Index and winding numbers on $T^2/{\mathbb{Z}}_N$ orbifolds with magnetic flux

Abstract

We analyze the number of independent chiral zero modes and the winding numbers at the fixed points on $T^2/{\mathbb{Z}}_N$ ($N=2,3,4,6$) orbifolds with magnetic flux. In the case of $N=2$, we derive the index formula $n_{+}-n_{-}=M/2+(-V_{+}+V_{-})/4=M/2-V_{+}/2+1$ by using the trace formula, where $n_{\pm }$ are the numbers of the $\pm$ chiral zero modes and $V_{\pm }$ are the sums of the winding numbers at the fixed points on $T^2/{\mathbb{Z}}_2$. We also obtain the formula $n_{+}-n_{-}=M/N+(-V_{+}+V_{-})/(2N)=M/N-V_{+}/N+1$ for $N=3,4,6$ under an assumption.

37 pages, 2 figures

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• Mass spectrum in a six-dimensional $SU(n)$ gauge theory on a magnetized torus
• Three-generation solutions of equations of motion in heterotic supergravity
• Trace conservation laws in $T^2/Z_m$ orbifold gauge theories