Zeta-Regularized Determinant Overview

• Zeta-regularized determinants are analytic constructs that assign finite values to divergent products of operator eigenvalues using spectral zeta functions.
• They extend to elliptic and pseudodifferential operators, yielding explicit formulas and anomaly corrections crucial for spectral geometry and quantum field theory.
• Their application in gluing formulas, comparison theorems, and discrete approximations unifies concepts across geometric analysis, noncommutative geometry, and mathematical physics.

A zeta-regularized determinant is an analytic construction that assigns a finite value to the multiplicative spectral invariant $\prod_j \lambda_j$ for a suitable elliptic or self-adjoint operator, regularizing the divergent infinite product of its nonzero eigenvalues by means of the associated spectral zeta function and its analytic continuation. This apparatus plays a central role in geometric analysis, spectral theory, mathematical physics, noncommutative geometry, and quantum field theory, and underlies a wide range of comparison, gluing, and anomaly formulas for spectral invariants.

1. Spectral Zeta Function and Regularization Framework

For a positive, self-adjoint operator $L$ with discrete spectrum $0<\lambda_1\leq\lambda_2\leq \dotsb\uparrow\infty$, the spectral zeta function is defined by

$\zeta(s;L)=\sum_{j=1}^\infty \lambda_j^{-s}$

for $\Re s$ sufficiently large. For classical Laplacians and broad classes of elliptic operators, $\zeta(s;L)$ admits meromorphic continuation to $\mathbb{C}$, is regular at $s=0$, and often satisfies additional analytic properties (e.g., "discrete dimension spectrum" (Hartmann et al., 2021)).

The zeta-regularized determinant is then

$\det_\zeta(L)=\exp\left(-\zeta'(0;L)\right)$

which is the exponential of the negative derivative of the meromorphic continuation at $s=0$. This reconciles the divergent product $\prod_j \lambda_j$ by interpreting

$-\zeta'(0;L) = -\sum_{j=1}^\infty \ln \lambda_j$

as a regularized sum, rendering the determinant well-defined for a large class of operators under suitable spectral and asymptotic hypotheses (Momeni et al., 2011, Hartmann et al., 2021, Gesztesy et al., 2017).

In the presence of zero modes (i.e., $\ker L\ne 0$), the reduced determinant is used, omitting the null eigenvalues.

2. Analytic Properties and Extension to Pseudodifferential Operators

The definition above extends naturally to classical elliptic pseudodifferential operators $A$ of positive order on closed manifolds, using the Seeley calculus of complex powers, the operator trace, and meromorphic continuation of $\zeta(s;A) = \operatorname{Tr}(A^{-s})$ (Friedlander, 2018, Maniccia et al., 2013). For classical SG-pseudodifferential operators acting on noncompact settings, the canonical trace of Kontsevich–Vishik and finite-part integrals furnish the required functional-analytic and symbolic inputs for zeta-regularization, with pole structure controlled by homogeneous symbol expansions in phase space (Maniccia et al., 2013).

For operator pencils or parameter-dependent families, the zeta-determinant extends by holomorphic parameter-tracing, and for infinite direct sums (e.g., in separation-of-variables decompositions or metric graphs), polyhomogeneous resolvent expansions yield regularized formulas relating the infinite sum of individual determinants with boundary-local correction terms (Lesch et al., 2013, Texier, 2010).

3. Comparison and Multiplicative Anomaly Formulas

A fundamental feature of zeta-regularized determinants is their behavior under modification of the underlying operator, particularly under singular perturbations or gluing changes. For self-adjoint extensions of Laplacians on punctured compact manifolds (pseudo-Laplacians), explicit comparison theorems of the form

$$\det_\zeta(\Delta_\alpha - \lambda) = C\,\sigma(\lambda,\alpha)\, \det_\zeta(\Delta_X - \lambda)$$

hold, where $\Delta_\alpha$ denotes an extension indexed by $\alpha$, $\sigma(\lambda,\alpha)$ encodes the effect of the local boundary condition, and $C$ is an explicit constant depending on the geometry and dimension (Aissiou et al., 2012).

For pairs $(A, T)$ of elliptic operators, the classical multiplicativity $\det(AB) = \det(A)\det(B)$ generally fails once Fredholm and zeta-regularization are present. The multiplicative anomaly is given by explicit finite-part trace or Guillemin–Wodzicki residue formulas, measuring the noncommutativity of spectral and symbolic data: $\log\det(A(I+T)) - \log\det A - \log\det_m(I+T)$ is expressed in terms of finite-part traces and local noncommutative residue integrals (Friedlander, 2018).

For regularized Fredholm and zeta-determinants, precise expansions and comparison formulas link the two notions: $$\frac{\det_\zeta(L+z)}{\det_\zeta(L)} = \exp\left(\sum_{j=1}^{p-1}\frac{z^j}{j!}\frac{d^j}{dz^j}\log\det_\zeta(L+z)\big|_{z=0}\right)\det_p(I+z L^{-1})$$ where the higher Taylor coefficients are controlled by heat kernel asymptotics and zeta values (Hartmann et al., 2021, Gesztesy et al., 2017).

