Index theorem on magnetized blow-up manifold of T2/ZN

Abstract

We investigate blow-up manifolds of ${T}^{2}/{\mathbb{Z}}_{N}(N=2,3,4,6)$ orbifolds with magnetic flux. First, we construct the blow-up manifolds and zero-mode wave functions on them more precisely. In particular, through an appropriate singular gauge transformation, winding numbers of wave functions on ${T}^{2}/{\mathbb{Z}}_{N}$ can be replaced with localized curvature and localized flux at orbifold fixed points. In addition, since the blow-up manifolds have no singularities, we apply the Atiyah-Singer index theorem to them; the chiral zero-mode number is given by the total magnetic flux. It can be also applied for ${T}^{2}/{\mathbb{Z}}_{N}$ orbifolds through the blow-up process, and then we find that it is consistent with the zero-mode counting formula in M. Sakamoto et al. [Zero-mode counting formula and zeros in orbifold compactifications, Phys. Rev. D 102, 025008 (2020).]. Furthermore, the Atiyah-Singer index theorem shows that an additional degree of freedom of localized flux gives new chiral zero modes. We study their wave functions and then we find that they correspond to localized modes at the orbifold singular points. We also calculate their Yukawa couplings.