Schedule for: 17w5167 - Field Theories and Higher Structures in Mathematics and Physics

The BV-BFV formalism unifies the BV formalism (which deals with the problem of fixing the gauge of field theories on closed manifolds) with the BFV formalism (which yields a cohomological resolution of the reduced phase space of a classical field theory). I will explain how this formalism arises and how it can be quantized.

There are various, quite different mathematical approaches to quantum field theories, among them functorial field theories in the sense of Atiyah and Segal and the factorization algebras of quantum observables of Costello and Gwilliam. In the talk I will describe a construction that produces a twisted functorial field theory from a factorization algebra, thus relating these two approaches. This is joint work with Bill Dwyer and Peter Teichner.

An algebraic quantum field theory (AQFT) presents a QFT on Lorentzian manifolds as an assignment of algebras to spacetimes, subject to physical axioms (e.g. Einstein causality). Such algebras are interpreted as quantizations of the function algebras on the moduli space of a classical field theory. In many cases, e.g. the stack of a gauge theory, moduli spaces encode "higher structures". As a consequence, functions on such spaces form "higher algebras", which can be analyzed by homotopical algebra (à la Quillen). Therefore, to investigate the quantization of such moduli spaces, one needs to infuse AQFT with homotopical algebra, resulting in "homotopical AQFT", i.e. the assignment of "higher algebras" to spacetimes. After motivating our approach with a concrete application of homotopical algebra to the Cauchy problem of the Yang-Mills stack, I will provide a "working definition" of homotopical AQFT, emphasize its role in relation to gauge theories and present two toy examples arising via homotopy Kan extensions. Based on [arXiv: 1503.08839, 1610.06071, 1704.01378].

I shall introduce a noncommutative version of the notion of a CohFT, where the role of Deligne-Mumford compactifications of moduli spaces is played by toric varieries of Loday's realisations of associahedra. The corresponding noncommutative analogues of Gerstenhaber and Batalin-Vilkovisky algebras will also be discussed. This is a joint work with S.Shadrin and B.Vallette.

Using local intersections, Goldman and Turaev defined a BV operator on the exterior algebra of homotopy classes of loops on a surface. On a genus zero surface with three boundary components the linearization problem of this structure is equivalent to the Kashiwara-Vergne problem in Lie theory. Motivated by this result a generalization of the Kashiwara-Vergne problem in higher genera is proposed and solutions are constructed in analogy with elliptic associators. This is joint work with A. Alekseev, N. Kawazumi and Y. Kuno.

We will present an example of a topological field theory living on cobordisms endowed with CW decomposition (this example corresponds to the so-called BF theory in its abelian and non-abelian variants), which satisfies the Batalin-Vilkovisky master equation, satisfies (a version of) Segal's gluing axiom w.r.t. concatenation of cobordisms and is compatible with cellularaggregations. In non-abelian case, the action functional of the theory is constructed out of local unimodular L-infinity algebras on cells; the partition function carries the information about the Reidemeister torsion, together with certain information pertaining to formal geometry of the moduli space of local systems. This theory provides an example of the BV-BFV programme for quantization of field theories on manifolds with boundary in cohomological formalism. This is a joint work with Alberto S. Cattaneo and Nicolai Reshetikhin.

In work with David Ben-Zvi and Adrien Brochier, we introduced a (would-be) 4-D topological field theory which relates to N=4 d=4 SYM in the same way that the Reshetikhin-Turaev 3-D theory relates to Chern-Simons theory. On surfaces it assigns certain explicit categories quantizing quasi-coherent sheaves on the character variety of the surface (along the Atiyah-Bott/ Goldman/Fock-Rosly Poisson bracket), and these in turn relate to many well-known constructions in quantum algebra. The parenthetical "would be" above means that, while the theory had an a priori definition on *surfaces* via factorization homology -- due to work of Ayala-Francis, Lurie, and Scheimbauer, these techniques do not apply to 3- and 4-manifolds. In this talk I'll explain work with Adrien Brochier and Noah Snyder, which constructs the 3-manifold invariants following the prescription of the cobordism hypothesis. This is in the spirit of Douglas-Schommer-Pries-Snyder's work on finite tensor categories -- but in the infinite setting -- and also echoes early ideas of Lurie and Walker. The resulting 3-manifold invariants quantize Lagrangians in the character variety of the boundary. They are not at all well-understood or computed explicitly in general, but they appear phenomenologically to relate to many emerging structures, such as quantum A-polyonomials, DAHA-Jones polynomials, and Khovanov-Rozansky knot homologies.

In this talk I will survey our investigations regarding quantum toric varieties (Katzarkov, Lupercio, Meersseman, Verjovsky), its relation to sandpiles, tropical geometry and complex systems (Guzman, Kalinin, Lupercio, Prieto, Shkolnikov) and Mirror Symmetry (Katzarkov, Kerr, Lupercio, Meerssemann).

During last years numerous results were obtained for the exact partition functions and other supersymmetric observables for supersymmetric gauge theories in diverse dimensions. Typically the result can be expressed through the matrix model which satisfy the different versions of Virasoro constraints (or its deformations). I will review the subject and provide some examples for 3D and 4D gauge theories.

