Large time limit of the heat operator for families of cocompact manifolds

par Sara Azzali (IFM, Potsdam)

vendredi 9 sept. 2016, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Institut Camille Jordan)

The heat kernel’s supertrace of a Dirac operator interpolates between the local geometry of a closed manifold and a global invariant, the index of the Dirac operator. This fundamental property, first applied by Atiyah, Bott and Patodi to give a proof of the Atiyah–Singer index theorem, is a crucial tool in local index theory. In the case of a fibre bundle and a family of fibrewise Dirac operators, heat kernel techniques are combined with the use of superconnections and were introduced by Quillen and Bismut: the heat operator’s supertrace is here a differential form on the parameter space (the base space of the fibration) and the computation of its large time limit provides a differential forms refinement of the cohomological Atiyah–Singer formula. The case when the fibres are noncompact is particularely tricky as in general the large time limit is not convergent: Heitsch, Lazarov and Benameur studied this problem on a foliated manifold and found regularity conditions on the spectrum which ensure the convergence at large time. In a joint work with Sebstian Goette and Thomas Schick, we have studied the particular case of a family of fibrewise signature operators for families of manifolds with a cocompact group action. We prove that there exists a way to carefully estimate all the terms appearing in the Volterra expansion of the heat operator’s supertace and compute explicitely its large time limit, without requiring extra regularity assumption on the spectrum. With some regularity conditions (involving the determinant class or the positivity of the Novikov-Shubin invariants) we obtain the L^2 local index formula.