Topological strings and Wilson loops

Abstract

We propose the refined topological string correspondence to the expectation values of half-BPS Wilson loop operators in 5d $$ \mathcal{N} $$ N = 1 gauge theory partition function on the Omega-deformed background $$ {\mathbb{R}}_{\upepsilon_{1,2}}^4 $$ ℝ ϵ 1 , 2 4 × S1. We provide the refined topological vertex method and the refined holomorphic anomaly equation method in the topological string theory, from which we have exact computations on the 5d Wilson loops partition functions in both A- and B-models. Finally, with the exact results we have in B-model, we recover the quantum periods of local ℙ1 × ℙ1 model and local ℙ2 model in the study of quantum geometry and we further give a refined generalization of A-period.

On the resurgent structure of quantum periods

“…Let us now turn to Wilson loops (more precisely, Wilson loop vevs). Originally, these quantities are defined for local CY geometries which engineer five and four dimensional gauge theories [42], and we refer to [24] and e.g. [37] for background and references.…”

“…In [24], a HAE is proposed for Wilson loops associated to local CY manifolds, i.e. to curves of the form (17).…”

“…It can be fixed (at least partially) by requiring the appropriate behavior at special points. It is argued in [24] that, in the case of toric CYs, the holomorphic limit ω n (z) in the conifold frame has to be regular at the conifold point. The HAE (34) were studied in detail in [24] for various local geometries, like local 2 or local 0 , and it was checked explicitly that regularity at the conifold is enough to fix the holomorphic ambiguities.…”

“…It is argued in [24] that, in the case of toric CYs, the holomorphic limit ω n (z) in the conifold frame has to be regular at the conifold point. The HAE (34) were studied in detail in [24] for various local geometries, like local 2 or local 0 , and it was checked explicitly that regularity at the conifold is enough to fix the holomorphic ambiguities. The HAE (34) should be valid as well for Schrödinger operators, and we have verified explicitly that this is the case for the double well potential in quantum mechanics [27,31].…”

“…For example, if the curve has genus one, there are two independent quantum periods, while the NS free energy only provides one quantity. It has been pointed out recently [24] that the Wilson loop vevs of the supersymmetric gauge theories associated to mirror curves are also governed by a set of HAE, and together with the NS free energy they provide a complete set of quantities to describe the quantum periods. In fact, Wilson loops are the functional inverses of the quantum flat coordinates or quantum mirror maps.…”