Multi-loop zeta function regularization and spectral cutoff in curved spacetime

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.     

Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

We investigate the existence of the meromorphic extension of the spectral zeta function of a Laplacian on self-similar fractals using the results of Kigami and Lapidus (based on renewal theory) and the newer results by Hambly and Kajino based on heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true for the examples that have been analyzed in the existing literature.     

Zeta function regularization of path integrals in curved spacetime

This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.     

On the ζ-function of a one-dimensional classical system of hard-rods

The -function of a one-dimensional classical hard-rod system with exponential pair interaction is defined as the generating function for the partition function of the system with periodic boundary conditions. It is shown, here, that the -function for this system is simply related to the traces of the restrictions of the Ruelle's transfer matrix, and related operators to a suitable function space. This -function does not, in general, extend to a meromorphic function.     

Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators

This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s=1, and further that it has a zero at all non-positive even integers, i.e. at s=0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.Work in part supported by Grant-in Aid for Scientific Research (B) No. 16340038, Japan Society for the promotion of ScienceWork in part supported by Grant-in Aid for Scientific Research (B) No. 15340012, Japan Society for the promotion of Science     

Heat Kernel Coefficients for Laplace Operators on the Spherical Suspension

In this paper we compute the coefficients of the heat kernel asymptotic expansion for Laplace operators acting on scalar functions defined on the so called spherical suspension (or Riemann cap) subjected to Dirichlet boundary conditions. By utilizing a contour integral representation of the spectral zeta function for the Laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure. Thanks to the well known relation between the zeta function and the heat kernel obtainable via Mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions. The particular case of a d-dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given.     

Meromorphic continuation approach to noncommutative geometry

Following an idea of Nigel Higson, we develop a method for proving the existence of a meromorphic continuation for some spectral zeta functions. The method is based on algebras of generalized differential operators. The main theorem states, under some conditions, the existence of a meromorphic continuation, a localization of the poles in supports of arithmetic sequences and an upper bound of their order. We give an application in relation to a class of nilpotent Lie algebras.     

The transfer-matrix of a one-sided subshift of finite type with exponential interaction

We discuss the generalized transfer matrix for a one-sided subshift of finite type with exponential interaction. Using a trace formula due to Ruelle we then calculate the function of this system. It is holomorphic in a neighbourhood of zero and can be extended to a meromorphic function in the complex z plane. This generalizes results of Bowen and Lanford who studied the function for subshifts without interactions and of Viswanathan who calculated the function of a classical lattice gas with exponential-polynomial interaction.Work supported by DFG fellowship Ma 633/1.     

Poles of Zeta and Eta Functions for Perturbations¶of the Atiyah–Patodi–Singer Problem

The zeta and eta functions of a differential operator of Dirac-type on a compact n-dimensional manifold, provided with a well-posed pseudodifferential boundary condition, have been shown in [G99] to be meromorphic on ℂ with simple or double poles on the real axis. Extending results from [G99] we show how perturbations of the boundary condition of order −J affect the poles; in particular they preserve a possible regularity of zeta at 0 and a possible simple pole of eta at 0 when J≥n. This applies to perturbations of spectral boundary conditions, also when the structure is non-product and the problem is non-selfadjoint. Received: 4 October 1999 / Accepted: 7 July 2000     

Algebro-geometric Constructions of Quasi Periodic Flows of the Discrete Self-dual Network Hierarchy and Applicationsℰ

In this paper we obtain the discrete integrable self-dual network hierarchy associated with a discrete spectral problem. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the discrete self-dual network hierarchy are straightened using the Abel-Jacobi coordinates. The meromorphic function and the Baker-Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the discrete self-dual network hierarchy are constructed with the help of the asymptotic properties and the algebra-geometric characters of the meromorphic function, the Baker-Akhiezer function and the hyperelliptic curve.     

Analytic Surgery of the Zeta Function

In this paper we study the asymptotic behavior (in the sense of meromorphic functions) of the zeta function of a Laplace-type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface, separating the manifold into two manifolds with infinite cylindrical ends. We also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary, forming a manifold with an infinite cylindrical end. Such singular deformations fall under the category of “analytic surgery”, developed originally by Hassell (Comm Anal Geom 6:255–289, 1998), Hassell et al. (Comm Anal Geom 3:115–222, 1995) and Mazzeo and Melrose (Geom Funct Anal 5:14–75, 1995) in the context of eta invariants and determinants.     

A note on the quasi-periodic solutions of the modified Boussinesq hierarchy

Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebro-geometric characters of the meromorphic function.     

A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.     

A class of solutions of the dispersionless KP equation

We construct a class of (complex-valued) solutions to the dispersionless KP equation using a meromorphic function on a plane algebraic curve as a variable dependent on suitable Abelian integrals.     

Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.     

Higher Selberg Zeta Functions

In the paper [KW2] we introduced a new type of Selberg zeta function for establishing a certain identity among the non-trivial zeroes of the Selberg zeta function and of the Riemann zeta function. We shall call this zeta function a higher Selberg zeta function. The purpose of this paper is to study the analytic properties of the higher Selberg zeta function z(s), especially to obtain the functional equation. We also describe the gamma factor of z(s) in terms of the triple sine function explicitly and, further, determine the complete higher Selberg zeta function with having a discussion of a certain generalized zeta regularization.Work in part supported by Grant-in Aid for Scientific Research (B) No.11440010, and by Grant-in Aid for Exploratory Research No.13874004, Japan Society for the Promotion of Science     

Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions

The modified Boussinesq hierarchy associated with the 3×3 matrix spectral problem is derived with the help of Lenard recursion equations. Based on the characteristic polynomial of Lax matrix for the modified Boussinesq hierarchy, we introduce an algebraic curve K_m−1 of arithmetic genus m−1, from which we establish the associated Baker-Akhiezer function, meromorphic function and Dubrovin-type equations. The straightening out of various flows is exactly given through the Abel map. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of solutions for the entire modified Boussinesq hierarchy.     

Spectral functions,special functions and the Selberg zeta function

The functional determinant of an eigenvalue sequence, as defined by zeta regularization, can be simply evaluated by quadratures. We apply this procedure to the Selberg trace formula for a compact Riemann surface to find a factorization of the Selberg zeta function into two functional determinants, respectively related to the Laplacian on the compact surface itself, and on the sphere. We also apply our formalism to various explicit eigenvalue sequences, reproducing in a simpler way classical results about the gamma function and the BarnesG-function. Concerning the latter, our method explains its connection to the Selberg zeta function and evaluates the related Glaisher-Kinkelin constantA.Member of CNRS     

The Resultant on Compact Riemann Surfaces

We introduce a notion of the resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.     

Milnor–Selberg zeta functions and zeta regularizations

By a similar idea for the construction of Milnor’s gamma functions, we introduce “higher depth determinants” of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor–Selberg zeta function. It is shown that the Milnor–Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.