Matrix Models from Operators and Topological Strings, 2

Abstract

The quantization of mirror curves to toric Calabi-Yau threefolds leads to trace class operators, and it has been conjectured that the spectral properties of these operators provide a non-perturbative realization of topological string theory on these backgrounds. In this paper, we find an explicit form for the integral kernel of the trace class operator in the case of local P 1 × P 1 , in terms of Faddeev's quantum dilogarithm. The matrix model associated to this integral kernel is an O(2) model, which generalizes the ABJ(M) matrix model. We find its exact planar limit, and we provide detailed evidence that its 1/N expansion captures the all genus topological string free energy on local P 1 × P 1 . arXiv:1505.02243v2 [hep-th] 1 Feb 2016One concludes that, in the 't Hooft limit (3.15), the modified grand potential has the asymptotic expansion, J 't Hooft (ζ, ξ, ) = ∞ g=0 J g (ζ, ξ) 2−2g , (3.26)

Instanton effects in rank deformed superconformal Chern-Simons theories from topological strings

“…It is also desirable to establish a closed string formalism for all of four boundary deformations. This may encode the target Calabi-Yau threefold of the topological string theory in a direct way [52]. It may also enable us to study the membrane instantons from the WKB expansion [53].…”

The short pulse equation by a Riemann–Hilbert approach

“…The small κ expansion of the spectral determinant describing the operator (3.34) is given by 38) where Z(N, ) can be written as a matrix model of the form [39] Z(N, ) = e…”

“…Moreover the standard 't Hooft expansion of (3.50) describes Seiberg-Witten theory in the magnetic frame and not the electric one as it was the case in previous proposals. In turns, these differences are related to the fact that our model arises as a four-dimensional limit of the matrix model describing topological string on toric CYs [24,39]. Other proposals instead are more related to topological string theory on the Dijkgraaf-Vafa types of manifold [89] which can also be used to engineer Seiberg-Witten theory in four dimension [17,26,[75][76][77][78].…”

“…In the following we will often use the "renormalized" mass parameter introduced in [39] log m = 2π log m F 0 .…”

“…Even though the conjecture of [24] has been successfully tested in many examples [24,35,39,48,[54][55][56][57][58], a proof is still missing. In this paper we perform a first step in this direction by proving the conjecture in a four-dimensional limit of local P 1 × P 1 .…”