Analysis, Geometry and Topology of Elliptic Operators

The following sections are included:

• Contents

• Preface

To honor and to please our friend Krzysztof P. Wojciechowski I will review the milestones of his mathematical work. 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 Calderón 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 influential that many different proofs of the gluing formula for the eta-invariant were published. Finally turning to the new millennium we will look at the zeta-determinant. Compared to eta this is a much more rigid spectral invariant which is technically challenging.

We review the gluing formulae of the spectral invariants - the ζ-regularized determinant of a Laplace type operator and the eta invariant of a Dirac type operator. In particular, we explain the crucial role of the Cauchy data spaces in these gluing formulae.

The Atiyah-Singer index formula states that the analytic index of an elliptic pseudo-differential operator equals the topological index of the K-theoretic class of its asymptotic symbol. In the embedding proof of the index formula, it is shown that both indices obey certain axioms and which guarantee that they are the same. For the most part, the verification that the topological index obeys the axioms is fairly direct, once one has the relevant background in K-theory. However, the verification is not as straightforward for the analytic index. This is particularly true for the multiplication axiom which is used to show how the index behaves under embedding. Complications arise when the normal bundle of the embedding is nontrivial. Our aim is to clarify the verification of the multiplication axiom for the analytic index by defining the participating elliptic pseudo-differential operators in terms of global symbols invariantly defined on the cotangent bundles of the relevant base spaces.

In this survey we describe situations when analytic invariants of an isolated complex surface singularity can be computed by topological methods.

We define Chern-Weil forms c_k(A) associated to a superconnection A using ζ-regularisation methods extended to ΨDO valued forms. We show that they are cohomologous in the de Rham cohomology to tr(A^2k π_P) involving the projection π_P onto the kernel of the elliptic operator P to which the superconnection A is associated. A transgression formula shows that the corresponding Chern-Weil cohomology classes are independent of the scaling of the superconnection. When P is a differential operator of order p with scalar leading symbol, the k-th Chern-Weil form corresponds to the regularised k-th derivative at t = 0 of the Chern character ch(tA) and it has a local description

in terms of the Wodzicki residue extended to ΨDO -valued forms.

We study second-order elliptic partial differential operators acting on sections of vector bundles over a compact manifold with boundary with a non-scalar positive definite leading symbol. Such operators, called non-Laplace type operators, appear, in particular, in gauge field theories, string theory as well as models of non-commutative gravity theories, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays the role of a “non-commutative” metric. It is well known that there is a small-time asymptotic expansion of the trace of the corresponding heat kernel in half-integer powers of time. We initiate the development of a systematic approach for the explicit calculation of these coefficients, construct the corresponding parametrix of the heat equation and compute explicitly the first two heat trace coefficients.

For a compact manifold with boundary, M, there are well known local boundary conditions that make the de Rham operator A = d+δ elliptic, namely the absolute and relative boundary conditions. We study the eta invariants of such elliptic boundary value problems under the metric deformation

where x ɛ C^∞(M) is, near the boundary, the geodesic distance to the boundary, and g is a Riemannian metric on M which is of product type near the boundary. Under certain acyclicity condition we show that when M is odd dimensional where the subscript a (r) indicates the absolute (relative) boundary condition, and η_b(A₀) is the b-eta invariant of the limiting operator A₀. If M is even dimensional then Most of the analysis extends to analytic torsion, yielding when dim M is odd, and when dim M is even. Here the sign ± depends on the choice of the boundary condition and r₁ , r₂ vanishes at ɛ = 0.

Since the heat kernel of the sub-Laplacian on Heisenberg group was constructed in an explicit integral form by A. Hulanicki, we have several ways to construct the heat kernel for the sub-Laplacian and the Laplacian on 2-step nilpotent Lie groups. In this note we explain a method effectively employed by Beals-Gaveau-Greiner, the so called complex Hamilton-Jacobi theory, and illustrate the construction of the heat kernel for general 2-step cases. We discuss the solution of the generalized Hamilton-Jacobi equation and a quantity similar to the van Vleck determinant and their roles in the integral expression of the heat kernel. We expect this method will work also for 3-step cases to construct the heat kernel together with the theory of elliptic functions. So as an example, we consider the solution of the generalized Hamilton-Jacobi equation for the lowest dimensional 3-step nilpotent Lie group (Engel group). Then we discuss a hierarchy of heat kernels for the three dimensional Heisenberg group and Heisenberg manifolds as a simple example.

In a recent work, Paycha and Scott establish formulas for all the Laurent coefficients of Tr(AP^−s) at the possible poles. In particular, they show a formula for the zero'th coefficient at s = 0, in terms of two functions generalizing, respectively, the Kontsevich-Vishik canonical trace density, and the Wodzicki-Guillemin noncommutative residue density of an associated operator. The purpose of this note is to provide a proof of that formula relying entirely on resolvent techniques (for the sake of possible generalizations to situations where powers are not an easy tool). — We also give some corrections to transition formulas used in our earlier works.

