Ralph Blumenhagen: `Lectures on Non-geometric Strings and Noncommutative Geometry'

Abstract:

Lecture 1 : String theory basics, open strings in gauge field backgrounds, noncommutativity

Lecture 2: Non-constant fluxes, closed strings in 3-form flux backgrounds, nonassociativity, N-tri-products

Lecture 3: Non-geometric fluxes in double field theory, towards nonassociative gravity

Peter Bouwknegt: `A First Introduction to T-duality'

Abstract: In these lectures I will introduce the basics of T-duality, both from a physical and mathematical point of view. The lectures will focus on both the differential geometry and algebraic topology aspects of T-duality. Connections to noncommutative geometry will be discussed in the follow-up lectures by Mathai Varghese.

Shahn Majid: `Reconstruction and quantisation of Riemannian manifolds', `Cosmological constant from quantum spacetime', `Hodge operator in noncommutative geometry as a braided Fourier transform'

Abstract: (I:Reconstruction and quantisation of Riemannian manifolds) Based on my paper arXiv:1307.2778 (math.QA), we explain how a Riemannian structure is equivalent to equivalent to a certain type of cocycle on the exterior algebra Omega(M) of a manifold M, controlling its cleft extensions within noncommutative differential algebras. This new point of view gives a new Cartan-type formula for the Levi-Civita connection and a new point of view on the Ricci tensor. It can also be used to induce noncommutative Riemannian structures and bimodule connections as equivalent to cleft central extensions of an already noncommutative differential algebra Omega(A).

(II:Cosmological constant from quantum spacetime) Based on my recent paper Phys. Rev. D 91 (2015) 124028 (12pp) with W.-Q. Tao, We show that a hypothesis that spacetime is quantum with coordinate algebra [x_i,t]=i \lambda x_i, and spherical symmetry under rotations of the xi, essentially requires in the classical limit that the spacetime metric is the Bertotti-Robinson metric, i.e., a solution of Einstein's equations with a cosmological constant and a non-null electromagnetic field.

(III:Hodge operator in noncommutative geometry as a braided Fourier transform) We reinterpret the classical hodge operator as fourier transform long the exterior algebra regarded as a superspace. This point of view generalises to a Hodge operator on any quantum group equipped with a bicovariant calculus and we explain what it amounts to for the permutation group S_3 as a discrete noncommutative geometry and for C_q[SU_2].

Christian Saemann: `Basics of Higher Gauge Theory', `Twistor Descriptions of Higher Gauge Theories', `Quantized Multisymplectic Manifolds and Categorified Matrix Models'

Abstract: (I: Basics of Higher Gauge Theory) I give an overview over the motivation behind Higher Gauge Theory and summarize some misconceptions about its non-abelian version. I then present the necessary mathematical background such as higher gauge groups and higher gauge algebras as well as higher principal bundles.

(II: Twistor Descriptions of Higher Gauge Theories) Twistor spaces allow for a nice description of classical gauge theories. I show how this description can be extended to higher gauge theories. In particular, I demonstrate how candidates for six-dimensional superconformal field theories can be obtained in this way.

(III: Quantized Multisymplectic Manifolds and Categorified Matrix Models) The IKKT matrix model has been considered as a background independent formulation of string theory and recently, much effort has gone into showing that this matrix model gives rise to a noncommutative version of general relativity. There is evidence, however, that this approach is limited and that a full description requires categorification of the IKKT model. I discuss in detail how to obtain such a model and how quantized multisymplectic manifolds arise naturally as its solutions.

Peter Schupp: `Beyond noncommutative geometry'

Abstract: In these lectures we will review structures that are one step up from noncommutative geometry on the geometric/algebraic ladder and shall discuss aspects of their quantization: Generalized geometry, with its underlying Courant algebroid, is a higher version of Poisson geometry in at least two aspects: Its structures derive from a Courant sigma model, which is a higher version of the Poisson-sigma model. Its Dorfman bracket is a higher analog of the Schouten bracket, from the derived bracket point of view. Nambu-Poisson structures generalize Poisson manifolds in another way, by introducing higher brackets and multi-Hamiltonian dynamics. All these structures naturally appear in the study of non-geometric fluxes in string theory (which are another topic of this workshop). They seem to be also useful in the context of gravity theories. The quantization of these structures introduces non-associativity, thus going beyond noncommutative geometry. We shall illustrate some of the ensuing challenges with the formulation of quantum mechanics in the presence of a 3-cocycle.

Urs Schreiber: `Prequantum field theory and Green-Schwarz WZW terms'

Abstract: In classical field theory it is traditionally assumed that the Lagrangian is a globally defined horizontal p+1-form on the jet bundle of the field bundle. However, for important classes of of field theories there is no globally defined Lagrangian form, instead there is a p-gerbe with connection on the jet bundle which constitutes a Kostant-Souriau prequantization of the Euler-Lagrange form. This applies notably the Green-Schwarz type sigma models for all the super p-branes on curved supergravity backgrounds. I discuss the general formulation of prequantum field theory, discuss how the Noether theorem is recovered in terms of symmetries of Euler-Lagrange p-gerbes and how, when applied to the GS-WZW models, this yields the brane-charge extensions of superisometry algebras.

Mathai Varghese: `T-duality via noncommutative geometry and applications'

Abstract: Slides for Talk I