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Complex-valued analytic torsion on compact bordisms

Maldonado, Osmar (2013) Complex-valued analytic torsion on compact bordisms.
Dissertation, University of Vienna. Fakultät für Mathematik
BetreuerIn: Haller, Stefan

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DOI: 10.25365/thesis.28995
URN: urn:nbn:at:at-ubw:1-29791.05728.893165-6

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Abstract in English

A compact Riemannian bordism is a compact manifold M of dimension m, with Riemannian metric g, whose boundary N is the disjoint union of two closed submanifolds V and W, with absolute boundary conditions on V and relative ones on W. This thesis is concerned with the complex-valued analytic torsion on compact Riemannian bordisms. Consider E, a flat complex vector bundle over M, with a Hermitian metric h. The Ray--Singer metric, defined with the use of self-adjoint Laplacians, acting on E-valued smooth forms satisfying the boundary conditions above, is a Hermitian metric on the determinant line of the relative cohomology groups (with respect to W). Assume E is endowed with a fiber-wise nondegenerate complex symmetric bilinear form b. The complex-valued analytic torsion considered as a nondegenerate bilinear form on the determinant line was first studied by Burghelea and Haller on closed manifolds in analogy with the Ray--Singer metric. In order to define this torsion one uses spectral theory of certain not necessarily self-adjoint Laplacians In few words, one looks at the restriction of the bilinear form to the generalized zero-eigenspace of the generalized Laplacian and considers the corresponding induced nondegenerate bilinear on the determinant line of the relative cohomology groups Thus, the complex-valued analytic torsion is the product of this bilinear form with the non-zero complex number obtained as zeta-regularized determinant of generalized Laplacians. The variation of the torsion with respect to smooth changes of the Riemannian metric and the bilinear form is encoded in the anomaly formulas. In order to obtain these formulas, we use the coefficient of the constant term in the heat trace asymptotic expansion for small time, associated to the generalized Laplacian. Our method uses the anomaly formulas for the Ray--Singer metric obtained by Bruening and Ma. CoEuler structures, the dual notion to Euler Structures of Turaev, were used by Burghelea and Haller to discuss the anomaly formulas for the torsion on closed manifolds. We extend the notion of coEuler structures to the situation of compact Riemannian bordisms. The space of coEuler structures is an affine space modeled by the relative cohomology group (wrt N) in degree m-1.

Schlagwörter in Englisch

analytic / torsion / bordisms / asymptotic / expansion / bilinear / hermitian / Laplacians / relative / absolute

Abstract in German

Wir berechnen diese Koeffizienten, indem wir die von Bruening und Ma gefundenen Formeln fuer die Ray--Singer Metrik benutzen. Schliesslich definieren wir coEuler Strukturen auf einem kompakten riemannschen Bordismus. Im Rahmen einer geschlossenen Mannigfaltigkeit sind CoEuler Strukturen von Burghelea und Haller studiert worden. In unserem Fall wird der Raum von coEuler Strukturen als ein affiner Raum ueber die relative (bzw. N) Kohomologie Gruppe im Grad m-1 von M definiert. Diese koennen als duale Objekte fuer die Euler-Strukturen von Turaev angesehen werden.

Schlagwörter in Deutsch

Analytische / Torsion / Bordismem / Randbedingungen / asymptotische / Expansion / Bilinearforme / Laplaceoperatoren / Hermitische

Item Type: Hochschulschrift (Dissertation)
Author: Maldonado, Osmar
Title: Complex-valued analytic torsion on compact bordisms
Umfangsangabe: XIII,131 S. : graph. Darst.
Institution: University of Vienna
Faculty: Fakultät für Mathematik
Publication year: 2013
Language: eng ... Englisch
Supervisor: Haller, Stefan
Assessor: Brüning, Jochen
2. Assessor: Braverman, Maxim
Classification: 31 Mathematik > 31.52 Differentialgeometrie
31 Mathematik > 31.55 Globale Analysis
AC Number: AC11101427
Item ID: 28995
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