Excision for deformation K-theory of free products

Associated to a discrete group $G$, one has the topological category of finite dimensional (unitary) $G$-representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated $K$-theory spectrum is Carlsson's deformation $K$-theory of G. The goal of this paper is to examine the behavior of this functor on free products. Our main theorem shows the square of spectra associated to $G*H$ (considered as an amalgamated product over the trivial group) is homotopy cartesian. The proof uses a general result regarding group completions of homotopy commutative topological monoids, which may be of some independent interest.Comment: 32 pages, 1 figure. Final version: The title has changed, and the paper has been substantially revised to improve clarit

Z2Z_2 Topological Order and the Quantum Spin Hall Effect

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel $Z_2$ topological invariant, which distinguishes it from an ordinary insulator. The $Z_2$ classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the $Z_2$ order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multi band and interacting systems.Comment: 4 pages RevTeX. Added reference, minor correction

Real secondary index theory

In this paper, we study the family index of a family of spin manifolds. In particular, we discuss to which extend the real index (of the Dirac operator of the real spinor bundle if the fiber dimension is divisible by 8) which can be defined in this case contains extra information over the complex index (the index of its complexification). We study this question under the additional assumption that the complex index vanishes on the k-skeleton of B. In this case, using local index theory we define new analytical invariants \hat c_k\in H^{k-1}(B;\reals/\integers). We then continue and describe these invariants in terms of known topological characteristic classes. Moreover, we show that it is an interesting new non-trivial invariant in many examples.Comment: LaTeX2e, 56 pages; v2: final version to appear in ATG, typos fixed, statement of 4.5.5 improve

A path integral derivation of χy\chi_y-genus

The formula for the Hirzebruch $\chi_y$-genus of complex manifolds is a consequence of the Hirzebruch-Riemann-Roch formula. The classical index formulae for Todd genus, Euler number, and Signature correspond to the case when the complex variable $y=$ 0, -1, and 1 respectively. Here we give a {\it direct} derivation of this nice formula based on supersymmetric quantum mechanics.Comment: 5 page