[Paper Review] Torsion type invariants of singularities

Huijun Fan, Hao Fang|arXiv (Cornell University)|Mar 21, 2016

Geometry and complex manifolds22 references3 citations

TL;DR

This paper establishes the existence of a heat kernel expansion for the Schrödinger operator associated with a non-degenerate quasi-homogeneous singularity on $\mathbb{C}^n$, proving a local index formula that expresses the Milnor number as a Gaussian-type integral. It introduces new torsion-type spectral invariants derived from the heat trace, providing a novel analytic approach to singularity invariants in the context of the LG/CY correspondence.

ABSTRACT

Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on ${\mathbb C}^n$ by a quasi-homogeneous polynomial $f$. Under some mild assumption on $f$, we show that the small time heat kernel expansion of the corresponding Schrödinger operator exists and is a series of fractional powers of time $t$. Then we prove a local index formula which expresses the Milnor number of $f$ by a Gaussian type integral. Furthermore, the heat kernel expansion provides spectral invariants of $f$. Especially, we define torsion type invariants associated to a singularity. These spectral invariants provide a new direction to study the singularity.

Motivation & Objective

• To develop local index theory for the $\bar{\partial}_f$-Laplacian on singularities defined by quasi-homogeneous polynomials.
• To establish the existence of a small-time heat kernel expansion in fractional powers of time for non-compact, matrix-type Schrödinger operators with unbounded potentials.
• To define and study torsion-type spectral invariants for singularities, motivated by the BCOV torsion in Calabi-Yau mirror symmetry.
• To prove a local index formula expressing the Milnor number as a Gaussian integral via heat kernel trace asymptotics.
• To explore the behavior of these invariants under direct sums of singularities, particularly in the $A_r$ case.

Proposed method

• Prove that the heat kernel $e^{-t\Delta_f}$ is of trace class under mild tameness and weight conditions on the quasi-homogeneous polynomial $f$.
• Construct a parametrix $P_k$ for the operator $L = \partial_t + \Delta_f$ using iterative error terms $R_k$, ensuring convergence of the series $P = \sum_{i=0}^\infty (-1)^i P^i_k$.
• Establish the heat kernel expansion $\operatorname{tr}(e^{-t\Delta_f}) \sim \sum_{j=0}^\infty a_j t^{j/2}$ as $t \to 0^+$, with coefficients depending on local geometry and potential growth.
• Derive the local index formula by relating the $L^2$ $\bar{\partial}_f$-cohomology Euler characteristic to the Gaussian integral of the heat kernel trace.
• Define the spectral torsion $T^2(f)$ via the zeta-regularized trace of the heat kernel, using $\Theta^2_f(s) = \sum_{i=1}^\infty \lambda_i^{-s}$ for eigenvalues $\lambda_i$ of $\Delta_f^0$.
• Analyze the behavior of $T^2(f)$ under direct sums of singularities, proving a multiplicative anomaly formula involving Milnor numbers and alternating signs.

Experimental results

Research questions

• RQ1 Does the heat kernel of the $\bar{\partial}_f$-Laplacian on $\mathbb{C}^n$ admit an expansion in fractional powers of time under mild conditions on $f$?
• RQ2 Can the Milnor number of a non-degenerate quasi-homogeneous singularity be expressed as a Gaussian-type integral via spectral data?
• RQ3 What are the properties of torsion-type invariants derived from the $\bar{\partial}_f$-Laplacian's spectrum, and how do they behave under direct sums of singularities?
• RQ4 How does the spectral zeta function $\Theta^2_f(s)$ behave for $A_r$-type singularities, and what is the resulting analytic torsion?
• RQ5 Is there a multiplicative anomaly formula for the torsion invariant $T^2(f_1 \oplus f_2)$ in terms of the Milnor numbers and $T^2(f_1), T^2(f_2)$?

Key findings

• The heat kernel $P(\mathbf{z}, \mathbf{w}, t)$ for $\partial_t + \Delta_f$ exists and converges as a series of fractional powers of $t$ under the condition $q_M - q_m < \frac{1}{3}$.
• The Milnor number $\mu(f)$ is given by the Gaussian integral $\int_{\mathbb{C}^n} \operatorname{tr}(P(\mathbf{z}, \mathbf{z}, t)) \, d\mathbf{z}$ as $t \to 0^+$, providing a local index formula.
• The spectral torsion $T^2(f)$ is defined via the zeta-regularized trace of the heat kernel, and for the $A_r$ singularity $f = \frac{\tau z^2}{2}$, it evaluates to $T^2(f) = (2|\tau|)^{-1/12} e^{-\zeta'(-1)}$.
• The torsion invariant satisfies the multiplicative anomaly formula $\log T^2(f_1 \oplus f_2) = (-1)^{n_1}\mu(f_1)\log T^2(f_2) + (-1)^{n_2}\mu(f_2)\log T^2(f_1)$ for direct sums of singularities.
• The heat kernel expansion is valid for $k > \frac{3l_0 + n + 1 + 2\sum q_i}{\delta}$ with $\delta = \frac{1 - 3(q_M - q_m)}{3(1 - q_M)}$, ensuring convergence and smoothness up to specified derivatives.
• For the 1D harmonic oscillator case ($f = \frac{\tau z^2}{2}$), the spectral zeta function is $\Theta^2_f(s) = (2|\tau|)^{-s} \zeta(s-1)$, with $\zeta(s)$ the Riemann zeta function.