Abstract: Donaldson-Thomas theory on Calabi-Yau 4-folds (DT4) is a complexification of Donaldson theory on 4-manifolds. In this talk, we will discuss a complexification of Donaldson-Witten TQFT. This establishes a degeneration formula of DT4 invariants and a Gromov-Witten type theory for critical loci (quivers with potentials).
Abstract: We study one-parameter families of Calabi-Yau threefolds with fourth order Picard-Fuchs equations that are Hadamard products of two second order Picard-Fuchs equations. We identify 3-cycles on the fibred product of elliptic surfaces and use the fourth order Hadamard product to identify these 3-cycles with 3-cycles on a Calabi-Yau threefold. We illustrate this in detail in several examples. Finally, we look for analogues of special Lagrangian submanifolds in the fibred product of elliptic surfaces. This leads to a construction that is reminiscient of spectral/exponential networks that have previously appeared in string theory literature.
Abstract: In this series of two talks I’ll give an overview on the Eynard-Orantin topological recursion (2007), initially motivated by random matrix models and then applied in several contexts including Enumerative Geometry (specifically Gromov-Witten theory of CY 3-folds, among other examples), on the relation between spectral curves and quantum curves, the wave and partition functions, as well as the relation with moduli spaces of curves and some more recent developments and generalisations of the theory.
Abstract: The topological string free energy provides the generating function for higher genus Gopakumar-Vafa (GV) invariants, which are related to the enumeration of holomorphic embeddings of curves in a Calabi-Yau threefold. On physical grounds one expects that the data of the GV invariants can be used to define certain Maldacena-Strominger-Witten (MSW) indices, and moreover the MSW generating function is a vector valued modular form for SL(2,Z), thus realising highly nontrivial modularity properties of the enumerative data. Recent work of Alexandrov, Feyzbakhsh, Klemm, Pioline, and Schimannek provides rigorous formulae giving precise relations between the GV and MSW indices, which require as input GV invariants of high genus, thus restricting attention to the case of hypergeometric threefolds for which the necessary higher-genus invariants are known. In this talk, we discuss ongoing work to compute MSW indices in non-hypergeometric examples, by a careful choice of examples and the computation (using the previously existing direct integration method) of novel sets of GV invariants.
Abstract: I will define the quantum trace for hyperbolic knots and how it relates to the asymptotics of the colored Jones polynomial of links.
Abstract: We propose a conjecture on the log-concavity from E-polynomials of character varieties over Riemann surfaces. Via some 'BPS calculus', we explain an idea of reducing the conjecture to a local one: log-concavity from Severi strata of a versal deformation of planar algebraic curve singularities. In the case when the singularity link is a torus knot, we verify the local conjecture via a connection to the HOMFLY-PT polynomials. Joint work in progress with Chenglong Yu + ?.
Abstract: Higher dimensional Heegaard Floer homology (HDHF) is a higher dimensional analogue of Heegaard Floer homology in dimension three. It's partly used to study contact topology in higher dimensions. In a special case, it's related to symplectic Khovanov homology. In this talk, we discuss HDHF of cotangent fibers of the cotangent bundle of an oriented surface and show that it is isomorphic to various Hecke algebras. This is a joint work with Ko Honda and Tianyu Yuan.
Abstract: Based on the work of Eynard-Orantin and Marino, the Remodeling Conjecture was proposed in the papers of Bouchard-Klemm-Marino-Pasquetti in 2007 and 2008. The Remodeling Conjecture can be viewed as an all genus open-closed mirror symmetry for toric Calabi-Yau 3-orbifolds. In this talk, I will explain an all genus mirror symmetry for descendant Gwomov-Witten nvariants of toric Calabi-Yau 3- orbifolds. The B-model is given by the Laplace transform of the Chekhov-Eynard-Orantin invariants of the mirror curve. This talk is based on ongoing joint work with Bohan Fang, Melissa Liu, and Song Yu.