[Paper Review] Asymptotics of the Hilbert-Smith norm of curve operators in TQFT

Jørgen Ellegaard Andersen|ArXiv.org|May 11, 2006

Geometric and Algebraic Topology16 references4 citations

TL;DR

This paper establishes a precise asymptotic equivalence between the Hilbert-Smith norm of curve operators in Reshetikhin-Turaev TQFTs for $SU(n)$ and the $L^2$-inner product of holonomy functions on the moduli space of flat connections. Using Toeplitz operator theory and geometric quantization, it proves that the normalized trace of TQFT curve operators converges to the symplectic volume integral of products of holonomy functions as the level $k \to \infty$, generalizing a result by March{\'e} and Narimanejad.

ABSTRACT

Applying standard techniques from Toeplitz operator theory, we analyze the asymptotics of the Hilbert-Smith norms of the TQFT operators coming from isotopy classes of one dimensional oriented submanifolds on a closed oriented surface. We thereby obtain a Toeplitz operator interpretation and generalization of the asymptotic formula obtained recently by Marche and Narimannejad.

Motivation & Objective

• To establish the large-level asymptotics of the Hilbert-Smith norm of curve operators in $SU(n)$ TQFTs.
• To connect the quantum TQFT framework with classical geometric quantization and holonomy functions on moduli spaces.
• To generalize the asymptotic formula of March{\'e} and Narimanejad using Toeplitz operator theory.
• To demonstrate that the normalized trace of TQFT operators converges to the $L^2$-inner product of holonomy functions on the moduli space.

Proposed method

• Uses geometric quantization at level $k$ on the moduli space $M$ of flat $SU(n)$-connections with fixed holonomy around a marked point.
• Applies Toeplitz operator theory to the space of holomorphic sections of the $k$-th power of the determinant line bundle over $M$, with respect to a polarization.
• Defines Toeplitz operators $T_{f,\sigma}^{(k)}$ as compositions of multiplication by smooth functions $f$ and orthogonal projections onto holomorphic sections.
• Employs asymptotic results from Bordemann, Meinrenken, and Schlichenmaier: operator norm convergence to $\sup |f|$, product approximation $T_f T_g \approx T_{fg}$ up to $O(k^{-1})$, and trace normalization to $\int_M f \omega^m / m!$.
• Leverages Theorem 3 from Andersen (2006) to relate the TQFT curve operators $Z^{(k)}(\gamma,\lambda)$ to the Toeplitz operators $T_{h_{\gamma,\lambda}}^{(k)}$ in the semiclassical limit.
• Combines trace convergence and operator norm control to show that $k^{-m} \operatorname{Tr}(Z^{(k)}(\gamma_1,\lambda_1) Z^{(k)}(\gamma_2,\lambda_2)^*) \to \langle h_{\gamma_1,\lambda_1}, h_{\gamma_2,\lambda_2} \rangle$.

Experimental results

Research questions

• RQ1 How do the Hilbert-Smith norms of curve operators in $SU(n)$ TQFTs behave in the large level limit?
• RQ2 Can the asymptotic formula for curve operator norms be derived from Toeplitz operator theory?
• RQ3 Is there a precise correspondence between the quantum TQFT trace and the classical $L^2$-inner product of holonomy functions?
• RQ4 Does the normalized trace of TQFT operators converge to the symplectic volume integral of holonomy products as $k \to \infty$?
• RQ5 What is the role of geometric quantization and polarization in relating quantum and classical invariants in TQFT?

Key findings

• The normalized trace of the product of curve operators in the $SU(n)$ TQFT converges to the $L^2$-inner product of the corresponding holonomy functions: $\lim_{k \to \infty} k^{-(g-1)(n^2-1)} \langle Z^{(k)}(\gamma_1,\lambda_1), Z^{(k)}(\gamma_2,\lambda_2) \rangle = \langle h_{\gamma_1,\lambda_1}, h_{\gamma_2,\lambda_2} \rangle$.
• The asymptotic equivalence is established via the convergence of Toeplitz operators: $\|T_{h_{\gamma,\lambda}}^{(k)} - Z^{(k)}(\gamma,\lambda)\| \to 0$ as $k \to \infty$.
• The trace normalization factor $k^{-m}$ with $m = (g-1)(n^2-1)$ matches the dimension growth of the space of holomorphic sections, ensuring convergence to the symplectic volume integral.
• The operator norm of a Toeplitz operator converges to the supremum norm of the symbol: $\lim_{k \to \infty} \|T_{f,\sigma}^{(k)}\| = \sup_M |f|$.
• The product of Toeplitz operators approximates the Toeplitz operator of the product up to $O(k^{-1})$, ensuring algebraic consistency in the semiclassical limit.
• The proof relies on the faithfulness of the Toeplitz operator assignment and the convergence of traces to symplectic integrals, as established in [BMS].