I am going to present a construction of an infinity stable category associated to a closed symplectic manifold whose symplectic form has integer periods. The category looks like the Fukaya category of M with coefficients in a certain local system. One first define an infinity category C_{rR} associated to the product of two symplectic balls B_r times B_R whose objects are (roughly) graphs of symplectomorphic embeddings B_r to B_R and homs are positive isotopies (it is defined via listing axioms which characterize it). We have a composition C_{r_1r_2} times C_{r_2r_3} to C_{r_1r_3} so that we have an infinity 2-category C whose 0- objects are balls and the category of morphisms between B_r and B_R is C_{rR} One has a functor F_M} from C to the infinity 2 category of infinity categories, where F_M(B_r) is the category of symplectic embeddings B_r—>M. One also has another functor P between the same infinity categories and one defines the micro local category on M as hom(P,F_M).

I will talk about ongoing progress in understanding Morse theory as a deformation problem encoded by (a sheaf living on) the stack of broken lines. This is joint work with Jacob Lurie.

We discuss fermionization of quantum gauge theories in 3 and 4 dimensions, with momentum cut offs. The corresponding Fermionic quantum field theories are nonlocal and have convergent perturbation expansions. We discuss some conjecturesthat arise from studying the structure of these theories.

Lie ∞-groupoids are simplicial manifolds which satisfy conditions similar to the horn filling conditions for Kan simplicial sets. Lie ∞-groupoids are to non–negatively graded dg manifolds, or L∞-algebroids, as Lie groups are to Lie algebras. In particular, there is an integration procedure based on a smooth analog of Sullivan’s realization functor from rational homotopy theory that pro- duces a Lie ∞-groupoid from dg– manifold. There is also a differentiation functor due to Sˇevera, which uses supergeometry to construct the 1-jet of a simplicial manifold. In this talk, I will present joint work (arXiv:1609.01394) in progress with Chenchang Zhu in which we study the relationship between these integration and differentiation procedures, in analogy with Lie’s Second Theorem. A crucial first step involves constructing a user–friendly homotopy theory for Lie ∞-groupoids. This is a subtle problem, due to the fact that the category of manifolds lacks limits. I will describe how results of Behrend and Getzler can be generalized to develop a homotopy theory for Lie ∞-groups/groupoids that is compatible with the well–known homotopy theory of L∞- algebras/algebroids. If time permits, I will mention some possible applications to AKSZ σ-models via Kotov–Strobl’s theory of characteristic classes for (non–trivial) Q-bundles.

This subject combines ideas from quantum field theory on noncommutative spaces with technologies developed for matrix models to rigorously compute all renormalised correlation functions of these toy models. There is a natural projection of matricial correlation functions to Schwinger functions of an ordinary Euclidean quantum field theory. The main question we are working on is whether or not these functions satisfy Osterwalder-Schrader reflection positivity. If so the model would define a true relativistic (but very simple) quantum field theory in four dimensions. We have partial results that this could be the case.

Geometric quantization attaches vector spaces to symplectic manifolds equipped with extra data. In this talk I will discuss how higher, or categorified, geometric quantization looks like. The input this time will be a shifted symplectic space. As an example, in shift (-1) one discovers BV quantization.

Generalisations of gauge theory and gravity suggested by supersymmetry and string theory are explored. Their physical and mathematical structure are discussed and superconformal symmetry in six dimensions is seen to play an interesting role.

Working towards the aim of axiomatizing realistic quantum field theories in a TQFT-type framework we focus on the simplest class of examples: linear field theories and their perturbation theory. In order to understand quantization it turns out to be useful to introduce an axiomatization of classical field theory also, on manifolds with boundary. We show how geometric quantization together with the Feynman path integral then leads to a quantization functor from (augmented) classical field theories to quantum field theories. We discuss scope, applications, limitations and future directions of this approach.

I will report on progress of a joint project with Alan Weinstein in which we try to understand the initial value constraints of field theories with external symmetries as momenta of a Lie algebroid action on the presymplectic manifold of fields. Having shown earlier how a Lie algebroid symmetry naturally appears in General Relativity, we are currently studying a notion of hamiltonian momentum map for Lie algebroids. I will explain how hamiltonian Lie algebroids generally arise in lagrangian field theories with spacetime diffeomorphism symmetry.

The classical BV-BFV formulation of the Poisson sigma model with boundary produces isotropic evolution relations, i.e. immersed submanifolds of products of the spaces of boundary fields. In this talk we will prove that such relations are in fact split Lagrangian, i.e. isotropic with isotropic complements, and they form a groupoid object in an extended version of the symplectic category.

Consider the BV cohomology associated to a solution of the BV master equation for field theory on a d-dimensional world sheet. When d=0, Felder and Kazhdan have suggested the additional axiom that the cohomology for a physically relevant theory vanish below dimension 0. The natural generalization of this to d>0 is that the cohomology vanish below dimension d. In earlier work, we have shown that this axiom is violated for the spinning particle, which is a toy model of a supersymmetric field coupled to supergravity in d=1. Sean Pohorence and I have shown that in contrast, the axiom holds for the superparticle, which is a toy model of the Green- Schwartz superstring in d=1. This theory exhibits some interesting features: there is an infinite tower of ghosts, so it is important to work with the correct completion of the space of local observables, and also one must work with Cech cochains at every stage of the calculation.