The detailed definition and construction of nonperturbative quantum field theory still remains one of the outstanding major problems in mathematical and theoretical physics. However, there has been a lot of progress on geometric and topological approaches to this problem via functorial field theories.
These come with interesting connections to other areas of mathematics and mathematical physics, including knot theory, tensor categories, low-dimensional topology, and structures arising in conformal field theory.
The goal of this meeting is to bring together experts in these areas to discuss recent developments and make progress towards the above big question.
Title: On algebraisation of low-dimensional Topology
Abstract: Categories of n-cobordisms (for n=2,3 and 4) are among the most studied objects in low dimensional topology.
For n=2 we know that 2Cob is a monoidal category freely generated by its commutative Frobenius algebra object: the circle.
This result also classifies all TQFT functors on 2Cob. In this talk I will present similar classification results for special categories of 3- and 4-cobordisms. They were obtained in collaboration with Marco De Renzi and are based on the work of Bobtcheva and Piergallini. Frobenius algebra in these cases will be replaced by a braided Hopf algebra.
I plan to finish by relating our results with the famous problem in combinatorial group theory — the Andrews–Curtis conjecture.
Title: How do field theories detect the torsion in topological modular forms?
Abstract: Since the mid 1980s, there have been hints of a deep connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) in which cocycles are 2-dimensional supersymmetric field theories. Basic properties of these field theories lead to expected integrality and modularity properties, but the abundant torsion in TMF has always been mysterious. In this talk, I will describe deformation invariants of 2-dimensional field theories that realize certain torsion classes in TMF. This leads to a description of the generator of \pi_3(TMF)=Z/24 in terms of the supersymmetric sigma model with target S^3.
Title: Heisenberg homologies of surface configurations
Abstract: The Heisenberg group of a surface is the central extension of its one-dimensional homology associated with the intersection cocycle. We show that a representation of the Heisenberg group defines local coefficients on the unordered configuration space of points in the surface. We study the corresponding homologies, the Mapping Class Group action and the connection with quantum constructions. This is based on joint work with Awais Shaukat and Martin Palmer.
Abstract: In the swampland program one tries to delineate effective theories consistent with quantum gravity from those which are not by so-called swampland conjecture. As a consequence of the absence of global symmetries in QG, one such conjecture is saying that the cobordism group has to vanish. In mathematics very often these groups do not vanish right away. Physics tells us that this can be ameliorated by either gauging or breaking of the corresponding global symmetries.
First, we show how the gauging fits into some known constraints in string theory, the so-called tadpole cancellation conditions. Mathematically, this is reflecting a well known connection between certain K-theory and cobordism groups. Second, we report on new results related to the breaking of a global symmetry via codimension one defects. In fact, going beyond topology a very similar mechanism arises for (for a long time puzzling) rolling solutions in string theory, giving rise to the notion of a dynamical cobordism conjecture.
Abstract: According to the cobordism hypothesis with singularities, fully extended topological quantum field theories with defects are equivalently described in terms of coherent full duality data for objects and (higher) morphisms as well as appropriate homotopy fixed point structures. We discuss the 2-dimensional oriented case in some detail and apply it to truncated affine Rozansky-Witten models, which are under very explicit computational control. This is joint work with Ilka Brunner, Pantelis Fragkos, and Daniel Roggenkamp.
Title: Quasi-alternating links, Examples and obstructions
Abstract: Quasi-alternating links represent an important class of links that has been introduced by Ozsváth and Szabó while studying the Heegaard Floer homology of the branched double-covers of alternating links. This new class of links, which share many homological properties with alternating links, is defined in a recursive way which is not easy to use in order to determine whether a given link is quasi-alternating. In this talk, we shall review the main obstruction criteria for quasi-alternating links. We also discuss how new examples of quasi-alternating links can constructed.
Abstract: Roughly speaking, a weight system is a function from a space of chord diagrams to the complex numbers. Weight systems can be used to recover invariants for the relevant kind of knotted object (eg. knots, links, braids etc.) from the Kontsevich integral. The work of Sati and Schreiber highlighted the connection between (horizontal) chord diagrams and higher observables in quantum brane physics. This motivates the question: "which weight systems are quantum states?" Corfield, Sati and Schreiber showed that all gl(n) weight systems associated to the defining representation are indeed quantum states. In this talk I will present an extension of their result to more general weight systems.
The plan of the talk is the following; first, I will introduce the mathematical problem. Then, I will review the proof given by Corfield, Sati and Schreiber that gl(n) weight systems associated to the defining representation are quantum states. Finally, I will show how this result can be extended to weight systems associated to exterior and symmetric powers of the defining representation.
