CONFERENCE SPEAKERS & ABSTRACTS:
Anton Alekseev (University of Geneva)
Title: Symplectic structure in Jackiw-Teitelboim gravity
Abstract: Recent works on Jackiw-Teitelboim gravity (e.g. Saad-Shenker-Stanford) suggest that moduli spaces of hyperbolic 0-metrics on surfaces with boundary carry canonical symplectic structures. Here 0-metrics are metrics which have a singularity of the form y^-2 at the boundary, and y is a boundary defining function. An example of such a metric is the standard hyperbolic metric on the Poincaré half-plane.
In this talk, I will explain a construction of symplectic forms on spaces of hyperbolic 0-metrics modulo diffeomorphisms which preserve the boundary. This construction involves an interesting instance of symplectic reduction, and the resulting spaces carry a Hamiltonian action of the Bott-Virasoro group.
The talk is based on a joint work with Eckhard Meinrenken.
Colleen Delaney ( University of California at Berkeley)
Title: An efficient* classical algorithm for some quantum invariants of 3-manifolds
Abstract: We will share some recent results that are instructive for approaching the classification of 3d TQFTs and topological order by computational complexity. We show that the Turaev-Viro-Barrett-Westbury state sum TQFT invariants of 3-manifolds that arise from certain “near-group” fusion categories can actually be computed in polynomial time on a classical computer, provided that there is a bound on the first Betti number. On the other hand, if we don’t insist on a bound on the first Betti number, then the invariants should be NP-hard to compute. This illustrates a distinction between the contribution to the intrinsic complexity of a fully-extended 3d TQFT from its source (topology) versus the target (algebra).
Theo Johnson-Freyd (Dalhousie University, Perimeter Institute)
Title: Quantum homotopy groups
Abstract: An open-closed tqft is a tqft with a choice of boundary condition. Example: the sigma model for a sufficiently finite space, with its Neumann boundary. Slogan: every open-closed tqft is (sigma model, Neumann boundary) for some “quantum space”. In this talk, I will construct homotopy groups for every such “quantum space” (and recover usual homotopy groups). More precisely, these “groups” are Hopf algebras in some category. Given a “quantum fibre bundle” (a relative open-closed tqft), I will construct a Puppe long exact sequence. Retracts in 3-categories and a higher Beck–Chevalley condition will make cameo appearances. This project is joint work in progress with David Reutter.
Si Li (Tsingua University)
Title: Stochastic process and noncommutative geometry
Abstract: We explain a stochastic approach to topological field theory and present a case study of quantum mechanical model and its relation to noncommutative geometry.
Matthias Ludewig (University of Regensburg)
Title: The stringor bundle and the spinor bundle on loop space
Abstract: We explain the construction of the stringor bundle on a string manifold recently given in joint work with Peter Kristel and Konrad Waldorf. We start by discussing the spinor bundle on the loop space of a string manifold, together with its fusion product. Then we explain how the stringor bundle on the manifold itself can be obtained using a regression procedure.
Laura Murray (Providence College)
Title: Principal 2-group bundles and the Freed-Quinn line bundle
Abstract: A 2-group is a categorification of a group: it is a category with a multiplication operation that satisfies the group axioms only up to coherent isomorphisms. For any 2-group, the isomorphism classes of its objects form an underlying, ordinary group. Performing this categorification process in the setting of bundles gives the notion of a principal 2-group bundle over a manifold (or stack) X. The bicategory of these principal 2-group bundles lives over the category of principal G-bundles Bun_G(X), for G the underlying group of the 2-group.
For G a finite group and X a Riemannian manifold, we prove that this gives a categorification of the Freed-Quinn line bundle, a mapping class group-equivariant line bundle over Bun_G(X). The global sections of this line bundle give the state spaces of Dijkgraaf-Witten theory (Chern-Simons theory for the finite group G). Our characterization of principal 2-group bundles is also amenable to explicit geometric constructions, such as characterizing flat string structures.
