Secondary invariants of elliptic operators

Atiyah, Patodi and Singer wrote a remarkable series of three papers in Mathematical Proceedings of the Cambridge Philosophical Society that were published in the mid seventies, on non-local elliptic boundary value problems, which have been intensely studied ever since, both in the mathematics and physics literature. They applied their theory in particular to the important case of the signature operator on an oriented, compact manifold with boundary, where they identified the boundary operator which is now known as the odd signature operator, which is self-adjoint and elliptic, having spectrum in the real numbers. For this and the Dirac operator, they introduced the eta invariant which measures the spectral asymmetry of the operator and is a spectral invariant. Coupling with flat bundles, they introduced the closely related rho invariant, which has the striking property that it is independent of the choice of Riemannian metric needed in its definition. In [BM2] we generalize the construction of Atiyah-Patodi-Singer to the twisted signature complex with an odd-degree differential form as flux and with coefficients in a flat vector bundle. We establish the homotopy invariance of the twisted rho invariant under some hypotheses on the fundamental group of the manifold. In [BM1], [BM3], we study the twisted Dirac operator and its twisted rho invariant as well as the relation to positive scalar curvature.

In a series of papers in the early 1970s, Ray and Singer introduced analytic torsion and its holomorphic counterpart. They conjectured that it was equal to Reidemeister-Franz torsion defined using simplicial complexes. This was proved by Cheeger and Mueller in the late 1970s, with a direct analytic proof given by Bismut-Zhang in the early 1990s. These invariants have been intensely studied ever since, both in the mathematics and physics literature. In [MW1, MW3], we study twisted analytic torsion for the twisted de Rham complex with odd-degree differential form as flux and with coefficients in a flat vector bundle. We show that it is independent of the choice of Riemannian needed in its definition. In certain cases, we show that it is equal to twisted Reidemeister-Franz torsion defined using simplicial complexes. We relate it to T-duality in String Theory. We also define and study the holomorphic analog of our twisted analytic torsion in [MW2], for flat superconnections and finally for general Z2-graded elliptic complexes in the same paper.

References

  • [BM3] M-T. Benameur and V. Mathai,
    Spectral sections, twisted rho invariants and positive scalar curvature
    Journal of Noncommutative Geometry,
    8, no. 3 (2015) 821-850, [1309.5746]

  • [BM2] M-T. Benameur and V. Mathai,
    Index type invariants for twisted signature complexes and homotopy invariance,
    Mathematical Proceedings of the Cambridge Philosophical Society,
    156 no.3 (2014) 473-503, [1202.0272]

  • [BM1] M-T. Benameur and V. Mathai,
    Conformal invariants of twisted Dirac operators and positive scalar curvature
    Journal of Geometry and Physics,
    70 (2013) 39-47, [1210.0301]
    Erratum, Journal of Geometry and Physics,
    76 (2014) 263-264,

  • [MW3] V. Mathai and S. Wu,
    Analytic torsion for twisted de Rham complexes,
    Journal of Differential Geometry,
    88 (2011) 297-332, [0810.4204]

  • [MW2] V. Mathai and S. Wu,
    Analytic torsion of Z2-graded elliptic complexes,
    Contemporary Mathematics,
    546 (2011) 199-212. [1001.3212]

  • [MW1] V. Mathai and S. Wu,
    Twisted Analytic Torsion,
    Science China Mathematics,
    53 no. 3 (2010) 555-563 [0912.2184]