Mina Aganagic
Knot Homology from Refined Chern-Simons Theory
Abstract
We formulate a refinement of $SU(N)$ Chern-Simons theory via the refined topological string. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the $S$ and $T$ matrices of Chern-Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern-Simons theory are similar in many ways; for example, the Verlinde formula holds in both. We obtain new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the knot invariants we compute are the Poincare polynomials of the $sl(n)$ knot homology theory. The latter includes the Khovanov-Rozansky knot homology, as a special case. The conjecture passes a number of nontrivial checks. We show that, for a large number of torus knots colored with the fundamental representation of $SU(N)$, our knot invariants agree with the Poincare polynomials of Khovanov-Rozansky homology. As a byproduct, we show that our theory on $S^3$ has a dual description in term of the refined topological string on $X=\mathcal{O}(-1)\oplus \mathcal{O}(-1) \to \mathbb{P}^1$. This supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on $X$ to $sl(n)$ knot homology.
Paul Aspinwall
A Study of (0,2) Deformations of (2,2) Theories
Abstract
It has long been known that tangent bundle deformations are related to the "standard embedding" in the heterotic string. We consider issues of worldsheet instanton corrections and a kind of McKay correspondence for the tangent bundle.
David Ben-Zvi
Geometric Character Theory
Abstract
Topological field theory provides a very general viewpoint on the notions of dimension, character, trace and charge. I'll explain (based on joint work with David Nadler) how to understand results such as Riemann-Roch theorems, the Atiyah-Bott fixed point theorem, Lefschetz trace formula, character formulas of Frobenius, Weyl, and Harish-Chandra, and the Fundamental Lemma from this perspective.
Lev Borisov
Recent developments in the vertex algebra approach to toric mirror symmetry
Abstract
Vertex algebra approach to mirror symmetry aims to describe mirror symmetry at the level of N=2 vertex (chiral) algebras which arise from the half-twisted sigma models. I will describe the known results in this direction, as well as recent developments. Specifically, I will talk about the applications of vertex algebra approach to the unification of Berglund-Hubsch and Batyrev's versions of mirror symmetry. I will also describe the application of the vertex algebra approach to (0,2) sigma models on the quintic.
Peter Bouwknegt
Courant algebroids and generalizations of geometry
Abstract
In recent years there has been a flurry of interest in so-called 'generalized geometry' -- as formalized by Hitchin and his students -- motivated by its applications in String Theory. At an algebraic level this kind of generalized geometry arises from exact Courant algebroids. In this talk I will review some aspects of generalized geometry and discuss even more general geometries, including what are known in the physics literature as 'exceptional generalized geometries' arising from certain (non-exact) Courant algebroids, and Leibniz algebroids. I will also give a brief discussion of T-duality in this context.
Andrei Caldararu
Understanding brane intersections from a Lie theoretic viewpoint
Abstract
The simplest example of a D-brane is a smooth holomorphic submanifold of a smooth complex manifold. In the B-model, to compute the space of open string states between two such branes one needs to understand their intersection, understood in the sense of derived algebraic geometry. I will discuss recent results with Damien Calaque and Junwu Tu which express this derived intersection in terms of natural Lie algebroids associated to the subvarieties. Our work generalizes classic work of Kapranov and Kontsevich on a metric-free definition of Rozansky-Witten invariants.
Miranda Cheng
$M_{24}$, K3 String Theory, and Holographic Moonshine
Abstract
A theory of moonshine connects a sporadic group and a set of modular objects, and string theory has been proven crucial in the understanding of such an unexpected moonshine phenomenon. Recently, a conjectural relation has been proposed between the elliptic cohomology of K3 surfaces and the sporadic group Mathieu 24. This proposal has passed various non-trivial checks and points to a novel theory of $M_{24}$ moonshine. Moreover, via the BPS spectrum of type II string theory on $K3\times T^2$, it gets connected to another version of $M_{24}$-moonshine that has been previously proven. In this talk I will summarize the intricate web of objects with (conjectured) $M_{24}$-symmetry including cusp forms, weak Jacobi forms, Mock modular forms, automorphic forms, and a generalized Kac-Moody algebra, with the link between them provided by K3-compactified string theory.
