Exploring W ∞ $$ {\mathcal{W}}_{\infty } $$ in the quadratic basis

Abstract

We study the operator product expansions in the chiral algebra W ∞ , first using the associativity conditions in the basis of primary generating fields and then using a different basis coming from the free field representation in which the OPE takes a simpler quadratic form. The results in the quadratic basis can be compactly written using certain bilocal combinations of the generating fields and we conjecture a closed-form expression for the complete OPE in this basis. Next we show that the commutation relations as well as correlation functions can be easily computed using properties of these bilocal fields. In the last part we verify the consistency with results derived previously by studying minimal models of W ∞ and comparing them to known reductions of W ∞ to W N . The results we obtain illustrate nicely the role of triality symmetry in the representation theory of W ∞ .

Instanton R-matrix and $$ \mathcal{W} $$-symmetry

“…A very surprising property found by [47] was the triality symmetry of the algebra: parametrizing W ∞ in terms of the central charge c and the rank-like parameter λ, for each value of c there are generically three values of λ which give the same structure constants. This has important consequences for the representation theory of the algebra and points towards to the connection to topological strings and affine Yangian picture [69,53].…”

“…This algebra is by definition u(1) × W N ≡ Y 0,0,N . The currents U j (z) do not transform as primary fields under conformal transformations, but perhaps surprisingly their operator product expansions have purely quadratic non-linearity [56,69]. This is one of signs of the connection to integrability, where the algebras with quadratic non-linearity appear naturally.…”

“…We can split the N free bosons in (2.2) into two groups of N 1 and N 2 bosons. Multiplying two Miura operators associated to N 1 and N 2 bosons and passing the derivatives to the right, we find a coproduct in W 1+∞ which embeds [69] Y 0,0,…”

“…This fusion operation fixes the value of α 0 parameter, i.e. the ratios of λ j parameters introduced later stay the same and the vector (λ 1 , λ 2 , λ 3 ) is thus additive under the fusion [69]. Another important point to mention is that the way W N is embedded in the bosonic Fock space depends on the ordering of free fields in (2.2).…”

“…Using the compatible coproduct guarantees that these identifications will hold also for any MacMahon representations. Note that the coproduct used in the R-matrix description of the algebra is the same one as the one used in [69] as product of Miura operators. In [53] this coproduct was studied in the language of Tsymbaliuk's generators.…”