# Topological Quantum Field Theory Seminar

## Past sessions

### QFT V

In the final lecture of our gentle introduction to quantum field theory, we discuss the renormalization of phi cubed theory at one loop.
Please note that the seminar lasts an hour and a half, 11h00 - 12h30.

### QFT IV

We will introduce Feynman diagrams by studying finite-dimensional Gaussian integrals and their perturbations, leading up to phi-cubed theory.

### QFT III

Last time, we talked about quantization of the free scalar field by replacing the modes of the field by quantum oscillators. Now, we put this field into the form used by physicists, and talk about the Wightman axioms, which allow a rigorous treatment of free fields.

### QFT II

We continue our gentle introduction to quantum field theory for mathematicians. We discuss the Klein-Gordon equation, and how it decomposes into oscillators. We quantize this system by quantizing the oscillators, obtaining the free scalar field, the simplest quantum field there is.

### QFT I

This series of lectures will be a gentle introduction to quantum field theory for mathematicians. In our first lecture, we give a lightning introduction to quantum mechanics and discuss the simplest quantum system: the harmonic oscillator. We then sketch how this system is used to quantize the free scalar field.
Note: room 4.35 is on the 4th floor at the end of the main corridor

### Anomalies IV

We will introduce the notion of stable isomorphism for gerbes, and talk about how stable isomorphism classes are in one-to-one correspondence with Deligne cohomology classes. We define WZW branes and discuss how the basic gerbe on a group trivializes when restricted to the brane.

### Categorification of Spin Foam Models

We briefly review spin foam state sums for triangulated manifolds and motivate the introduction of state sums based on 2-groups. We describe 2-BF gauge theories and the construction of the corresponding path integrals (state sums) in the case of Poincaré 2-group.

#### References

• J. F. Martins and A. Mikovic, Lie crossed modules and gauge-invariant actions for 2-BF theories, Adv. Theor. Math. Phys. 15 (2011) 1059, arxiv:1006.0903
• A. Mikovic and M. Vojinovic, Poincaré 2-group and quantum gravity, Class. Quant. Grav. 291 (2012) 165003, arxiv:1110.4694
Room 3.10 is confirmed

### Anomalies III

We continue examining Gawedzki and Reis's paper:

WZW branes and gerbes, http://arxiv.org/abs/hep-th/0205233

We define a gerbe, and show gerbes can be "transgressed" to give line bundles over loop space. Trivial gerbes give trivial bundles on loop space, whose sections are thus mere functions. Any compact, simply connected Lie group comes with a god-given gerbe whose curvature is the canonical invariant 3-form. Restricting this gerbe to certain submanifolds, we get trivial gerbes who thus transgress to trivial line bundles, "cancelling" the anomaly of a nontrivial line bundle.

Room 3.10 is now confirmed

### Anomalies II

We continue our informal discussion of anomalies by talking about global anomalies on branes, and their relationship with gerbes.
Room 3.10 is confirmed

### Introduction to anomalies

In physics, an "anomaly" is the failure of a classical symmetry at the quantum level. Anomalies play a key role in assessing the consistency of a quantum field theory, and link up with cohomology in mathematics, a general tool by which mathematicians understand whether a desired construction is possible. In this informal series of talks, we aim to understand what physicists mean by an "anomaly" and their mathematical interpretation.

### Observables in 2D BF theory

BF theory in two dimensions has been the subject of intensive study in the last twenty five years. I will readdress it by highlighting the TQFT interpretation of the spinfoam approach to its quantisation. I will also introduce the mathematical model that allows us to treat surfaces with inbuilt topological defects and how we expect them to relate to operators in the quantum field theory.

### Non-Commutative Worlds and Classical Constraints

This talk shows how discrete measurement leads to commutators and how discrete derivatives are naturally represented by commutators in a non-commutative extension of the calculus in which they originally occurred. We show how the square root of minus one ($i$) arises naturally as a time-sensitive observable for an elementary oscillator. In this sense the square root of minus one is a clock and/or a clock/observer. This sheds new light on Wick rotation, which replaces $t$ (temporal quantity) by $it$. In this view, the Wick rotation replaces numerical time with elementary temporal observation. The relationship of this remark with the Heisenberg commutator $\left[P,Q\right]=i\hslash$ is explained. We discuss iterants - a generalization of the complex numbers as described above. This generalization includes all of matrix algebra in a temporal interpretation. We then give a generalization of the Feynman-Dyson derivation of electromagnetism in the context of non-commutative worlds. This generalization depends upon the definitions of derivatives via commutators and upon the way the non-commutative calculus mimics standard calculus. We examine constraints that link standard and non-commutative calculus and show how asking for these constraints to be satisfied leads to some possibly new physics.

https://www.math.ist.utl.pt/seminars/qci/index.php.en?action=show&id=3243
Note also another seminar session by the same speaker on Friday 30th November

### Classifying Extended TQFT and the Cobordism Hypothesis

An overview of the concept of extended field theories, and a look at the role of the Cobordism Hypothesis (now more accurately the Cobordism Theorem) in classification of such theories. Given time the talk will touch on Jacob Lurie's proof of the Cobordism Hypothesis.

