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arXiv 2005.12642hep-thhep-phmath.NT
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磁化环面和轨道紧化中的模对称性

Revisiting modular symmetry in magnetized torus and orbifold compactifications

Shota Kikuchi, Tatsuo Kobayashi, Shintaro Takada, Takuya H. Tatsuishi, Hikaru Uchida

备注：
29 pages
发表信息：
Phys. Rev. D 102, 105010 (2020)
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摘要：

我们研究了$T^2$中的模对称性和磁通量对轨道的影响。在磁通量为$M$的$T^2$上存在$|M|$零模。它们的波函数和质量模表现为权重$1/2$的模形式，并表示$Γ\equiv SL(2,\mathbb{Z})$的双覆盖群，$\widetildeΓ \equiv \widetilde{SL}(2,\mathbb{Z})$。$T^2$上的每个波函数在磁通量为$M$的情况下变换到$\widetildeΓ(2|M|)$下，$\widetildeΓ(2|M|)$是$\widetilde{SL}(2,\mathbb{Z})$的正规子群。那么，$|M|$零模是商群$\widetildeΓ'_{2|M|} \equiv \widetildeΓ/\widetildeΓ(2|M|)$的表示。我们还研究了扭曲和移位轨道折叠$T^2/\mathbb{Z}_N$上的模对称性。波函数被分解成更小的表示的扭曲和移位的本征值。它们为我们提供了$T^2$上可约表示的约化。

英文摘要：

We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $Γ\equiv SL(2,\mathbb{Z})$, $\widetildeΓ \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetildeΓ(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetildeΓ'_{2|M|} \equiv \widetildeΓ/\widetildeΓ(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds $T^2/\mathbb{Z}_N$. Wavefunctions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on $T^2$.

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