Deformed boson-fermion correspondence, Q-bosons, and topological strings on the conifold

TL;DR

This paper explores how deformations of fermionic and bosonic systems, parametrized by Q, relate to topological string partition functions on the conifold, revealing a deep connection between algebraic structures and string theory.

Contribution

It demonstrates the equivalence of two Q-deformed systems and their relation to topological string partition functions on the conifold, linking algebraic deformations to geometric string theory models.

Findings

Deformed generating functions reproduce topological string partition functions on the conifold.

Q-parameter corresponds to the size of P^1 in the conifold geometry.

Deformation of fermion one-point functions yields A-brane partition functions.

Abstract

We consider two different physical systems for which the basis of the Hilbert space can be parametrized by Young diagrams: free complex fermions and the phase model of strongly correlated bosons. Both systems have natural, well-known deformations parametrized by a parameter Q: the former one is related to the deformed boson-fermion correspondence introduced by N. Jing, while the latter is the so-called Q-boson, arising also in the context of quantum groups. These deformations are equivalent and can be realized in the same way in the algebra of Hall-Littlewood symmetric functions. Without a deformation, these reduce to Schur functions, which can be used to construct a generating function of plane partitions, reproducing a topological string partition function on $C^3$. We show that a deformation of both systems leads then to a deformed generating function, which reproduces topological string partition function of the conifold, with the deformation parameter Q identified with the size of $P^1$. Similarly, a deformation of the fermion one-point function results in the A-brane partition function on the conifold.