GAP XI — Pittsburgh
Schedule and Abstracts
Schedule and Abstracts
Homotopy theory with chain complexes
Differential spectra
Classical and non-classical examples
The goal of the mini course is to present differential cohomology theories as refinements of cohomology theories applied to smooth manifolds. A differential cohomology theory will be given by a sheaf of spectra on the category smooth manifold which extends its underlying homotopy invariant sheaf by a sheaf of spectra with an HR-module structure.
We will start with explaining the main homotopy theoretic notions in the case of chain complexes. The main example here is smooth Deligne cohomology.
We then move to spectra in order to capture the general cohomology theories. We discuss the construction and classification of differential extensions of a cohomology theory. We will place the classical examples of differential cohomology theories in this new framework.
In the final talk I plan to discuss a new version of differential K-theory which is universal for capturing invariants for vector bundles with connections.
I will begin by giving some background about string topology, and in particular I will describe the topological fietd theory that has been produced with the chains of the loop space of a manifold, C∗(LM) as the state space. We refer to this underlying DGA (actually ring spectrum) as the “string topology spectrum of M,” S(M). More generally, given any principal bundle G→P→M, there is an analogous string topology spectrum S(P). In the universal case (when P is contractible) one recovers C∗(LM)=S(M). One of our main results is to identify the group of units of S(P), and to relate it to the gauge group of the original bundle, G(P). We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections, to the setting of “DGA - line bundles over a manifold” and show how it allows us to do explicit calculations. We also show how, in the universal case, an action of a Lie group on a manifold yields a representation of the loop group on the string topology spectrum. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum S(P) of a principal bundle is the “linearization” of the gauge group G(P).
Factorization algebras encode the structure of observables in perturbative quantum field theory. The aim of these lectures will be to develop a refined version of Noether's theorem in the language of factorization algebras, and to explain how this theorem leads naturally to a local version of the index theorem. I'll start at a quite basic level, by describing explicitly the factorization algebra associated to a free field theory. These lectures are based on joint work with Owen Gwilliam.
Deformation quantization is a procedure which assigns a formal deformation of the associative algebra of functions on a variety to a Poisson structure on this variety. Such a procedure can be obtained from Kontsevich's formality quasi-isomorphism and, it is known that, there are many homotopy inequivalent formality quasi-isomorphisms. I propose a framework in which all homotopy classes of formality quasi-isomorphisms can be described. This description is related to very interesting objects: Kontsevich's graph complex and the Grothendieck-Teichmueller group introduced in 1990 by Vladimir Drinfeld. I plan to devote a good portion of my talk to these objects.
A notion of relative quantum field theory has been recently porposed by Freed and Teleman within the context of 3-tier extended topological field theories. The aim of this talk is to show how, from the point of view of fully extended tqfts, Freed-Teleman notion of relative quantum field theory is naturally recovered as a boundary TQFT and relative fields configurations are identified with Sati-Schreiber-Stasheff notion of twisted G-structures.
I want to explain a way to find a system of virtual fundamental chains (in de Rham theory) of the moduli space of constant maps from bordered Riemann surface to a (cotangent bundle of a) manifold. They should be
leading order term of Lagrangian Floer theory of arbitrary genus
related to perturbative Chern Simons theory
de Rham version of string topology.
A point which is new (and not in my previous talks in various places) is now we can include the case when there is no marked point on certain connected component of the boundary.
We will describe centralizers of En-algebras in terms of factorization algebras and will discuss some applications.
I will give an overview of how to compute obstructions to BV quantization, as it appears in Costello's approach. The goal will be to explain how this machinery recovers well-known anomalies for some 2D conformal field theories.
I will report on a recent joint work with L. Katzarkov and M. Kontsevich on the smoothness and the existence of special coordinates on the moduli space of Landau-Ginzburg models. I will describe a Tian-Todorov theorem for the deformations of Landau-Ginzburg models and will explain the new Hodge theoretic statements needed in the proof. I will also discuss the various definitions of Hodge numbers for non-commutative Hodge structures of Landau-Ginzburg type and the role they play in mirror symmetry.
The first part of the talk indicates how topological local prequantum field theory is naturally presented by higher correspondences in the slice of a cohesive (∞,1)-topos over an ∞-group of units of some E-∞ ring E. These are correspondences of moduli stacks of fields (spaces of field trajectories) where the correspondence space is equipped with a cocycle in bivariant twisted E-cohomology. The second part of the talk indicates how quantization of such prequantum data is given by pull-push in twisted E-cohomology, sending correspondences to morphisms of E-∞-modules. The existence condition on the required orientation in generalized cohomology are the quantum anomaly cancellation conditions. Finally we indicate a list of examples of holographic quantization of boundary field theories this way: the Poisson manifold on the boundary of the Poisson sigma-model, the charged particle at the boundary of the open string and the heterotic string at the boundary of the M2-brane.
The first part is joint work with Domenico Fiorenza and Hisham Sati. The second part is joint work with Joost Nuiten. Further pointers are available at http://ncatlab.org/schreiber/show/Motivic+quantization+of+prequantum+field+theory
Motivated by physics, Graeme Segal defined the notion of a d-dimensional field theory over a manifold X. It turns out that concordance classes of certain kinds of field theories over X are in bijective correspondence to generalized cohomology classes. For example, supersymmetric field theories lead to de Rham cohomology for d=0 (this is joint work with Hohnhold, Kreck and Teichner), to K-theory for d=1 (joint with Teichner), and conjecturally to the cohomology theory known as topological modular form theory for d=2.
In this mini-course I will concentrate on d=0 field theories, but extend the statement above to the equivariant setting by showing that supersymmetric gauged field theories over a G-manifold X lead to the equivariant de Rham cohomology of X (this is joint work with Schommer-Pries and Teichner).
TBA