariants which are virtual enumerations of holomorphic disks in X with boundary conditions in L. Qualitative and quantitative properties of these open Gromov-Witten invariants play important roles in the symplectic geometry and mirror symmetry of X. Open Gromov-Witten invariants are difficult to compute because standard methods in Gromov-Witten theory (such as virtual localization) are not applicable. In this talk we explain a geometric method that leads to a complete calculation of these open Gromov-Witten invariants for compact semi-Fano toric manifolds. This is joint work with K. Chan, S.-C. Lau, N. C. Leung.

• 23/9/2013 - Sabin Cautis (UBC) @ UBC: Categorical Heisenberg actions on Hilbert schemes of points [PDF]
• We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE spaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and derived categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

• 25/9/2013 - David Favero (UAlberta) @ UAlberta: Homological projective duality via variation of geometric invariant theory quotients (I)
• I will briefly Homological Projective Duality, an analog of projective duality in the setting of derived categories due to Kuznetsov. I will explain the relationship between this duality, GLSMs, and variation of geometric invariant theory quotients as part of joint work with M. Ballard, D. Deliu, U. Isik, and L. Katzarkov.

• 30/9/2013 - David Favero (UAlberta) @ UAlberta: Homological projective duality via variation of geometric invariant theory quotients (II)
• Continuation of talk on Sep. 25.

• 2/10/2013 - Simon Wood (IPMU) @ UAlberta: On the extended W-algebra of type sl_2 at positive rational level
• The extended W-algebra of type sl_2 at positive rational level is a vertex operator algebra that is of great interest in logarithmic conformal field theory. In this talk I will give an overview of how it is constructed as a subvertex operator algebra of a lattice vertex operator algebra by means of so called screening operators. I will also explain how the screening operator formalism allows one to prove c_2 cofiniteness, compute relations in Zhu's algebra and classify all simple modules of the extended W-algebra of type sl_2 at positive rational level.

• 7/10/2013 - Martijn Kool (UBC/PIMS) @ UBC: Curves on surfaces [PDF]
• The Hilbert scheme of curves in class \beta on a smooth projective surface S carries a natural virtual cycle. In many cases this cycle is zero (often when S has a holomorphic 2-form and \beta is not sub-canonical). However, in these cases one can often remove part of the obstruction bundle and obtain a non-trivial reduced virtual cycle. Both cycles have interesting applications. (1) Both are related to Pandharipande-Thomas' stable pair invariants on the total space of the canonical bundle over S. (2) The reduced virtual cycle is related to Severi degrees and classical curve counting on S. (3) The non-reduced virtual cycle is related to the Seiberg-Witten invariants of S (by work of Duerr-Kabanov-Okonek and Chang-Kiem).

• 28/10/2013 - Amin Gholampour (Maryland) @ UBC: Stable pair theory of K3 fibrations [PDF]
• I will talk about my joint work with A. Sheshmani and Y. Toda. We study the stable pair theory of K3 fibrations over curves with possibly nodal fibers. We express the stable pair invariants of the fiberwise irreducible classes in terms of the famous Kawai-Yoshioka formula for the Euler characteristics of moduli space of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the 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 of the fiberwise curve classes in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.

• 30/10/2013 - Francois Gay-Balmaz (ENS) @ UAlberta: Integrable PDEs on semisimple Lie algebras
• Motivated partly by previous work on the zero curvature representation (ZCR) of completely integrable chiral models and partly by the underlying Hamiltonian structures of ideal complex fluids, we derive systems of partial differential equations, called G-strands, that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. Using the root space decomposition, the G-strand equations can be formulated explicitly for the compact real form and the normal real form of any semisimple Lie algebra. We present several particular examples, including the exceptional group G_2. We also determine the general form of Hamilton's principles and Hamiltonians for these systems, and analyze the linear stability of their equilibrium solutions.

• 4/11/2013 - Paul Johnson (Colorado State) @ UBC: Topology and combinatorics of Hilbert schemes of points on orbifolds [PDF]
• The Hilbert scheme of n points on C^2 is a smooth manifold of dimension 2n. The topology and geometry of Hilbert schemes have important connections to physics, representation theory, and combinatorics. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology and combinatorics of the situation are well understood, but much less is known for general G. After outlining the well-understood situation, I will discuss some conjectures in the general case, and a combinatorial proof that their homology stabilizes.

• 6/11/2013 - Alan Thompson (San Diego) @ UAlberta: Families of lattice polarized K3 surfaces with monodromy
• The concept of lattice polarization for a K3 surface was first introduced by Nikulin. I will discuss ways in which his definition can be extended to families of K3 surfaces over a (not necessarily simply connected) base curve, with the aim of gaining control over the action of monodromy upon the Néron-Severi lattice of a general fibre. I will then present an application of this to the study of Calabi-Yau threefolds that admit fibrations by Kummer surfaces.

• 18/11/2013 - Dragos Oprea (San Diego) @ UBC: The Chern classes of the Verline bundle
• The Verlinde bundles over the moduli space M_g of smooth curves have as fibers spaces of generalized theta functions i.e., spaces of global sections of determinant line bundles over moduli of parabolic bundles. I will discuss a formula for the Chern classes of the Verlinde bundles, as well as extensions over the compactification \overline M_g.

• 27/11/2013 - Stefan Mendez-Diez (UAlberta) @ UAlberta: Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras
• The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincar\'{e} algebra is closely related to the study of a class of decorated graphs known as Adinkras. We will discuss how these combinatorial objects possess a form of emergent supergeometry: Adinkras are equivalent to very special super Riemann surfaces with divisors. The method of proof critically involves Grothendieck's theory of "dessins d'enfants'', work of Cimasoni-Reshetikhin expressing spin structures on Riemann surfaces via dimer models, and an observation of Donagi-Witten on parabolic structure from ramified coverings of super Riemann surfaces.

• 27/11/2013 - Michel van Garrel (Fields) @ UAlberta: Integrality of relative BPS state counts of toric Del Pezzo surfaces
• This is joint work with Tony Wong and Gjergji Zaimi. Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert and conjectured to be integers. For toric Del Pezzo surfaces, a proof of this conjecture will be presented.

• 2/12/2013 - Atsushi Kanazawa (UBC) @ UBC: BCOV theory and Calabi-Yau 3-folds with two large complex structure limits.
• In their famous paper in 1994, Bershadsky, Cecotti, Ooguri and Vafa derived a set of equations called the BCOV holomorphic anomaly equations. The BCOV theory presents a generalization of the classical g=0 mirror symmetry (Hodge theory) and is capable of computing higher genus Gromov-Witten invariants. The key ingredient is the special Kaehler geometry of the moduli space of Calabi-Yau 3-folds. In this talk, I will explain the basic idea of the BCOV theory and show some interesting computations.

• 9/12/2013 - Ziv Ran (Riverside) @ UBC: Lagrangian-like submanifolds in Sympletic-like Poisson manifolds
• The deformation theory of holomorphic symplectic manifolds and their Lagrangian submanifolds is well known to be well behaved, thanks to the influence of Hodge theory. We will report on recent work extending some of these results to certain Poisson manifolds and their Lagrangian-like submanifolds, using mixed Hodge theory.