Theta Functions and Adiabatic Curvature on a Torus

Abstract

Let $M$ be a complex torus, $L_{\hat{\mu }}\to M$ be positive line bundles parametrized by $\hat{\mu }\in {\mathrm{Pic}}^0(M)$, and $E\to {\mathrm{Pic}}^0(M)$ be a vector bundle with $E{|}_{\hat{\mu }}\cong H^0(M,L_{\hat{\mu }})$. We endow the total family $\{L_{\hat{\mu }}{\}}_{\hat{\mu }}$ with a Hermitian metric that induces the $L^2$-metric on $H^0(M,L_{\hat{\mu }})$ hence on $E$. By using theta functions $\{{\theta }_m{\}}_m$ on $M\times M$ as a family of functions on the first factor $M$ with parameters in the second factor $M$, our computation of the full curvature tensor ${\Theta }_E$ of $E$ with respect to this $L^2$-metric shows that ${\Theta }_E$ is essentially an identity matrix multiplied by a constant $2$-form, which yields in particular the adiabatic curvature $c_1(E)$. After a natural base change $M\to \hat{M}$ so that $E{\times }_{\hat{M}}M:=E^{\prime }$, we also obtain that $E^{\prime }$ splits holomorphically into a direct sum of line bundles each of which is isomorphic to $L_{\hat{\mu }=0}^{\ast }$. Physically, the spaces $H^0(M,L_{\hat{\mu }})$ correspond to the lowest eigenvalue with respect to certain family of Hamiltonian operators on $M$ parametrized by $\hat{\mu }$ or in physical notation, by wave vectors $\mathbf{k}$.

References in corpus (1)

• Effect of Landau quantization on linear magnetoresistance of periodically modulated two-dimensional electron gas