Free Field Realization of $\mathcal{W}$-Algebra Associated with Exceptional Lie Algebras

Abstract

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 $\mathcal{W}B{C}_r$ algebras is also studied. We then realize the $\mathcal{W}{F}_4$ algebra based on the $\mathcal{W}B{C}_3$ algebra. Furthermore, the $\mathcal{W}$-charges of the generators of the $\mathcal{W}{E}_{6,7}$, $\mathcal{W}B{C}_{2,3}$, and $\mathcal{W}{F}_4$ algebras are calculated and expressed in terms of the Casimir invariants.

40 pages; v2: format changed, references added, minor changes