abstract: We compare two approaches to make path integrals on curved spaces mathematically rigorous: The first one based on Wiener measure and the second one based on a generalization of measure theoretic integrals which we call “renormalized integrals”. We present a rigorous path integral formula or the heat kernel. We conclude by pointing out a strange phenomenon for the determinant of the Dirac operator on the sphere which can be intepreted as a Gaussian functional integral.

abstract: The main goal of this talk is description of the recurrent (by n) construction of Baxter Q-operator for sl(n)-invariant Heisenberg spin chain.

abstract:I present a generalization of the framework of locally covariant field theory to accommodate fields charged under a gauge group. The background metric and the background gauge connection are treated on equal footing, as well as isometries and gauge transformations. As an application, I discuss a locally covariant definition of the current density.

abstract: Presently quantum field theory is undergoing significant changes which started with the solution of an old problem: the quantum field theory behind the third positive energy Wigner representation class (“infinite spin”) in terms of of string-localized generating fields. This in turn led to foundational changes for the other two classes: the m>0 and the m=0 finite helicity class The stringlocal description of these two classes leads among other things to the extension of renormalizability for any spin, and the “adiabatic equivalence principle” makes sure that the physical content is the same as in the nonrenormalizable pointlike description i.e. stringlocal fields play the role of a different “field coordinatization” of the same QFT. The affiliation to the same Borchersclass attributes to the pointlike description that of a nontemperate (Jaffe) field which depends on the same finite set of coupling parameters as the renormalizable string like description. The special case m>0, s=1 amounts to a Hilbert space analog of the BRST operator gauge description in Krein space in which the subalgebra of pointlike fields coincides with the gauge invariant observables. The stringlike approach leads to new insights into the Higgs mechanism and confinement. Last not least it reveals that the noncompact localization properties of the “third Wigner class” matter as those which astrophysicists ascribe to dark matter.

abstract: The discovery of black hole radiation (Hawking, 1975) was soon followed by the conjecture that a free scalar quantum field on the extended Schwarzschild black hole (Kruskal space-time) admits a unique ground state, which restricts to a thermal state at the Hawking temperature in the exterior region (Hartle and Hawking, 1976; Israel, 1976). This conjecture was later extended to more general static black holes (Jacobson, and the state is known as the Hartle-Hawking-Israel state (HHI-state). It is formally defined by a Wick rotation, starting from a corresponding Euclidean ground state. Since 1975, the study of quantum fields in curved spacetimes has made much progress, but a full proof of the existence of the HHI-state and its regularity (i.e. Hadamard property) across the horizon has remained elusive. We will present a proof of the existence of the HHI-state on a class of static black holes, focussing especially on the mathematical techniques needed to analyse its properties near the bifurcation surface of the spacetime.

abstract: Higher-spin gauge theories are extensions of gravity by massless higher-spin fields. I will review what higher-spin gauge fields are, why it is so difficult to formulate interacting theories, and what has been achieved so far. At the end I want to give a glimpse into the higher-spin AdS/CFT correspondence that has triggered much of the interest in higher-spin gauge theories in recent years.

abstract: an overview talk

abstract: The framework of Connes’ noncommutative geometry provides a noncommutative generalisation of Riemannian manifolds. Of particular interest to physics is the special case of almost-commutative manifolds, which can be used for a description of (classical) gauge theories. In particular, it allows for a derivation of the Standard Model in high energy physics from noncommutative-geometric principles, while at the same time including the coupling with gravity. In this talk I will discuss how these ideas could be extended to provide a spectral generalisation of Lorentzian manifolds as well.

abstract:A short review is given of some recent developments on non-perturbative phenomena in certain N=2 supersymmetric field theories. The main focus will be the relations to conformal field theory and applications to the study of electric-magnetic duality phenomena.

abstract: The thermal equilibrium states (KMS states) of the field algebra of a Moyal-deformed but Poincare covariant QFT are investigated. It is shown that to each temperature, there exist uncountably many KMS functionals. However, only one of these functionals allows for a consistent probability interpretation (positivity). It is shown how algebraic techniques can be successfully applied to solve this positivity question, whereas direct calculations run into problems. (joint work with Jan Schlemmer)

abstract: DeSitter spacetime plays an important role in the context of cosmology, both during the inflationary epoch (early universe), as well as during the present epoch of accelerated expansion. Quantum fields play an important role in particular in the early universe, since their fluctuations are thought to have left their imprint in the CMB, and to have also served as the seeds for the subsequent structure formation process. In the quantum field theoretic language, what one wants to study in this context are the correlation functions of fields (possibly with self-interaction), propagating in a deSitter spacetime. I review recent progress in this area, paying particular attention to the question of “IR-stability” and the “cosmic no-hair” phenomenon.

abstract:Quantum spin chains are mathematically described in terms of states/ positive linear functionals on quasi-local C*-algebras. From an operational point of view one is interested in their restrictions to local subalgebras. A general question is how to identify a state by means of locally performed measurements. We want to discuss some specific problems of discrimination between shift-invariant states. We should highlight how the solutions rely on or are related to corresponding results from mathematical statistics.

abstract:The KMS condition characterizes thermal equilibrium states in quantum statistical mechanics and quantum field theory. It is based on certain analytical and periodicity conditions of correlation functions. The characterization of non-equilibrium states which locally still have thermal properties constitues a challenge in quantum field theory. Analyzing the analyticity properties of KMS states, a proposal for a generalized KMS condition is made in the case of the free massless scalar field. The relations of that condition to a related proposal for characterizing local thermal equilibrium states by D. Buchholz et al. are investigated.

abstract: Q-systems are devices to characterize and define relatively local extensions of a given local QFT in terms of the superselection sectors of the latter. Various operations with Q-systems: decompositions, products, the “center”, etc, have simple algebraic interpretations in terms of the corresponding QFT extensions. These recent insights can be used for the construction of new extensions, for a better algebraic understanding of old constructions, and for a description and classification of phase boundaries between a pair of local QFTs.