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]

Recently, Kapustin, Willett and Yaakov have found, by using localization techniques, that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix model. We show that this matrix model is closely related to Chern-Simons theory on a lens space with a gauge supergroup. This theory has a topological string large N dual, and this makes possible to solve the matrix model exactly in the large N expansion. In particular, we find the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM theory, as a function of the 't Hooft parameters, and in the planar limit. This expression gives an exact interpolating function between the weak and the strong coupling regimes. The behavior at strong coupling is in precise agreement with the prediction of the AdS string dual. We also give explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and Trancanelli.

Abstract:We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N = (0, 2) theories, we obtain a number of results, which include new 3d N = 2 theories T [M 3 ] associated with rational homology spheres and new results for Vafa-Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0, 2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines / walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d N = (0, 2) theories and 3d N = 2 theories, respectively.

We provide a physical definition of new homological invariants H a (M 3 ) of 3manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M 3 times a 2-disk, D 2 , whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d N = 2 theory T [M 3 ]: D 2 × S 1 half-index, S 2 × S 1 superconformal index, and S 2 × S 1 topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern-Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of M 3 . The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.C. Categorification of the Turaev-Viro invariants 733 It is in fact (Tor H1(M3, Z)) * /Z2 that is canonically identified with components of abelian flat connections.However, as the distinction between Tor H1(M3, Z) and its dual is only important in section 2.2, we will use the same set of labels {a, b, . . .} for elements in both groups. 4 A related conjecture was made in [28]. However it did not include the S-transform, which is crucial for restoring integrality and categorification. 5 The constant positive integer c depends only on M3 and in a certain sense measures its "complexity". In many simple examples c = 0, and the reader is welcome to ignore 2 −c factor which arises from some technical subtleties. Its physical origin will be explained later in the paper. 6 Later in the text we will sometimes use slightly redefined quantities, Za(q) → q ∆ Za(q), where ∆ is a common, a independent rational number.7 Recall that Tor H1(M3, Z), as a finitely generated abelian group, can be decomposed into Tor H1(M3, Z) = i Zp i . We ask for a fairly weak condition that Z2 doesn't appear in this decomposition. In other words, M3 is a Z2-homology sphere. Equivalently, there is a unique Spin structure on M3, so that there is no ambiguity in specifying Nahm-pole boundary condition for N = 4 SU (2) SYM on M3 × R+ [18]. The general case, in principle, could also be worked out. We leave it as an exercise to an interested reader.8 The presence of 1/2 factors that produce 2 −c overall factor in (2.7) can be interpreted as presence of factors ∼ = C[x] with deg q x = 0 in Ha[M3]. The q-graded Euler charecteristic of C[x] is naively divergent: 1+1+1+. . ., but its zeta-regularization gives 1/2.