Intertwining operator and integrable hierarchies from topological strings. (English) Zbl 1466.81074
Summary: In [T. Nakatsu and K. Takasaki, Commun. Math. Phys. 285, No. 2, 445–468 (2009; Zbl 1160.81026)], 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 \(W_{1+ \infty}\) 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.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 83E30 | String and superstring theories in gravitational theory |
Citations:
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