q-Difference Kac-Schwarz Operators in Topological String Theory

Abstract

Abstract. The perspective of Kac-Schwarz operators is introduced to the authors' previous work on the quantum mirror curves of topological string theory in strip geometry and closed topological vertex. Open string amplitudes on each leg of the web diagram of such geometry can be packed into a multi-variate generating function. This generating function turns out to be a tau function of the KP hierarchy. The tau function has a fermionic expression, from which one finds a vector |W in the fermionic Fock space that represents a point W of the Sato Grassmannian. |W is generated from the vacuum vector |0 by an operator g on the Fock space. g determines an operator G on the space V = C((x)) of Laurent series in which W is realized as a linear subspace. G generates an admissible basis {Φ j (x)} ∞ j=0of W . q-difference analogues A, B of Kac-Schwarz operators are defined with the aid of G.The lowest equation AΦ 0 (x) = Φ 0 (x) reproduces the quantum mirror curve in the authors' previous work.

4D limit of melting crystal model and its integrable structure

“…1. Quantum curves: One can derive a quantum spectral curve of the melting crystal model by the method of our work on quantum mirror curves in topological string theory [34]. This quantum curve is formulated as the q-difference equation (3.16) for the single-variate specialization Z(x) of Z(t).…”

“…As we shall see in the next section, the combinatorial expression (3.2) of Z(x) has a desirable form from which one can derive the equation of quantum curve of Dunin-Barkowski et al [6]. To apply the method of our previous work [34], however, it is more convenient to have Γ + (x) rather than Γ ′ + (x) in the fermionic expression (3.3) of Z(x). This problem can be settled by the following transformation rule of matrix elements of fermionic operators under conjugation of partitions [38]:…”

“…Having obtained the fermionic expression (3.4) containing Γ + (x), we now remove the other Γ + 's from (3.4). This is the last step for applying the method of out previous work [34].…”

“…Since the structure of the generating operator (3.13) resembles those of tau functions in topological string theory [34], we define the Kac-Schwarz operator A in essentially the same form, 2 namely,…”

“…Combinatorial and fermionic expressions of the deformed partition function Z(t) are introduced. The fermionic expression is further converted to a form that fits into the method of our work on quantum mirror curves of topological string theory [34]. Section 3 presents the quantum spectral curve of the melting crystal model.…”