Title: Twisted string bordism and a potential anomaly in $E_8 \times E_8$ heterotic string theory
Abstract: Quantum field theories can have an inconsistency called an anomaly, formulated as an invertible field theory in one dimension higher. Theorems of Freed-Hopkins-Teleman and Freed-Hopkins classify invertible field theories in terms of bordism groups. In this talk, I'll apply this to the low-energy approximation of $E_8 \times E_8$ heterotic string theory; Witten proved anomaly cancellation in a restricted setting, but we perform a twisted string bordism computation to show that the relevant group of 11-dimensional invertible field theories does not vanish, and therefore there could be an anomaly in $E_8 \times E_8$ heterotic string theory. Standard computational techiques for twisted string bordism do not work for this problem; I will also discuss work joint with Matthew Yu using Baker-Lazarev's R-module Adams spectral sequence to simplify a large class of twisted spin and string bordism computations.
Title: String bordism invariants in dimension 3 from U(1)-valued TQFTs
Abstract: The third string bordism group is known to be Z/24Z. Using Waldorf's notion of a geometric string structure on a manifold, Bunke--Naumann and Redden have exhibited integral formulas involving the Chern-Weil form representative of the first Pontryagin class and the canonical 3-form of a geometric string structure that realize the isomorphism between the third string bordism group and Z/24Z (these formulas have been recently rediscovered by Gaiotto--Johnson-Freyd--Witten). In the talk I will show how these formulas naturally emerge when one considers the U(1)-valued 3d TQFTs associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure. Joint work with Eugenio Landi (arXiv:2209.12933).
Title: Deformation classes of invertible field theories and the Freed--Hopkins conjecture
Abstract: In their seminal paper, Freed and Hopkins proved a classification theorem that identifies deformation classes of certain topological invertible field theories with the torsion subgroup of some generalized cohomology of a Thom spectrum. They conjectured that the identification continues to hold for non-topological field theories, if one passes from the torsion subgroup to the full generalized cohomology group of the Thom spectrum. In this talk, I will discuss a result which provides an affirmative answer to this conjecture. The method of proof uses recent joint work with Dmitri Pavlov on the geometric cobordism hypothesis.
Title: Correspondence between automata and one-dimensional Boolean topological theories and TQFTs
Abstract: Automata are important objects in theoretical computer science. I will describe how automata emerge from topological theories and TQFTs in dimension one and carrying defects. Conversely, given an automaton, there's a canonical Boolean TQFT associated with it. In those topological theories, one encounters pairs of a regular language and a circular regular language that describe the theory.
Title: Universal construction, foams and link homology
Abstract: In this series of three talks we will explain the foam
approach to link homology. Bigraded link homology theories categorify
the Jones polynomial and other Reshetikhin-Turaev link invariants, such
as the HOMFLYPT polynomial. Foams, which are polyhedral 2D complexes
embedded in 3-space allow to construct state spaces for planar graphs
which are then used to define link homology groups. The most explicit
and efficient way to define graph state spaces is via evaluation of the
closed foams (Robert-Wagner formula).
A) This formula will be first explained in the less technical unoriented
SL(3) case. Resulting graph state spaces are then related to the
Four-Color Theorem and Kronheimer-Mrowka homology for 3-orbifolds.
B) A step in that construction requires building a topological theory (a
lax TQFT) from an evaluation of closed objects, such as closed
n-manifolds. We will explain the setup for topological theories,
including in two dimensions, recovering the Deligne categories and their
generalizations. In one dimension and adding defects, these topological
theories are related to noncommutative power series, pseudocharacters,
and, over the Boolean semiring, to regular languages and automata.
C) Robert-Wagner GL(N) foam evaluation and its application to
constructing link homology theories will be explained as well.
Title: Symmetry breaking and homotopy types for link homologies.
Abstract: I will describe how the spaces that record symmetry breaking data in a U(n)-gauge theory (for arbitrary n) can be used to construct homotopy types that are invariants for links in R^3. In particular, I will show how one may recover Khovanov-Rozansky Link homology and sl(n) link homology by evaluating this homotopy type under suitable cohomology theories.
Abstract: I will explain my recent joint work with Daniel Grady on locality of functorial field theories (arXiv:2011.01208) and the geometric cobordism hypothesis (arXiv:2111.01095).
The latter generalizes the Baez–Dolan cobordism hypothesis to nontopological field theories, in which bordisms can be equipped with geometric structures,
such as smooth maps to a fixed target manifold, Riemannian metrics, conformal structures, principal bundles with connection, or geometric string structures.
Applications include a generalization of the Galatius–Madsen–Tillmann–Weiss theorem, a solution to a conjecture of Stolz and Teichner on representability of concordance classes of functorial field theories, and a construction of power operations on the level of field theories (extending the recent work of Barthel-Berwick-Evans-Stapleton).
I will illustrate the general theory by constructing the prequantum Chern-Simons theory as a fully extended nontopological functorial field theory.
If time permits, I will discuss the ongoing work on defining quantization of field theories in the setting of the geometric cobordism hypothesis.