4. Explicit Formulas and Special Geometries

Zeta-regularized determinants admit closed formulas in a variety of geometric contexts:

• Automorphic Laplacians and Selberg Zeta: On compact hyperbolic surfaces or for automorphic Laplacians, the determinant of shifted Laplacians is explicitly related to the Selberg zeta function and various gamma and scattering matrix data:

$$\det_\zeta(\Delta - s(1-s)) \propto Z_\Gamma(s) \times \text{(elliptic, parabolic, identity factors)}$$

with function-theoretic implications and arithmetic factorization reflecting the underlying group symmetries (Momeni et al., 2011, Kurokawa et al., 2010).

• Metric Graphs: For quantum graphs, the zeta-regularized determinant is given via periodic-orbit expansions, Roth trace formulas, or explicit vertex/bond matrix expressions. In the case of general self-adjoint boundary conditions, one conjectures universal determinant expressions involving characteristic matrices and local Wronskian data (Texier, 2010).
• Polygonal and Fractal Domains: In polygonal domains, the zeta-determinant appears as the constant term in the large-lattice-size expansion of discrete Laplacian determinants (e.g., for domains $\Omega_L$ discretizing a continuum $\Omega$). For fractals with spectral decimation, the regularized Laplacian determinant is identified as the constant term in the expansion of the graph Laplacian determinant after subtraction of exponential and logarithmic divergence terms (Greenblatt, 2021, Tsougkas, 2023, Chen et al., 2016).
• One-Dimensional Laplace-Type Operators: On the circle or intervals, explicit heat kernel/Volterra expansions combined with analytic continuation yield closed or asymptotic formulas for functional determinants, underpinning links with integrable hierarchies (e.g., generalized KdV Hamiltonians) and explicit computations for Sturm–Liouville operators (Avramidi, 2014, Fucci et al., 2021, Gesztesy et al., 2017).
• Manifolds with Singularities: For Laplacians on surfaces with conical points or deformed spheres (Riemann caps, conical tori), the zeta-determinant incorporates explicit dependencies on moduli or singularity parameters, with precise formulas derived via heat kernel asymptotics, contour integral methods, or uniform expansions for special functions (Kalvin et al., 2017, Flachi et al., 2010).

5. Relations to Discrete Determinants and Approximation Limits

The asymptotic behavior of discrete Laplacian determinants on lattice approximations or finite graphs is intimately connected to zeta-regularized determinants in the continuum or limiting fractal. For example, for graphs approximating a torus,

$$\lim_{n\to\infty}\log\det\Delta_n = -\zeta'_\Delta(0) = \log\det_\zeta \Delta$$

with the precise regularization performed via Hadamard partie finie integrals and careful matching of polyhomogeneous expansions for the resolvent traces (Vertman, 2015, Tsougkas, 2023). On fractals, the discrete graph determinant expansion is of the form

$$\log\det\Delta_n = c\,m^n + n\,j\,\log\lambda + \log\det_\zeta(\Delta)$$

with $c$ and $j$ explicit, and the remainder vanishing as $n\to\infty$ (Tsougkas, 2023, Chen et al., 2016). Thus, the zeta-determinant encodes the universal "constant-term limit" in the sequence of discrete invariants.

6. Advanced Topics: Higher-Depth Determinants and Dynamical Systems

Generalizations to higher-depth determinants involve analytic continuations at negative integers $w=1-r$: $\Det_r(\Delta+z) = \exp\left(-\partial_w \zeta_\Delta(w,z)|_{w=1-r}\right)$ such that depth-$r$ Milnor–Selberg zeta functions appear naturally in higher-genus settings, realizing functional equations, Euler-product expansions, and further intricate connections with Barnes’ multiple gamma and sine functions (Kurokawa et al., 2010).

In Riemannian foliated dynamical systems, zeta-regularized determinants of infinitesimal generators acting on reduced leafwise cohomology provide determinantal formulas for dynamical zeta functions, as conjectured by Deninger, with spectral Lefschetz trace techniques underpinning the analytic continuation and functional equations (López et al., 2024).

7. Applications and Significance in Geometric Analysis and Mathematical Physics

Zeta-regularized determinants serve as canonical spectral invariants with deep connections to quantum field theory (as one-loop effective actions, Polyakov anomalies), spectral geometry (analytic torsion, Ray–Singer invariants), moduli spaces (determinant line bundles), arithmetic geometry (Selberg and Milnor zeta functions), and noncommutative residue theory (multiplicative anomalies, gluing formulas) (Aissiou et al., 2012, Momeni et al., 2011, Friedlander, 2018).

They provide a bridge between discrete combinatorial invariants (spanning tree counts, graph entropies), continuum spectral data, and arithmetic or representation-theoretic zeta functions, enabling comparisons across geometric, analytic, and arithmetic moduli. The general framework unifies diverse phenomena, from heat kernel asymptotics and regularized traces to functional equations and quantum anomalies.