[Paper Review] The asymptotics of the analytic torsion on CR manifolds with $S^1$ action

Chin-Yu Hsiao, Rung-Tzung Huang|arXiv (Cornell University)|May 24, 2016

Holomorphic and Operator Theory15 references3 citations

TL;DR

This paper establishes an asymptotic formula for the Fourier components of the analytic torsion on compact, strongly pseudoconvex CR manifolds equipped with a transversal CR $S^1$-action, generalizing Bismut and Vasserot's asymptotic formula for holomorphic torsion on positive line bundles to the CR setting via heat kernel and spectral methods.

ABSTRACT

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the $S^1$-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray-Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR $S^1$-action.

Motivation & Objective

• To extend the asymptotic formula for holomorphic analytic torsion—previously known for positive line bundles on complex manifolds—to the setting of strongly pseudoconvex CR manifolds with transversal $S^1$-action.
• To define and analyze the Fourier components of the Ray-Singer analytic torsion on such CR manifolds using spectral theory.
• To establish an asymptotic expansion for the analytic torsion in the high-weight limit, analogous to the complex geometry case.
• To provide a framework for studying analytic torsion on CR manifolds that arise as circle bundles over complex orbifolds, offering a smoother alternative to general orbifold methods.

Proposed method

• Introduce the Fourier decomposition of differential forms on the CR manifold $X$ with respect to the $S^1$-action, defining $\Omega^{0,\bullet}_m(X)$ as the $m$-th eigenspace of the infinitesimal generator $T$ of the $S^1$-action.
• Establish a canonical isomorphism $A_m: \Omega^{0,\bullet}_m(X) \to \Omega^{0,\bullet}(M, L^m)$, linking the CR Dolbeault complex on $X$ to the Dolbeault complex on the base complex manifold $M$ with values in high powers of a positive line bundle $L$.
• Use the heat kernel method to analyze the Kohn Laplacian $\Box_{b,m}$ on $X$ and relate its spectral properties to the Kodaira Laplacian $\Box_m$ on $M$, leveraging the isomorphism $A_m$.
• Derive the asymptotic expansion of the analytic torsion via the Mellin transform of the super-trace of the heat operator, using spectral zeta function techniques.
• Apply the heat kernel expansion and curvature trace identities to compute the leading-order term in the asymptotic formula, involving the curvature form $R^L$ and the Ricci curvature $\dot{\mathcal{R}}$ of the CR structure.
• Use the Riemann zeta function and its derivative to evaluate the spectral zeta function at zero, leading to the final asymptotic expression involving $\log \det(\frac{m \dot{R}^L}{2\pi})$.

Experimental results

Research questions

• RQ1 How does the analytic torsion behave asymptotically for high-weight Fourier components on CR manifolds with transversal $S^1$-action?
• RQ2 Can the Bismut-Vasserot asymptotic formula for holomorphic torsion on positive line bundles be generalized to the CR setting?
• RQ3 What is the precise leading-order term in the asymptotic expansion of the analytic torsion on such CR manifolds?
• RQ4 How do the curvature and spectral data of the CR manifold influence the asymptotic behavior of the torsion?

Key findings

• The paper derives an asymptotic formula for the analytic torsion's Fourier components on a compact, strongly pseudoconvex CR manifold with transversal $S^1$-action, valid as the weight $m \to \infty$.
• The leading-order term in the asymptotic expansion is given by $\frac{\operatorname{rk}F}{2} \int_M \log\left(\det\left(\frac{m\dot{R}^L}{2\pi}\right)\right) e^{m\frac{\sqrt{-1}}{2\pi}R^L} + o(m^n)$, matching the Bismut-Vasserot formula in the complex case.
• The spectral zeta function $\widetilde{\zeta}(z)$ is expressed in terms of the curvature operator $\dot{\mathcal{R}}$ and the Riemann zeta function, enabling evaluation of $\widetilde{\zeta}'(0)$ via $\zeta(0) = -1/2$ and $\zeta'(0) = -\frac{1}{2}\log(2\pi)$.
• The analytic torsion is shown to be equivalent to the $S^1$-invariant part of the heat kernel trace, with the $m$-th Fourier component of the torsion corresponding to the $L^m$-twisted Dolbeault cohomology on the base manifold.
• The curvature term $\dot{R}^L$ arises from the restriction of the Levi form to the contact distribution, and its determinant appears in the logarithmic correction to the torsion.
• The result confirms that the analytic torsion on the CR manifold $X$ (the circle bundle) reproduces the known holomorphic torsion on the base complex manifold $M$, establishing a deep link between CR geometry and complex geometry.