Spectral Zeta Function Research Articles

Evaluation of spectral zeta-functions with the renormalization group

We evaluate spectral zeta-functions of certain network Laplacians that can be treated exactly with the renormalization group. As specific examples we consider a class of Hanoi networks and those hierarchical networks obtained by the Migdal–Kadanoff bond moving scheme from regular lattices. As possible applications of these results we mention quantum search algorithms as well as synchronization, which we discuss in more detail.

One-loop tests of the supersymmetric higher spin AdS4/CFT3 correspondence

We compute one loop free energy for D=4 Vasiliev higher spin gravities based on Konstein-Vasiliev algebras hu(m;n|4), ho(m;n|4) or husp(m;n|4) and subject to higher spin preserving boundary conditions, which are conjectured to be dual to the U(N), O(N) or USp(N) singlet sectors, respectively, of free CFTs on the boundary of $AdS_4$. Ordinary supersymmetric higher spin theories appear as special cases of Konstein-Vasiliev theories, when the corresponding higher spin algebra contains $OSp({\cal N}|4)$ as subalgebra. In $AdS_4$ with $S^3$ boundary, we use a modified spectral zeta function method, which avoids the ambiguity arising from summing over infinite number of spins. We find that the contribution of the infinite tower of bulk fermions vanishes. As a result, the free energy is the sum of those which arise in type A and type B models with internal symmetries, the known mismatch between the bulk and boundary free energies for type B model persists, and ordinary supersymmetric higher spin theories exhibit the mismatch as well. The only models that have a match are type A models with internal symmetries, corresponding to $n=0$. The matching requires identification of the inverse Newton's constant $G_N^{-1}$ with $N$ plus a proper integer as was found previously for special cases. In $AdS_4$ with $S^1\times S^2$ boundary, the bulk one loop free energies match those of the dual free CFTs for arbitrary $m$ and $n$. We also show that a supersymmetric double-trace deformation of free CFT based on OSp(1|4) does not contribute to the ${\cal O}(N^0)$ free energy, as expected from the bulk.

SPECTRAL ZETA FUNCTIONS FOR THE QUANTUM RABI MODELS

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at$s=1$. As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.

Exceptional F (4) higher-spin theory in AdS6 at one-loop and other tests of duality

We study the higher-spin gauge theory in six-dimensional anti-de Sitter space $AdS_6$ that is based on the exceptional Lie superalgebra $F(4)$. The relevant higher-spin algebra was constructed in arXiv:1409.2185 [hep-th]. We determine the spectrum of the theory and show that it contains the physical fields of the Romans $F(4)$ gauged supergravity. The full spectrum consists of an infinite tower of unitary supermultiplets of $F(4)$ which extend the Romans multiplet to higher spins plus a single short supermultiplet. Motivated by applications to this novel supersymmetric higher-spin theory as well as to other theories, we extend the known one-loop tests of $AdS/CFT$ duality in various directions. The spectral zeta-function is derived for the most general case of fermionic and mixed-symmetry fields, which allows one to test the Type-A and B theories and supersymmetric extensions thereof in any dimension. We also study higher-spin doubletons and partially-massless fields. While most of the tests are successfully passed, the Type-B theory in all even dimensional anti-de Sitter spacetimes presents an interesting puzzle: the free energy as computed from the bulk is not equal to that of the free fermion on the CFT side, though there is some systematics to the discrepancy.

Zeta functions of the Dirac operator on quantum graphs

We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices. The regularized spectral determinant of the Dirac operator is also obtained as the derivative of the zeta function at a special value. In each case the zeta function is formulated using a contour integral method, which extends results obtained for Laplace and Schrödinger operators on graphs.

A functional relation for L-functions of graphs equivalent to the Riemann Hypothesis for Dirichlet L-functions

In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an asymptotic functional equation for these L-functions and the corresponding case of the Generalized Riemann Hypothesis. We also establish a relation between the positivity of such functions and the existence of real zeros in the critical strip of the classical Dirichlet L-functions with the same character.

Functional determinants and Casimir energy in higher dimensional spherically symmetric background potentials

In this paper we analyze the spectral zeta function associated with a Laplace operator acting on scalar functions on an N-dimensional Euclidean space in the presence of a spherically symmetric background potential. The obtained analytic continuation of the spectral zeta function is then used to derive very simple results for the functional determinant of the operator and the Casimir energy of the scalar field.

On a conjecture of Kimoto and Wakayama

We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apéry-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p p -adic analysis.

On complex zeros off the critical line for non-monomial polynomial of zeta-functions

In this paper, we show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler–Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelof hypothesis for the Riemann zeta-function is equivalent to the Lindelof hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler–Zagier–Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

The Casimir effect for thick pistons

In this work, we analyze the Casimir energy and force for a thick piston configuration. This study is performed by utilizing the spectral zeta function regularization method. The results we obtain for the Casimir energy and force depend explicitly on the parameters that describe the general self-adjoint boundary conditions imposed. Numerical results for the Casimir force are provided for specific types of boundary conditions and are also compared to the corresponding force on an infinitely thin piston.

The local counting function of operators of Dirac and Laplace type

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the residues of the spectral zeta function as well as certain Wodzicki residues. We then use the Wodzicki residue to obtain results about the local counting function of operators of Dirac and Laplace type. In particular, we express the second term of the mollified spectral counting function of Dirac type operators in terms of geometric quantities and characterize those Dirac type operators for which this coefficient vanishes.