Kevin Costello
Mirror symmetry at higher genus
Abstract
In 1994, Bershadsky, Cecotti, Ooguri and Vafa argued that one could understand the B-model topological string on a Calabi-Yau manifold X as the partition function of a certain quantum field theory on X. I'll describe joint work with Si Li, where we provide a rigorous construction of the quantum BCOV theory on certain Calabi-Yaus. I'll also explain a result of Li: the partition function of the BCOV theory on an elliptic curve recovers the Gromov-Witten theory of the mirror curve.
Emanuel Diaconescu
Stable pairs and knots
Abstract
Michael Douglas
Foundation of Quantum Field Theory
Abstract
One of the long-standing and central problems of the math-physics interface is to make good definitions of quantum field theory, which are nonperturbative and can make contact with the many nonperturbative methods used in physics, can be made rigorous if desired, and are as simple to work with as possible. We discuss questions and problems for which it is important to have such foundations, and survey some of the approaches to making them.
Dan Freed
Remarks on fully extended 3-dimensional topological field theories
Abstract
Reshetikhin and Turaev, motivated by Witten's work on Chern-Simons theory, constructed 1-2-3 field theories from modular tensor categories. In ongoing joint work with Constantin Teleman we aim to construct an invertible 4-dimensional field theory from a modular tensor category, and then use the cobordism hypothesis to construct a companion 0-1-2-3 theory which extends the Reshetikhin-Turaev theory as an anomalous field theory. I will explain these ideas in a more general context and indicate the techniques we are using.
Davide Gaiotto
BPS states in 4d and 2d
Abstract
In this talk I will review some properties of BPS states in two-dimensional systems with (2,2) supersymmetry, possibly coupled to four-dimensional N=2 gauge theories.
Sergei Gukov
Branes and Geometric Representation Theory
Abstract
Marco Gualtieri
Poisson structures in Generalized Kahler geometry
Abstract
I will describe the various real and holomorphic Poisson structures, Lie algebroids, and gerbes which feature in our current understanding of generalized Kahler geometry. Then I will describe how these structures help us to construct and understand new examples. Some examples will be constructed by birational transformations. The first part of the talk will be based on http://arxiv.org/abs/1007.3485.
Kentaro Hori
Duality In Two-Dimensional (2,2) Supersymmetric Non-Abelian Gauge Theories
Abstract
We study the low energy behaviour of N=(2,2) supersymmetric gauge theories in 1+1 dimensions, with orthogonal and symplectic gauge groups and matters in the fundamental representation. We observe supersymmetry breaking in super-Yang-Mills theory and in theories with small numbers of flavors. For larger numbers of flavors, we discover duality between regular theories with different gauge groups and matter contents, where regularity refers to absence of quantum Coulomb branch. The result is applied to study families of superconformal field theories that can be used for superstring compactifications, with corners corresponding to three-dimensional Calabi-Yau manifolds. This work is motivated by recent development in mathematics concerning equivalences of derived categories.
Sheldon Katz
(0,2) Quantum Cohomology
Abstract
In this talk, a mathematical definition is given of the topological correlation functions of a (0,2) gauged linear sigma model in the geometric phase of a neighborhood of the (2,2) locus in moduli. The geometric data determining the model is a smooth toric variety X and a deformation E of its tangent bundle TX. This definition is consistent with the known results of physics and leads to a proof of the existence of a quantum cohomology ring in complete generality, extending known results of physics. This is joint work with Ron Donagi, Josh Guffin, and Eric Sharpe.
Ludmil Katzarkov
Degenerations, dimension spectra, and wall-crossing
Abstract
Albrecht Klemm
Omega backgrounds and generalized holomorphic anomaly equation
Abstract
We derive an anomaly equation which incorporates the general Omega background in the B-model. We discuss applications to topological string theory on Calabi-Yau backgrounds and N = 2 gauge theory with massive flavors. Using geometric engineering on the Enriques Calabi-Yau we derive Seiberg-Witten curves for the conformal cases, which are compatible with Nekrasovs partition function.
