PhD Thesis Defence

Baran Serajelahi

Aug 12, 13:0 - 14:0

MC 107

We will be interested in quantization in a setting where the algebraic structure on $C^{\infty}(M)$ is given by an m-ary bracket $\{.,\dots,.\}:\otimes^m C^{\infty}(M)\rightarrow C^{\infty}(M)$. Quantization in this context is the same as in the symplectic case, where we have a bracket of just two functions except that now we are interested in a correspondence $\{.,\dots,.\}\rightarrow [.,\dots,.]$, between an m-ary bracket and a generalizeation of the commutator. In particular we will be interested in two situations where the m-ary bracket comes from an $(m-1)$-plectic form defined on M (i.e. a closed non-degenerate $m$-form), $\Omega$, for $m\ge 1$. The case $m=1$ is when $\Omega$ is symplectic. Let $(M,\omega)$ be a compact connected integral K\"ahler manifold of complex dimension $n$. In both of the cases that we will be looking into, the $(m-1)$-plectic form $\Omega$ on $(M,\omega)$ is constructed from a K\"ahler form (or forms):
(I) $m=2n$, $\Omega = \frac{\omega^n}{n!}$
(II) $M$ is, moreover, hyperk\"ahler, $m=4$, $$ \Omega = \omega_1\wedge \omega_1 + \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3 $$ where $\omega_1, \omega_2, \omega_3$ are the three K\"ahler forms on $M$ given by the hyperk\"ahler structure.
It is well-known (and easy to prove) that a volume form on an oriented $N$-dimensional manifold is an $(N-1)$-plectic form, and that the $4$-form above is a $3$-plectic form on a hyperk\"ahler manifold.
It is intuitively clear that in these two cases the classical multisymplectic system is essentially built from Hamiltonian system(s) and it should be possible to quantize $(M,\Omega)$ using the (Berezin-Toeplitz) quantization of $(M,\omega)$.