AI 中文总结
本文针对卡罗尔共形族缺失的后代问题，将$K^0$连续降阶至原初算子的独立后代链纳入标准共形族，构建了完整的卡罗尔共形表示，推导了相关局域算子、轨道结构等，可用于描述平坦全息中的质量态。
AI 中文摘要
在卡罗尔共形代数中，关系$[P^0,K^0]=0$意味着$K^0$会湮灭所有由原初算子经时间平移$P^0$生成的算子。由平移后代定义的标准共形族不包含$K^0$连续降阶至原初算子的独立后代链。我们通过将该链及其所有平移后代纳入其中，构建了完整的卡罗尔共形表示。我们推导了对应的局域算子、轨道结构，并在二维和三维中检验了它们的卡西米尔量。对于每个轨道构型，全局两点沃德恒等式确定了磁非接触分支和电接触分支的运动学因子与选择规则。与普通共形对称性不同，卡罗尔共形对称性通常仅将关联函数确定为卡罗尔不变量的任意函数，而非常数。该完整表示包含二维中满足$\boldsymbol{\textit{C}_2=m^2=\boldsymbol{\textit{\beta}}\rho-\beta^2>0}$、三维中满足$\boldsymbol{\textit{C}_2=m^2=\boldsymbol{\textit{\beta}}\rho-\boldsymbol{\beta}^2>0}$的部分，可描述平坦全息中的质量态。
英文摘要
In the Carrollian conformal algebra, the relation $[P^0,K^0]=0$ implies that $K^0$ annihilates all operators generated from a primary by temporal translation $P^0$. The standard conformal family defined by translation descendants does not contain the independent descendant chain that $K^0$ lowers successively to the primary. We construct the complete Carrollian conformal representation by including this chain together with all of its translation descendants. We derived the corresponding local operators, orbit structures, and checked their Casimirs in 2D and 3D. For each orbit configuration, the global two-point Ward identities fix the kinematic factors and selection rules for both magnetic non-contact branches and electric contact branches. Unlike ordinary conformal symmetry, Carrollian conformal symmetry generally determines the correlators only up to arbitrary functions of Carrollian invariants rather than constants. The complete representation contains sectors with $\mathcal{C}_2=m^2=κρ-β^2>0$ for 2D and $\mathcal{C}_2=m^2=κρ-\vecβ^{\,2}>0$ for 3D, which may describe massive states in flat holography.
Comments42 pages, 2 figures
