Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
The cobordism hypothesis classifies extended topological quantum field theories (TQFTs) in terms of algebraic information in the target category. One of the core principles in quantum field theory - unitarity - says that state spaces are not just vector spaces, but Hilbert spaces. Recently in joint work with many others, we have defined unitarity for extended TQFTs, inspired by Freed and Hopkins. Our main technical tool is a higher-categorical version of dagger categories; categories $C$ equipped with a strict anti-involution $\dagger: C \to C^{op}$ which is the identity on objects. I explain joint work in progress with Theo Johnson-Freyd, Cameron Krulewski and Lukas Müller in which we prove a version of the cobordism hypothesis for unitary TQFTs. The main observation is that the \emph{stably} framed bordism $n$-category is freely generated as a symmetric monoidal dagger $n$-category with unitary duals by a single object: the point.The spin-statistics theorem asserts that in a unitary quantum field theory, the spin of a particle—characterized by its transformation under the central element of the spin group, which corresponds to a 360-degree rotation—determines whether it obeys bosonic or fermionic statistics. This relationship can be formalized mathematically as equivariance for a geometric and algebraic action of the 2-group ${\rm B}{\bf Z}_2$. In my talk, I will present a refinement of these actions, extending from ${\rm B}{\bf Z}_2$ to appropriate actions of the stable orthogonal group ${\rm O}$, and demonstrate that every unitary invertible quantum field theory intertwines these ${\rm O}$-actions.In this talk I will discuss the differential topology of non-linear proper Fredholm mappings. In applications these mappings arise as non-linear PDE problems (of elliptic type). I will discuss work with Lauran Toussaint that relates these mappings to the stable homotopy groups of spheres, and if time permits, I will discuss an ongoing project on defining a new homology theory of singular type for infinite dimensional spaces. This is joint work with Alberto Abbondandolo, Michael Jung and Lauran Toussaint.After briefly recalling how the analog of Dirac charge quantization in exotic (effective, higher) gauge theories, providing their global topological completion, is encoded in a choice of classifying space 𝒜 whose rationalization reflects the flux Bianchi identities, I explain how the choice 𝒜 ≔ S^2 (“flux quantization in 2-Cohomotopy”) implements effective corrections to ordinary Dirac flux quantization, which over surfaces yields exactly the topological quantum observables of fractional quantum Hall systems, traditionally described by abelian Chern-Simons theory. I close by briefly indicating how this situation is geometrically engineered on probe M5-branes if the M-theory C-field is flux-quantized in 4-Cohomotopy (“Hypothesis H”). This is joint work with Hisham Sati; for more pointers see ncatlab.org/schreiber/show/ISQS25.The infinitesimal counterpart of a Lie group(oid) is its Lie algebra(oid). I will show that the differentiation procedure works in any category with an abstract tangent structure in the sense of Rosicky, which was later rediscovered by Cockett and Cruttwell. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category $C$ with a scalar $R$-multiplication, where $R$ is a ring object of $C$. Examples include differentiation of infinite-dimensional Lie groups, elastic diffeological groupoids, etc. This is joint work with Christian Blohmann.A key aspect of quantum theory its insistence that states evolve via unitary transformations. In order to understand the symmetries of higher dimensional quantum field theory, we need to develop higher dimensional analogues of unitarity. The language and theory of higher categories has greatly clarified the way we express these higher symmetries, but unfortunately this language imposes a certain dogma seems to be in conflict with various attempts at describing unitarity. In the nLab for example, there is a great debate over whether or not unitary structures on a (higher) category are evil; at term which is both dogmatic and technically precise.
Various attempts have been made to force these structures to play nice with one another, to varying degrees of success. In this talk I will present our most recent contribution to these efforts: defining the notion of a 3-Hilbert space. Our work aims to encode a kind of evaluation on spheres of every dimension that plays nicely with duality structures that are imposed by the cobordism hypothesis. I will show how this compatibility is stronger than simply having daggers at all levels, thus differentiating our construction from previous attempts at higher unitarity. If time permits, we will discuss a roadmap for unitarity in any dimension via a unitary version of condensation completion.
Two-dimensional chiral conformal field theories (CFTs) admit three distinct mathematical formulations: vertex operator algebras (VOAs), conformal nets, and Segal (functorial) chiral CFTs. With the broader aim to build fully extended Segal chiral CFTs, we start with the input of a conformal net.
In this talk, we focus on presenting three equivalent constructions of the category of solitons, i.e. the category of solitonic representations of the net, which we propose is what theory (chiral CFT) assigns to a point. Solitonic representations of the net are one of the primary class of examples of bicommutant categories (a categorified analogue of a von Neumann algebras). The Drinfel’d centre of solitonic representations is the representation category of the conformal net which has been studied before, particularly in the context of rational CFTs (finite-index nets). If time permits, we will briefly outline ongoing work on bicommutant category modules (which are the structures assigned by the Segal Chiral CFT at the level of 1-manifolds), hinting towards a categorified analogue of Connes fusion of von Neumann algebra modules.
(Bicommutant categories act on W*-categories analogous to von Neumann algebras acting on Hilbert spaces.)There is a classical notion of elliptic genera, which assigns Jacobi forms to SU-manifolds. In this talk, I explain my work with Ying-Hsuan Lin (arXiv:2412.02298) to give its homotopy-theoretical refinements and variants, which we call “topological elliptic genera”. The codomain becomes genuinely equivariant twisted Topological Modular Forms. In this talk, I explain the construction and physical idea behind, and discuss an application where we derive an interesting divisibility result of Euler numbers for Sp-manifolds. Also I explain a recent update with Tilman Bauer (in preparation) proving the surjectivity results of topological elliptic genera.