Speaker: Can Görmez

Title: Contact Geometry, Quantization, and the Seven Sphere

Abstract: The dynamics of a classical mechanical system can be described by a contact form on an odd-dimensional manifold. Quantization of this description is achieved by suitably associating a Schrödinger equation to paths in the contact manifold, using a contact analog of Fedosov’s formal connection on symplectic spinor bundles. Our main result is the quantization of the standard contact seven sphere, viewed as a homogeneous space of the quaternionic unitary group. This reveals an interplay between homogeneous spaces and the passage from formal to bona fide quantization. We show that requiring convergence of the formal connection naturally filters the symplectic spinor bundle and yields an exact flat connection on each corresponding subbundle. As a result, we obtain quantum dynamical systems parametrized by certain unitary representations of the model’s principal group. These representations are controlled by a generalization of the Holstein–Primakoff mechanism. This is joint work with Subhobrata Chatterjee and Andrew Waldron.