Abstract: The Tomita-Takesaki theory of modular automorphisms has found many uses in quantum field theory, in particular in computing relative entropies. However, the modular Hamiltonian is only known in some cases. I present results for the full modular data (flow, Hamiltonian and involution) for conformal scalar fields in the de Sitter (Bunch-Davies) vacuum state, restricted to diamonds. I further show that in the limit where the diamonds become large and fill the full static patch, the modular Hamiltonian generates time translations in the static patch, recovering the results of Sewell and Kay for this case.

Abstract: In the talk, an algebraic topological framework for studying state spaces of quantum lattice spin systems is presented, using the frame- work of algebraic quantum mechanics. We first provide some old and new results about the state space of the quasi-local algebra of a quantum lattice spin system when endowed with either the natural metric topology or the weak *-topology. Switching to the algebraic topological side we then determine the homotopy groups of the unitary group of a UHF algebra and then show that the pure state space of any UHF algebra is simply connected. We finally indicate how these and related results may lead to a framework for constructing Kitaev’s loop spectrum of bosonic invertible gapped phases of matter. The talk is on joint work with A. Beaudry, M. Hermele, J. Moreno, M. Qi and D. Spiegel.

Abstract: The heat kernel is a central object for quantum field theory in Euclidean signature, both for one-loop perturbation theory and non-perturbative functional renormalization group methods. On generic curved backgrounds, however, the link between Euclidean and Lorentzian signature QFT via a Wick rotation is not fully understood. In this talk I will present a proposal for Wick rotating on generic globally hyperbolic manifolds by analytically continuing the lapse function in the ADM decomposition. This keeps the coordinates real and can be shown to define “admissible” complex metrics even under foliation changing diffeomorphisms.  A proof of existence of the associated Wick-rotated heat semi-group (and its kernel) will be sketched, together with a conjecture on the strict Lorentzian limit. Some aspects are illustrated with the Wick rotated de Sitter heat kernel.

Abstract: The calculation of expectation values of polynomials and moments is a fundamental problem in many areas of physics and mathematics. Depending on the applications one has in mind, the probability measure is usually the Haar measure, comes from the Brownian motion, or from the Wilson action. In the case of the Haar measure and the unitary group, such expectation values can be calculated in terms of the Weingarten functions. Recently, there has been interest in extending these results to other measures than the Haar measure, and to other compact Lie groups, such as the orthogonal group and $G_2$. In this talk, I will present a powerful framework for calculating such expectation values on compact Lie groups in a unified way. The framework is based on elementary representation-theoretic arguments and an integration by parts formula. I then discuss how this framework can be applied in the setting of lattice gauge theory. Specifically, our results generalize expectation value formulas for products of unitary Wilson loops by Chatterjee and Jafarov to arbitrary compact Lie groups. Moreover, it extends classical results by Collins and Lévy for expectation value under the Haar measure and for Brownian motion. This is joint work with Lukas Miaskiwskyi.