Simplicity of vacuum modules and associated varieties (Q1996824)

The paper is concerned with the associated variety of the simple affine vertex algebras \(L^k(\mathfrak{g})\) and of the corresponding quantum Drinfeld-Sokolov reductions. For any vertex algebra \(V\), Zhu's \(C_2\)-algebra, a certain quotient \(R_V=V/C_2(V)\), comes naturally equipped with the structure of a Poisson algebra [\textit{Y. Zhu}, J. Am. Math. Soc. 9, No. 1, 237--302 (1996; Zbl 0854.17034)]. The associated variety \(X_V\) is then defined as the reduced scheme corresponding to \(R_V\) [\textit{T. Arakawa}, Math. Z. 270, No. 1--2, 559--575 (2012; Zbl 1351.17031)]. The associated variety has proved to be a useful invariant of the vertex algebra \(V\) and plays an important role in the context of 4D \(\mathcal{N}=2\) superconformal field theories [\textit{C. Beem} and \textit{L. Rastelli}, J. High Energy Phys. 2018, No. 8, Paper No. 114, 72 p. (2018; Zbl 1396.81191)]. For a complex, finite-dimensional, simple Lie algebra \(\mathfrak{g}\) and level \(k\in\mathbb{C}\) let \(V^k(\mathfrak{g})\) denote the corresponding universal affine vertex algebra. In general, it is not simple and there is a unique simple graded quotient \(L_k(\mathfrak{g})\). Sometimes, \(V^k(\mathfrak{g})\) is already simple, in which case \(V^k(\mathfrak{g})=L_k(\mathfrak{g})\). Gorelic and Kac characterise exactly for which levels \(k\) this occurs [\textit{M. Gorelik} and \textit{V. Kac}, Adv. Math. 211, No. 2, 621--677 (2007; Zbl 1112.17023)]. The associated variety of the universal affine vertex algebra \(V^k(\mathfrak{g})\) is just the affine space \(\mathfrak{g}^*\) with Kirillov-Kostant Poisson structure. The associated variety of the simple affine vertex algebra \(L_k(\mathfrak{g})\) is a Poisson subscheme of \(\mathfrak{g}^*\) and in general much harder to describe (e.g., as closures of nilpotent orbits). However, if \(V^k(\mathfrak{g})\) is already simple so that it coincides with \(L_k(\mathfrak{g})\), then clearly \(X_{L_k(\mathfrak{g})}=\mathfrak{g}^*\). The authors of the article show that the converse is also true, i.e.\ that if \(V^k(\mathfrak{g})\neq L_k(\mathfrak{g})\), then \(X_{L_k(\mathfrak{g})}\subsetneq\mathfrak{g}^*\). The article is also concerned with the associated variety of certain vertex algebras called \(\mathcal{W}\)-algebras. For a nilpotent element \(f\) of \(\mathfrak{g}\) the \(\mathcal{W}\)-algebra \(\mathcal{W}^k(\mathfrak{g},f)=H_{\mathrm{DS},f}^0(V^k(\mathfrak{g}))\) is defined as the generalised quantised Drinfeld-Sokolov reduction of \(V^k(\mathfrak{g})\) associated with \(f\). Again, \(\mathcal{W}^k(\mathfrak{g},f)\) is usually not simple and there is a unique simple graded quotient \(\mathcal{W}_k(\mathfrak{g},f)\) of \(\mathcal{W}^k(\mathfrak{g},f)\). Conjecturally, \(\mathcal{W}_k(\mathfrak{g},f)=H_{\mathrm{DS},f}^0(L_k(\mathfrak{g}))\), provided that \(H_{\mathrm{DS},f}^0(L_k(\mathfrak{g}))\neq\{0\}\). The associated variety \(X_{\mathcal{W}^k(\mathfrak{g},f)}\) can be shown to equal the Slodowy slice \(\mathcal{S}_f\) at \(f\). The associated variety of \(H_{\mathrm{DS},f}^0(L_k(\mathfrak{g}))\) is a Poisson subvariety of \(\mathcal{S}_f\). In analogy to and as a consequence of the above result for affine vertex algebras the authors show that \(X_{H_{\mathrm{DS},f}^0(L_k(\mathfrak{g}))}=\mathcal{S}_f\) if and only if \(\mathcal{W}^k(\mathfrak{g},f)=H_{\mathrm{DS},f}^0(L_k(\mathfrak{g}))\) (and if and only if \(V^k(\mathfrak{g})\) is simple).