Geometric quantization of symplectic maps and Witten's asymptotic conjecture

Abstract

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of the mapping class group.

51 pages, to appear in Adv. Math

References in corpus (5)

• Toeplitz operators on symplectic manifolds
• Semi-classical properties of geometric quantization with metaplectic correction
• Teichmueller space via Kuranishi families
• Donaldson's $Q$-operators for symplectic manifolds
• Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula