We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum KodairaSpencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using W-algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the Calabi-Yau. In the highly effective fermionic/brane formulation this leads to a free fermion description of the amplitudes. Furthermore we argue that topological strings on Calabi-Yau geometries provide a unifying picture connecting non-critical (super)strings, integrable hierarchies, and various matrix models. In particular we show how the ordinary matrix model, the double scaling limit of matrix models, and Kontsevich-like matrix model are all related and arise from studying branes in specific local Calabi-Yau three-folds. We also show how A-model topological string on P^1 and local toric threefolds (and in particular the topological vertex) can be realized and solved as B-model topological string amplitudes on a Calabi-Yau manifold.Comment: 82 pages, harvmac, 1 figur

The partition function of N = 6 supersymmetric Chern-Simons-matter theory (known as ABJM theory) on S 3 , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super-matrix model is closely related to a matrix model describing topological Chern-Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS 4 × CP 3 and gives the correct N 3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in CP 3 . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi-Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi-Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two 't Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi-Yau, and leads to an expansion around topological Chern-Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.We present the matrix model for the ABJM theory and that for CS theory on the lens space L(2, 1) = S 3 /Z 2 in the next section. The matrix model of ABJM has an underlying U (N 1 |N 2 ) symmetry while that of the lens space has U (N 1 + N 2 ) symmetry, which in both cases are broken to U (N 1 ) × U (N 2 ). It is easy to see that the expressions for them are related by analytical continuation of N 2 → −N 2 , or analogously a continuation of the 't Hooft coupling N 2 /k → −N 2 /k (which may be attributed to the negative level of the CS coupling of this group in the ABJM theory). We can then go on to study the lens space model and analytically continue to ABJM at the end.Conveniently, the lens space matrix model has been studied in the past [10,13,14,8]. The planar resolvent is known in closed form and the expressions for its periods are given as power series at special points in moduli space. We review the details of this matrix model and its solution in Sections 2 and 3.The matrix model of ABJM theory was derived by localization: it captures in a finite dimensional integral all observables of the full theory which preserve certain supercharges. At the time it was derived in [4], the only such observable (apart for the vacuum) was the 1/6 BPS Wilson loop constructed in [15,16,17] and 1/2 BPS vortex loop operators [18]. Indeed, the expectation value of the 1/6 BPS Wilson loop can be expressed as an observable in the ABJM matrix model, and by analytical continuation in the lens space model.Anothe...

The partition function on the three-sphere of many supersymmetric Chern-Simonsmatter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a nontrivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N 3/2 behavior for ABJM theory and N = 3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background. arXiv:1110.4066v3 [hep-th]

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