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We establish a mod k index theorem for signature operators. [ABSTRACT FROM AUTHOR]

Let X be a closed Riemannian manifold and let H !→ X be an embedded hypersurface. Let X = X_{+} ∪_H X_{−} be a decomposition of X into two manifolds with boundary, with X_{+} ∩X_{−} = H. In this expository article, surgery – or gluing – formulæ for several geometric and spectral invariants associated to a Dirac-type operator ðX on X are presented. Considered in detail are: the index of ðX, the index bundle and the determinant bundle associated to a family of such operators, the eta invariant and the analytic torsion. In each case the precise form of the surgery theorems, as well as the different techniques used to prove them, are surveyed.

Abstract: We consider geometric and analytical aspects of -theory on a manifold with boundary . The partition function of the -field requires summing over harmonic forms. When is closed, Hodge theory gives a unique harmonic form in each de Rham cohomology class, while in the presence of a boundary the Hodge–Morrey–Friedrichs decomposition should be used. This leads us to study the boundary conditions for the -field. The dynamics and the presence of the dual to the -field gives rise to a mixing of boundary conditions with one being Dirichlet and the other being Neumann. We describe the mixing between the corresponding absolute and relative cohomology classes via Poincaré duality angles, which we also illustrate for the M5-brane as a tubular neighborhood. Several global aspects are then considered. We provide a systematic study of the extension of the bundle and characterize obstructions. Considering as a fiber bundle, we describe how the phase looks like on the base, hence providing dimensional reduction in the boundary case via the adiabatic limit of the eta invariant. The general use of the index theorem leads to a new effect given by a gravitational Chern–Simons term on whose restriction to the boundary would be a generalized WZW model. This suggests that holographic models of -theory can be viewed as a sector within this index-theoretic approach. [Copyright &y& Elsevier]

Abstract: We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type associated to the spinor bundle Σ on an odd dimensional convex co-compact hyperbolic manifold . As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant associated to the Dirac operator D over a convex co-compact hyperbolic manifold and that , thus extending Millson''s formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group Γ, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil–Petersson metric on Schottky space. [Copyright &y& Elsevier]