I. M. Singer

Spectral asymmetry and Riemannian Geometry. I

1. Introduction . The main purpose of this paper is to present a generalization of Hirzebruchs signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula: where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X . In particular if, near the boundary, X is isometric to the product Y x R + , the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H 2 ( X , R) by an integral formula where p 1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p 1 = (2π) −2 Tr R 2 . It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general

R-Torsion and the Laplacian on Riemannian manifolds

Let W be a compact oriented Riemannian manifold of dimension N, and let K be a simplicial complex which is a smooth triangulation of W. The Reidemeister-Franz torsion (or R-torsion) 7 of K is a function of certain representations of the fundamental group of K. Since it is a combinatorial invariant, and since smooth triangulations of W are equivalent, this torsion is a manifold invariant. We raise the question as to how to describe this manifold invariant in analytic terms. Arnold Shapiro once suggested that there might be a formula for the torsion in terms of the Laplacian d acting on differential forms on W. Our candidate T involves the zeta function for appropriate Laplacians. Though we have been unable to prove that T = T, we show in this paper that T is a manifold invariant and present some evidence that T = 7. If one thinks of analytic torsion as an invariant associated to the De Rham complex, it is natural to ask whether there are analogous invariants for other elliptic complexes. For complex manifolds and the &complex, there is indeed such a holomorphic invariant, which will be the subject of a subsequent paper. In Section 1 we give a short exposition of Reidemeister-Franz torsion and motivate our definition of the analytic torsion T. In Section 2 are collected the main results of the paper. First we prove that T = T, is independent of the metric of W, for W closed. Next we prove three results which are formal analogs of known properties of the ReidemeisterFranz torsion, namely, T, = 1 if W is closed and has even dimension; Twlxlv, = ( Twl)X(Wz), x( W,) being the Euler characteristic of W, , if

Some remarks on the Gribov ambiguity

The set of all connections of a principal bundle over the 4-sphere with compact nonabelian Lie group under the action of the group of gauge transformations is studied. It is shown that no continuous choice of exactly one connection on each orbit can be made. Thus the Gribov ambiguity for the Coloumb gauge will occur in all other gauges. No gauge fixing is possible.

The index of elliptic operators on compact manifolds

© Association des collaborateurs de Nicolas Bourbaki, 1964, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Index theory for skew-adjoint fredholm operators

© Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Special Quantum Field Theories¶in Eight and Other Dimensions

Abstract:We build nearly topological quantum field theories in various dimensions. We give special attention to the case of eight dimensions for which we first consider theories depending only on Yang–Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU(4) and Spin(7). The choice of SU(4) gives a quantum field theory for a Calabi–Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M- and F-theories. Relations to the eight dimensional supersymmetric Yang–Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg–Witten equations. The latter are thus related to pure Yang–Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang–Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions.

Some comments on Chern-Simons gauge theory

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.

Gravitational anomalies and the family's index theorem

We discuss the use of the familys index theorem in the study of gravitational anomalies. The geometrical framework required to apply the familys index theorem is presented and the relation to gravitational anomalies is discussed. We show how physics necessitates the introduction of the notion oflocal cohomology which is distinct from the ordinary topological cohomology. The recent results of Alvarez-Gaumé and Witten are derived by using the familys index theorem.

The topological sigma model

We obtain the invariants of Wittens topologicalσ-model by gauge fixing a topological action and using BRST symmetry. The fields and the BRST formalism are interpreted geometrically.

The index of projective families of elliptic operators.

An index theory for projective families of elliptic pseudodifferential operators is developed when the twisting, i.e. Dixmier-Douady, class is decomposable. One of the features of this special case is that the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic index of a projective family of elliptic operators both take values in the twisted K-theory of the parameterizing space. The main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern-Weil theory.