W-algebras and integrability

Abstract

This is a short non-technical review focusing on the $\mathcal{W}_N$ family of $\mathcal{W}$-algebras and on their relation to quantum integrability. It is a summary of recently given seminars and workshop contributions.

Figures (8)

• Figure 1: An example of plane partition with $23$ boxes representing a state in the vacuum representation.
• Figure 2: Truncation curves of $\mathcal{W}_{\infty}$ algebra. Every curve corresponds to locus in the parameter space where the vacuum Verma module is reducible. Double intersections correspond to minimal models of $\mathcal{W}_\infty$. The triality symmetry is manifest in this diagram.
• Figure 3: A typical plane partition with non-trivial Young diagram asymptotics along the coordinate directions corresponding to a non-trivial primary.
• Figure 4: Periodic plane partition corresponding to the vacuum (primary state with $\Delta=0$) of the Ising model, $c=\frac{1}{2}$.
• Figure 5: The difference between the Yangian description and the description in terms of local fields.