Programme
Printable book of abstracts of workshop talks and further mini-courses
Venue: Room QA1.2 or QA1.4, Torre Sul - campus map
School (provisional)
Planned minicourses
Laurent Freidel - Aspects of the new spin foam models for quantum gravity (to be confirmed)
Lecture 1, part 1: Why higher gauge theory? Categorification in mathematics and physics. 2-groups as an example.
Lecture 1, part 2: Nontrivial 2-bundles without connection. U(1) gerbes as an example.
Lecture 2, part 1: 2-connections on trivial 2-bundles.
Lecture 2, part 2: Nontrivial 2-bundles with 2-connection. U(1) gerbes with connection as an example.
The theme of this short introduction to higher gauge theory is "categorification", a process by which one replaces set-based mathematics with mathematics based on categories, and then mathematics based on categories with mathematics based on 2-categories. Using categorification, the mathematics of gauge theory, used to describe the parallel transport of point particles, becomes the mathematics of higher gauge theory, useful for describing higher-dimensional objects, like strings and membranes. In the first lecture, we will see how categorifying the definition of a bundle leads to the concept of a 2-bundle, which includes U(1) gerbes as a special case. In the second lecture, we will see how categorifying the definition of a connection leads to the concept of a 2-connection, which includes U(1) gerbes with connection as a special case.
Homotopy Quantum field theories were introduced by Turaev in about 1999. They are a variant of TQFTs but which the manifolds and cobordisms have extra structure given by a characteristic map to a background space B.
The other ingredient is theory of homotopy n-types and their algebraic models (crossed modules, 2-crossed modules, n-hypergroupoids etc.)
There will be two lectures.
Lecture 1. Introduction to Yetter's construction of TQFTs, and relative versions. Intro. to HQFTs, examples and the variation with B.
Lecture 2. n-types and their algebraic models (crossed modules, 2-crossed modules, n-hypergroupoids etc.) crossed complexes and a hint of higher things. Finally the link between them.
There is a cut down version of the Crossed menagerie notes (which themselves are way too long), which provides much more than we will look at, and are available here. The prerequisites are some knowledge of basic simplicial sets and homotopy theory, plus basic category theory.
Hisham Sati - Higher spin structures and quantum gravity
Spin structures play an important role in quantum gravity, mainly in the presence of fermions. There are other variants and generalizations of a Spin structure which appear in the context of string theory and M-theory. I will explain these structures and briefly describe how they show up in physics.
I will introduce the basic notions, starting with a Spin structure, which is a lift of the structure group of the tangent bundle from the special orthogonal group SO to the covering group, the Spin group. I will discuss how the process of killing higher homotopy groups leads to generalizations of a Spin structure, namely String and Fivebrane structures. I will also discuss the twists for such structures, as well as generalizations along the line of getting Spin^c starting from Spin.
Lecture 1: Part 1: TQFTs and the Cobordism Hypothesis in Dimension One
Lecture 1: Part 2: The Cobordism Hypothesis in Dimension Two
Lecture 2: Part 1: Symmetric Monoidal Bicategories: Skeletalization
Lecture 2: Part 2: Presentations of Symmetric Monoidal Bicategories
Lecture 3: Part 1: Jet Transversality and Cerf Theory
Lecture 3: Part 2: Singularities in dimensions 2 and 3, Proof of the Main Theorem
Lecture 4: Part 1: Adding Structure Groups
Lecture 4: Part 2: The Cobordism Hypothesis Revisited
Abstract: The study of Topological Quantum Field Theories (TQFTs) has often revealed extraordinary connections between both geometric/topological structures and algebraic structures. One of the most profound examples is provided by the Cobordism Hypothesis which provides a classification of extended topological field theories in terms of simple algebraic data. In this mini-course we will prove the cobordism hypothesis in dimensions one and two using a direct "generators and relations" approach to classifying extended TQFTs.
Most of the material that I will teach will be from my Thesis: Christopher Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories.
For preparation, it would be good to be familiar with some basic material on bicategories. For a brief introduction I recommend,
"Basic Bicategories" T. Leinster. http://arxiv.org/abs/math/9810017.
Students should familiarize themselves with the definitions and material in Section 1. They don't need to know about coherence.
It would be good to skim the following to get acquainted with the ideas:
"Higher-dimensional Algebra and Topological Quantum Field Theory" - J. Baez, J. Dolan. http://arxiv.org/abs/q-alg/9503002
Of course J. Lurie's "On the Classification of Topological Field Theories" http://arxiv.org/abs/0905.0465 is related, but it is at a level which is beyond what I intend to aim for. We'll be taking a more hands-on approach.
I would also recommend that students review some of the basics of Morse Theory. Namely the definition of Morse functions, the proof that Morse functions are dense in the space of all functions, and the derivation of the Morse Lemma. These can be found in Section 2 of Milnor's "Lectures on the h-Cobordism Theorem" or any other standard text. It would also be helpful to be familiar with the proof of the transversality theorem.
