Limits of quantization from mixed to real polarizations on toric varieties

Abstract

Let $(M, \omega, J)$ be a $2n$-dimensional toric variety determined by a Delzant polytope $P$, whose $T^{n}$-symmetry determines a real polarization $\mathcal{P}_{\mathbb{R}}$. Let $K \subset T^{n}$ be a subtorus. By a construction due to Leung and the first author, the $K$-action induces a mixed polarization $\mathcal{P}_{K}$. This paper investigates the relationship between the quantum Hilbert spaces $\mathcal{H}_{K}$ and $\mathcal{H}_{\mathbb{R}}$ associated with the polarizations $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$. Starting from $\mathcal{P}_{K}$, we use an imaginary-time flow to construct a one-parameter family of mixed polarizations $\mathcal{P}_{K,t}$ on $M$ interpolating between $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$, with $\mathcal{P}_{K,0}=\mathcal{P}_{K}$ and $\lim_{t\to\infty}\mathcal{P}_{K,t}=\mathcal{P}_{\mathbb{R}}$. For the corresponding quantum Hilbert spaces $\mathcal{H}_{K,t}$, we lift the imaginary-time flow to the prequantum line bundle to obtain a $T^{n}$-equivariant isomorphism $\mathcal{H}_{K}\cong\mathcal{H}_{K,t}$. We finally show that $\mathcal{H}_{K,t}$ converges to $\mathcal{H}_{\mathbb{R}}$ as $t\to\infty$.