Melissa Liu
Open Gromov-Witten invariants of toric Calabi-Yau 3-folds
Abstract
Open Gromov-Witten invariants count holomorphic maps from bordered Riemann surfaces to a Kahler manifold with Lagrangian boundary conditions. We will describe some conjectures and results on generating functions of open Gromov-Witten invariants of smooth toric Calabi-Yau 3-folds. This is partly based on joint work with Bohan Fang.
Matilde Marcolli
Motives in quantum field theory
Abstract
Greg Moore
Surface defects and the BPS spectrum of 4d N=2 theories
Abstract
We review how the study of line and surface defects leads to wall-crossing formula for various kinds of BPS states. We also describe techniques in a special class of 4d N=2 theories (obtained from compactification of the six dimensional (2,0) theory on a Riemann surface with punctures) that might lead to an algorithm for computing the BPS spectrum.
Andrew Neitzke
A 2d-4d wall-crossing formula
Abstract
I will state a new wall-crossing formula which arose in the study of N=2 supersymmetric field theories coupled to supersymmetric surface defects (joint work with Davide Gaiotto and Greg Moore). This wall-crossing formula is a hybrid between formulas previously written by Cecotti-Vafa and Kontsevich-Soibelman. It should play a role in a yet-undeveloped theory of "open" Donaldson-Thomas invariants. I will briefly explain some examples and applications of the formula.
Yongbin Ruan
Towards a global mirror symmetry
Abstract
During last twenty years, the mirror symmetry has been a driving force in geometry and physics. Many incredible results have been obtained in mathematics. However, a brief investigation reveals that the current form of mirror symmetry in mathematics is only a "local" form of mirror symmetry concerning about so called large complex structure limit. An interesting problem is if we can gain more information by moving away from large complex structure limit. Namely, is there an interesting theory of "global" mirror symmetry? In this talk, we will cover some of exciting developments in this direction.
Sakura Schafer-Nameki
F-theory: global aspects and phenomenology
Abstract
Paul Seidel
Families of objects in Fukaya categories
Abstract
In any category, one can try to move objects in the direction indicated by some global deformation class. An example would be to move points on a manifold in the direction of a vector field. As this example shows, the existence of global deformations is a tricky issue involving some form of compactness, but uniqueness is more robust an easier to obtain. I will apply this idea to Fukaya categories, and compare the result with known phenomena in symplectic topology.
Yuji Tachikawa
On 2d TQFTs whose values are hyperkähler cones
Abstract
We describe a class of 2d TQFTs whose values are hyperkähler cones. Namely, to a Riemann surface with boundaries is associated a hyperkähler cone with isometry $G$, and gluing corresponds to the hyperkähler quotient with respect to the diagonal $G$ action. The possible `values' include flat hyperkähler spaces and moduli spaces of $E_n$ instantons. Various properties of this TQFT are `calculable' in terms of an M-theory construction. Mathematicians are urged to construct them rigorously.
Cumrun Vafa
Complete N=2 Gauge Theories in 4 Dimensions
Abstract
We point out the power of BPS quivers in characterizing N=2 gauge theories in 4d. The notion of complete N=2 gauge theories are defined and shown to correspond to mutation finite quivers. This is then used to classify and identify the corresponding gauge theories.
Martijn Wijnholt
Higgs bundles and String Phenomenology
Abstract
String phenomenology is the branch of string theory concerned with making contact with particle physics. The original models involved compactifying ten-dimensional supersymmetric Yang-Mills theory on an internal Calabi-Yau three-fold. In recent years, this picture has been extended to compactifications of supersymmetric Yang-Mills theory in seven, eight or nine dimensions. We review some of this progress and explain the fundamental role played by Higgs bundles in this story.
Shing-Tung Yau
NonKahler Calabi-Yau manifolds
Abstract