Quantization Commutes with Reduction of Chern-Simons Gauge Theory

TL;DR

This work establishes an infinite‑dimensional analogue of the Guillemin–Sternberg quantization–reduction principle for genus‑one Chern–Simons gauge theory. The authors prove an isomorphism between equivariant holomorphic sections on the space of connections $H^0_{\\Sigma G}(\\mathcal{A},\\mathcal{L}^k)$ and the holomorphic sections of the descended line bundle on the moduli $H^0(M_G[\\tau],\\widetilde{\\mathcal{L}}^k)$, using the Yang–Mills flow to implement reduction in the infinite‑dimensional setting. Key technical contributions include an equivariant isometry between determinant/Pfaffian line bundles and the Chern–Simons line bundle at genus one, and a demonstration that the zeta‑regularized determinant is nondecreasing along the flow, with a detailed analysis of regular/ poly‑stable loci. The results link geometric quantization, moduli of holomorphic bundles on elliptic curves, and loop‑group representations, yielding a rigorous infinite‑dimensional QR theorem with implications for genus‑one conformal blocks and the Verlinde framework.

Abstract

We prove an infinite-dimensional version of "quantization commutes with reduction" in the framework of geometric quantization of Chern-Simons gauge theory, focusing on the genus one case. The proof is complex-analytic and relies on the Atiyah-Bott stack and the Chern-Simons line bundle.

Theorems & Definitions (60)

• Theorem 1.1
• Proposition 2.1: Atiyah1983TheYE
• Definition 2.2
• Proposition 2.3: Freed1995ClassicalCT
• Definition 2.4
• Theorem 2.5
• Proposition 2.6
• proof
• Theorem 2.7
• proof