1st talk: An introduction to infinite-dimensional Lie groups and Lie algebras
In this talk we lay the basic notions of infinite-dimensional Lie theory, assuming some pre-knowledge in ordinary differential calculus and the Hahn-Banach Theorem for locally convex spaces. From this we introduce the setting of differentiable calculus, leading then naturally to the notion of Lie groups and algebras. In the end, we will line out the construction of Lie group structures on mapping and diffeomorphism groups.
2nd talk: Ordinary gauge theory in Lie theory
This talk explains how ordinary gauge theory (i.e., the theory of principal bundles and connections thereon, which we assume familiarity with) enters into Lie theoretic questions via the concept of central extensions. This leads for instance to a basic construction of central extensions of loop groups and to a natural construction of integrating Lie groups to given finite-dimensional Lie algebras. Moreover, we will also deduce a proof that in infinite dimensions these integrating Lie groups do not always exist. Moreover, we elaborate on the corresponding constructions in terms of Lie group- and Lie algebra cohomology in dimension 2.
3rd talk: Higher gauge theory in Lie theory
We explain how higher gauge theory arises naturally as a framework for extending the constructions from the previous talk to higher cohomological dimensions, in particular to dimension 3. There are some natural classes in these 3rd cohomology groups which allow to extend the integration of Lie algebras to Lie 2-groups (no misprint here) in the case that integrating Lie groups don't exist. Moreover, these constructions have a close relation to the construction of String 2-group models. We will also line out some limitations and open questions of the existing theory.
Workshop (provisional)
Paolo Aschieri - Noncommutative Gerbes
In deformation quantization there is an equivalence between commutative and noncommutative gauge theories, given by the Seiberg-Witten map. This map can be seen as an output of Kontsevich formality theory and has been used to describe deformation quantization of line bundles by Jurco, Schupp, Wess. We exploit it in order to construct noncommutative (abelian) gerbes, they turn out to be special cases of nonabelian gerbes. Noncommutative gerbes arising in deformation quantization are presented, they correspond to quantization of twisted Poisson structures.
Benjamin Bahr - Spin foam operators (slides)
In this talk the spin foam operator approach is reviewed. This is a generalized framework, into which most of the existing state sum models fit naturally. It provides a convenient way of accessing the 'space of all spin foam models', and issues like holonomy formulation, cylindrical consistency and renormalization of these models will be discussed.
For any topological space X, there is a well known pair of adjoint functors between a category Bun(X) of bundles over X, and a category Set^(O(X)^op) of presheaves over X, which restricts to an adjoint equivalence between a category Sh(X) of sheaves over X, and a category Et(X) of etale spaces over X. The right adjoint is a cross-section functor which assigns to every etale space over X a sheaf of its cross-sections, and the left adjoint is a stalk functor which assigns to every presheaf over X its etale space of germs. Stalks of stacks were less familiar so far, since they appeared as filtered bicolimits over (an opposite of) a category O_x(X) of open neighborhoods of a fixed point x in X. Stalks defined in a such way were categories with too many objects and it was not possible to introduce a sensible topology on them. We introduce a new notion of a stalk of a stack, using the smallest equivalence relation generated by a restriction relation on objects, which allows to introduce topology on both objects and morphisms. This construction corresponds to a filtered pseudocolimit over a category O_x(X), and its universal property is defined up to an isomorphism of categories, unlike the case of filtered bicolimits, whose universal property is defined up to an equivalence of categories. In this way, we extend above pair of adjoint functors to a pair of biadjoint 2-functors between a 2-category Fib(X) of fibered categories over X, and a 2-category 2Bun(X) of 2-bundles over X, which restricts to an adjoint biequivalence between a 2-category St(X) of stacks over X, and a 2-category 2Et(X) of etale 2-spaces over X. Furthermore, such adjoint biequivalence restricts to the one between a 2-category Gerb(X) of gerbes over X, and a 2-category EtGpg(X) of etale groupoids over X, providing an alternative proof of Duskin's theorem which associates to any gerbe G over a Grothendieck topos E, corresponding nonempty connected groupoid G~ in E, which is called a bouquet by Duskin, or more recently a bundle gerbe by Murray.
Aristide Baratin - 2-Group Representations for State Sums
Just as 3d state sum models, including 3d quantum gravity, can be built using categories of group representations, 2-categories of 2-group representations may provide interesting state sum models for 4d quantum topology, if not quantum gravity. In this talk, I will describe the construction of such a model from the representations of 'Euclidean 2-group', built from the rotation group SO(4) and its action on the translation group of Euclidean space. I will show that this model gives a new way to compute Feynman integrals for ordinary quantum field theories on 4d Euclidean spacetime.
Inspired by the Chas-Sullivan String Topology and Turaev's homotopy quantum field theories we introduce the notion of homological quantum field theory, and show that it is a fertile ground for applications of higher dimensional parallel transport.
Reference:
E. Castillo, R. Diaz, Homological Quantum Field Theory, in M. Levy (Ed.) Mathematical Physics Research Developments, Nova, New York, 2009, 189-236.
http://arxiv.org/PS_cache/math/pdf/0509/0509532v2.pdf
Laurent Freidel - The Principle of Relative Locality
Björn Gohla - A Mapping Space for Gray-Categories
In order to give an algebraic description of the moduli space of connections on a manifold M valued in a Lie 2-crossed module G, one can look at the collection of Gray-functors from the fundamental Gray-groupoid of M to the Gray-groupoid G corresponding to G, and their transformations. As the canonical internal hom for enriched categories is not adequate, we have to give a construction from the ground up. The principal obstacle is the lack of a unique horizontal composite in Gray-categories.
Branislav Jurco - Nonabelian bundle 2-gerbes
John Huerta - Higher Supergroups for String and M-theory
Starting from the four normed division algebras, there is a systematic way to construct finite-dimensional models of a Lie 2-supergroup which categorifies the Poincare supergroup in spacetimes of dimension 3, 4, 6 and 10, and of a Lie 3-supergroup doing the same thing in dimensions 4, 5, 7 and 11. In the first sequence of dimensions, one can define superstrings, and in the second, one can define super-2-branes. This is not a coincidence, and some possible implications are discussed.
Thomas Krajewski - Quasi-Quantum Groups from Strings (slides)
Motivated by string theory on the orbifold ${\cal M}/G$ in presence of a Kalb-Ramond field strength $H$, we define the operators that lift the group action to the twisted sectors. These operators turn out to generate the quasi-quantum group $D_{\omega}[G]$, introduced in the context of conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche, with $\omega$ a 3-cocycle determined by a series of cohomological equations in a tricomplex combining de Rham, \u{C}ech and group cohomologies. We further illustrate some properties of the quasi-quantum group from a string theoretical point
of view.
João Martins - Three-Dimensional Holonomy
I will report on recent work with Roger Picken and Björn Gohla on three-dimensional holonomy based on a Lie 2-crossed module.
Aleksandar Mikovic - Finiteness and semi-classical limit of EPRL-FK spin foam models
In this talk I will describe a construction of Extended TQFT from gauge theories using biadjoint pairs of functors. Each manifold is associated to a groupoid of connections and gauge transformations, and there are restriction maps between groupoids associated to inclusions of submanifolds, and in particular to boundaries. The restriction and induction functors for groupoid representations are two-sided adjoints, and this allows the construction of ETQFT's in codimension 2 as 2-functors into 2Vect. I will also give a sketch of higher versions of this construction giving codimension-(n+1) TQFT's for n-groups via representations of n-groupoids.
For a finite group G there is a well known Quantum field theory called Dijkgraaf-Witten theory. This can be described as an extended 3d TFT. From that theory one can extract an interesting tensor category which can also be described as the representation category of a Quantum group D(G) (the Drinfel'd double of G). We present an equivariant extension of Dijkgraaf-Witten theory. This leads us to equivariant generalizations of D(G) and its represenation categories. We furthermore discuss the issue of modularity and the orbifold theory.
We will look at the models and classifications of HQFTs so far known and some of the interpretations of them. If there is time, I will discuss what might be expected and, perhaps more importantly what is there next to do following on from the paper by Porter and Turaev, and Turaev's recent book.
Hisham Sati - M-theory and Generalized Cohomology
Urs Schreiber - Infinity-connections and their Chern-Simons functionals
The notion of principal bundles/gerbes with connection makes sense in quite some generality in higher geometry. I give an explicit realization of the general concept in the smooth context in terms of Čech cocycles with coefficients in L-infinity algebra valued differential forms. This allows for an explicit realization of the higher Chern-Weil homomorphism on these connections and I briefly indicate one or two familiar examples of the resulting higher Chern-Simons functionals.
Related material for this is given here: http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+a+cohesive+topos
In this talk we wish to discuss an explicit interpretation of a class of n-gerbes with multi-layered connections in terms of transgression of a fibrewise closed q-form on a fibration to a closed (q+1)-form on the base manifold, with the basic example of the Euler class of an oriented vector bundle
in mind. Picken's and Ferreira-Gothendieck's n-gerbopoles are discussed from this point of view. Furthermore, string structures (à la Cocquereaux-Pilch and à la S.-Wurzbacher) are briefly addressed and recast within the proposed framework.
Jamie Vicary - 123 TQFTs
I will present some new results on classifying 123 TQFTs, using a 2-categorical approach. The invariants defined by a TQFT are described using a new graphical calculus, which makes them easier to define and to work with. Some new and interesting physical phenomena are brought out by this perspective, which we investigate. We finishing by banishing some TQFT myths! This talk is based on joint work with Bruce Bartlett, Chris Schommer-Pries and Chris Douglas.
Certain gauge theories on loop spaces arise from string theories (with B-field) by transgression. I will characterize these gauge theories by imposing conditions and reqiuring addtional structure, most prominently a "fusion" product for the gauge field. Further, I will define regression for such gauge theories: a procedure that converts them to a string theory (with B-field) and that constitutes the inverse of transgression.
Derek Wise - 2-group representations and geometry
Just as work in gauge theory often requires knowledge of group representations, 2-group representations are naturally important for higher gauge theory. I will review some results on the representation theory of 2-groups, with particular emphasis on its geometric aspects.