Geometry and Topology

Eli Hawkins

Aug 11, 14:30 - 15:30

MC 107

In order to study deformations of associative algebras, in 1963, Gerstenhaber constructed operations on the Hochschild complex of an algebra. These operations induce a product and bracket on cohomology, which generalize those on the space of multivector fields over a manifold. This is now known as a Gerstenhaber algebra structure.
Now consider a functor from a small category to a category of algebras. In 1988, Gerstenhaber and Schack showed indirectly that the Hochschild cohomology of such a functor is also a Gerstenhaber algebra. I will describe my efforts to construct operations on the complex that induce this structure.
My motivation comes from algebraic quantum field theory. My methods rely on operad theory.

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 $(M,\omega)$.