Kronecker’s limit formula and the height of Euclidean lattices

The height \({h(\Lambda)}\) of a Euclidean lattice \({\Lambda}\) is defined as the derivative at the point 0 of the spectral zeta function of the associated flat torus \({{\bf R}^n /\Lambda}\). The quest of the lattices which realise a minimum of h is an open problem in dimension higher than 3. In this paper, we show that a lattice which realises a local minimum of the height is irreducible, i.e. it cannot be decomposed into the orthogonal sum of two non-zero sublattices. The proof is based on a formula found by Terras, which generalises Kronecker’s limit formula for the Epstein zeta function.

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$. We relate $\zeta_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $\xi_{\mathbb{Z}}(1-s)=\xi_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $\zeta_{\mathbb{Z}^d}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $\zeta(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_1$ via an Euler-type integral formula due to Picard.

Spectral zeta function and non-perturbative effects in ABJM Fermi-gas

The exact partition function in ABJM theory on three-sphere can be regarded as a canonical partition function of a non-interacting Fermi-gas with an unconventional Hamiltonian. All the information on the partition function is encoded in the discrete spectrum of this Hamiltonian. We explain how (quantum mechanical) non-perturbative corrections in the Fermi-gas system appear from a spectral consideration. Basic tools in our analysis are a Mellin-Barnes type integral representation and a spectral zeta function. From a consistency with known results, we conjecture that the spectral zeta function in the ABJM Fermi-gas has an infinite number of "non-perturbative" poles, which are invisible in the semi-classical expansion of the Planck constant. We observe that these poles indeed appear after summing up perturbative corrections. As a consequence, the perturbative resummation of the spectral zeta function causes non-perturbative corrections to the grand canonical partition function. We also present another example associated with a spectral problem in topological string theory. A conjectured non-perturbative free energy on the resolved conifold is successfully reproduced in this framework.

Asymptotic and Exact Expansions of Heat Traces

We study heat traces associated with positive unbounded operators with compact inverses. With the help of the inverse Mellin transform we derive necessary conditions for the existence of a short time asymptotic expansion. The conditions are formulated in terms of the meromorphic extension of the associated spectral zeta-functions and proven to be verified for a large class of operators. We also address the problem of convergence of the obtained asymptotic expansions. General results are illustrated with a number of explicit examples.

Spectral zeta function on pseudo H-type nilmanifolds

We explain the explicit integral form of the heat kernel for the sub-Laplacian on two step nilpotent Lie groups G based on the work of Beals, Gaveau and Greiner. Using such an integral form we study the heat trace of the sub-Laplacian on nilmanifolds L\G where L is a lattice. As an application a common property of the spectral zeta function for the sub-Laplacian on L\G is observed. In particular, we introduce a special class of nilpotent Lie groups, called pseudo H-type groups which are generalizations of groups previously considered by Kaplan. As is known such groups always admit lattices. Here we aim to explicitly calculate the heat trace and the spectrum of the (sub)-Laplacian on various low dimensional compact nilmanifolds including several pseudo H-type nilmanifolds L\G, i.e. where G is a pseudo H-type group. In an appendix we discuss a zeta function which typically appears as the Mellin transform for these heat traces.

Spectral functions for regular Sturm-Liouville problems

In this paper, we provide a detailed analysis of the analytic continuation of the spectral zeta function associated with one-dimensional regular Sturm-Liouville problems endowed with self-adjoint separated and coupled boundary conditions. The spectral zeta function is represented in terms of a complex integral and the analytic continuation in the entire complex plane is achieved by using the well-known Liouville-Green (or WKB) asymptotic expansion of the eigenfunctions associated with the problem. The analytically continued expression of the spectral zeta function is then used to compute the functional determinant of the Sturm-Liouville operator and the coefficients of the asymptotic expansion of the associated heat kernel.

QFT Over the Finite Line. Heat Kernel Coefficients, Spectral Zeta Functions and Selfadjoint Extensions

Following the seminal works of Asorey–Ibort–Marmo and Muñoz–Castañeda–Asorey about selfadjoint extensions and quantum fields in bounded domains, we compute all the heat kernel coefficients for any strongly consistent selfadjoint extension of the Laplace operator over the finite line [0, L]. The derivative of the corresponding spectral zeta function at s = 0 (partition function of the corresponding quantum field theory) is obtained. To compute the correct expression for the a 1/2 heat kernel coefficient, it is necessary to know in detail which non-negative selfadjoint extensions have zero modes and how many of them they have. The answer to this question leads us to analyze zeta function properties for the Von Neumann–Krein extension, the only extension with two zero modes.

Casimir pistons with general boundary conditions

In this work we analyze the Casimir energy and force for a scalar field endowed with general self-adjoint boundary conditions propagating in a higher dimensional piston configuration. The piston is constructed as a direct product I×N, with I=[0,L]⊂R and N a smooth, compact Riemannian manifold with or without boundary. The study of the Casimir energy and force for this configuration is performed by employing the spectral zeta function regularization technique. The obtained analytic results depend explicitly on the spectral zeta function associated with the manifold N and the parameters describing the general boundary conditions imposed. These results are then specialized to the case in which the manifold N is a d-dimensional sphere.

Sharp spectral bounds on starlike domains

We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber–Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizes the spectral zeta function and heat trace. The normalization on the domain incorporates volume and a computable geometric factor that measures the deviation of the domain from roundness, in terms of moment of inertia and a support functional introduced by Pólya and Szegó. Additional functionals handled by our method include finite sums and products of eigenvalues. The results hold on convex and starlike domains, and for Dirichlet, Neumann or Robin boundary conditions.