Twisted analytic torsion

We review the Reidemeister, Ray-Singer’s analytic torsion and the Cheeger-Mller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle and the flux form and the radius of the circular fiber with its inverse, the twisted torsion of invariant forms are inverse to each other for any dimension.     

Quaternionic analytic torsion

We define an (equivariant) quaternionic analytic torsion for anti-self-dual vector bundles on quaternionic Kähler manifolds, using ideas by Leung and Yi. We do so by constructing a Laplace operator associated to a complex defined by Salamon. We compute this torsion for vector bundles on quaternionic homogeneous spaces with respect to any isometry in the component of the identity, in terms of roots and Weyl groups.     

The analytic torsion of a disc

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145–210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger–Müller theorem. We use a formula proved by Brüning and Ma (GAFA 16:767–873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Lück, J Diff Geom 37:263–322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695–714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529–533, 2009).     

Functoriality of real analytic torsion forms

In this paper, we prove the functoriality of the analytic torsion forms of Bismut and Lott [BLo] with respect to the composition of two submersions.     

Scattering matrix resonances and quasi-levels

In the present paper, the relationship between resonances of the scattering matrix and quasi-levels in the abstract scattering theory is analyzed.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 16–23, January, 1996.     

Ray-Singer type theorem for the refined analytic torsion

We show that the refined analytic torsion is a holomorphic section of the determinant line bundle over the space of complex representations of the fundamental group of a closed oriented odd-dimensional manifold. Further, we calculate the ratio of the refined analytic torsion and the Farber-Turaev combinatorial torsion. As an application, we establish a formula relating the eta-invariant and the phase of the Farber-Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology.     

On the gluing formula for the analytic torsion

In this paper, we derive the Cheeger–Müller/Bismut–Zhang theorem for manifolds with boundary and the gluing formula for the analytic torsion of flat vector bundles in full generality, i.e., we do not assume that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary.     

紧群作用下的扭化解析挠率

在紧群中的非单位元的作用只有孤立的不动点的情形下, 我们研究了扭化de Rham 复形上的解析挠率的等变情形. 在一些条件下, 我们也研究了Z₂- 分次的椭圆复形上的离域的(delocalized) L²-解析挠率.     

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimensions 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.     

Scattering matrix in conformal geometry

Inventiones mathematicae -     

Perturbation of analytic hermitian matrix functions

The behaviour of real eigenvalues of selfadjoint analytic matrix valued functions under small selfadjoint analytic perturbations is studied. Attention is paid mainly to the case when the perturbation is definite (or semidefi-nite). Earlier results of the authors concerning matrix polynomials of first degree are extended to the case of analytic functions.     

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.     

Witten deformation of analytic torsion and the spectral sequence of a filtration

LetF be a flat vector bundle over a compact Riemannian manifoldM and letf :M be a Morse function. Letg ^F be a smooth Euclidean metric onF, letg _t ^F =e ^–2tf g ^F , and let ^RS (t) be the Ray-Singer analytic torsion ofF associated with the metricg _t ^F . Assuming that f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log ^RS (t) fort+ of the forma ₀+a ₁ t+blog(t/)+o(1), where the coefficientb is a half-integer depending only on the Betti numbers ofF. In the case where all the critical values off are rational, we calculate the coefficientsa ₀ anda ₁ explicitly in terms of the spectral sequence of a filtration associated with the Morse function. These results are obtained as applications of a theorem by Bismut and Zhang.The research was supported by grant No. 449/94-1 from the Israel Academy of Sciences and Humanities.     

On the perturbation of analytic matrix functions

In this paper behaviour of the spectrum of matrix-valued functions depending analytically on two parameters is studied. Generalizations of the Rellich theorem on analytic dependence of the spectrum and complete regular splitting of multiple eigenvalues are established.This work is partially supported by Natural Sciences and Engineering Research Council of Canada. R. H. also acknowledges appointment as a Post Doctoral Fellow of the Pacific Institute for Mathematical Sciences.     

Semidefinite perturbations of analytic Hermitian matrix functions

In this paper we consider the behaviour of a real eigenvalue of an analytic Hermitian matrix valued function under perturbation with a positive semidefinite analytic Hermitian matrix valued function. We extend previous results on perturbation with positive definite functions to the positive semidefinite case.Partially supported by an NSF grant.