Abstract
The partition function of ABJM theory on the three-sphere has non-perturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi-Yau manifold known as local P1xP1, in the Nekrasov-Shatashvili limit. Our result can be interpreted as a first-principles derivation of the full series of non-perturbative effects for the closed topological string on this Calabi-Yau background. Based on this, we make a proposal for the non-perturbative free energy of topological strings on general, local Calabi-Yau manifolds.
Highlights
Well the non-perturbative structure for the free energy of topological string theory on the Calabi-Yau manifold known as local P1 × P1, since both problems are formally identical
We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi-Yau manifold known as local P1 × P1, in the NekrasovShatashvili limit
Notice that the Gopakumar-Vafa representation of the free energy is precisely what is needed for the M-theory expansion: it resums the genus expansion order by order in the exponentiated parameter e−T, it leads to an expansion at large N in ABJM theory, but which is exact in k at each order in e−μ/k
Summary
As it was shown in [5], the partition function of ABJM theory on the three-sphere, Z(N, k), is given by the matrix integral. The genus g free energies Fg(λ) can be calculated exactly as a function of λ, and order by order in the genus expansion, by using matrix model techniques [6] They contain non-perturbative information in α , since they involve exponentially small corrections of the form. It was conjectured in [6] that these terms correspond to worldsheet instantons wrapping a two-cycle CP1 ⊂ CP3, which were first considered in [27]. The Fermi gas approach makes it possible to determine both the subleading 1/N corrections and non-perturbative corrections due to D2-brane instantons. J(p)(μ, k) leads to the Airy function result for Z(N, k) first obtained in [25]
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