# Topological Quantum Field Theory Seminar

## Past sessions

### Trivalent TQFT and applications

MOY calculus has been introduced in the 90s to compute combinatorially the quantum link invariant associated with the Hopf algebra $U_q(\mathfrak{sl}(N))$. It associates to any decorated graph a Laurent polynomial in $q$. I will describe a TQFT-like functor which categorifies the MOY calculus and provides a new description of the $\mathfrak{sl}(N)$-homology.

(joint work with L.-H. Robert)

### 24/05/2017, 14:15 — 15:15 — Room P3.10, Mathematics BuildingPaul Wedrich, Imperial College, London

Link homology theories are powerful generalizations of classical (and quantum) link polynomials, which are being studied from a variety of mathematical and physical viewpoints. Besides providing stronger invariants, these theories are often functorial under link cobordisms and carry additional topological information. The focus of this talk is on the Khovanov-Rozansky homologies, which categorify the Chern-Simons/Reshetikhin-Turaev $\mathfrak{sl}(N)$ link invariants and their large $N$ limits. I will survey recent results about their behaviour under deformations as well as their stability at large $N$, which together lead to a rigorous proof of a package of conjectures originating in string theory.

### A diagrammatic categorification of the higher level Heisenberg algebras

Khovanov defined a diagrammatic 2-category and conjectured (and partially proved) that it categorifies the level-one Heisenberg algebra. Since then, several interesting generalizations and applications have been found, e.g. Cautis and Licata's generalization involving Hilbert schemes and their construction of categorical vertex operators. However, these are all for level one. In my talk, I will explain Alistair Savage and my results on a generalization of Khovanov's original results for higher level Heisenberg algebras. This is work in progress.

### Topological Complexity of the Klein Bottle

The notion of topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. Surprisingly, the determination of this invariant for non-orientable surfaces has turned out to be difficult. A. Dranishnikov has recently established that the topological complexity of the non-orientable surfaces of genus at least 4 is maximal. In this talk, we will determine the topological complexity of the Klein bottle and extend Dranishnikov's result to all the non-orientable surfaces of genus at least 2. This is a work in collaboration with Daniel C. Cohen.

### M-theory from the superpoint revisited

The last talk we gave on this topic (in the meeting Iberian Strings 2017) was largely about the physics; here we focus on the mathematics. No prior knowledge will be assumed.

We define the process of invariant central extension: taking central extensions by cocycles invariant under a given subgroup of automorphisms of a Lie superalgebra. We give conditions that allow us to carve out the Lorentz group inside the automorphisms of Minkowski superspacetime, and prove that by successive invariant central extensions of the superpoint, we construct all superspacetimes up to dimension 11.

### Operads of genus zero curves and the Grothendieck-Teichmuller group

In Esquisse d’un programme, Grothendieck made the fascinating suggestion that the absolute Galois group of the rationals could be understood via its action on certain geometric objects, the (profinite) mapping class groups of surfaces of all genera. The collection of these objects, and the natural relations between them, he called the "Teichmuller tower”.

In this talk, I plan to describe a genus zero analogue of this story from the point of view of operad theory. The result is that the group of automorphisms of the (profinite) genus zero Teichmuller tower agrees with the Grothendieck-Teichmuller group, an object which is closely related to the absolute Galois group of the rationals. This is joint work with Geoffroy Horel and Marcy Robertson.

### Hamiltonian analysis of the BFCG theory for a generic Lie 2-group

We perform a complete Hamiltonian analysis of the BFCG action for a general Lie 2-group by using the Dirac procedure. We show that the resulting dynamical constraints eliminate all local degrees of freedom which implies that the BFCG theory is a topological field theory.

Room 3.10 now confirmed.

### EPRL/FK Asymptotics and the Flatness Problem: a concrete example

Spin foam models are a "state-sum" approach to loop quantum gravity which aims to facilitate the description of its dynamics, an open problem of the parent framework. Since these models' relation to classical Einstein gravity is not explicit, it becomes necessary to study their asymptotics — the classical theory should be obtained in a limit where quantum effects are negligible, taken to be the limit of large triangle areas in a triangulated manifold with boundary.

In this talk we will briefly introduce the EPRL/FK spin foam model and known results about its asymptotics, proceeding then to describe a practical computation of spin foam and asymptotic geometric data for a simple triangulation, with only one interior triangle. The results are used to comment on the "flatness problem" — a hypothesis raised by Bonzom (2009) suggesting that EPRL/FK's classical limit only describes flat geometries in vacuum.

### Duality in String/M-theory from Cyclic cohomology of Super Lie $n$-algebras

I discuss how, at the level of rational homotopy theory, all the pertinent dualities in string theory (M/IIA/IIB/F) are mathematically witnessed and systematically derivable from the cyclic cohomology of super Lie $n$-algebras. I close by commenting on how this may help with solving the open problem of identifying the correct generalized cohomology theory for M-flux fields, lifting the classification of the RR-fields in twisted K-theory.

This is based on joint work with D. Fiorenza and H. Sati arxiv:1611.06536

https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes

### 2+1 TQFTs with Defects

We will give an overview of TQFTs with defects along the lines of the preprint 3-dimensional defect TQFTs and their tricategories by Carqueville, Meusburger and Schaumann. In this paper ordinary Atiyah type functorial TQFTs on 2+1 cobordisms are generalized to 2+1 stratified cobordisms with decorations in a certain graph structure. The decoration graph structure together with a given TQFT of this type then give rise to a linear Gray-category with duals. This provides a unifying framework for well known 2+1 TQFTs.

