Lectures SS 2026

Speaker: Michael Preeg, Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by K.-H. Neeb

Abstract: In quantum field theory the CCR and CAR algebras play an important role. The are defined as the algebra generated by elements of complex Hilbert space $\mathcal{H}$ with respect to their corresponding relations either involving the symplectic form or the symmetric bilinear form defined by the inner product of $\mathcal{H}$. Both algebras have a natural way of acting on elements of the corresponding Fock space (symmetric or antisymmetric) defined via creation and annihilation operators. Since orthogonal and symplectic operators on $\mathcal{H}$ (understood as the underlying real Hilbert space) are preserving the corresponding CCR or CAR relations, one could ask whether the representations of the algebras are unitary equivalent before and after application of these operators.

This is answered by D. Shale and W. F. Stinespring. They are unitarily equivalent if and only if the orthogonal or symplectic operators are restricted in the sense that their antilinear part with respect to the complex strucure defined by the multiplication by $i \in \mathbb{C}$ is Hilbert-Schmidt. They are Banach-Lie groups, but this is not the appropriate topology. In this talk we will define the strongly restricted topology, explore them as topological groups and show the continuity of the projective unitary representation via the implementation on their corresponding Fock spaces.

Speaker: Vincenzo Morinelli, Università degli Studi di Roma Tor Vergata – Invited by K.-H. Neeb

Abstract: If M is asymptotically flat and conformal to Minkowski space, is it possible to define a quasi-free Hadamard state for a massless conformally coupled scalar field by holography from the unique BMS-invariant quasi-free state at null lightlike infinity.

For specific regions called deformed cones, the associated one-particle spaces are standard and decompose as direct integrals of $U(1)$ current half-line spaces. This decomposition naturally extends to the modular Hamiltonian, enabling the explicit computation of relative entropy between coherent states and the vacuum, which ultimately yields a proof of the QNEC.

(Joint work with C. Dappiaggi, V. Morinelli and A. Ranallo.)

Speaker: Patrick Kinnear, Universität Hamburg – Invited by C. Meusburger

Abstract: In the study of topological symmetry, defects are embedded submanifolds labelled by algebraic data; the evaluation of defect TQFTs (i.e. TQFTs which make sense for bordisms supporting defects) implements topological symmetry. The (2+1) TQFT of Reshetikhin and Turaev can be enhanced to a defect TQFT, using a construction called orbifolding, due to work of Carqueville–Runkel–Schaumann. This construction is a generalized state sum, and requires that the defects are labelled by semisimple data; the state spaces for surfaces are given by a projector from the state space of the original RT theory. On the other hand, given a 3-manifold whose boundary is a surface with defects, the usual definitions of skein theory can be extended to define a defect skein module. In joint work with Ingo Runkel, we define the defect skein module and prove that it is isomorphic to the defect RT state space of its boundary. This generalizes the well-known fact that the state spaces of RT theory are skein modules. Our work points to non-semisimple generalizations of defect RT theories using skein methods, while in the other direction it gives a state sum construction for defect skein modules labelled by semisimple data.

Speaker: Lorenz Schwachhöfer, Technische Universität Dortmund – Invited by K.-H. Neeb

Abstract: In the classical theory of Lie groups, Kirillov’s coadjoint orbit method establishes a profound link between algebra and geometry, equipping orbits in the dual of a Lie algebra with a canonical symplectic structure. In this talk, we explore a parallel construction developed for Jordan algebras and their $Z_2$-graded extensions, Jordan superalgebras.

We demonstrate how the Jordan product can be used to define a generalized distribution on the dual space, inducing canonical pseudo-Riemannian metrics on the resulting orbits. For formally real (Euclidean) Jordan algebras, we show that these orbits possess a natural Riemannian structure that recovers fundamental objects of information geometry: specifically, the Fisher–Rao metric in the classical setting and the Bures–Helstrom metric in the quantum setting.

This is based on joint work with Florio Ciaglia, Shuhan Jiang and Jürgen Jost.