On the KP Hierarchy, W^∞\hat{W}_{\infty} Algebra, and Conformal SL(2,R)/U(1) Model: II. The Quantum Case

This paper is devoted to constructing a quantum version of the famous KP hierarchy, by deforming its second Hamiltonian structure, namely the nonlinear $\hat{W}_{\infty}$ algebra. This is achieved by quantizing the conformal noncompact $SL(2,R)_{k}/U(1)$ coset model, in which $\hat{W}_{\infty}$ appears as a hidden current algebra. For the quantum $\hat{W}_{\infty}$ algebra at level $k=1$, we have succeeded in constructing an infinite set of commuting quantum charges in explicit and closed form. Using them a completely integrable quantum KP hierarchy is constructed in the Hamiltonian form. A two boson realization of the quantum $\hat{W}_{\infty}$ currents has played a crucial role in this exploration.Comment: 33

Bulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the Z2-invariant in topological insulators

We study the bulk-edge correspondence in topological insulators by taking Fu-Kane spin pumping model as an example. We show that the Kane-Mele invariant in this model is Z2 invariant modulo the spectral flow of a single-parameter family of 1+1-dimensional Dirac operators with a global boundary condition induced by the Kramers degeneracy of the system. This spectral flow is defined as an integer which counts the difference between the number of eigenvalues of the Dirac operator family that flow from negative to non-negative and the number of eigenvalues that flow from non-negative to negative. Since the bulk states of the insulator are completely gapped and the ground state is assumed being no more degenerate except the Kramers, they do not contribute to the spectral flow and only edge states contribute to. The parity of the number of the Kramers pairs of gapless edge states is exactly the same as that of the spectral flow. This reveals the origin of the edge-bulk correspondence, i.e., why the edge states can be used to characterize the topological insulators. Furthermore, the spectral flow is related to the reduced eta-invariant and thus counts both the discrete ground state degeneracy and the continuous gapless excitations, which distinguishes the topological insulator from the conventional band insulator even if the edge states open a gap due to a strong interaction between edge modes. We emphasize that these results are also valid even for a weak disordered and/or weak interacting system. The higher spectral flow to categorize the higher-dimensional topological insulators are expected.Comment: 9 page, accepted for publication in Nucl Phys