### Invariants and TQFTs for cut cellular surfaces from finite groups and $2$-groups

We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of $0$-, $1$- and $2$-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the $1$-cells with elements of a finite group $G$, subject to a “flatness” condition for each $2$-cell. These invariants are also described in a TQFT setting, which is not the same as the usual $2$-dimensional TQFT framework. We study the properties of functions which arise in this context,associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology. One such property states that the number of conjugacy classes of $G$ equals the commuting fraction of $G$ times the order of $G$.

We will comment on the extension of these invariants to 2-groups and their (higher) gauge theory interpretation.

This is work done in collaboration with Diogo Bragança (Dept. Physics, IST).

https://arxiv.org/abs/1512.08263

### On the representations of 2-groups in Baez-Crans 2-vector spaces

In this talk, we review the notions of 2-group, Baez-Crans 2-vector space, and 2-representations of 2-groups. We will study the irreducible and indecomposable 2-representations, and finally we will show that for a finite 2-group $G$ and base field $k$ of characteristic zero, this theory essentially reduces to the representation theory of the first homotopy group of $G$.

### Poincaré Duality as Duality of Categories

We give a construction that associates a small category $\mathcal{C}(X)$ to a CW-decomposition $X$ of a manifold. We obtain interesting families of finite categories from spheres and projective spaces as examples. Under some conditions this category $\mathcal{C}(X)$ seems to represent the homotopy type of $X$. Interestingly, for finite dimensional $X$ the Poincaré dual $\hat{X}$ has associated to it the opposite category $(\mathcal{C}(X))^{\rm{op}}=\mathcal{C}(\hat{X})$.

This is part of a joint project with Benjamin Heredia.

Please note the new date of 19th October - this seminar was originally scheduled for 12th October.

### M-theory from the superpoint

One mysterious facet of M-theory is how a 10-dimensional string theory can "grow an extra dimension" to become 11-dimensional M-theory. Physically, the process is understood via brane condensation. Mathematically, Fiorenza, Sati, and Schreiber have proposed that brane condensation coincides with extending superspacetime, viewed as a Lie superalgebra, by the cocycle in Lie algebra cohomology which encodes the brane's WZW term. The resulting extension can be regarded as an "extended superspacetime" where still other super p-branes may live, whose condensates yield further extensions, and so on. In this way, all the super p-branes of string theory and M-theory fit into a hierarchy called "the brane bouquet". In this talk, we show how the brane bouquet grows out of the simplest kind of supermanifold, the superpoint.

### Deformations of Super-Riemann Surfaces

In this talk I will detail aspects of the deformation theory of super-Riemann surfaces, in the spirit of Kodaira and Spencer. The goal is to argue a relation between: (1) deformations of super-Riemann surfaces; and (2) the obstruction theory of these deformations (to be thought of as complex supermanifolds). With this relation the consequences of a vanishing Kodaira-Spencer map of a deformation, in a particular instance, will become apparent and will lead on to further questions.

### $L_\infty$-algebra of sphere-valued supercocycles in M-theory

In this talk we describe how super $L_\infty$-algebras capture the dynamics of the various fields and branes encoded in supercocycles associated with super Minkowski spacetimes at the rational level. We illustrate how rational 4-sphere-valued supercocycles in M-theory descend to supercocycles in type IIA string theory and capture the dynamics of the Ramond-Ramond fields predicted by the rational image of twisted K-theory, with the twist given by the usual B-field. We explicitly derive the M2/M5 $\leftrightarrow$ F1/Dp/NS5 correspondence via dimensional reduction of sphere valued $L_\infty$ supercocycles in rational homotopy theory. These results highlight, in the context of M-theory, the observations that supercocycles are still rich even in flat superspace and spectra are still rich even rationally. This is joint work with Domenico Fiorenza and Urs Schreiber.

### M-theory via rational homotopy theory

M-theory has proven to be very rich both from physics and mathematics points of view. We explain how the fields and their dynamics in M-theory can be succinctly captured by the 4-sphere, viewed via spectra and via cohomotopy. Working rationally allows us to have a mathematical handle on the sphere and to bring in interesting techniques and results from rational homotopy theory. Using spheres, we also describe the dynamics arising from various special points in the moduli space of M-theory, including reduction to type IIA and to heterotic string theory, inclusion into the bounding theory, inclusion of M-branes, and lifts. The detailed agreement with the dynamics expected fromthese theories is tantalizing and suggests an emerging deep picture on the mathematical structure of M-theory. This is joint work with Domenico Fiorenza and Urs Schreiber.

### Simplicial Cocommutative Hopf Algebras

In this talk, we define the Moore complex of any simplicial cocommutative Hopf algebra by using Hopf kernels which are defined quite different from the kernels of groups or various well-known algebraic structures. Furthermore, we will see that these Hopf kernels only make sense in the case of cocommutativity. We also introduce the notion of 2-crossed modules of cocommutative Hopf algebras and continue to talk about its categorical properties such as its relations with simplicial objects, Lie algebras, groups and also the Milnor-Moore theorem, as long as time allows.

Joint work with: João Faria Martins.

### Structure Theory for Higher WZW Terms VI

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde G$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde G$ and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

1. Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
2. Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

### Structure Theory for Higher WZW Terms V

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde G$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde G$ and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

1. Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
2. Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)