Events

In this talk, Mayuko Yamashita will explain works with Y. Tachikawa to study anomalies in heterotic string theory via homotopy theory, especially the theory of Topological Modular Forms (TMF). TMF is an E-infinity ring spectrum which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics. In the previous work (link), we proved the vanishing of anomalies in heterotic string theory mathematically using TMF. Additionally, we have a recent update on the previous work (link). Due to the vanishing result, we can consider a secondary transformation of spectra, which coincides with the Anderson self-duality morphism of TMF. This allows us to detect subtle torsion phenomena in TMF by differential-geometric methods.

Speaker: Mayuko Yamashita (Kyoto University, Japan)

Twisted K-theory is a variant of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments of twisted K-theory. In this talk I will discuss an operator-algebraic model for equivariant higher (i.e. non-classical) twists over SU(n) induced by exponential functors on the category of vector spaces and isomorphisms. These twists are represented by Fell bundles and the C*-algebraic picture allows a full computation of the associated K-groups at least in low dimensions. I will also draw some parallels of our results with the FHT theorem. This is joint work with D. Evans.

Speaker: Ulrich Pennig (Cardiff University, UK)

Urs Schreiber will recall some of the theory of "quantum channels" and then explain how it is captured by "monadic computation" with the linear version of the "State monad" — the "QuantumState Frobenius monad."

Speaker: Urs Schreiber (NYU Abu Dhabi)

By recalling the textbook description of a (variational) classical field theory and its critical locus of on-shell fields, I will list desiderata for a category in which this can rigorously take place.  This category will consist of generalized smooth spaces, completely determined by `how they may be smoothly probed by finite dimensional manifolds'. By expanding on this intuition, I will describe how one naturally arrives at the definition of a smooth set as a "sheaf over the site of Cartesian spaces''. I will then explain how the sheaf topos of smooth sets satisfies the desiderata of (variational) classical field theory. Time permitting, I will indicate how the setting naturally generalizes to include the description of fermionic fields, and (gauge) fields with internal symmetries.

Speaker: Grigorios Giotopoulos (NYU Abu Dhabi)

In modern homotopy theory, spaces are represented by combinatorial models called simplicial sets. Their elegant formulation gives them great expressive power to capture spaces up to homotopy. Simplicial distributions are basic mathematical objects that mix simplicial sets with probabilities. That is, they model probability distributions on spaces. In my talk, I will show how simplicial distributions provide a framework for studying a central quantum feature associated with probabilities, known as contextuality. A typical measurement scenario consists of a set of measurements and outcomes, whereas simplicial distributions can be defined for spaces of measurements and outcomes. Our approach unifies and goes beyond two earlier approaches: the sheaf-theoretic (Abramsky-Brandenburger) and group cohomological (Okay-Roberts-Bartlett-Raussendorf.

Speaker: Cihan Okay, Bilkent University, Turkey

In their 1967 book Calculus of Fractions and Homotopy Theory, P. Gabriel and M. Zisman introduced calculus of fractions as a tool for understanding the localization of a category at a class of weak equivalences. While powerful, the condition of calculus of fractions is quite restrictive and it is rarely satisfied in various homotopical settings, like model categories or Brown's categories of fibrant objects, where one instead has homotopy calculus of fractions.

This talk is based on a recent preprint arXiv:2306.02218, which aims to reconcile the two. We define calculus of fractions for quasicategories and give a workable model for marked quasicategories satisfying our condition. Although we have already found several applications of this result, we would be very interested in getting feedback from the audience and exploring new applications from diverse areas.

Speaker: Chris Kapulkin, Western University, Canada

In their 1967 book Calculus of Fractions and Homotopy Theory, P. Gabriel and M. Zisman introduced calculus of fractions as a tool for understanding the localization of a category at a class of weak equivalences. While powerful, the condition of calculus of fractions is quite restrictive and it is rarely satisfied in various homotopical settings, like model categories or Brown's categories of fibrant objects, where one instead has homotopy calculus of fractions.

This talk is based on a recent preprint arXiv:2306.02218, which aims to reconcile the two. We define calculus of fractions for quasicategories and give a workable model for marked quasicategories satisfying our condition. Although we have already found several applications of this result, we would be very interested in getting feedback from the audience and exploring new applications from diverse areas.

