We compute the large N limit of Wilson loop expectation values for a broad class of N=2 supersymmetric gauge theories defined on a general class of background three-manifolds M_3, diffeomorphic to S^3. We find a simple closed formula which depends on
the background geometry only through a certain supersymmetric Killing vector field. The supergravity dual of such a Wilson loop is an M2-brane wrapping the M-theory circle, together with a complex curve in a self-dual Einstein manifold M_4, whose conformal boundary is M_3. We show that the regularized action of this M2-brane also depends only on the supersymmetric Killing vector, precisely reproducing the large N field theory computation.
We present the gravity dual of large N supersymmetric gauge theories on a squashed five-sphere. The one-parameter family of solutions is constructed in Euclidean Romans F(4) gauged supergravity in six dimensions, and uplifts to massive type IIA super
gravity. By renormalizing the theory with appropriate counterterms we evaluate the renormalized on-shell action for the solutions. We also evaluate the large N limit of the gauge theory partition function, and find precise agreement.
We consider supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere. The three-manifold is always equipped with an almost contact structure and an associated Reeb vector field. We show that the partition functi
on depends only on this vector field, giving an explicit expression in terms of the double sine function. In the large N limit our formula agrees with a recently discovered two-parameter family of dual supergravity solutions. We also explain how our results may be applied to prove vortex-antivortex factorization. Finally, we comment on the extension of our results to three-manifolds with non-trivial fundamental group.
We study a general class of supersymmetric AdS_4 x Y_7 solutions of M-theory that have large N dual descriptions as N = 2 Chern-Simons-matter theories on S^3. The Hamiltonian function h_M for the M-theory circle, with respect to a certain contact str
ucture on Y_7, plays an important role in the duality. We show that an M2-brane wrapping the M-theory circle, giving a fundamental string in AdS_4, is supersymmetric precisely at the critical points of h_M, and moreover the value of this function at the critical point determines the M2-brane action. Such a configuration determines the holographic dual of a BPS Wilson loop for a Hopf circle in S^3, and leads to an effective method for computing the Wilson loop on both sides of the correspondence in large classes of examples. We find agreement in all cases, including for several infinite families, and moreover we find that the image h_M(Y_7) determines the range of support of the eigenvalues in the dual large N matrix model, with the critical points of h_M mapping to points where the derivative of the eigenvalue density is discontinuous.
We study the matrix model for N M2-branes wrapping a Lens space L(p,1) = S^3/Z_p. This arises from localization of the partition function of the ABJM theory, and has some novel features compared with the case of a three-sphere, including a sum over f
lat connections and a potential that depends non-trivially on p. We study the matrix model both numerically and analytically in the large N limit, finding that a certain family of p flat connections give an equal dominant contribution. At large N we find the same eigenvalue distribution for all p, and show that the free energy is simply 1/p times the free energy on a three-sphere, in agreement with gravity dual expectations.
This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
We construct three-dimensional N=2 Chern-Simons-quiver theories which are holographically dual to the M-theory Freund-Rubin solutions AdS_4 x V_{5,2}/Z_k (with or without torsion G-flux), where V_{5,2} is a homogeneous Sasaki-Einstein seven-manifold.
The global symmetry group of these theories is generically SU(2) x U(1) x U(1)_R, and they are hence non-toric. The field theories may be thought of as the n=2 member of a family of models, labelled by a positive integer n, arising on multiple M2-branes at certain hypersurface singularities. We describe how these models can be engineered via generalized Hanany-Witten brane constructions. The AdS_4 x V_{5,2}/Z_k solutions may be deformed to a warped geometry R^{1,2} x T^* S^4/Z_k, with self-dual G-flux through the four-sphere. We show that this solution is dual to a supersymmetric mass deformation, which precisely modifies the classical moduli space of the field theory to the deformed geometry.