Title: On the Finiteness of Quasi-alternating Links
Abstract: The generalization of alternating links to quasi-alternating links raises some natural questions that have affirmative answer in the class of alternating links.
In this talk, we discuss these questions and then we give an affirmative answer to one question without any assumption. As a consequence, we prove that one of these questions is solved in the affirmative iff Green's conjecture on the finiteness of quasi-alternating links of a given determinant holds. Also, we prove that another question is solved in the affirmative implies Green's conjecture on the finiteness of quasi-alternating links of a given determinant holds.
Title: Cobordism in Quantum M-Theory 1: M/F-Theory as Mf-Theory
Abstract: In the quest for mathematical foundations of M-theory, the “Hypothesis H" that fluxes are quantized in Cohomotopy theory, implies that M-brane charges on flat spacetimes locally organize into equivariant homotopy groups of spheres. This leads generally to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory Mf. (Based on arxiv.org/abs/2103.01877).
Title: Quantum field theories on Lorentzian manifolds
Abstract: This talk provides an introduction and survey of recent developments in algebraic QFT on Lorentzian manifolds. I will outline an axiomatization of such QFTs in terms of operad theory and illustrate this formalism through classification results in low dimensions. One of the central axioms is a certain local constancy condition, called the time-slice axiom, that encodes a concept of time evolution. Using model categorical localization techniques, I will show that this i.g. homotopy-coherent time evolution admits a strictification in many relevant cases. I will conclude this talk by explaining similarities and differences between algebraic QFT and other approaches such as factorization algebras and functorial field theories.
Abstract: Based on a graphical calculus for pivotal bicategories, we develop a
string-net construction of a modular functor. We show that a rigid
separable Frobenius functor between strictly pivotal bicategories induces
a linear map between the corresponding bicategorical string-net spaces
that is compatible with the mapping class group actions and with sewing.
This result implies that correlators of two-dimensional conformal field
theories factorize over string-net spaces constructed from defect data.
Title: Cobordism in Quantum M-Theory 2: Topological Quantum Gates in HoTT
Abstract: Recent results on defect branes in string/M-theory and on their holographically dual anyonic defects in condensed matter theory allow for the specification of realistic topological quantum gates, operating by anyon defect braiding in topologically ordered quantum materials. This has a surprisingly slick formulation in parameterized point-set topology, which is so fundamental that it lends itself to certification in modern homotopically typed programming languages, such as cubical Agda. (Based on arxiv.org/abs/2303.02382).
Abstract: The stringor bundle plays the role of the spinor bundle, but in string
theory instead of quantum mechanics. It has been anticipated in
pioneering work of Stolz and Teichner as a vector bundle on loop space.
I will talk about joint work with Matthias Ludewig and Peter Kristel
that provides a fully rigorous and neat presentation of the stringor
bundle as an associated 2-vector bundle, via a representation of the
string 2-group on a von Neumann algebra
Title: Non-semisimple TFT and U(1,1) Chern-Simons theory
Abstract: Chern--Simons theory, as introduced by Witten, is a three dimensional quantum gauge theory associated to a compact simple Lie group and a level. The mathematical model of this theory as a topological field theory was introduced by Reshetikhin and Turaev and is at the core of modern quantum topology. The goal of this talk is to explain a non-semisimple modification of the construction of Reshetikhin and Turaev which realizes Chern--Simons theory with gauge supergroup U(1,1), as studied in the physics literature by Rozansky-Saleur and Mikhaylov. The key new algebraic structure is a relative modular structure on the category of representations of the quantum group of gl(1,1). Based on joint work with Nathan Geer.
Day One: Wednesday, March 15, 2023
Hisham Sati – Welcome & Opening Remarks
Mikhail Khovanov (Columbia University)
Nitu Kitchloo (Johns Hopkins University)
Sergei Gukov (DIAS, Dublin and Caltech)
Nils Carqueville (University of Vienna)
Arun Debray (Purdue University)
Matthew Young (Utah State University)
Day Two: Thursday, March 16, 2023
Mikhail Khovanov (Columbia University)
Mee Seong Im (United States Naval Academy)
Alexander Schenkel (University of Nottingham)
Konrad Waldorf (University of Greifswald)
Domenico Fiorenza (Sapienza University of Rome)
Day Three: Friday, March 17, 2023
Mikhail Khovanov (Columbia University)
Christian Blanchet (University of Paris)
Nafaa Chbili (United Arab Emirates University)
Khaled Qazaqzeh (Kuwait University)
Daniel Grady (Wichita State University)
Dmitri Pavlov (Texas Tech University)
Daniel Berwick-Evans (University of Illinois Urbana-Champaign)
Day Four: Saturday, March 18, 2023
Christoph Schweigert (Hamburg University)
Anna Beliakova (University of Zurich)
Carlo Collari (University of Pisa)
Ralph Blumenhagen (Max-Planck-Institute for Physics, Munich)
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