Speaker: Daniel Carranza, Johns Hopkins University, USA

Ultra Unification, and Noninvertible Symmetry of the Standard Model from Gravitational Anomaly Abstract: In the Standard Model, the total "sterile right-handed" neutrino number n_{νR} is not equal to the family number Nf. The anomaly index (-Nf+n_{νR}) had been advocated to play an important role in the previous work on Cobordism and Deformation Class of the Standard Model [arxiv:2112.14765, arxiv:2204.08393] and Ultra Unification [arxiv:2012.15860], in order to predict new highly entangled sectors beyond the Standard Model. Ultra Unification would combine the Standard Model and Grand Unification, particularly for the models with 15 Weyl fermions per family, without the necessity of right-handed sterile neutrinos, by adding new gapped topological phase sectors (in 4d or 5d) or new gapless interacting conformal sectors (in 4d) consistent with the nonperturbative global anomaly cancellation and cobordism constraints (especially from the mixed gauge-gravitational anomaly, such as a ℤ_{16} class anomaly, associated with the baryon minus lepton number B−L and the electroweak hypercharge Y). 

Moreover, for the Standard Mode alone, the invertible B−L symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons, hypothetically pertinent for leptogenesis in the very early universe. In specific, we show that a noninvertible categorical counterpart of the B−L symmetry still survives in gravitational backgrounds. In general, we construct noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by ℤ and ℤ_{16} respectively.

For this construction, we utilize the anomaly inflow concept, the 4d Pontryagin class and the gravitational Chern-Simons 3-form, the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory with a framing anomaly corresponding to a 2d rational conformal field theory with an appropriate chiral central charge, and the 4d Z_4^{TF}-time-reversal symmetric topological superconductor with 3d boundary topological order [arxiv:2302.14862].

Speaker: Juven Wang, Harvard University

Hilbert spaces form more than a category: their morphisms maps can be composed, but also every morphism $f : X \to Y$ has a distinguished "adjoint" $f^\dagger : Y \to X$, making it into a "dagger category." This extra data is important for axiomatizing functional analysis, quantum mechanics, quantum information theory... However, the assignment $f \mapsto f^\dagger$ is unsatisfying from a higher category theorist's perspective because it is "evil," i.e., it violates the principle of equivalence: a category equivalent to a dagger category may not admit a dagger structure. This in particular interferes with generalizing the notion of dagger category to the (non-strict) higher categories necessary for axiomatizing fully-local quantum field theory. In this talk, I will propose a manifestly non-evil definition of "dagger $(\infty,n)$-category." The same machinery also produces non-evil definitions of "pivotal $(\infty,n)$-category" and helps to clarify the relationship between reflection positivity and spin-statistics. This is based on joint work with B. Bartlett, G. Ferrar, B. Hungar, C. Krulewski, L. Müller, N. Nivedita, D. Penneys, D. Reutter, C. Scheimbauer, L. Stehouwer, and C. Vuppulury.

Speaker: Theo Johnson-Freyd, Dalhousie University and Perimeter Institute for Theoretical Physics, Canada

Manifold diagrams are the higher categorifications of string diagrams. They lie at the intersection of several interesting topics, such as:

1. The tangle and cobordism hypotheses,
2. The constructive description of free higher categorical structures,
3. The combinatorialization of differential structures and singularities

Nonetheless, the precise role of manifold diagrams in these topics remains largely mysterious. In this talk, we will focus on describing the basic interplay between (stratified) geometry, combinatorics, and (directed) cell complexes, exposed by the mathematical framework of manifold diagrams. This will include, in particular, two equivalent definitions of manifold diagrams, one geometric and one combinatorial, as well as a discussion of how these relate to the above topics

Speaker: Christoph Dorn, Oxford University, UK

The BV formalism and its shifted versions in field theory have a nice compatibility with boundary structures. Namely, one such structure in the bulk induces a shifted (possibly degenerated) version on its boundary, which can be interpreted as aPoisson structure (up to homotopy). Cattaneo presented the results for some field theories, in particular, 4D BF theory and 4D gravity.

Speaker: Alberto Cattaneo, Zurich University, Switzerland

I will begin by explaining the construction of a category CofCos, whose objects are topological spaces and whose morphisms are cofibrant cospans. Here the identity cospan is chosen to be of the form $X\to X\times [0,1]\rightarrow X$, in contrast with the usual identity in the bicategory Cosp(V) of cospans over a category V. The category CofCos has a subcategory HomCob in which all spaces are homotopically 1-finitely generated. There exist functors into HomCob from a number of categorical constructions which are potentially of use for modelling particle trajectories in topological phases of matter: embedded cobordism categories and motion groupoids for example. Thus, functors from HomCob into Vect give representations of the aforementioned categories.

