On the Gluing Formula of Real Analytic Torsion Forms

We report on the construction of secondary invariants in connection with the Atiyah-Singer index theorem for families, and the Theorem of Riemann-Roch-Grothendieck. The local families index theorem plays an important role in the construction. In complex geometry, the corresponding objects are the analytic torsion forms and the analytic torsion currents. These objects exhibit natural functorial properties with respect to composition of maps. Gillet and Soulé have used these objects to prove a Riemann-Roch theorem in Arakelov geometry. Also we state a Riemann-Roch theorem for flat vector bundles, and report on the construction of corresponding higher analytic torsion forms.

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.

This chapter describes a general group-theoretical property enjoyed by fundamental groups of compact Kahler manifolds. In general, the condition that an infinitesimal deformation be tangent to an analytic path is equivalent to the existence of a formal power series deformation with the infinitesimal deformation as its leading term. If a k-jet of a deformation exists, the condition that this k-jet of a deformation extend to a -jet of a deformation is given by the vanishing of a certain cohomology class, which depends on the previous k terms of the series. The chapter discusses the Lemma C is proved, the canonical isomorphisms between de Rham cohomology and Eilenberg-MacLane cohomology, Theorem A is reduced to Theorem D. The proofs of Theorems A and C involve the harmonic theory of M with coefficients in a flat vector bundle. The flat vector bundle we shall need is the complexification of the flat vector bundle ad P, denote this flat complex vector bundle by E.

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.

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.

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D -invariant subbundle (resp. if it is the D -invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to the existence of a so-called harmonic metric in E . In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X,g) . We propose new notions of g -(poly-)stability of such bundles, and of g -Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if g is a Kähler metric, but are different in general. Our main result is that the g -polystability in our sense is equivalent to the existence of a g -Hermitian-Einstein metric. Our notion of a g -Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].

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

We summarize the global geometric formulation of Einstein-Scalar-Maxwell theories twisted by flat symplectic vector bundle which encodes the duality structure of the theory. We describe the scalar-electromagnetic symmetry group of such models, which consists of flat unbased symplectic automorphisms of the flat symplectic vector bundle lifting those isometries of the scalar manifold which preserve the scalar potential. The Dirac quantization condition for such models involves a local system of integral symplectic spaces, giving rise to a bundle of polarized Abelian varieties equipped with a symplectic flat connection, which is defined over the scalar manifold of the theory. Generalized Einstein-Scalar-Maxwell models arise as the bosonic sector of the effective theory of string/M-theory compactifications to four-dimensions, and they are characterized by having non-trivial solutions of "U-fold" type.

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.

The direct image of a flat fibration with complex fibers

In a previous paper, the authors together with L. Vrancken initiated the study of $3$-dimensional CR submanifolds of the nearly K\" ahler homogeneous $\mathbb S^3\times \mathbb S^3$. As is shown by Butruille this is one of only four homogeneous $6$-dimensional nearly K\"ahler manifolds. Besides its almost complex structure $J$ it also admits a canonical almost product structure $P$. Along a $3$-dimensional CR submanifold the tangent space of $\mathbb S^3\times\mathbb S^3$ can be naturally split as the orthogonal sum of three $2$-dimensional vector bundles $\mathcal D_1$, $\mathcal D_2$ and $\mathcal D_3$. We study the CR submanifolds in relation to the behavior of the almost product structure on these vector bundles.

Toeplitz operators and the full asymptotic torsion forms

The Partition Function for Topological Field Theories

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.

We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev settings, using an argument of Alvarez Lopez-Kordyukov.