Local index theory and higher analytic torsion

The purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de Rham theory. This difference is the sum of the integral of a Chern-Simons current and of a new invariant, the V -invariant of an odd dimensional manifold equipped with an action of a compact Lie group. The V -invariant localizes on the critical manifolds of invariant Morse-Bott functions. The results in this paper are shown to be compatible with results of Bunke, and also our with previous results on analytic torsion forms.

In this note we prove a vanishing theorem for the analytic torsion of a locally symmetric space.

In this paper we extend first the Bismut-Lott's analytic torsion form for flat vector bundles to the boundary case, then we establish its gluing formula on a smooth fibration under the assumption that a fiberwise Morse function exists. We assume that the metrics have product structures near the cutting hypersurface.

This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes–Chern character of an equivariant Dirac spectral triple. The formula that we obtain is used in the first paper of the series. The computation has two main steps. The first step is the justification that the CM cocycle represents the Connes–Chern character. The second step is the computation of the CM cocycle as a byproduct of a new proof of the local equivariant index theorem of Donnelly–Patodi, Gilkey and Kawasaki. The proof combines the rescaling method of Getzler with an equivariant version of the Greiner–Hadamard approach to the heat kernel asymptotics. Finally, as a further application of this approach, we compute the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple.

An explicit formula for analytic torsion forms for fibrations by projective curves is given. In particular one obtains a formula for direct images in Arakelov geometry in the corresponding setting. The main tool is a new description of Bismut’s equivariant Bott–Chern current in the case of isolated fixed points.

We intend to produce a theory which generalises topological spaces; we call it non-commutative topology. In non-commutative differential geometry the basic homology theory is the periodic cyclic homology, based on the bi-complex (b, B). In non-commutative topology this structure will be replaced by the bi-complex \((\tilde {b}, d)\); the boundary \(\tilde {b}\) is called topological Hochschild boundary. These ideas combine Connes’ work (Noncommutative geometry, Academic Press, 1994) with ideas of the articles by Teleman and Teleman (St Cerc Math Tom 18(5):753–762, Bucarest, 1966; Rev Roumaine Math Pures Applic 12:725–731, 1967; Central European journal of mathematics. C*-algebras and elliptic theory II. Trends in mathematics, 251–265, Birkhauser, 2008; Local3 index theorem. arXiv: 1109.6095v1, math.KT, 28 Sep. 2011; textitLocal Hochschild homology of Hilbert-Schmidt operators on simplicial spaces. arXiv hal-00707040, Version 1, 11 June 2012; Local algebraic K-theory, 26 Aug 2013 - arXiv.org > math > arXiv:1307.7014, 2013; The local index theorem, HAL-00825083, arXiv 1305.5329, 22 May 2013; \(K_ {i}^{loc}(\ mathbb {C}) \), i = 0, 1 10 set 2013 - arXiv:1309.2421v1 [math.KT] 10 Sep 2013).

We show an equality between the analytic torsion and the absolute value at zero of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author’s previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.

Local atiyah-singer index theorem for families of dirac operators

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed’s odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.

Superconnection and family Bergman kernels

We present a new proof, as well as a C/Q extension (and also certain C/Z extension), of the Riemann–Roch–Grothendieck theorem of Bismut–Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of η-invariants associated to the so-called sub-signature operators. We further show that the Bismut–Lott analytic torsion form can be derived naturally from transgressions of η-forms appearing in the adiabatic limit computations.

Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes Weiping Zhang’s index theorem for sub-signature operators.

We study the asymptotic expansion of the determinants of the graph Laplacians associated to discretizations of a half-translation surface endowed with a unitary flat vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the partition function of cycle-rooted spanning forests, weighted by the monodromy of the unitary connection of the vector bundle, to the corresponding zeta-regularized determinants.