On the extended \(W\)-algebra of type \(\mathfrak{sl}_2\) at positive rational level

Constructs, using screening operators on lattice vertex operator algebras, a family of vertex operator algebras that extend the Virasoro minimal models and serve as symmetries of logarithmic conformal field theories. Proves they are \(C_2\)-cofinite, determines their Zhu algebras and classifies their simple modules.

Abstract

The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu’s c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu’s algebra) and classify all simple M_{p_+,p_-}-modules.

Recommended citation: A. Tsuchiya and S. Wood, Int. Math. Res. Not., 2015, 5357–5435 (2015)
Download Paper