We extend the definition, in the extended cohomology framework, of the L²- analytic torsion for covering spaces due to Braverman-Carey-Farber-Mathai to the case of manifolds with boundary, and prove an associated anomaly formula. Our main result may be viewed as a common generalization of the anomaly formula for Ray-Singer analytic torsion for manifolds with boundary proved by Brüning-Ma, as well as the anomaly formula for L²-analytic torsions for closed manifolds proved by Zhang. It generalizes also an earlier result of Lück-Schick, without the assumptions on the unitary representations as well as the technical “determinant class condition”.

Using Laurent expansions of canonical traces of holomorphic families of classical pseudodifferential operators, we define functionals on the space of Riemannian metrics and investigate their conformal properties, thereby giving a unified description of several conformal invariants and anomalies.

The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow first analysed in the classical setting by Getzler and extended to Breuer-Fredholm operators by some of the current authors. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (non-compact) covering spaces of a compact spin manifold using the adiabatic method. This answers a question of Mathai connecting Atiyah's L²-index theorem to our analytic spectral flow. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on R^N using some ideas of Shubin. This provides us with an opportunity to describe the deepest part of the theory namely the semifinite local index theorem in noncommutative geometry. This theorem, which gives a formula for spectral flow was recently proved by some of the present authors. It provides a far-reaching generalisation of the original 1995 result of Connes and Moscovici.

We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of “elliptic operators” as defined by Colin de Verdiere [4]. For such operators, we discuss analogs of inequalities of Cheeger and Harnack and of the maximum principle (in both elliptic and parabolic versions), and apply them to study spectral theory, the ground state and the heat semigroup associated to these operators.

Index theory has had profound impact on many branches of mathematics. In this note we discuss the context for a new kind of index theorem. We begin, however, with some operator-theoretic results. Berger and Shaw [11] established that finitely cyclic hyponormal operators have trace-class self-commutators. Berger [9] and Voiculescu [31] extended this result to operators whose self-commutators can be expressed as the sum of a positive and a trace-class operator. In this note we show this result can't be extended to operators whose self-commutators can be expressed as the sum of a positive and a S_p-class operator. Then we discuss a conjecture of Arveson [4] on homogeneous submodules of the m-shift Hilbert space and propose some refinements of it.

Further, we show how a positive solution would enable one to define K-homology elements for subvarieties in a strongly pseudo-convex domain with smooth boundary using submodules of the corresponding Bergman module. Finally, we discuss how the Chern character of these classes in cyclic cohomology could be defined and indicate what we believe the index to be.

We study noncommutative versions of holomorphic and harmonic functions on the unit disk.

The purpose of the present note is two-fold. First, to show that deformations of algebras of smooth functions can be used to construct topologically nontrivial standard central extensions of loop groups. Second, to use noncommutative geometry as a regularization of current algebras in higher dimensions with the aim of constructing representations of current algebras.

We complete a gap in the proof of a crucial result of Pressley and Segal's book on “Loop groups”. It is well-known that given a polarized, separable complex Hilbert space of infinite dimension (K = K₊ ⊕ K₋), the elements of the restricted linear group GL_res (K, K₊) can be written as two-by-two matrices of operators and that notably the upper left entry of these is a Fredholm operator from K₊ to itself. The resulting map from the restricted linear group to the space of Fredholm operators on K₊ is a homotopy equivalence. We complete the proof of this proposition (6.2.4 in “Loop groups”) relying on a simple “Boardman-Vogt type lemma”. We then remark on some applications of this result to classifying spaces of principal bundles, to geometric quantization in infinite dimensions and to string structures on loop spaces.

We study the topology of T-duality for pairs of U(1)-bundles and three-dimensional integral cohomology classes over orbispaces.

A general method exists for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry. In the de Donder gauge, however, the resulting boundary-value problem for the Laplace type operator acting on h is known to be self-adjoint but not strongly elliptic.

The present paper shows that, on the Euclidean four-ball, only the scalar part of perturbative modes for quantum gravity is affected by the lack of strong ellipticity. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is “confined” to the remaining fourth sector. The integral representation of the resulting ζ-function asymptotics on the Euclidean four-ball is also obtained; this remains regular at the origin by virtue of a peculiar spectral identity obtained by the authors. There is therefore encouraging evidence in favour of the ζ(0) value with fully diff-invariant boundary conditions remaining well defined, at least on the four-ball, although severe technical obstructions remain in general.

In this paper, we prove a Morse index theorem for the index form of even order linear Hamiltonian systems on the closed interval with self-adjoint boundary conditions. The highest order term is assumed to be nondegenerate.