Geometry/Topology Seminar
Swarnava Mukhopadhyay
TIFR (Mumbai, India)
Hitchin Connection for parabolic bundles
Abstract: Given an family of polarized abelian varieties, let $\mathcal{L}$ be a relatively ample line bundle, consider the family effective divisors on the fibers or equivalently the space of theta-functions induced by $\mathcal{L}$. Mumford-Welters work endowed this family with a flat projective connection which is realized via the heat equations. Hitchin's paper on {\em Flat connections and geometric quantizations, 1990} generalizes the above construction. Namely, for a complex simply connected Lie group $G$ and a surface $\Sigma$, the Hitchin connection is a flat projective connection on a vector bundle over $\mathcal{M}_g$, whose fiber over a compact Riemann surface $C$ ($\Sigma$ and a complex structure) is the space of holomorphic sections of natural line bundle $\mathcal{L}$ on the moduli space of principal $G$ bundles on $C$. Hitchin's work can be realized as an initial step in the direction of Witten's proposal of quantization of Chern-Simons theory via the method of geometric quantization.
In this talk, we will discuss the construction of a Hitchin-type connection for the moduli of parabolic bundles. We will start with a more general framework of constructing connections from heat operators with a given symbol map due to van Geeman-de Jong. We will also identify our connection with the Wess-Zumino-Witten/Tsuchiya-Ueno-Yamada connection in conformal field theory generalizing a result of Laszlo. If time permits, we will make further comments on the abelian case. This is a joint work with Indranil Biswas and Richard Wentworth.
Friday April 4, 2025 at 1:30 PM in 512 SEO