This talk is based on joint work with Daniel Berwick-Evans, Emily Cliff, Apurva Nakade, and Emma Phillips.
Avia Raviv-Moshe (Simons Center for Geometry and Physics, Stonybrook)
Title: Line Defects in CFTs: from Magnets and Impurities to Wilson Lines
Abstract: Line defects (one-dimensional defects) in CFTs provide a framework that is of relevance to describe various physical systems, with applications ranging from condensed matter physics to conformal gauge theories. I will review some general properties of line defects in CFTs, especially in the context of RG flows on line defects, where recent progress has been achieved. Then, we will focus on three types of applications: external field defects, models of spin impurities, and Wilson lines in conformal gauge theories.
Kasia Rejzner (University of York)
Title: Quantization of gauge theories in perturbative algebraic quantum field theory
Abstract: In this talk I will summarise some key results in quantization of gauge theories using the perturbative algebraic quantum field theory (pAQFT) framework. This framework allows for a rigorous construction of interacting QFT models using methods of deformation quantization and homological algebra. Recent result include treatment of boundaries and a non-perturbative formulation using C*-algebras.
Michele Schiavina (University of Pavia)
Title: Hamiltonian gauge theory with corners
Abstract: I will discuss the Hamiltonian formulation of local field theories on manifolds with corners in the particular, yet common, case in which they admit an equivariant momentum map. In the presence of corners, the momentum map naturally splits into a part encoding “Cauchy data” or constraints, and a part encoding the “flux” across the corner.
This decomposition plays an important role in the construction of the reduced phase space, which is conveniently handled via symplectic reduction by stages, adapted for the occasion to local group actions.
The output of this analysis are natural Poisson structures associated to corner submanifolds (both on- and off-shell), leading to the concept of (classical) flux superselection sectors as their symplectic leaves, and providing a roadmap to quantum superselection. Moreover, in terms of higher structures, embedding “on shell” poisson manifolds in ambient configuration spaces of corner structures leads to the emergence of “Poisson infinity data”.
This general approach interprets both the Peter-Weyl theorem and Verlinde formula as descending from the superselection factorisation for the appropriate gauge theory, and has modern applications ranging from electrodynamics to gravitation, where other factorisation formulas are expected, owing to the general structure of reduced phase spaces.
Pavol Severa (University of Geneva)
Title: Non-topological boundary conditions of perturbative Chern-Simons and AKSZ models
Abstract: In Witten’s CS/WZW correspondence the chiral WZW model appears from the chiral boundary condition imposed on the CS theory.
I will describe how similar boundary conditions of the CS theory and of its analogs (given by the AKSZ construction) produce many other interesting examples and how they explain, in particular, Poisson-Lie T-duality. Unlike the CS/WZW correspondence, this generalization is only perturbative; on the other hand, the perturbative renormalization leads to (what is called) the generalized Ricci tensor and hopefully still hides many interesting things. The main calculation tool is a CS propagator compatible with the chiral boundary conditions. Based on joint works with J. Pulmann, F. Valach, and D. Youmans.
Pelle Steffens (Technical University of Munich)
Title: A global classical BV formalism in derived geometry.
Abstract: The classical BV formalism of mathematical physics is about the formal geometry of solution spaces of Euler-Lagrange PDEs on spacetime manifolds, regarded in a formal neighbourhood of a particular solution, or a familiy thereof. Such are formal derived spaces over the infinite jet bundle associated to the field bundle on the manifold, equipped with (-1)-shifted symplectic structures. I will propose a global version of the BV formalism in the framework of derived C^oo-geometry, that allows one to access the full solution space of a PDE. This will be a derived and differential geometric version of D-geometry of Beilonson and Drinfeld; more precisely, the basic objects in this theory are derived C^oo-schemes and -stacks that locally admit a closed immersion into the infinite jet bundle of some fibre bundle on spacetime. As we will see, such a formalism is indispensable for extending (-1)-shifted symplectic structures from a formal neighbourhood to the entire solution space of an (elliptic) Euler-Lagrange PDE.
