Tag - Mirror symmetry

Yoel Groman: Wrapped Floer theory and homological mirror symmetry for toric Calabi-Yau manifolds

9th October, 2017

Consider a Lagrangian torus fibration à la SYZ over a non-compact base. Using techniques from this arXiv paper, I will discuss the construction of wrapped Floer theory in this setting. Note that this setting is generally not exact even near infinity. The construction allows the formulation of a version of the homological mirror symmetry conjecture for open manifolds which are not exact near infinity. According to time constraints, I will apply this to prove homological mirror symmetry in the case where the A-model is the complement of an anti-canonical divisor in a toric Calabi Yau manifold.

Tristan Collins: The deformed Hermitian-Yang-Mills equation

2nd May, 2017

Mirror symmetry predicts that the moduli space of complex structures/special Lagrangians on one Calabi-Yau is dual to the moduli space of complexified forms/stable bundles on the mirror Calabi-Yau. However, the precise definition of a complexified Kähler form/stable bundle has remained mysterious. I will discuss these notions in the setting of Strominger-Yau-Zaslow mirror symmetry, the connection to fully non-linear PDEs and algebro-geometric stability.

Denis Auroux: Speculations about homological mirror symmetry for affine hypersurfaces

2nd May, 2017

The wrapped Fukaya category of an algebraic hypersurface H in (ℂ*)n is conjecturally related via homological mirror symmetry to the derived category of singularities of a toric Calabi-Yau manifold X, whose moment polytope is determined by the tropicalization of H. In this talk we will first explain the statement, and illustrate it for the case of the pair of pants; then we will outline some more speculative ideas about "relative" homological mirror symmetry for pairs ((ℂ*)n, H) and wrapped Fukaya categories of higher-dimensional pairs of pants.

Si Li: Vertex algebras, quantum master equation and mirror symmetry

30th April, 2017

We develop the effective Batalin-Vilkovisky quantization theory for chiral deformation of 2-dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. As an application, we explain a universal approach to KdV type integrable hierarchies via B-twisted topological string field theory. This leads to an exact solution of quantum B-model (BCOV theory) in complex one dimension that solves the higher genus mirror symmetry conjecture on elliptic curves.

Chiu-Chu Melissa Liu: Mirror Symmetry and Topological Recursion

28th April, 2017

The Remodelling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti provides a precise correspondence between open-closed Gromov-Witten invariants of a symplectic toric Calabi-Yau threefold and the invariants of the mirror curve defined by Eynard-Orantin topological recursion. It can be viewed as a version of all genus open-closed mirror symmetry. I will present a proof of the conjecture and describe its implications on the structure of higher genus Gromov-Witten invariants, based on joint work with Bohan Fang and Zhengyu Zong.

Ailsa Keating: Homological Mirror Symmetry for singularities of type Tpqr

10th March, 2016

We present some homological mirror symmetry statements for the singularities of type Tp,q,r. Loosely, these are one level of complexity up from so-called 'simple' singularities, of types A, D and E. We will consider some symplectic invariants of the real 4-dimensional Milnor fibres of these singularities, and explain how they correspond to coherent sheaves on certain blow-ups of the projective space P2, as suggested notably by Gross-Hacking-Keel. We hope to emphasize how the relations between different 'flavours' of invariants (e.g., versions of the Fukaya category) match up on both sides.

James Pascaleff: Equivariant structures in mirror symmetry

17th October, 2014

When a variety X is equipped with the action of an algebraic group G, it is natural to study the G-equivariant vector bundles or coherent sheaves on X. When X furthermore has a mirror partner Y, one can ask for the corresponding notion of equivariance in the symplectic geometry of Y. The infinitesimal notion (equivariance for a single vector field) was introduced by Seidel and Solomon (GAFA 22 no. 2), and it involves identifying a vector field with a particular element in symplectic cohomology. I will describe the analogous situation for a Lie algebra of vector fields, and discuss the application of this theory to mirror symmetry of flag varieties. In this situation, we expect to find a close connection to the canonical bases of Gross-Hacking-Keel.

Mohammed Abouzaid: On Floer cohomology and non-archimedian geometry

14th February, 2014

Ideas of Kontsevich-Soibelman and Fukaya indicate that there is a natural rigid analytic space (the mirror) associated to a symplectic manifold equipped with a Lagrangian torus fibration. I will explain a construction which associates to a Lagrangian submanifold a sheaf on this space, and explain how this should be the mirror functor.

Tim Perutz: Calabi-Yau mirror symmetry: from categories to curve-counts

15th November, 2013

I will report on joint work with Nick Sheridan concerning structural aspects of mirror symmetry for Calabi-Yau manifolds. We show (i) that Kontsevich's homological mirror symmetry (HMS) conjecture is a consequence of a fragment of the same conjecture which we expect to be much more amenable to proof; and, in ongoing work, (ii) that from HMS one can deduce (some of) the expected equalities between genus-zero Gromov-Witten invariants of a CY manifold and the Yukawa couplings of its mirror.