  For the KPZ equation on a torus with a $1+1$ spacetime white noise, it was
shown in \cite{GK21,ADYGTK22} that the height function satisfies a central
limit theorem, and the variance can be written as the expectation of an
exponential functional of Brownian bridges. In this paper, we consider another
physically relevant quantity, the winding number of the directed polymer on a
cylinder, or equivalently, the displacement of the directed polymer endpoint in
a spatially periodic random environment. It was shown in \cite{YGTK22} that the
polymer endpoint satisfies a central limit theorem on diffusive scales. The
main result of this paper is an explicit expression of the effective
diffusivity, in terms of the expectation of another exponential functional of
Brownian bridges. Our argument is based on a combination of tools from
Malliavin calculus, homogenization, and diffusion in distribution-valued random
environments.
