Topological Aspects of Matters and Langlands Program

Abstract

In the framework of Langlands program, we offer a unified description of the integer and fractional quantum Hall effect as well as the fractal nature of energy spectra of 2d Bloch electrons. We categorify topological invariants on the Brillouin Zone and address the several dualities in a coherent manner where analogs of the classical Fourier transform provide an essential crux of the matter. Based on the Langlands philosophy, we elucidate the duality of topological computation and that of Ising models in the same context.