We study Wilson loop operators in three-dimensional, N = 6 superconformal Chern-Simons theory dual to IIA superstring theory on AdS 4 × CP 3 . Novelty of Wilson loop operators in this theory is that, for a given contour, there are two linear combinations of Wilson loop transforming oppositely under time-reversal transformation. We show that one combination is holographically dual to IIA fundamental string, while orthogonal combination is set to zero. We gather supporting evidences from detailed comparative study of generalized time-reversal transformations in both D2-brane worldvolume and ABJM theories. We then classify supersymmetric Wilson loops and find at most 1 6 supersymmetry. We next study Wilson loop expectation value in planar perturbation theory. For circular Wilson loop, we find features remarkably parallel to circular Wilson loop in N = 4 super Yang-Mills theory in four dimensions. First, all odd loop diagrams vanish identically and even loops contribute nontrivial contributions. Second, quantum corrected gauge and scalar propagators take the same form as those of N = 4 super Yang-Mills theory. Combining these results, we propose that expectation value of circular Wilson loop is given by Wilson loop expectation value in pure Chern-Simons theory times zerodimensional Gaussian matrix model whose variance is specified by an interpolating function of 't Hooft coupling. We suggest the function interpolates smoothly between weak and strong coupling regime, offering new test ground of the AdS/CFT correspondence. dτ n (6.1)We shall perform perturbative evaluation in Euclidean spacetime R 3 . In this case, the exponent of the Wilson loop is changed to A 0 (τ)dτ → A m (x(τ))ẋ m (τ)dτ, M I J → i M I J . (6.2) Computations of W N [C, M] , W N [C, M] or W N [C, M] etc. proceed exactly the same.

Using localization, matrix model and saddle-point techniques, we determine exact behavior of circular Wilson loop in N = 2 superconformal (quiver) gauge theories in the large number limit of colors. Focusing at planar and large 't Hooft couling limits, we compare its asymptotic behavior with well-known exponential growth of Wilson loop in N = 4 super Yang-Mills theory with respect to 't Hooft coupling. For theory with gauge group SU(N) coupled to 2N fundamental hypermultiplets, we find that Wilson loop exhibits non-exponential growth -at most, it can grow as a power of 't Hooft coupling. For theory with gauge group SU(N) × SU(N) and bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups.We find Wilson loop in untwisted sector grows exponentially large as in N = 4 super Yang-Mills theory. We then find Wilson loop in twisted sector exhibits non-analytic behavior with respect to difference of the two 't Hooft coupling constants. By letting one gauge coupling constant hierarchically larger/smaller than the other, we show that Wilson loops in the second type theory interpolate to Wilson loops in the first type theory. We infer implications of these findings from holographic dual description in terms of minimal surface of dual string worldsheet.We suggest intuitive interpretation that in both classes of theory holographic dual background must involve string scale geometry even at planar and large 't Hooft coupling limit and that new results found in the gauge theory side are attributable to worldsheet instantons and infinite resummation therein. Our interpretation also indicates that holographic dual of these gauge theories is provided by certain non-critical string theories. Reduction to One-Matrix ModelThe work [6] provided a proof for the conjecture [4,5] that the evaluation of the half-BPS Wilson loop in N = 4 super Yang-Mills theory [2, 3] is reduced to a related problem in a Gaussian Hermitian one-matrix model. In this section, we show that the similar reduction also works for N = 2 superconformal gauge theories of general quiver type. The resulting matrix model is, however, not Gaussian but includes non-trivial vertices due to nontrivial one-loop determinant.

Abstract:Recently it was found that the density matrix for a certain orthosymplectic Chern-Simons theory matches with that for the ABJM theory with the odd chiral projection. We prove this fact for a general case with the inclusion of fractional branes. We also identify the first few diagonal Gopakumar-Vafa invariants for the grand potential constructed from the chirally projected density matrix.

We investigate another supersymmetric Chern-Simons theory called the orientifold ABJM theory, which replaces the unitary supergroup structure of the ABJM theory with an orthosymplectic one. Its non-perturbative structure is completely clarified by considering the duplication of the quiver.

The planar solution is discussed for an N = 3 Chern-Simons-matter theory constructed recently by Gaiotto and Tomasiello. The planar resolvent is obtained in terms of contour integrals. If the sum of two Chern-Simons levels k 1 , k 2 is small, the expectation value of a supersymmetric Wilson loop grows exponentially with the total 't Hooft coupling, as is expected from AdS/CFT correspondence. If one of the Chern-Simons levels, say k 2 , is taken to infinity, for which one of the 't Hooft coupling constants becomes zero, then the exponential behavior disappears.