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 function 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.the R-charge of the matter field is denoted r, and s β (z) denotes the double sine function.Notice that we may absorb a factor of 1/ √ b 1 b 2 into σ 0 , which is then integrated over, and thus we see that the partition function (1.2) depends on the background geometry only through the single parameter β 2 = b 1 /b 2 . We shall also present a sketch of a proof for why (1.2) continues to hold in the case that K generates only a U(1) action on S 3 , and comment on the extension of our results to three-manifolds with non-trivial fundamental group.

We construct the gravity duals of large N supersymmetric gauge theories defined on squashed five-spheres with SU (3) × U (1) symmetry. These five-sphere backgrounds are continuously connected to the round sphere, and we find a oneparameter family of 3/4 BPS deformations and a two-parameter family of (generically) 1/4 BPS deformations. The gravity duals are constructed in Euclidean Romans F (4) gauged supergravity in six dimensions, and uplift to massive type IIA supergravity. We holographically renormalize the Romans theory, and use our general result to compute the renormalized on-shell actions for the solutions. The results agree perfectly with the large N limit of the dual gauge theory partition function, which we compute using large N matrix model techniques. In addition we compute BPS Wilson loops in these backgrounds, both in supergravity and in the large N matrix model, again finding precise agreement. Finally, we conjecture a general formula for the partition function on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector.S 4 [12]. In particular the computation of [5] effectively determines the six-dimensional Newton constant. Having constructed supergravity solutions that have squashed fivesphere conformal boundaries, we compute the holographic free energy F = − log Z by holographically renormalizing the on-shell Euclidean action. More specifically, we construct families of solutions with different numbers of preserved supercharges. Two of these families are shown to be dual to the 1/4 BPS and 3/4 BPS gauge theories defined

In this paper we continue the study of the superconformal index of fourdimensional N = 2 theories of class S in the presence of surface defects. Our main result is the construction of an algebra of difference operators, whose elements are labeled by irreducible representations of A N −1 . For the fully antisymmetric tensor representations these difference operators are the Hamiltonians of the elliptic Ruijsenaars-Schneider system. The structure constants of the algebra are elliptic generalizations of the Littlewood-Richardson coefficients. In the Macdonald limit, we identify the difference operators with local operators in the two-dimensional TQFT interpretation of the superconformal index. We also study the dimensional reduction to difference operators acting on the three-sphere partition function, where they characterize supersymmetric defects supported on a circle, and show that they are transformed to supersymmetric Wilson loops under mirror symmetry. Finally, we compare to the difference operators that create 't Hooft loops in the fourdimensional N = 2 * theory on a four-sphere by embedding the three-dimensional theory as an S-duality domain wall.

M-theory on AdS 7 × S 4 admits a description where the AdS 7 factor is constructed as a timelike Hopf fibration over a non-compact three dimensional complex projective spaceCP 3 [1]. We consider the worldvolume theory for M5-branes at a fixedCP 3 radius which, after reduction along the timelike fibre, is given by an Ω-deformed Yang-Mills theory with eight supercharges. Taking the radius to infinity then induces a classical RG flow. We construct the fixed point action which has an enhanced 24 supercharges and which can be understood as the (2, 0) theory of M5-branes on flat space reduced along a compact null Killing

In arXiv:1007.2982 a novel system of equations which propagate in one null and four space directions were obtained as the on-shell conditions for the six-dimensional (2, 0) superalgebra. In this paper we show how this system reduces to one-dimensional motion on instanton moduli space. Quantization leads to the previous light-cone proposal of the (2, 0) theory, generalized to include a potential that arises on the Coulomb branch as well as couplings to background gauge and self-dual two-form fields.