Hiro Lee Tanaka (Texas State University)
Title: When do Sigma models lift to spectra?
Abstract: Some Sigma models, perhaps most famously the Fukaya category, are expected to give rise not only to linear invariants, but invariants that are linear over the topology of spheres. A theorem of Abouzaid-Blumberg (a categorification of the Pontrjagin-Thom theorem) gives one satisfying answer to when theories with analogues of “instanton moduli” enjoy such linearity. In this talk I’ll try to present a framework for computing such theories — as spectrally enriched theories — through deformation theory and stacks of broken objects. Parts of this is joint with Jacob Lurie.
Brian Williams (Boston University)
Title: The higher Segal—Sugawara construction
Abstract: We introduce higher dimensional Virasoro algebras which are central extensions of the dg Lie algebra which is comprised of derived sections of the tangent sheaf on punctured affine space. We then prove a free-field realization of this algebra in the spirit of the Segal—Sugawara construction in conformal field theory. I will discuss the string and M-theoretic origins of this construction in particular cases.
Maxim Zabzine (Uppsala University)
Title: Color-kinematics duality: BV point of view
Abstract: The color-kinematic duality was observed in the study of scattering amplitudes over 15 years ago and it produces big excitement in the field. I will review the color-kinematic duality and explain at the simple example of the Chern-Simons theory. I will offer more abstract approach to this problem within BV formalism and outline different aspects.
Anton Zeitlin (Louisiana State University)
Title: On the BV double of a Courant Algebroid
Abstract: I will discuss the homotopy BV algebra associated with vertex/Courant algebroids and how it fits in the algebraic structures of Double Field Theory and Gravity, viewed as a double copy of Yang-Mills theory.
Micro Talks:
Gert Vercleyen
Title: Anyonica and the AnyonWiki
Abstract: In this talk, two tools that aid in research on Anyons and fusion categories will be presented.
On the one hand, there is the Anyonica software package that contains data on fusion rings and
categories together with various tools for working with these. On the other hand, there is the
AnyonWiki, a website that presents all the data gathered together with theoretical results on anyons
and fusion categories. In particular, there will be a showcase of how Anyonica can help work with data
of fusion rings and categories and how it can be used to find new fusion categories.
Grigorii Taroian
Title: Equivalent models of derived stacks
Abstract: In my talk I will explain why given a certain flavor of geometry (i.e. algebraic, holomorphic or differential) in characteristic 0 there exists a unique notion of a derived stack for this geometry. The talk is based on my paper of the same name (arXiv:2303.12699).
Filippos Sytilidis
Title: Presentations of bordism categories via surgery
Abstract: Morse – Cerf theory has been a very powerful tool in obtaining presentations of bordism categories with generators and relations that come from surgery. I will briefly outline how that has manifested for $Cob_d$ in Juhasz’s work and give an overview of work in progress for bordism bicategories.
Cory Gillette
Title: Towards a Structure Theory of E_n-Monoidal k-Linear Categories
Abstract: Possibly modulo adjectives, the structure theories of Lie algebras, associative algebras, and rings have been well-known since the beginning of the 20th century. The proofs of these results have been clarified by the later inventions of monadic cohomology theories, which translate questions about outer derivations and solvable/nilpotent extensions to questions about low-dimensional cocycles. Analogies between these proofs lead one to suspect that there ought to exist one proof to rule them all, and to hope that this proof may be sufficiently agnostic to the (possibly higher) category of values for an operadic algebra so as to provide a structure theory for tensor categories. In this talk, we will briefly attempt to indicate some tentative progress in enlarging the domain of discourse in which Hochschild’s 1945 arguments for the structure theory of associative algebras hold, and gesture at the desired applications to TFTs which motivate this work.
Nzaganya Edson
Title: Coherent loop states in Geometric applications.
Abstract: I will discuss coherent states and use the asymptotics of their pairing to study Wigner small d matrices.