与例外李代数相关的$\mathcal{W}$-代数的自由场实现
Free Field Realization of $\mathcal{W}$-Algebra Associated with Exceptional Lie Algebras
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中文总结 AI 辅助
本文通过递归构造，利用自由玻色子与屏蔽电荷，实现了与例外李代数相关的W-代数，并验证了不同构造的等价性，计算了生成元的W-荷。
中文摘要 AI 辅助
我们研究了与例外李代数$E_6$、$E_7$、$E_8$和$F_4$相关的$\mathcal{W}$-代数的自由场实现。我们发展了一种递归构造，其中秩为$r$的$\mathcal{W}$-代数由秩为$r-1$的$\mathcal{W}$-代数与一个自由玻色子得到。$\mathcal{W}$-流由与屏蔽电荷的零对易关系构造。$\mathcal{W}E_6/\mathcal{W}E_7$代数由$\mathcal{W}D_5/\mathcal{W}D_6$代数构造，并证明其与由$\mathcal{W}A_5/\mathcal{W}E_6$代数实现的代数相同，仅相差自由场基的变换。$\mathcal{W}E_8$代数的自旋-8生成元由$\mathcal{W}D_7$代数构建。我们还研究了$\mathcal{W}BC_r$代数的递归构造。然后我们基于$\mathcal{W}BC_3$代数实现了$\mathcal{W}F_4$代数。此外，计算了$\mathcal{W}E_{6,7}$、$\mathcal{W}BC_{2,3}$和$\mathcal{W}F_4$代数生成元的$\mathcal{W}$-荷，并用Casimir不变量表示。
英文摘要
We study the free field realization of the $\mathcal{W}$-algebra associated with the exceptional Lie algebras $E_6$, $E_7$, $E_8$, and $F_4$. We develop a recursive construction in which a $\mathcal{W}$-algebra of rank $r$ is obtained from a $\mathcal{W}$-algebra of rank $r-1$ together with a free boson. The $\mathcal{W}$-currents are constructed from the zero commutation relation with the screening charges. The $\mathcal{W}E_6/\mathcal{W}E_7$ algebra is constructed from the $\mathcal{W}D_5/\mathcal{W}D_6$ algebra and is shown to be the same as that realized from the $\mathcal{W}A_5/\mathcal{W}E_6$ algebra, up to a change of the free field basis. The spin-$8$ generator of the $\mathcal{W}E_8$ algebra is built from the $\mathcal{W}D_7$ algebra. The recursive construction of the $\mathcal{W}BC_r$ algebras is also studied. We then realize the $\mathcal{W}F_4$ algebra based on the $\mathcal{W}BC_3$ algebra. Furthermore, the $\mathcal{W}$-charges of the generators of the $\mathcal{W}E_{6,7}$, $\mathcal{W}BC_{2,3}$, and $\mathcal{W}F_4$ algebras are calculated and expressed in terms of the Casimir invariants.
发表机构
- Institute of Science Tokyo（东京科学大学）
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