We discuss the decomposition of the zeta-determinant of the square of the Dirac operator into contributions coming from the different parts of the manifold. The easy case was worked in the previous paper of authors. Due to the assumptions made on the operators in the previous paper, we were able to avoid the presence of the small eigenvalues which provide the large time contibution to the determinant. In the present work we analyze the general case. We offer a detailed analysis of the contribution made by the small eigenvalues.
We prove an adiabatic decomposition formula of the zeta-determinant of the Laplace type operator with respect to Dirichlet boundary condition. We allow the non-invertible tangential operator. As a result, our adiabatic decomposition formula involves the scattering matrix over the manifold with cylindrical end. We also describe the adiabatic limit of the zeta-determinant of the Dirichlet to Neumann operator, which plays the essential role of Burghelea-Friedlander-Kappeler's Meyer-Vietoris type formula of the zeta determinant.
we discuss the decomposition of the zeta-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold in the case of an invertible tangential operator.
Given two unitary involutions $σ_{1}$ and $σ_{2}$ satisfying $G σ_{i} = - σ_{i} G$ on $ker B$ on a compact manifold with cylindrical end, M. Lesch, K. Wojciechowski ([LW]) and W. Müller ([M]) established the formula describing the difference of two eta-invariants with the APS boundary conditions associated with $σ_{1}$ and $σ_{2}$. In this paper we establish the analogous formula for the zeta-determinants of Dirac Laplacians. For the proof of the result we use the Burghelea-Friedlander-Kappeler's gluing formula for zeta-determinants and the scattering theory developed by W. Müller in [M]. This…
The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under the assumption of the product structure near boundary. As applications of this result,we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.
We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm-Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the zeta-determinant of an infinite direct sum of Sturm-Liouville operators. The Laplace-Beltrami operator on a surface of revo…
In this paper we first establish the relation between the zeta-determinant of a Dirac Laplacian with the Dirichlet boundary condition and the APS boundary condition on a cylinder. Using this result and the gluing formula of the zeta-determinant given by Burghelea, Friedlander and Kappeler with some assumptions, we prove the adiabatic decomposition theorem of the zeta-determinant of a Dirac Laplacian. This result was originally proved by J. Park and K. Wojciechowski in [11] but our method is completely different from the one they presented.
We study the zeta determinant of global boundary problems of APS-type through a general theory for relative spectral invariants. In particular, we compute the zeta determinant for Dirac-Laplacian boundary problems in terms of a scattering Fredholm determinant over the boundary.
We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detailed analysis of the unipotent terms give the relation between the eta invariant and the Selberg zeta function of odd type. This allows us to show the vanishing of the unipotent contribution. We show that the eta invariant and the Selberg zeta function of odd type satisfy certain …
We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over the boundary for a class of higher-order elliptic boundary value problems.
The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing form…
We review the work of the authors and their collaborators on the decomposition of the zeta-determinant of the Dirac operator into the contribution coming from different parts of a manifold.
On an open manifold, the spaces of metrics or connections of bounded geometry, respectively, split into an uncountable number of components. We show that for a pair of metrics or connections, belonging to the same component, relative $ζ$-functions, determinants, torsion for pairs of generalized Dirac operators are well defined.
We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we provide a technique for obtaining the first terms in the Laurent expansion at zero of the zeta function associated to a double sequence. We particularize this technique to the case of sums of sequences of spectral type, and we give two applications: the first concerning some special functions appearing in number the…
We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions $η(δ_t,s)$ and $tζ(δ_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. After no…
The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac operators, we prove for the pair $(D,D')$ relative index theorems, define relative $ζ$-- and $η$--functions, relative determinants and in the case of $D=Δ$ relative analytic torsion. To define relative $ζ$-- and $η$--functions, we assume additionally that the essential spectrum of $…
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with …
We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4π$.
For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the classical single operator spectral zeta function and determinant. In the case where F is the curvature of a superconnection the zeta form is exact, extending to families the Atiyah-Bott-Seeley zeta function formula for the pointwise index, and equivalent to the transgression formu…
We obtain explicit expressions for the determinants of the Laplacians on zero and one forms for an infinite class of three dimensional lens spaces $L(p,q)$. These expressions can be combined to obtain the Ray-Singer torsion of these lens spaces. As a consequence we obtain an infinite class of formulae for the Riemann zeta function $ζ(3)$. The value of these determinants (and the torsion) grows as the size of the fundamental group of the lens space increases and this is also computed. The triviality of the torsion for just the three lens spaces $L(6,1)$, $L(10,3)$ and $L(12,5)$ is also noted. (…
We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the "radius" of the fiber goes to zero. We explain the lines of the calculations precisely for three dimensional cases and state the corresponding results for five dimensional Heisenberg manifolds. We s…
We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansions, we derive adiabatic splitting formulas for regularized determinants of Laplace type operators.
We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.
Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large $p$ behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for zeta(3): the ordinary Riemann zeta function evaluated at s=3. The torsion turns out to be trivial for the cases L(6,1), L((10,3) and L(12,5) and is, in general, gr…
In this note we specialize and illustrate the ideas developed in the paper math.DG/0201112 of the first author ("Index theory, eta forms, and Deligne cohomology ") in the case of the determinant line bundle. We discuss the surgery formula in the adiabatic limit using the adiabatic decomposition formula of the zeta regularized determinant of the Dirac Laplacian obtained by the second author and K. Wojciechowski.
We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, or trivialization of the bundle near the boundary. An application to the computation of bosonization constants follows directly from these formulas.
The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the noncommutative residue. The ambiguity is here seen to disappear in the massive case, giving rise to…
We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.
