On a quantization of (semi-infinite) Hodge theory

Topological string theory at genus zero corresponds
to Hodge theory in mathematics, and the higher genus theory
is considered as a quantization of the genus zero theory.
In 1993, Witten proposed that the B-model partition function of
a Calabi-Yau 3-fold X should be viewed as an element of
the Fock space produced from the symplectic vector space
H^3(X,C) via canonical (Schroedinger) quantization.
On the A-model side, Givental introduced a quantization based on
an infinite-dimensional symplectic vector space (loop space)
to describe various relationships between higher genus
Gromov-Witten potentials.

In this talk, I describe joint work with Tom Coates on
how to produce an analytic version of Givental's Fock space
from a semi-infinite Hodge theory.
As an application, we can formulate (and prove in some cases)
the crepant resolution conjecture for higher genus
Gromov-Witten theory in a precise way.

The talk will start from an elementary level:
I do not assume any knowledge on (semi-infinite)
Hodge theory or Givental's formalism.