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Refined analytic torsion as analytic function on the representation variety and applications

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Abstract

We prove that refined analytic torsion on a manifold with boundary is a weakly holomorphic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a unitary point. Finally we provide a new proof of Brüning-Ma gluing formula for the Ray–Singer torsion associated to a non-Hermitian connection. Our proof is quite different from the one given by Brüning and Ma and uses a temporal gauge transformation.

Résumé

On démontre que la torsion analytique fine sur une variété à bord est une section faiblement holomorphe du fibré déterminant au-dessus de la variété des reprsentations. Ce résultat nous permet d’établir une formule de recollement pour la torsion analytique fine sur les composantes connexes de l’espace des représentations complexes contenant un point unitaire. Finalement, nous donnons une nouvelle preuve de la formule de recollement de Brüning-Ma pour la torsion de Ray–Singer associée á une connexion non-hermitienne. Notre preuve est très diffrente de celle de Brüning et Ma et utilise une transformation de jauge temporelle.

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Notes

  1. The notion and properties of temporal gauge are recalled in the “Appendix”.

  2. There is no product structure assumption on g and \(h_j, j=0,1\).

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Maxim Braverman

  2. Mathematisches Institut, Universität Bonn, 53115, Bonn, Germany

    Boris Vertman

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  1. Maxim Braverman
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  2. Boris Vertman
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Correspondence to Maxim Braverman.

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Braverman, M., Vertman, B. Refined analytic torsion as analytic function on the representation variety and applications. Ann. Math. Québec 41, 67–96 (2017). https://doi.org/10.1007/s40316-016-0062-x

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