Narrow Results

For parity-conserving fermionic chains, we review how to associate ${\mathbb{Z}}_2$-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the ${\mathbb{Z}}_2$-valued spectral flow provides a topological obstruction for two systems to have the same ${\mathbb{Z}}_2$-index. A rudimentary definition of a ${\mathbb{Z}}_2$-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

In this paper, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group $H$ of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow and equivariant Maslov indices is established. We also study equivariant $\eta$-invariants which play a fundamental role in the equivariant analog of Getzler’s spectral flow formula. As a consequence, we establish a relation between equivariant $\eta$-invariants and equivariant Maslov triple indices in the splitting of manifolds.

We solve the massless Schwinger model exactly in Hamiltonian formalism on a circle. We construct physical states explicitly and discuss the role of the spectral flow and nonperturbative vacua. Different thermodynamical correlation functions are calculated and after performing the analytical continuation are compared with the corresponding expressions obtained for the Schwinger model on the torus in Euclidean path integral formalism obtained before.

In this paper we study some aspects of closed string theories in the Nappi–Witten space–time. The effects of spectral flow on the geodesics are studied in terms of an explicit parametrization of the group manifold. The worldsheets of the closed strings under the spectral flow of the geodesics can be classified into four classes, each with a geometric interpretation. We also obtain a free field realization of the Nappi–Witten affine Lie algebra in the most general conditions using a different but equivalent parametrization of the group manifold.

Based on the spectral flow and the stratification structures of the symplectic group Sp(2n,C), the Maslov-type index theory and its generalization, the ω-index theory parameterized by all ω on the unit circle, for arbitrary paths in Sp(2n,C) are established. Then the Bott-type iteration formula of the Maslov-type indices for iterated paths in Sp(2n,C) is proved, and the mean index for any path in Sp(2n,C) is defined. Also, the relation among various Maslov-type index theories is studied.

The spectral flow for paths of admissible operators in arbitrary Banach space is defined, and some properties of the spectral flow is studied.

We relate the spectral flow to the index for paths of selfadjoint Breuer–Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin–Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.

In this paper, we give a comprehensive treatment of a “Clifford module flow” along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in ${\mathrm{\text{KO}}}_{\ast }(ℝ)$ via the Clifford index of Atiyah–Bott–Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that

We present an analytic proof of the Poincaré-Hopf index theorem. Our proof makes use of an old idea of Atiyah and works for the case where the isolated zeros of the vector field can be degenerate.

In this review we discuss the local index formula in noncommutative geometry (NCG) from the viewpoint of two new proofs that are given in [16, 17] and [18] respectively. These proofs are partly inspired by the approach of Higson [35], especially that in [18], but they differ in several fundamental aspects, in particular they apply to semifinite spectral triples for a *-subalgebra of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem, and reduce the hypotheses of the theorem to those necessary for its statement.

These proofs rely on the introduction of a function valued cocycle which is ‘almost’ a (b, B)-cocycle in the cyclic cohomology of . They do not need the ‘discrete dimension spectrum’ assumption of the original Connes-Moscovici proof [25], only a much weaker condition on the analytic continuation of certain zeta functions, and this only for part of the statement.

In this article we also explain the relationship of the pairing between K-theory and semifinite spectral triples to KK-theory and the Kasparov product. This discussion shows that semifinite spectral triples are a specific kind of representative of a KK-class, and the analytically defined index is compatible with the Kasparov product.