Cobordism invariance shows that the index, in K-theory, of a family of pseudodifferential operators on the boundary of a fibration vanishes if the symbol family extends to be elliptic across the whole fibration. For Dirac operators with spectral boundary condition, Dai and Freed \cite{dai-freed1} gave an explicit version of this at the level of the determinant bundle. Their result, that the eta invariant of the interior family trivializes the determinant bundle of the boundary family, is extended here to the wider context of pseudodifferential families of cusp type.
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.
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 study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included)…
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…
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.
The infinite matrix `Schwartz' group $G^{-\infty}$ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on $G^{-\infty}.$ We show that while the higher (even, Schwartz) loop groups of $G^{-\infty},$ again classifying for odd K-theory, do \emph{not} carry multiplicative determinants generating the first Chern class, `dressed' extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the et…
This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an explicit formula for the regularized determinants of the flat metrics with fixed volume on the elliptic curves. The following formula holds in this case; the regularized determinant is the product of the imaginary part of the complex number in the Siegel upper half plane with the De…
Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topology 4, 2000, 171-178
We extend the theory of the universal eta-invariant to the case of relative bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal eta-invariant. We obtain an interpretation of Laures' f-invariant as an example of this general construction. As an aside we improve a recent result of Han-Zhang on the modularity of a certain power series of eta invariants.
This is a note for the conference proceedings Topological and Geometrical Problems related to Quantum Field Theory. We summarize our joint work with Dai about eta invariants on manifolds with boundary. Then we apply these results to prove the curvature and holonomy formulas for the natural connection on the determinant line bundle of a family of Dirac operators. These were originally proved by Bismut and the author--the proofs here are much simpler.
In previous work, we introduced eta invariants for even dimensional manifolds. It plays the same role as the eta invariant of Atiyah-Patodi-Singer, which is for odd dimensional manifolds. It is associated to $K^1$ representatives on even dimensional manifolds and is closely related to the so called WZW theory in physics. In fact, it is an intrinsic interpretation of the Wess-Zumino term without passing to the bounding 3-manifold. Spectrally the eta invariant is defined on a finite cylinder, rather than on the manifold itself. Thus it is an interesting question to find an intrinsic spectral int…
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 Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low dimensions these refinements correspond to the exponentiated eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed connection, and Lott's index gerbe with connection and curving. We give a unified treatement of these cases as well as their higher genera…
We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the Laplacian. In particular we prove that contact versions of the relative eta-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.
We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of $D_m$ encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the foliation is endowed with a holonomy invariant transverse measure, we explain the compatibility of various traces and determinants. We extend Atiyah's index theorem on Galois coverings to these foliat…
Several proofs have been published of the Mod Z gluing formula for the eta-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the eta-invariant is left obscure in the literature. In this article we present a gluing formula for the eta-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderon projectors of the two parts of the decomposition. The main ingredients of our presentation are the Scott-Wojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and…
The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} λ_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons and simplices with the topology of a disc in the Euclidean plane. The domains we consider are hence piece--wise flat with corners on the boundary and in the interior. Our results are complementary to earlier investigations of the determinants on smooth…
This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic torsion) that occur in more refined index theorems, such as the determinant line bundle setting and the index theorem for families of manifolds with boundary. We also discuss the higher torsion forms of Bismut and Lott and their conjectured relation to the rational homotopy of th…
Let $A(t)$ be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration $ϕ$ with base $Y.$ The standard example is $A+it$ where $A$ is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and $t\in\bbR$ is the `suspending' parameter. Let $π_{\cA}:\cA(ϕ)\longrightarrow Y$ be the infinite-dimensional bundle with fibre at $y\in Y$ consisting of the Schwartz-smoothing perturbations, $q,$ making $A_y(t)+q(t)$ invertible for all $t\in\bbR.$ The total e…
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta inva…
I review the milestones of the mathematical work of Krzysztof P. Wojciechowski. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderon projector and the topology of space of elliptic boundary problems. The theme of the 90s is the eta invariant. The paper with Douglas was fundamental for establishing spectral invariants for manifolds with boundary and for the investigation of the behavior of spectral invariants under analytic surgery. This was so influ…
Let X be a compact manifold with boundary, and suppose that the boundary is the total space of a fibration with base Y and fibre Z. Let D be a generalized Dirac operator associated to a Phi-metric g on X. Under the assumption that D is fully elliptic we prove an index formula for D. The proof is in two steps: first, using results of Melrose and Rochon, we show that the index is unchanged if we pass to a certain b-metric h(s), with s a positive real number. Next we write the b- (i.e. the APS) index formula for h(s); the Phi-index formula follows by analyzing the limiting behaviour as s goes to …
In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator $\mathcal{B}$ in the sense of Seeley. We then show that the zeta-determinants of $\mathcal{B}^2$ and eta-invariants of $\mathcal{B}$ subject to these boundary conditions are well defined by using the method of the asymptotic expansions of the traces of the heat kernels. We use these facts to define the refined …
Let X be a closed Riemannian manifold and let H\hookrightarrow X be an embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two manifolds with boundary, with X_+ \cap X_- = H. In this expository article, surgery -- or gluing -- formulæfor several geometric and spectral invariants associated to a Dirac-type operator ð_X on X are presented. Considered in detail are: the index of ð_X, the index bundle and the determinant bundle associated to a family of such operators, the eta invariant and the analytic torsion. In each case the precise form of the surgery theorems, as well as…
The goal of this paper is to apply the universal gerbe of \cite{CMi1} and \cite{CMi2} to give an alternative, simple and more unified view of the relationship between index theory and gerbes. We discuss determinant bundle gerbes \cite{CMMi1} and the index gerbe of \cite{L} for the case of families of Dirac operators on odd dimensional closed manifolds. The method also works for a family of Dirac operators on odd dimensional manifolds with boundary, for a pair of Melrose-Piazza's $Cl(1)$-spectral sections for a family of Dirac operators on even dimensional closed manifolds with vanishing index …
This article is a follow up of the previous article of the authors on the analytic surgery of eta- and rho-invariants. We investigate in detail the (Atiyah-Patodi-Singer)-rho-invariant for manifolds with boundary. First we generalize the cut-and-paste formula to arbitrary boundary conditions. A priori the rho-invariant is an invariant of the Riemannian structure and a representation of the fundamental group. We show, however, that the dependence on the metric is only very mild: it is independent of the metric in the interior and the dependence on the metric on the boundary is only up to its ps…
