Mohammed Abouzaid
Homological Mirror Symmetry for T4.
Abstract
I shall explain how the pseudo-holomorphic quilt techniques of Wehrheim and Woodward yield of proof of Homological mirror symmetry for products, assuming mirror symmetry is known for both factors, and stringent conditions are imposed on the behaviour of holomorphic curves. The current conditions are so stringent that the only application is a proof of Homological mirror symmetry for T4. An application to Lagrangian embeddings in T4 will be deduced. This work was done jointly with Ivan Smith.
Paul Aspinwall
Matrix factorizations and D-branes.
Paul Aspinwall
Probing Geometry with D-Branes and Stability.
Abstract
The notion that the geometry of spacetime is given by the moduli space of 0-branes is examined in some examples of Calabi-Yau threefolds. An important consideration when determining the moduli space of D-branes is the stability condition and this is key in our analysis. I will consider a hybrid model of a Landau--Ginzburg fibration over a CP1. Using the technology of matrix factorizations we find a D-brane probe whose moduli space is this CP1 but it is not a 0-brane and is not stable at the large radius limit of the Calabi-Yau manifold. If time permits I will also consider an exoflop where the linear sigma model implies a CP1 external to the Calabi-Yau threefold is part of the geometry. The 0-brane probe sees no such external CP1 and furthermore exhibits a surprising discontinuity when following an extremal transition associated to the exoflop.
Jonathan Block
Aspects of noncommutative geometry.
Abstract
Noncommutative geometry means different things to different people. We describe some of these different approaches with an emphasis on examples, phenomena and interesting problems in the subject.
Andrei Caldararu
A conjecture of Duflo and the Ext algebra of branes.
Abstract
The Duflo theorem is a statement in Lie theory which allows us to compute the ring structure of the center of the universal enveloping algebra of a finite-dimensional Lie algebra. A categorical version of it was used by Maxim Kontsevich to give a spectacular proof of the so-called "Theorem on complex manifolds," which computes the multiplicative structure of Hochschild cohomology of a complex manifold in terms of the algebra of polyvector fields. In Lie theory the multiplicative structure of Hochschild cohomology of a complex manifold in terms of the algebra of polyvector fields. In Lie theory there are also more general Duflo-type statements (mostly conjectural), which study the case of a pair (Lie algebra, Lie subalgebra). I will explain how these translate into conjectures about the multiplicative structure of the Ext-algebra of the structure sheaf of a complex submanifold of a complex manifold, and how from this interaction we can hope to gain new insights into both algebraic geometry and Lie theory. (Based on discussions with Damien Callaque.)
Emanuel Diaconescu
Chamber structure and wall-crossing for ADHM invariants.
Abstract
A construction of ADHM invariants of curves will be presented depending on a real stability parameter. These are residual invariants defined by equivariant virtual integration on certain moduli spaces of quiver sheaves. It will be shown that wallcrossing formulas can be derived using recent results of Joyce and Song. Applications to local stable pair theory and local BPS invariants will also be discussed.
Ron Donagi
F-theory Compactifications.
Abstract
This will be an introduction for mathematicians to F-theory, a "12 dimensional variant of string theory", and its global and local compactifications. The past year has seen tremendous progress towards F-theory based phenomenology. I will discuss the issue of local versus global in F-theory (and strings), and explore connections to the geometry of del Pezzo surfaces, Higgs bundles, and Noether-Lefschetz loci.
John Francis
En-geometry and topological field theory.
Abstract
En-geometry, for 1 < n < ∞, is a form of less commutative geometry, interpolating, with n, between noncommutative and commutative derived algebraic geometry. Certain structural features of topological quantum field theories lead to consideration of En--geometry. I will discuss how one may try to construct TQFTs using En--geometry, with starting point a complex symplectic manifold
Sheldon Katz
D-branes and BPS invariants of Calabi-Yau threefolds.
Andrew Neitzke
Hyperkahler geometry and BPS states.