I will also construct a family of functors $Z_G\colon HomCob\to Vect$, one for each finite group $G$, and show that topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten, generalise to functors from HomCob. I will construct this functor in such a way that it is clear the images are finite dimensional vector spaces, and the functor is explicitly calculable. I will also give example calculations throughout.

Speaker: Fiona Torzewska, University of Leeds, UK

Classically, since Z/p is a field, any module over the Eilenberg—MacLane spectrum HZ/p splits as a wedge of suspensions of HZ/p itself. Equivariantly, the module theory of G-equivariant Eilenberg—MacLane spectra is much more complicated. For the cyclic group G=C_p and the constant Mackey functor Z/p, there are infinitely many indecomposable HZ/p-modules. Previous work together with Dugger and Hazel classified all indecomposable HZ/2-modules for the group G=C_2. The isomorphism classes of indecomposables fit into just three families. By contrast, we show for G=C_p with p an odd prime, the classification of indecomposable HZ/p-modules is wild. This is joint work in progress with Grevstad.

Speaker: Clover May, Norwegian University of Science and Technology

Over a field of positive characteristic p, restricted Lie algebras are of prime interest, mainly due to their link to algebraic groups and their role in representation theory and classification. The cohomology associated with restricted Lie algebras is considerably more complicated than the ordinary Chevalley-Eilenberg cohomology and explicit formulas are only known up to order 2. In this talk, I will explain how to build the first and second restricted cohomology groups for restricted Lie superalgebras in characteristic p greater than 3, modifying a previous construction. I will explain how these groups capture some algebraic structures, such as restricted extensions. Further, I will show how to apply this construction to classify p-nilpotent restricted Lie superalgebras up to dimension 4 over an algebraically closed field of characteristic p greater than 3. This is a joint work with Sofiane Bouarroudj (NYU Abu Dhabi).

Speaker: Quentin Ehert, NYU Abu Dhabi

Our understanding of M-theory is based on a duality web connecting different limits of the theory. I'll discuss the extension of this duality web to a wide variety of decoupling limits related by duality to the null reduction of M-theory (and hence to the proposal that M-theory can be described by Matrix theory). From a modern perspective, these limits involve non-relativistic geometries, leading to new variants of supergravity in 11- and 10-dimensions. I'll discuss how to systematically explore these corners of M-theory, following the roadmap of https://arxiv.org/abs/2311.10564

Speaker: Chris Blair, Universidad Autonoma de Madrid, Spain

This talk is based on arXiv:2211.08549, in which we explore the Lie higher-algebraic and higher-geometric structures that arise from a procedure we dub “gauging the gauge”, and study the resulting 2- and 3-gauge theory. The homotopy weakening of algebraic properties afforded by Lie L_n-algebras is tied to violation of geometric higher-Bianchi conditions. In this talk, I will focus on the application of this observation to two specific cases, and examine its consequences. First, for the monopole U(1) gauge theory, the above observation gives rise to the 2-group Green-Schwarz anomaly cancellation (Benini-Cordova-Intriligator 2019), and the associated dipole conservation laws exhibit mobility restriction akin to fractonic matter (Slagel-Kim 2017). Second, for the string 2-gauge theory, we find a 3-group worth of charges exhibiting novel and intricate mobility restrictions and, under certain assumptions, the resulting 3-gauge theory achieves a higher-monopole charge which matched the fractional Pontrjagyn class 1/2p_1. This gives a way in which the string structure of a spin manifold X can be probed dynamically by a 3-group gauge theory.

Speaker: Hank Chen, University of Waterloo, Canada

I will talk about Liouville conformal blocks with degenerate primaries and one operator in an irregular representation of the Virasoro algebra. Using an algebraic approach, we derive modified BPZ equations satisfied by such blocks and subsequently construct corresponding integral representations based on integration over non-compact Lefschetz cycles. The integral representations are then used to derive novel types of flat connections on the irregular conformal block bundle.

Speaker: Babak Haghighat, Tsinghua University, China

We formalize Feynman's construction of the path integral in the context of Lie algebroid-valued sigma models. To do this, we use the pair groupoid and the van Est map to define integration on manifolds in a coordinate-free way. We discuss applications to Brownian motion and the Poisson sigma model.

Speaker: Joshua Lackman, Peking University, China

In this talk, we will explore a novel approach to study supersymmetric quantum field theories using tools from stable homotopy theory. We will explain how this approach leads to new invariants that can be used to detect subtle differences between phases that escape the detection of more conventional invariants.

Speaker: Du Pei, Centre for Quantum Mathematics, University of Southern Denmark