Abstract
We discuss the modular symmetry and zeros of zero-mode wave functions on two-dimensional torus T2 and toroidal orbifolds T2/ℤN (N = 2, 3, 4, 6) with a background homogeneous magnetic field. As is well-known, magnetic flux contributes to the index in the Atiyah-Singer index theorem. The zeros in magnetic compactifications therefore play an important role, as investigated in a series of recent papers. Focusing on the zeros and their positions, we study what type of boundary conditions must be satisfied by the zero modes after the modular transformation. The consideration in this paper justifies that the boundary conditions are common before and after the modular transformation.
Highlights
Issue is still to elucidate the relation between the internal geometry and its implications on flavor physics
Focusing on the zeros and their positions, we study what type of boundary conditions must be satisfied by the zero modes after the modular transformation
A reasonable expectation in various aspects is that a background homogeneous magnetic field may keep the modular symmetry, which has partially been confirmed in [49]
Summary
We consider a six-dimensional (6d) gauge theory compactified on T 2 and T 2/ZN. For a complex coordinate z ≡ y1 + τ y2 (τ ∈ C, Im τ > 0), the 2d torus T 2 is obtained by an identification under torus translations, i.e., z ∼ z + 1 ∼ z + τ. The 1-form defined above is not invariant under lattice translations. As stated in [13], the gauge transformation on T 2 is well-defined only when the homogeneous flux f is quantized for a given U(1) charge q as qf 2π ≡ M ∈ Z. The localized fluxes contribute to Wilson loops around the fixed points and are not ignorable. It turned out in [46] that the transition functions (2.2) and the quantization condition (2.3) are well-defined even on T 2/ZN. The gauge transformation induced by lattice translations requires the 2d Weyl spinors to satisfy the (pseudo-)periodicity conditions ψ±,n,j(z + 1) = U1(z)ψ±,n,j(z), ψ±,n,j(z + τ ) = U2(z)ψ±,n,j (z),. They are restricted to specific values on T 2/ZN , as explained later
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