# Universität AugsburgInstitut für Mathematik

Oberseminar Differentialgeometrie

Herr Miguel Pereira
Universität Augsburg

spricht am

Montag, 18. Mai 2020

um

16:00 Uhr

im

Zoom

über das Thema:

## »Geometric quantization of the cotangent bundle of a Lie group«

 Abstract: Geometric quantization is a procedure that given a symplectic manifold $(M,\omega)$ (and some additional choices of geometric data on the manifold), produces a Hilbert space $\mathcal{H}$. This assignment should satisfy some set of axioms which come from physical considerations. One of the limitations of this construction is that it depends on choices of geometric data on the symplectic manifold, namely the choice of a polarization (a distribution on $TM \otimes \mathbb{C}$ which is Lagrangian and involutive). An interesting question is then to understand the dependence of the resulting Hilbert space on this choice of polarization. For two different polarizations $\mathcal{P}, \mathcal{P}'$ (satisfying some conditions), there is a canonically defined pairing map between the Hilbert spaces. This map is a linear isomorphism, but it may not be unitary. In this talk, we are going to give an outline of the construction of geometric quantization, as well as present some original results regarding the application of geometric quantization to the cotangent bundle of a Lie group $G$. More specifically, we give an explicit family of polarizations $\mathcal{P}_{\tau,\sigma}$, for $\tau,\sigma \in \mathbb{C}$. The main results are the following. Depending of the values of $\tau,\sigma$, the polarization $\mathcal{P}_{\tau,\sigma}$ defines a foliation of $T^*G$ by Kähler manifolds, or gives $T^*G$ the structure of a Kähler manifold. We compute the Hilbert spaces associated to these polarizations. For two polarizations $\mathcal{P}_{0,\sigma}, \mathcal{P}_{0,\sigma'}$, the pairing map is unitary. The "original results" portion of this talk is joint work with José Mourão and João Nunes.

 Hierzu ergeht herzliche Einladung. Prof. Dr. Kai Cieliebak

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