Analytic Torsion Research Articles

On the Gluing Formula of Real Analytic Torsion Forms

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.

The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume

In this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds. We consider sequences of coverings Xi of a fixed hyperbolic orbifold X0. Our main result is that for certain sequences of coverings and strongly acyclic flat bundles the analytic torsion divided by the index of the covering converges to the L2-torsion. Our results apply to certain sequences of arithmetic groups, in particular to sequences of principal congruence subgroups of SO0(d,1)(Z) and to sequences of principal congruence subgroups or Hecke subgroups of Bianchi groups.

Erratum to: On the gluing formula for the analytic torsion

In this note, we want to correct a mistake in the proof of [1, Theorem 2.2] which does not affect the statement of the theorem, though. In [1, (2.7)], we incorrectly state an isomorphism of complexes, thus we need to modify the corresponding argument, by first replacing the last six lines on p. 1098, beginning with “we also have · · · ”, with the following text. “We also have a Z2-equivariant short exact sequence of complexes 0 → C•(W u/W u Y1 , F)⊗ C− γ −→ C•( W u, F) γ −→ C•(W u, F)⊗ C+ → 0, (2.7)

Low Energy Resolvent for the Hodge Laplacian: Applications to Riesz Transform, Sobolev Estimates, and Analytic Torsion

On an asymptotically conic manifold (M,g), we analyze the asymptotics of the integral kernel of the resolvent Rq(k):=(Δq+k2)−1 of the Hodge Laplacian Δq on q-forms as the spectral parameter k approaches zero, assuming that 0 is not a resonance. The first application we give is an Lp Sobolev estimate for d+δ and Δq. Then we obtain a complete characterization of the range of p>1 for which the Riesz transform for q-forms |$T_q=(d+\delta)\Delta _q^{-1/2}$| is bounded on Lp. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian manifolds (Ωϵ,gϵ) degenerating to a compact manifold (Ω0,g0) with a conic singularity as ϵ→0.

Modular curvature for noncommutative two-tori

Modular curvature for noncommutative two-tori

La torsion analytique holomorphe généralisée des fibrés en droites intégrables

Let X be a compact Kähler manifold. We extend the notion of Quillen metric to the integrable line bundles L¯ on X. In particular, we show that the notion of holomorphic analytic torsion extends to integrable line bundles L¯ satisfying Hq(X,L)=0 for q⩾1.

Burghelea-Haller analytic torsion of $\mathbb {Z}_{2}$-graded elliptic complexes

Burghelea-Haller analytic torsion of $\mathbb {Z}_{2}$-graded elliptic complexes

Determinantal Point Processes and Fermions on Complex Manifolds: Large Deviations and Bosonization

We study determinantal random point processes on a compact complex manifold X associated to a Hermitian metric on a line bundle over X and a probability measure on X. Physically, this setup describes a gas of free fermions on X subject to a U(1)-gauge field and when X is the Riemann sphere it specializes to various random matrix ensembles. Our general setup will also include the setting of weighted orthogonal polynomials in $${\mathbb{C}^{n}}$$ , as well as in $${\mathbb{R}^{n}}$$ . It is shown that, in the many particle limit, the empirical random measures on X converge exponentially towards the deterministic pluripotential equilibrium measure, defined in terms of the Monge–Ampère operator of complex pluripotential theory. More precisely, a large deviation principle (LDP) is established with a good rate functional which coincides with the (normalized) pluricomplex energy of a measure recently introduced in Berman et al. (Publ Math de l’IHÉS 117, 179–245, 2013). We also express the LDP in terms of the Ray–Singer analytic torsion. This can be seen as an effective bosonization formula, generalizing the previously known formula in the Riemann surface case to higher dimensions and the paper is concluded with a heuristic quantum field theory interpretation of the resulting effective boson–fermion correspondence.

Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of \(\mathrm{SL }_2(\mathbb {C})\) to the corresponding Reidemeister torsion. While the analytic torsion is a spectral invariant of the manifold, the Reidemeister torsion is of combinatorial nature. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.

Generalized holomorphic analytic torsion

In this paper we extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for projective morphisms. The extension of the holomorphic analytic torsion classes of Bismut and K¨ohler is obtained as the theory of generalized analytic torsion classes associated to –R=2 , R being the R -genus. As an application of the axiomatic characterization, we give new simpler proofs of known properties of holomorpic analytic torsion classes, we give a characterization of the R -genus, and we construct a direct image of hermitian structures for projective morphisms.

The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms

In this paper we generalize the arithmetic Grothendieck-Riemann- Roch Theorem to projective morphisms between arithmetic varieties that are not necessarily smooth over the complex numbers. The main ingredient is the theory of generalized holomorphic analytic torsion classes.

The hypoelliptic Laplacian, analytic torsion and Cheeger–Müller Theorem

The purpose of this Note is to prove a formula relating the hypoelliptic Ray–Singer metric and the Milnor metric on the determinant of the cohomology of a compact Riemannian manifold by a Witten-like deformation of the hypoelliptic Laplacian in de Rham theory.

