Quantum $W_{1+\infty}$ subalgebras of BCD type and symmetric polynomials

Abstract

The infinite affine Lie algebras of type ABCD, also called gl(∞), o(∞), sp(∞), are equivalent to subalgebras of the quantum W 1+∞ algebras. They have well-known representations on the Fock space of either a Dirac fermion ( Â∞ ), a Majorana fermion ( B∞ and D∞ ) or a symplectic boson ( Ĉ∞ ). Explicit formulas for the action of the quantum W 1+∞ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the vertical presentation of the quantum toroidal gl(1) algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the horizontal presentation. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum W 1+∞ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of B∞ , a q-difference equation is obtained for Q-Schur polynomials indexed by strict partitions.

Intertwining operator and integrable hierarchies from topological strings

“…The quantum W 1+∞ algebra has a representation of levels (1, 0) on the Fock space F of a Dirac fermion. We refer the reader to [39] for a recent review of this representation. The Fock space is built from a vacuum state |∅ annihilated by the positive modes of the fermionic fields 2…”

“…Finally, the action of the generators W m,n on the Schur basis can be written explicitly. When m = 0, these operators add or remove strips of |m| boxes to the Young diagram labeling the states [39]. On the other hand, the action of the modes W 0,n is diagonal and read…”

“…Then, we observe that the action of the modes W m,n with m > 0 on the Schur states |λ is to remove strips of m boxes to the partition λ, and, in particular, these modes annihilate the vacuum |∅ [39]. Applying this property to the S-dual modes S • W m,n = q −(m+1)n W −n,m in the previous equation, we obtain an exchange relation for the vacuum component λ = ∅,…”

“…We refer the reader to the appendix of ref. [39] for a short derivation of these well-known formulas. Then, the sum over partitions η in the expression 2.21 of the topological vertex can be performed using the closure relation of the Schur basis, leading to…”

“…In the self-dual limit t → q (or q 1 q 2 → 1), the original formulation of topological strings is recovered, the parameter q is identified with the exponentiated string coupling constant e gstr and the algebra reduces to the quantum W 1+∞ algebra [35,36]. The latter is equivalent to the gl(∞) algebra mentioned earlier upon a linear transformation of the generators [37][38][39], it is thus naturally expected to be involved in the relation with integrable hierarchies, which was indeed observed in [34].…”