Index formulas and charge deficiencies on the Landau levels

TL;DR

This paper explores charge deficiencies on Landau levels in complex n-dimensional space using KK-theory and an Atiyah-Singer index theorem, providing explicit calculations for higher Landau levels.

Contribution

It introduces a KK-theoretic approach to charge deficiency and applies an Atiyah-Singer index theorem to compute these deficiencies on Landau levels in ^n.

Findings

Calculated charge deficiencies for higher Landau levels in ^n

Established a connection between charge deficiency and index theory

Extended previous notions of charge deficiency to complex multi-dimensional settings

Abstract

The notion of charge deficiency from Avron, Seiler, Simon (Charge deficiency, charge transport and comparison of dimensions, Comm. Math. Phys. 159) is studied from the view of $KK$-theory and is applied to the Landau levels in $\C^n$. We calculate the charge deficiencies of the higher Landau levels in $\C^n$ by means of an Atiyah-Singer type index theorem.