The Metric Anomaly of Analytic Torsion on Manifolds with Conical Singularities

In this paper we study the analytic torsion of a manifold with isolated conical singularities. First we show that the analytic torsion is invariant under deformations of the metric which are of higher order near the singularities. Then we identify the metric anomaly of analytic torsion for a bounded generalized cone at its regular boundary in terms of spectral information of the cross-section. In view of previous computations of analytic torsion on cones, this leads to a detailed geometric identification of the topological and spectral contributions to analytic torsion, arising from the conical singularity. The contribution exhibits a torsion-like spectral invariant of the cross-section of the cone, which we study under scaling of the metric on the cross-section.

The comparison of two constructions of the refined analytic torsion on compact manifolds with boundary

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman (Vertman, 2009, 2008) and the authors (Huang and Lee, 2010, 2012) but these two constructions are completely different. Vertman used a double of de Rham complexes consisting of the minimal and maximal closed extensions of a flat connection and the authors used well-posed boundary conditions P−,L0, P+,L1 for the odd signature operator. In this paper we compare these two constructions by using the BFK-gluing formula for zeta-determinants, the adiabatic method for stretching cylinder part near boundary and the result for comparison of eta invariants in Huang and Lee (2012) when the odd signature operator comes from a Hermitian flat connection.

Reidemeister torsion for flat superconnections

We use higher parallel transport—more precisely, the integration \(\mathsf{A }_\infty \)-functor constructed in Arias Abad and Schatz (The \(\mathsf{A}_\infty \) de Rham theorem and the integration of representations up to homotopy. Int Math Res Not, 2013) and Block and Smith (A Riemann–Hilbert correspondence for infinity local systems. arXiv:0908.2843, 2012)—to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger–Muller theorem, namely that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu (Contemp Math 546, 199–212, 2011).

Analytic torsion and $L^2$-torsion of compact locally symmetric manifolds

In this paper we study the analytic torsion and the $L^2$-torsion of compact locally symmetric manifolds. We consider the analytic torsion with respect to representations of the fundamental group which are obtained by restriction of irreducible representations of the group of isometries of the underlying symmetric space. The main purpose is to study the asymptotic behavior of the analytic torsion with respect to sequences of representations associated to rays of highest weights.

Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume

Abstract We study Selberg zeta functions Z(s,σ) associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete oriented odd-dimensional hyperbolic spin manifold of finite volume. We assume a certain condition on the fundamental group of the manifold. A priori, the Selberg zeta functions are defined only for s in some right half-plane of ℂ. We will prove that for any locally homogeneous bundle the functions Z(s,σ) have a meromorphic continuation to ℂ and we will give a complete description of their singularities in terms of spectral data of the underlying manifold. Our work generalizes results of Bunke and Olbrich to the non-compact situation. As an application of our results one can compare the normalized Reidemeister torsions on hyperbolic 3-manifolds with cusps which were introduced by Menal-Ferrer and Porti to the corresponding regularized analytic torsions.

A gluing formula for the analytic torsion on singular spaces

We prove a gluing formula for the analytic torsion on non-compact (i.e. singular) riemannian manifolds. Let M= U\cup M_1, where M_1 is a compact manifold with boundary and U represents a model of the singularity. For general elliptic operators we formulate a criterion, which can be checked solely on U, for the existence of a global heat expansion, in particular for the existence of the analytic torsion in case of the Laplace operator. The main result then is the gluing formula for the analytic torsion. Here, decompositions M=M_1\cup_W M_2 along any compact closed hypersurface W with M_1, M_2 both non-compact are allowed; however product structure near W is assumed. We work with the de Rham complex coupled to an arbitrary flat bundle F; the metric on F is not assumed to be flat. In an appendix the corresponding algebraic gluing formula is proved. As a consequence we obtain a framework for proving a Cheeger-M\"uller type Theorem for singular manifolds; the latter has been the main motivation for this work. The main tool is Vishik's theory of moving boundary value problems for the de Rham complex which has also been successfully applied to Dirac type operators and the eta invariant by J. Br\"uning and the author. The paper also serves as a new, self--contained, and brief approach to Vishik's important work.

Anomaly formulas for the complex-valued analytic torsion on compact bordisms

We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bilinear form are used to define non-selfadjoint Laplacians acting on vector-valued smooth forms under absolute and relative boundary conditions. In order to define the complex-valued analytic torsion in this situation, we study spectral properties of these generalized Laplacians. Then, as main results, we obtain so-called anomaly formulas for this torsion. Our reasoning takes into account that the coefficients in the heat trace asymptotic expansion associated to the boundary value problem under consideration, are locally computable. The anomaly formulas for the complex-valued Ray–Singer torsion are derived first by using the corresponding ones for the Ray–Singer metric, obtained by Brüning and Ma on manifolds with boundary, and then an argument of analytic continuation. In odd dimensions, our anomaly formulas are in accord with the corresponding results of Su, without requiring the variations of the Riemannian metric and bilinear structures to be supported in the interior of the manifold.

Analytic torsion on manifolds under locally compact group actions

Analytic torsion on manifolds under locally compact group actions