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Register a new userZero-mode counting formula and zeros in orbifold compactifications
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We thoroughly analyze the number of independent zero modes and their zero points on the toroidal orbifold $T^2/mathbb{Z}_N$ ($N = 2, 3, 4, 6$) with magnetic flux background, inspired by the Atiyah-Singer index theorem. We first show a complete list for the number $n_{eta}$ of orbifold zero modes belonging to $mathbb{Z}_{N}$ eigenvalue $eta$. Since it turns out that $n_{eta}$ quite complicatedly depends on the flux quanta $M$, the Scherk-Schwarz twist phase $(alpha_1, alpha_2)$, and the $mathbb{Z}_{N}$ eigenvalue $eta$, it seems hard that $n_{eta}$ can be universally explained in a simple formula. We, however, succeed in finding a single zero-mode counting formula $n_{eta} = (M-V_{eta})/N + 1$, where $V_{eta}$ denotes the sum of winding numbers at the fixed points on the orbifold $T^2/mathbb{Z}_N$. The formula is shown to hold for any pattern.
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We discuss the modular symmetry and zeros of zero-mode wave functions on two-dimensional torus $T^2$ and toroidal orbifolds $T^2/mathbb{Z}_N$ ($N=2,3,4,6$) 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.
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Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to count the primes up to $x$. Perrons formula leads to sums over zeta zeros that can count the squarefree integers up to $x$, or tally Eulers $phi$ function and other arithmetical functions. This is largely a presentation of experimental results.
Relative orbifold Gromov-Witten theory is set-up and the degeneration formula is given.
We review the properties of orbifold operations on two-dimensional quantum field theories, either bosonic or fermionic, and describe the Orbifold groupoids which control the composition of orbifold operations. Three-dimensional TQFTs of Dijkgraaf-Witten type will play an important role in the analysis. We briefly discuss the extension to generalized symmetries and applications to constrain RG flows.
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