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.
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Bourgine, JE. Intertwining operator and integrable hierarchies from topological strings. J. High Energ. Phys. 2021, 216 (2021). https://doi.org/10.1007/JHEP05(2021)216
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DOI: https://doi.org/10.1007/JHEP05(2021)216