Abstract
In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of mathfrak{gl} (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.
Highlights
Is an intertwiner between a Fock representation F of the algebra and its tensor product V ⊗F with a level zero representation V
We investigate the role played by quantum algebras in the well-known relation between self-dual topological strings and integrable hierarchies [34]
Refined topological strings depend on two parameters (q, t−1), they are identified with the parameters (q1, q2) of the quantum toroidal gl(1) algebra
Summary
We review briefly here the definition of the quantum toroidal gl(1) algebra. We mostly follow the notations and conventions of [32], and refer to [27, 42, 43, 55] for more details on the correspondence with the (p, q)-brane construction of topological strings amplitudes. The algebra is usually formulated in terms of four Drinfeld currents, x±(z) = z−kx±k , ψ±(z) = z∓kψ±±k. They satisfy a set of exchange relations that can be found, e.g. in [27, 32], but we prefer to work here directly with the modes x±k , ψ±±k. The subalgebra generated by the elements ψ±±k is the analogue of the Cartan subalgebra of quantum affine algebras, it has an alternative formulation in terms of modes ak defined by exponentiation, ψ±(z) = ψ0± exp ± z∓ka±k , k>0. In the self-dual limit (q1, q2) → (q, q−1), the modes of the quantum toroidal gl(1) algebra satisfy the commutation relations of the quantum W1+∞ algebra. The roles of Wk,0, W0,k and Wk,−k is played in the refined case by ak, b−k and T · b−k respectively.
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