# Topological Quantum Field Theory Seminar

## Past sessions

### 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)

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

### Structure Theory for Higher WZW Terms IV

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 III

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

### Moonshine, conformal field theory and operator algebras

The Moonshine conjecture is about mysterious relations between the Monster group and elliptic modular functions.  It has been solved in the context of vertex operator algebras, which give an algebraic axiomatization of chiral conformal field theory. Another axiomatization is given in terms of operator algebras.  We present our new result going from the former framework to the latter and back.  No knowledge of these topics is assumed.

### Structure Theory for Higher WZW Terms II

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 I

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

### Homological knot invariants, A-polynomial and integrality properties

The theory of homological knot invariants - the categorification of polynomial knot invariants - appeared 15 years ago, and has been very active ever since. As in the case of the the quantum polynomial knot invariants, they turned out to be related with numerous different fields of mathematics (including topology, quantum groups, representation theory, homological algebra, von Neumann algebras, etc.). In this talk I'll present a basic overview of this categorification in the case of the HOMFLY-PT invariants - both concerning their definition and their properties. Finally, a particular recent application will be shown related to the physics interpretation via BPS invariants, which implies some surprising integrality properties of a pure number theoretical interest.

### Homological knot invariants, A-polynomial and integrality properties

The theory of homological knot invariants - the categorification of polynomial knot invariants - appeared 15 years ago, and has been very active ever since. As in the case of the the quantum polynomial knot invariants, they turned out to be related with numerous different fields of mathematics (including topology, quantum groups, representation theory, homological algebra, von Neumann algebras, etc.). In this talk I'll present a basic overview of this categorification in the case of the HOMFLY-PT invariants - both concerning their definition and their properties. Finally, a particular recent application will be shown related to the physics interpretation via BPS invariants, which implies some surprising integrality properties of a pure number theoretical interest.

### Actions of 2-groups, moduli spaces in higher gauge theory, and TQFT's

In the context of higher gauge theory (HGT) based on a 2-group, I will discuss how the language of double categories provides a natural description of 2-group actions on a category. One of the motivations is to understand moduli spaces of flat connections modulo gauge transformations in HGT, and this goal is achieved for some simple manifolds. I will also relate these ideas to a class of Topological Quantum Field Theories (TQFT's) for surfaces, obtained from finite groups and 2-groups.

This talk is based on callaborations with João Faria Martins, Jeffrey Morton and Diogo Bragança. It is also intended as preparation for the visit by Urs Schreiber, 15 Jan—14 Feb, 2016.

### Bicategories, classifying spaces and homotopy pullbacks

In this continuation of my last talk I will give an introduction to the homotopy theory of bicategories. First I will present the way to convert bicategories to spaces, and then I will use this to describe homotopy pullbacks of homomorphisms of bicategories.

### Homotopy theory using categories

This is an introductory talk about the homotopy theory of categories. I will present the classifying space of a category and the classical results of Thomason and Quillen for obtaining categorical descriptions of homotopy colimits and homotopy pullbacks.

### Positive energy unitary irreducible representations of $\operatorname{osp}(1|2n)$ superalgebras

Orthosymplectic $\operatorname{osp}(1|2n)$ superalgebras are being considered as alternatives to $d$-dimensional Poincaré/conformal superalgebras and thus have significant potential relevance in various subfields of High Energy Physics and Astrophysics. Yet, due to mathematical difficulties, even the classification of their unitary irreducible representations (UIR's) has not been entirely accomplished. This is also true for the physically most important subclass of positive energy UIR's.

In this talk I will first demonstrate this classification for the $n=4$ case (that corresponds to four dimensional space-time). The classification is obtained by careful analysis of the Verma module structure, which is particularly subtle due to the existence of subsingular vectors. Based on these results I will then conjecture their generalization to the case of arbitrary $n$ (thus also including cases relevant in the string/brane context). In addition, I will show an elegant explicit realization of these UIR's that exists for (half)integer values of the conformal energy and that makes manifest the mathematical connection existing between UIR's of orthogonal and orthosymplectic algebras. The existence of this realization, per se, proves a part of the conjecture.

### Geometrical diffeomorphism invariant observables for General Relativity and their applications

During the talk I will present a recent construction of observables for General Relativity invariant under spatial diffeomorphisms. The construction involves introducing a local structure representing the "observer" (based on arXiv:1403.8062). I will also present how those observables can be used to reduce the phase space of canonical General Relativity (based on arXiv:1506.09164).

If time permits, I will argue that the construction is particularly useful in spherically symmetric situations. This realization lead to a proposal of a scheme of reducing Loop Quantum Gravity to its spherically symmetric sector, which completes the standard, midisuperspace approach (based on arXiv:1410.5609).

swiezewski.pdf

### Physical Mathematics: old and new

In this talk  we will discuss some topics related  to the interaction of physical and mathematical theories that have led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. I coined the term Physical Mathematics to describe this new and fast growing area of research and used it in the title of my paper in Springer's book Mathematics Unlimited: 2001 and beyond.

We will discuss some recent developments in this area. General reference for this talk is my book Topics in Physical Mathematics, Springer (2010).

### Black hole entropy in loop quantum gravity

After briefly introducing the main ingredients of the loop quantum gravity approach, I show how it can applied to the calculation of black hole entropy. I review some well known results and open issues resulting from the interplay with Chern-Simons theory techniques. I then introduce a new analysis of the horizon degrees of freedom in terms of purely LQG methods, which turns out to be dual to a CFT description. I show how this unifying framework allows us to recover the semiclassical Bekenstein-Hawking entropy formula.

Pranzetti_Black_Hole_in_Loop_Quantum_Gravity_Slides_20150422.pdf

### Introduction to Loop Quantum Gravity (part 3)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

In the third lecture we will finish the construction of the spin-knot space and introduce the loop transform. Then we move on to the analysis of geometric observables (distance, area and volume) and the structure of the scalar constraint. Finally, matter coupling will be introduced. If time permits, we will also give a short review of two applications of the formalism: calculation of the black hole entropy, and the Big Bounce model of Loop Quantum Cosmology.

### Introduction to Loop Quantum Gravity (part 2)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The second lecture is devoted to the canonical quantization procedure within the LQG framework. We will begin by a short introduction to the notion of background independence, and differences between perturbative and nonperturbative quantization. We will then rewrite general relativity in the canonical space+time formulation and introduce Ashtekar variables, as preparation for the canonical quantization. Then the main step is the quantization itself, and the construction of the appropriate Hilbert space of the theory based on the notions of spin networks and spin-knots.

Please note the change of date from Wednesday 11th March to Wednesday 18th March.

### Introduction to Loop Quantum Gravity (part 1)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The first lecture will be devoted to the formulation of the problem of quantization of the gravitational field. We will give an overview of perturbative quantization, discuss the issue of nonrenormalizability, and provide a general classification of most prominent approaches to constructing a theory of quantum gravity. One such approach is Loop Quantum Gravity, which will be studied in more detail in subsequent lectures.

### Twisted actions of categorical groups

We develop a theory of twisted actions of categorical groups using the notion of semidirect product of categories. I will present many examples of semi-direct product of categories. If time permits I will also work-out an example of twisted action involving the Poincaré 2-group. Specializing to the case of representations, where the the category on which categorical group acts has some kind of a vector space structure, we will establish a categorical analogue of Schur's lemma.

This is a joint work with A. Lahiri and A. Sengupta.

Older session pages: Previous 4 5 6 7 8 9 10 11 12 13 Oldest

Current organizers: Roger Picken, Marko Stošić.

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