Index theorem on magnetized blow-up manifold of $T^2/\mathbb{Z}_N$

TL;DR

This paper derives a zero-mode counting formula for magnetized blow-up manifolds of orbifolds using the Atiyah-Singer index theorem, linking magnetic flux, winding numbers, and topological features.

Contribution

It establishes a new zero-mode counting formula on blow-up manifolds of orbifolds with magnetic flux, extending the Atiyah-Singer index theorem to these geometries.

Findings

Derived the zero-mode counting formula $n_{+}-n_{-}=(M-V_{+})/N+1$.

Clarified physical and geometrical meanings of the index formula.

Applied the Atiyah-Singer index theorem to smooth blow-up manifolds of orbifolds.

Abstract

We investigate blow-up manifolds of $T^2/{\mathbb{Z}}_N\,(N=2,3,4,6)$ orbifolds with magnetic flux $M$. Since the blow-up manifolds have no singularities, we can apply the Atiyah-Singer index theorem to them. Then, we establish the zero-mode counting formula $n_{+}-n_{-}=(M-V_{+})/N+1$, where $V_{+}$ denotes the sum of winding numbers at fixed points on the $T^2/{\mathbb{Z}}_N$ orbifolds, as the Atiyah-Singer index theorem on the orbifolds, and clarify physical and geometrical meanings of the formula.