Lorentzian complex powers and spectral zeta function densities
Montag, 4.7.22, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
The spectral theory of the Laplace–Beltrami operator on Riemannian manifolds is known to be intimately related to geometric invariants. This kind of relationships has inspired many developments in relativistic physics, but a priori it only applies to the case of Euclidean signature. In contrast, the physical setting of Lorentzian manifolds has remained problematic for very fundamental reasons. \n\nIn this talk I will present results that demonstrate that there is a well-posed Lorentzian spectral theory nevertheless, and moreover, it is related to Lorentzian geometry in a way that parallels the Euclidean case to a large extent. In particular, in a recent work with Nguyen Viet Dang (Sorbonne Université), we show that the scalar curvature can be obtained as the pole of a spectral zeta function density. The proof indicates that a key role is played by the dynamics of the null geodesic flow and its asymptotic properties. \n\nThe primary consequence is that gravity can be obtained from a spectral action; I will also outline furthermore motivation coming from Quantum Field Theory on curved spacetimes. \n
A Lie algebra constructed from BPS states of a torus conformal field theory
Montag, 11.7.22, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
Conformal field theories with extended supersymmetry contain a distinguished subspace of BPS (Bogomol'nyi–Prasad–Sommerfield) states. In 1995, physicists Jeffrey Harvey and Gregory Moore proposed a multiplication map to promote such spaces to algebras. I will present my attempt at formalizing this mathematically, specifically for the example theory emerging in the study of a heterotic string on a torus. The result is a Lie bracket on a space obtained from BPS states by a kind of subquotient construction. I intend to highlight, in particular, the very different roles played by the bosonic and fermionic side of the theory in this definition.
Well-posedness of the Laplacian with pure Neumann boundary conditions on domains with corners and cusps
Montag, 18.7.22, 16:15-17:15, Raum 127, Ernst-Zermelo-Str. 1
The Laplacian is invertible/fredholm on a smooth domain with Dirichlet/Neumann boundary conditions on the usual sobolev scale. This is still true for the Dirichlet case on a domain with corners, but no longer holds for the Neumann case. I will show that the statement can be recovered by introducing weighted sobolev spaces and furthermore I want to show that a similar statement holds for cusps.