费米规范场的球面路径积分
Sphere path integrals for fermionic gauge fields
- King’s College London（伦敦国王学院）
- UMONS（蒙斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文计算了四维球面上严格无质量费米规范场的单圈路径积分，用特征标表达并推广到任意自旋，证明对数发散由特征标编码且无虚相位，并确认无限塔费米子的体边抵消。
AI中文摘要：
我们考虑四维球面 $S^4$ 上自旋 $s \geq \frac{3}{2}$ 的复值严格无质量费米规范场，$S^4$ 是 de Sitter 时空 dS$_4$ 的欧几里得延拓。对于 $s=\frac{5}{2}$，在简要讨论 dS$_4$ 上理论量子化的不同选项后，我们计算了单圈球面路径积分。我们解释了如何将其表示为函数行列式，推广了先前 $s=\frac{3}{2}$ 的结果。我们进而将结果重写为 de Sitter 等距群 $Spin(4,1)$ 离散序列中的‘体’幺正 Harish-Chandra 特征标和‘边’特征标。然后，我们将球面路径积分的特征标表达式推广到任意复值严格无质量自旋 $s\geq\frac{3}{2}$ 费米规范势。此外，我们计算了所有自旋 $s \geq \frac{3}{2}$ 的 $S^4$ 路径积分对数发散系数，并证明它完全由体和边特征标编码。我们进一步证明，在单圈水平上，严格无质量费米规范势不会出现虚相位，这与玻色子情况相反。最后，我们确认了最近观察到的无限塔复无质量费米子（自旋 $s=\frac{1}{2}, \frac{3}{2},\dots$）情况下体与边贡献之间的精确单圈抵消。
英文摘要：
We consider spin-$s \geq \frac{3}{2}$ complex, strictly massless fermionic gauge fields on the four-sphere, $S^4$, which is the Euclidean continuation of de Sitter spacetime, dS$_4$. For $s=\frac{5}{2}$, after briefly discussing different options for the quantisation of the theory on dS$_4$, we compute the one-loop sphere path integral. We explain how it can be expressed in terms of functional determinants, generalising previous results for $s=\frac{3}{2}$. We proceed to rewrite the result in terms of `bulk' unitary Harish-Chandra characters in the discrete series of the de Sitter isometry group, $Spin(4,1)$, and `edge' characters. We then generalise the character expression of the sphere path integral for any complex, strictly massless spin-$s\geq\frac{3}{2}$ fermionic gauge potential. Additionally, we compute the coefficient of the logarithmic divergence of the $S^4$ path integral for all spin-$s \geq \frac{3}{2}$, and we show that it is fully encoded by bulk and edge characters. We further show that at one loop no imaginary phase appears for strictly massless fermionic gauge potentials, contrary to the case of bosons. Lastly, we confirm the exact one-loop cancellation between bulk and edge contributions for the case of an infinite tower of complex massless fermions of spins $s=\frac{1}{2}, \frac{3}{2},\dots$, observed recently.