### Indecomposable modules over a Kuperberg-Khovanov algebras

The ${\mathrm{𝔰𝔩}}_{3}$ polynomial is a quantum invariant for knots. It has been categorified by Khovanov in 2004 in a TQFT fashion. The natural way to extend this categorification to an invariant of tangle is construct a $0+1+1$ TQFT. From this construction emerge some algebras called Khovanov-Kuperberg algebras (or ${\mathrm{𝔰𝔩}}_{3}$-web algebras) and some particular projective modules called web-modules over these algebras. I will give a combinatorial caracterisation of indecomposable web-modules.
Categorification mini-workshop

### ${\mathrm{𝔰𝔩}}_{3}$ web algebras, cyclotomic KLR algebras and categorical quantum skew Howe duality

I will introduce ${\mathrm{𝔰𝔩}}_{3}$ web algebras $K\left(S\right)$, which involve Kuperberg's ${\mathrm{𝔰𝔩}}_{3}$ web space $W\left(S\right)$ and Khovanov ${\mathrm{𝔰𝔩}}_{3}$ foams with boundary in $W\left(S\right)$. These algebras are the ${\mathrm{𝔰𝔩}}_{3}$ analogues of Khovanov's ${\mathrm{𝔰𝔩}}_{2}$ arc algebras. I will show how the $K\left(S\right)$ are related to cyclotomic Khovanov-Lauda-Rouquier algebras (cyclotomic KLR algebras, for short) by a categorification of quantum skew Howe duality. This talk is closely related to the next one by Robert. In particular, I will show that the Grothendieck group of $K\left(S\right)$ is isomorphic to $W\left(S\right)$ and that, under this isomorphism, the indecomposable projective $K\left(S\right)$-modules, which Robert constructs explicitly, correspond precisely to the dual canonical basis elements in $W\left(S\right)$.
Categorification mini-workshop

### The endomorphism category of a cell 2-representation

Fiat 2-categories are 2-analogues of finite dimensional algebras with involutions. Cell 2-representations of fiat 2-categories are most appropriate analogues for simple modules over finite dimensional algebras. In this talk I will try to describe (under some natural assumptions) a 2-analogue of Schur's Lemma which asserts that the endomorphism category of a cell 2-representation is equivalent to the category of vector spaces. This is applicable, for example to the fiat category of Soergel bimodules in type A. This is a report on a joint work with Vanessa Miemietz.
Categorification mini-workshop

### Fermat-type equations of signature $$(13,13,p)$$ via Hilbert cuspforms

In this talk I will give an introduction to the modular approach to Fermat-type equations via Hilbert cuspforms and discuss how it can be used to show that certain equations of the form ${x}^{13}+{y}^{13}=C{z}^{p}$ have no solutions $\left(a,b,c\right)$ such that $\mathrm{gcd}\left(a,b\right)=1$ and $13\nmid c$ if $p>4992539$. We will first relate a putative solution of the previous equation to the solution of another Diophantine equation with coefficients in $Q\left(\sqrt{13}\right)$. Then we attach Frey curves $E$ over $Q\left(\sqrt{13}\right)$ to solutions of the latter equation. Finally, we will discuss on the modularity of $E$ and irreducibility of certain Galois representations attached to it. These ingredients enable us to apply a modular approach via Hilbert newforms to get the desired arithmetic result on the equation.
Duration 90 minutes or slightly less

### Diagrammatic categorification of extended Hecke algebra and quantum Schur algebra of affine type A

Joint work with Marco Mackaay.
Second of two talks in an Informal Categorication Afternoon about current research projects in the area of categorification

### $$\mathfrak{sl}_3$$ web algebras

This is joint work with Weiwei Pan and Daniel Tubbenhauer from Gottingen University, Germany.

First of two talks in an Informal Categorication Afternoon about current research projects in the area of categorification.

### Groupoidification and Khovanov's Categorification of the Heisenberg Algebra

The aim of this talk is to describe the connection between two approaches to categorification of the Heisenberg algebra. The groupoidification program of Baez and Dolan has been used to give a representation of the quantum harmonic oscillator in the category Span(Gpd) where the Fock space is represented by the groupoid of finite sets and bijections. This naturally gives a combinatorial interpretation of the (one-variable) Heisenberg algebra in the endomorphisms of this groupoid. On the other hand, Khovanov has given a categorification in which the integral part of the (many variable) Heisenberg algebra is recovered as the Grothendieck ring of a certain monoidal category described in terms of a calculus of diagrams. I will describe how an extension of the groupoidification program to a 2-categorical form of Span(Gpd) recovers the relations used by Khovanov's construction, and how to interpret them combinatorially in terms of the groupoid of finite sets.

Older session pages: Previous 7 8 9 10 11 12 13 14 Oldest

Current organizers: José MourãoRoger Picken, Marko Stošić

Mathseminars

FCT Projects PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry, PTDC/MAT-PUR/31089/2017, Higher Structures and Applications.