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Anomaly formulas for the complex-valued analytic torsion on compact bordisms
1Department of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria.
Differential Geometry and Its Applications
|April 19, 2016
Summary
This study extends complex-valued analytic torsion to compact Riemannian bordisms, yielding anomaly formulas for this geometric invariant. These findings are crucial for understanding torsion on manifolds with boundary.
Area of Science:
- Differential Geometry
- Geometric Analysis
- Topology
Background:
- The study builds upon the concept of complex-valued analytic torsion, initially defined for closed manifolds.
- It addresses the challenge of extending this invariant to manifolds with boundaries, specifically compact Riemannian bordisms.
Purpose of the Study:
- To generalize complex-valued analytic torsion to compact Riemannian bordisms.
- To derive anomaly formulas for this extended torsion.
- To investigate the spectral properties of non-selfadjoint Laplacians on vector-valued forms.
Main Methods:
- Consideration of flat complex vector bundles with fiberwise nondegenerate symmetric bilinear forms.
- Definition of non-selfadjoint Laplacians under absolute and relative boundary conditions.
- Analysis of spectral properties and heat trace asymptotic expansions.
Main Results:
- Successful extension of complex-valued analytic torsion to compact Riemannian bordisms.
- Derivation of anomaly formulas for the complex-valued analytic torsion.
- Demonstration that coefficients in heat trace expansions are locally computable.
Conclusions:
- The derived anomaly formulas align with existing results in odd dimensions under specific conditions.
- The methods provide a framework for studying analytic torsion on manifolds with boundary.
- The work contributes to a deeper understanding of geometric invariants in Riemannian geometry.
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