Analytic Torsion Research Articles

Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I

Let G be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let ρ be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation ρ. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright [5], and was exploited in [31] to define a localised eta function. Next, let Γ be a discrete co compact subgroup of G. We use the localised analytic torsion to define the relative analytic torsion of the pair (G,Γ), and we prove that the last coincides with the Lott L2 analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case G=H, the Heisenberg group.

A comparison of the absolute and relative real analytic torsion forms

A comparison of the absolute and relative real analytic torsion forms

On full asymptotics of real analytic torsions for compact locally symmetric orbifolds

On full asymptotics of real analytic torsions for compact locally symmetric orbifolds

Analytic torsion forms for fibrations by projective curves

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.

Toeplitz operators and the full asymptotic torsion forms

This paper aims to study the asymptotic expansion of analytic torsion forms associated with a certain series of flat bundles {Fp}p∈N⁎. We prove the existence of the full expansion and give a formula for the sub-leading term, while Bismut-Ma-Zhang have studied the first order expansion and expressed the leading term as the integral of a locally computable differential form.

The asymptotics of the holomorphic analytic torsion forms

AbstractThe purpose of this paper is to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with a sequence of direct images of increasing powers of a line bundle on a bigger manifold. To handle the growing dimension of the direct images, we use the theory of Toeplitz operators. This is the family versions of the results of Bimsut and Vasserot on the asymptotics of the holomorphic torsion, as well an extension for more general bundles.

Ray-Singer Torsion and the Rumin Laplacian on Lens Space

We express explicitly the analytic torsion functions associated with the Rumin complex on lens spaces in terms of the Hurwitz zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions. Moreover, we have a formula between this torsion and the Ray-Singer torsion.

Analytic torsion for arithmetic locally symmetric manifolds and approximation of L2-torsion

In this paper we define a regularized version of the analytic torsion for quotients of a symmetric space of non-positive curvature by arithmetic lattices. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat kernel. Then we study the limiting behavior of the analytic torsion as the lattices run through a sequence of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for sequences of principal congruence subgroups, which converge to 1 at a fixed finite set of places and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the L2-analytic torsion.

Analytic Torsion of Generic Rank Two Distributions in Dimension Five

We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray–Singer analytic torsion, up to a constant.

A Cheeger–Müller theorem for manifolds with wedge singularities

We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-M\"uller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.

A generalization of analytic torsion via differential forms on spaces of metrics

We introduce multi-torsion, a spectral invariant generalizing Ray–Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of multi-torsion is inspired by an interpretation of each of analytic torsion and the eta invariant as a regularized integral of a closed differential form on a space of metrics on a vector bundle or on a space of elliptic operators. We generalize the Stokes’ theorem argument explaining the dependence of torsion and eta on the geometric data used to define them to the local product setting to prove our metric-independence theorem for multi-torsion.

Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion

We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a tileable surface endowed with a flat unitary 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 on the vector bundle, to the corresponding zeta-regularized determinants. As a consequence, we establish open problems 2 and 4, formulated by Kenyon in 2000. The spectral theory on discretizations of flat surfaces, Fourier analysis on discrete square and the analytic methods used in the proof of Ray–Singer conjecture lie in the core of our approach.

Topological G2 and Spin(7) strings at 1-loop from double complexes

We study the topological G2 and Spin(7) strings at 1-loop. We define new double complexes for supersymmetric NSNS backgrounds of string theory using generalised geometry. The 1-loop partition function then has a target-space interpretation as a particular alternating product of determinants of Laplacians, which we have dubbed the analytic torsion. In the case without flux where these backgrounds have special holonomy, we reproduce the worldsheet calculation of the G2 string and give a new prediction for the Spin(7) string. We also comment on connections with topological strings on Calabi-Yau and K3 backgrounds.

Flat vector bundles and analytic torsion on orbifolds

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.

Analytic torsion for log-Enriques surfaces and Borcherds product

Abstract We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac {1}{4}(1,1)$ . The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature $(2,10-k)$ . We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$ . As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.

Complex Valued Analytic Torsion and Dynamical Zeta Function on Locally Symmetric Spaces

AbstractWe show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent series at the zero point is related to the regularised determinant of the flat Laplacian of Cappell–Miller. When the flat vector bundle is close to an acyclic and unitary one, we show that the dynamical zeta function is regular at the zero point and that its value is equal to the complex valued analytic torsion of Cappell–Miller. This generalises the author’s previous results for unitarily flat vector bundles as well as Müller and Spilioti’s results on hyperbolic manifolds.

Families torsion and Morse functions for covering spaces

In this paper we prove the Cheeger–Müller theorem for \(L^2\)-analytic torsion form under the assumption that there exists a fiberwise Morse function and the Novikov–Shubin invariants are positive.

Asymptotic equivariant real analytic torsions for compact locally symmetric spaces

In this paper, we study the asymptotics of the equivariant analytic torsions for a certain sequence of flat vector bundles over a compact locally symmetric space. Our approach is combining the twisted trace formula with an explicit geometric formula for the twisted orbital integrals. We show that the leading term of asymptotic equivariant analytic torsion is given in terms of W-invariants with oscillating coefficients.

Resolvent trace asymptotics on stratified spaces

Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian $\Delta$ is essentially self-adjoint. We establish the asymptotic expansion for the resolvent trace of $\Delta$. Our method proceeds by induction on the depth and applies in principle to a larger class of second-order differential operators of regular-singular type, e.g., Dirac Laplacians. Our arguments are functional analytic, do not rely on microlocal techniques and are very explicit. The results of this paper provide a basis for studying index theory and spectral invariants in the setting of smoothly stratified spaces and in particular allow for the definition of zeta-determinants and analytic torsion in this general setup.

Analytic Torsion, Dynamical Zeta Function, and the Fried Conjecture for Admissible Twists

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.