矩形上的6d狄拉克费米子;仔细检查边界条件、模式函数和频谱
6d Dirac fermion on a rectangle; scrutinizing boundary conditions, mode functions and spectrum
摘要：
在保持四维Lorentz结构的前提下，对矩形上六维Dirac费米子$Ψ$可能的边界条件进行了分类，并推导了在四维手征正分量为零和二维手征正分量为零这两种边界条件下零模和非零KK模的轮廓和谱.在（i）的情况下，出现了双重简并手征零模，其朝向由角度参数$θ$指向的矩形的特定方向局部化。这导致了一个新的方向，追求起源的三代在标准模型的物质领域的含义，即使三重简并零模式没有实现在六个维度。角参数$θ$的出现源于矩形额外维度上简并手征零模的旋转对称性，因为它们感觉不到边界。在（ii）的情况下，这种旋转对称性被提升到二维共形对称性，虽然没有手征无质量零模式出现。我们还详细讨论了我们的矩形模型和轨道模型之间的对应关系。
英文摘要：
We classify possible boundary conditions of a 6d Dirac fermion $Ψ$ on a rectangle under the requirement that the 4d Lorentz structure is maintained, and derive the profiles and spectrum of the zero modes and nonzero KK modes under the two specific boundary conditions, (i) 4d-chirality positive components being zero at the boundaries and (ii) 2d-chirality positive components being zero at the boundaries. In the case of (i), twofold degenerated chiral zero modes appear which are localized towards specific directions of the rectangle pointed by an angle parameter $θ$. This leads to an implication for a new direction of pursuing the origin of three generations in the matter fields of the standard model, even though triple-degenerated zero modes are not realized in the six dimensions. The emergence of the angle parameter $θ$ originates from a rotational symmetry in the degenerated chiral zero modes on the rectangle extra dimensions since they do not feel the boundaries. In the case of (ii), this rotational symmetry is promoted to the two-dimensional conformal symmetry though no chiral massless zero mode appears. We also discuss the correspondence between our model on a rectangle and orbifold models in some details.
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