Burghelea-Haller analytic torsion for manifolds with boundary

Su, Guangxiang

doi:10.1090/s0002-9939-09-10003-5

Cited by 5 publications

(12 citation statements)

References 19 publications

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“…The anomaly formulas given in Theorem 3 generalize the ones obtained by Burghelea and Haller in the closed situation in [4] , and also the ones in [25] by Su in odd dimensions: they do not longer require g and b to be constant in a neighborhood of the boundary and both kind of boundary conditions are considered at the same time.…”

Section: Introductionsupporting

confidence: 68%

“…for

. In [26] and [25] , given a bilinear form b , this involution has been exploited to study the bilinear Laplacian in terms of the Hermitian one associated to the compatible Hermitian form in (36) , in both cases with and without boundary. However, our approach is a little different since we do not use a Hermitian form globally compatible with

, but instead a local compatibility only, see Section 2.4 below.…”

Section: Heat Trace Asymptotic Expansion and Anomaly Formulasmentioning

confidence: 99%

“…The method described in [4, Section 6] leading to the infinitesimal variation of the torsion in the closed situation also holds in the situation with boundary; this was also used in [25] . In particular, by [4, formula (54)] , the problem of computing this infinitesimal variation boils down to computing

and

associated to

with

and

respectively given by (42) , (43) in Theorem 2 .…”

Section: Complex-valued Analytic Torsion On Compact Bordismsmentioning

confidence: 99%

“…In [25] , by using techniques from [26] , [27] , [10] , [19] , Su generalized the complex-valued analytic Ray–Singer torsion to the situation in which

(or

). Also in [25] , Su proved that in odd dimensions, the complex-valued analytic torsion does depend neither on smooth variations of the Riemannian metric nor on smooth variations of the bilinear form, as long as these are compactly supported in the interior of M . This section ends with Theorem 3 , which gives formulas for the variation of the complex-valued analytic Ray–Singer torsion with respect to smooth variations of the metric and the bilinear form.…”

Section: Introductionmentioning

confidence: 99%

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

Molina

2013

Differential Geometry and its Applications

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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.

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Section: Introductionsupporting

confidence: 68%

“…for

, but instead a local compatibility only, see Section 2.4 below.…”

Section: Heat Trace Asymptotic Expansion and Anomaly Formulasmentioning

confidence: 99%

and

associated to

with

and

respectively given by (42) , (43) in Theorem 2 .…”

Section: Complex-valued Analytic Torsion On Compact Bordismsmentioning

confidence: 99%

“…In [25] , by using techniques from [26] , [27] , [10] , [19] , Su generalized the complex-valued analytic Ray–Singer torsion to the situation in which

(or

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Molina

2013

Differential Geometry and its Applications

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“…In a subsequent paper [28] Vertman derived a gluing formula for the refined analytic torsion in this setting under the assumption that the Hermitian metric is flat. Inspired by the construction of [27], Su [23] extended the Burghelea-Haller analytic torsion to compact manifolds with boundary and compared it with the refined analytic torsion. In this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary and compare the Cappell-Miller analytic torsion with the refined analytic torsion.…”

Section: Introductionmentioning

confidence: 99%

Cappell-Miller analytic torsion for manifolds with boundary

Huang

2010

Proc. Amer. Math. Soc.

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Abstract. Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula for refined analytic torsion and the comparison theorem for the Cappell-Miller analytic torsion and the refined analytic torsion, we establish the gluing formula for the Cappell-Miller analytic torsion in the case where the Hermitian metric is flat.

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The refined analytic torsion and a well-posed boundary condition for the odd signature operator

Huang

Lee

2018

Journal of Geometry and Physics

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In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator B in the sense of Seeley. We then show that the zeta-determinants of B 2 and eta-invariants of B subject to these boundary conditions are well defined by using the method of the asymptotic expansions of the traces of the heat kernels. We use these facts to define the refined analytic torsion on a compact manifold with boundary and show that it is invariant on the change of metrics in the interior of the manifold. We finally describe the refined analytic torsion under these boundary conditions as an element of the determinant line.

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Burghelea-Haller analytic torsion for manifolds with boundary

Abstract: Abstract. In this paper, we extend the complex-valued Ray-Singer torsion, introduced by Burghelea-Haller, to compact connected Riemannian manifolds with boundary. We also compare it with the refined analytic torsion.

Cited by 5 publications

References 19 publications

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

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

Cappell-Miller analytic torsion for manifolds with boundary

The refined analytic torsion and a well-posed boundary condition for the odd signature operator

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