Abstract
I will describe some recent joint work with Davide Gaiotto and Greg Moore, in which we explain the origin of the wall-crossing formula of Kontsevich and Soibelman, in the context of N=2 supersymmetric field theories in four dimensions. The wall-crossing formula gives a recipe for constructing the smooth hyperkahler metric on the moduli space of the field theory reduced on a circle to 3 dimensions. In certain examples this moduli space is actually a moduli space of ramified Higgs bundles, so we obtain a new description of the hyperkahler structure on that space.
Masa-Hiko Saito
Deligne-Hitchin-Simpson twistor spaces and degenerations of Painlevé equations.
Abstract
In this talk, I will discuss Deligne-Hitchin-Simpson's twistor space related to the moduli space of λ-connections on a smooth projective curve with regular or irregular singularities of fixed (generalized) local exponents. All known examples of Painlevé equations can be obtained by isomonodromic deformations of linear connections with singularities. Therefore the phase spaces of Painlevé equations are given by families of moduli spaces of connections with singularities of fixed generalized local exponents. (I will briefly review about the classical Painlevé equations and the Riemann-Hilbert correspondences related to them.) By using the explicit description of moduli spaces of λ-connections, we can explain how these Painlevé equations are degenerating to the Hitchin's integrable systems in the twistor spaces. As an example, we will give an explicit calculation of the degeneration of Painlevé VI equations to Hitchin system.
Michael Schulz
Nongeometric compactifications, T-folds, and doubled geometry
Abstract
Nongeometric string theory compactifications do not have well defined compactification manifolds in the usual sense. The simplest examples are T-folds: generalizations of torus fibrations in which the transition functions are in the T-duality group of the fiber rather than its geometric subgroup. I will begin by motivating the T-fold construction in the context of a simple example, via T-duality to more conventional compactifications, and then provide another example with no such geometric dual. A useful geometric decription of T-folds is via a doubled fibration that includes both physical and dual fibers, or equivalently the spaces seen simultaneously by left plus right moving fields. It is natural to try to extend this description and basic construction to a fully doubled geometry. A motivation for doing so is that smooth compactifications on special holonomy manifolds yield supergravities with abelian gauge symmetry and no charged matter, while their nongeometric analogs would yield the most general nonabelian gaugings. In the latter part of the talk, I will describe insights toward this goal, as well as open problems.
Eric Sharpe
The topological A, B models from a physicists' perspective - a minicourse.
Abstract for the series
In these three lectures, my goal is to give a very fast and elementary introduction to some basic aspects of quantum field theory and how the A, B model topological field theories are built by physicists as special quantum field theories. In the first lecture, I'll give a quick introduction to quantum field theory, outlining where Feynman diagrams come from, and the idea of the renormalization group. In the second and third lectures, I'll specialize to nonlinear sigma models, talk about their definition, perturbation theory, and supersymmetric versions thereof, and then outline the A, B model topological field theories as twisted supersymmetric nonlinear sigma models. If time remains, I may briefly outline the open string B model and how derived categories arise in it.
Johannes Walcher
The real topological string on a local Calabi-Yau.
Abstract
The topological string becomes "real" when studied in orientifold backgrounds with D-branes on top of the orientifold plane, ie, at the fixed point locus of an anti-holomorphic involution (in the A-model). I will explain these notions, and present the solution of the real topological string on local P2 via the real topological vertex.
Katrin Wendland
The geometry of conformal field theory.
Abstract
This talk is meant to give an introduction and overview on the theme of conformal field theories and their geometric aspects. We start with a brief review of approaches to conformal field theory, both axiomatically and motivated from geometry by means of the chiral de Rham complex. Recalling toroidal conformal field theories, we will find examples of conformal field theories where the geometric content is particularly clear. We explain how these theories can be used to obtain explicit constructions of conformal field theories along with concrete geometric interpretations on more complicated geometries, like K3 surfaces and certain Calabi-Yau threefolds.