**
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
**Z**_{2}-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 **Z**_{2}-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] **