The matrix-extended $$ {\mathcal{W}}_{1+\infty } $$ algebra

Abstract

We construct a quadratic basis of generators of matrix-extended W 1+∞ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions defining the algebra. We study truncations of the algebra. An explicit calculation at low levels shows that these are parametrized in a way consistent with the gluing description of the algebra. It is perhaps surprising that in spite of the fact that the algebras are rather complicated and non-linear, the structure of their truncations follows very simple gluing rules.

The Grassmannian VOA

“…The first surprising feature of the algebra is that the Z 2 duality symmetry manifest in coset description (1.1) is enhanced to an S 3 × Z 2 symmetry, see also [35]. One way to see this enhancement is to observe that if we perform the coset (2.1) in two steps, the intermediate algebra is the matrix-valued W 1+∞ algebra (or the affine gl(M ) Yangian), which was studied previously in [36][37][38][39]. These algebras possess a conjugation duality which combined with the Z 2 duality of the coset (2.1) generates the triality symmetry S 3 .…”

“…We can attach to each of the punctures a copy of an affine Lie algebra. Attaching it to one of the punctures we find a matrix-valued W ∞ [36][37][38][39]. Attaching two affine Lie algebras to a Grassmannian results in an affine Lie algebra in the numerator (2.1).…”

“…One can choose a set of generators of the corresponding universal algebra such that for each conformal dimension there is a M × M matrix of generators transforming in the adjoint representation of the global u(M ) symmetry. This algebra was studied from a different perspective in [36,38,39,48] and one can think of it as a matrix extension of W 1+∞ . Just like W 1+∞ can be understood as the Yangian of affine gl(1), we can think of the matrix W 1+∞ as of the Yangian of affine gl(M ).…”

“…Passage from (2.6) to (2.1) is then equivalent to decoupling of spin 1 generators of u(M ) K subalgebra analogously to the M = 1 situation where one decouples the spin 1 current in W 1+∞ to get W ∞ . In [39], the matrix-valued W 1+∞ was parametrized by the parameters (M, N, κ). The parameter M is the rank of the u(M ) subalgebra.…”

“…The matrix algebra (2.6) has a duality symmetry generalizing the conjugation symmetry of u(M ) K [39]. It acts on the parameters of [39] as…”