Operator Counting and Eigenvalue Distributions for 3D Supersymmetric Gauge Theories

We give further support for our conjecture relating eigenvalue distributions of the Kapustin-Willett-Yaakov matrix model in the large N limit to numbers of operators in the chiral ring of the corresponding supersymmetric three-dimensional gauge theory. We show that the relation holds for non-critical R-charges and for examples with {\mathcal N}=2 instead of {\mathcal N}=3 supersymmetry where the bifundamental matter fields are nonchiral. We prove that, for non-critical R-charges, the conjecture is equivalent to a relation between the free energy of the gauge theory on a three sphere and the volume of a Sasaki manifold that is part of the moduli space of the gauge theory. We also investigate the consequences of our conjecture for chiral theories where the matrix model is not well understood.Comment: 27 pages + appendices, 5 figure

Integrability on the Master Space

It has been recently shown that every SCFT living on D3 branes at a toric Calabi-Yau singularity surprisingly also describes a complete integrable system. In this paper we use the Master Space as a bridge between the integrable system and the underlying field theory and we reinterpret the Poisson manifold of the integrable system in term of the geometry of the field theory moduli space.Comment: 47 pages, 20 figures, using jheppub.st

p-wave Holographic Superconductors and five-dimensional gauged Supergravity

We explore five-dimensional ${\cal N}=4$ $SU(2)\times U(1)$ and ${\cal N}=8$ SO(6) gauged supergravities as frameworks for condensed matter applications. These theories contain charged (dilatonic) black holes and 2-forms which have non-trivial quantum numbers with respect to U(1) subgroups of SO(6). A question of interest is whether they also contain black holes with two-form hair with the required asymptotic to give rise to holographic superconductivity. We first consider the ${\cal N}=4$ case, which contains a complex two-form potential $A_{\mu\nu}$ which has U(1) charge $\pm 1$. We find that a slight generalization, where the two-form potential has an arbitrary charge $q$, leads to a five-dimensional model that exhibits second-order superconducting transitions of p-wave type where the role of order parameter is played by $A_{\mu\nu}$, provided $q \gtrsim 5.6$. We identify the operator that condenses in the dual CFT, which is closely related to ${\cal N}=4$ Super Yang-Mills theory with chemical potentials. Similar phase transitions between R-charged black holes and black holes with 2-form hair are found in a generalized version of the ${\cal N}=8$ gauged supergravity Lagrangian where the two-forms have charge $q\gtrsim 1.8$.Comment: 35 pages, 14 figure

N=1 Chern-Simons theories, orientifolds and Spin(7) cones

We construct three dimensional N=1 Chern-Simons theories living on M2 branes probing Spin(7) cones. We consider Spin(7) manifolds obtained as quotients of Calabi-Yau four-folds by an anti-holomorphic involution, following a construction by Joyce. The corresponding Chern-Simons theories can be obtained from N=2 theories by an orientifolding procedure. These theories are holographically dual to M theory solutions AdS_4 \times H, where the weak G_2 manifold H is the base of the Spin(7) cone.Comment: 26 pages, 3 figures, reference added

The Large N Limit of Toric Chern-Simons Matter Theories and Their Duals

We compute the large N limit of the localized three dimensional free energy of various field theories with known proposed AdS duals. We show that vector-like theories agree with the expected supergravity results, and with the conjectured F-theorem. We also check that the large N free energy is preserved by the three dimensional Seiberg duality for general classes of vector like theories. Then we analyze the behavior of the free energy of chiral-like theories by applying a new proposal. The proposal is based on the restoration of a discrete symmetry on the free energy before the extremization. We apply this procedure at strong coupling in some examples and we discuss the results. We conclude the paper by proposing an alternative geometrical expression for the free energy.Comment: 40 pages, 7 figures, using jheppub.sty, references adde

Operator Counting for N=2 Chern-Simons Gauge Theories with Chiral-like Matter Fields

The localization formula of Chern-Simons quiver gauge theory on $S^3$ nicely reproduces the geometric data such as volume of Sasaki-Einstein manifolds in the large-$N$ limit, at least for vector-like models. The validity of chiral-like models is not established yet, due to technical problems in both analytic and numerical approaches. Recently Gulotta, Herzog and Pufu suggested that the counting of chiral operators can be used to find the eigenvalue distribution of quiver matrix models. In this paper we apply this method to some vector-like or chiral-like quiver theories, including the triangular quivers with generic Chern-Simons levels which are dual to in-homogeneous Sasaki-Einstein manifolds $Y^{p,k}(\mathbb{CP}^2)$. The result is consistent with AdS/CFT and the volume formula. We discuss the implication of our analysis.Comment: 23 pages; v2. revised version; v3. corrected typos and clarified argument

Refined Checks and Exact Dualities in Three Dimensions

We discuss and provide nontrivial evidence for a large class of dualities in three-dimensional field theories with different gauge groups. We match the full partition functions of the dual phases for any value of the couplings to underpin our proposals. We focus on two classes of models. The first class, motivated by the AdS/CFT conjecture, consists of necklace U(N) quiver gauge theories with non chiral matter fields. We also consider orientifold projections and establish dualities among necklace quivers with alternating orthogonal and symplectic groups. The second class consists of theories with tensor matter fields with free theory duals. In most of these cases the R-symmetry mixes with IR accidental symmetries and we develop the prescription to include their contribution into the partition function and the extremization problem accordingly.Comment: 38 pages, 3 figure, using jheppu

M5 spikes and operators in the HVZ membrane theory

In this note we study some aspects of the so-called dual ABJM theory introduced by Hanany, Vegh & Zaffaroni. We analyze the spectrum of chiral operators, and compare it with the spectrum of functions on the mesonic moduli space M=C^2\times C^2/Z_k, finding expected agreement for the coherent branch. A somewhat mysterious extra branch of dimension N^2 opens up at the orbifold fixed point. We also study BPS solutions which represent M2/M5 intersections. The mesonic moduli space suggests that there should be two versions of this spike: one where the M5 lives in the orbifolded C^2 and another where it lives in the unorbifolded one. While expectedly the first class turns out to be like the ABJM spike, the latter class looks like a collection of stacks of M5 branes with fuzzy S^3 profiles. This shows hints of the appearance of the global SO(4) at the non-abelian level which is otherwise not present in the bosonic potential. We also study the matching of SUGRA modes with operators in the coherent branch of the moduli space. As a byproduct, we present some formulae for the laplacian in conical CY_4 of the form C^n\times CY_{4-n}.Comment: 22 pages, 1 figure. Published version with corrected typos

From Necklace Quivers to the F-theorem, Operator Counting, and T(U(N))

The matrix model of Kapustin, Willett, and Yaakov is a powerful tool for exploring the properties of strongly interacting superconformal Chern-Simons theories in 2+1 dimensions. In this paper, we use this matrix model to study necklace quiver gauge theories with {\cal N}=3 supersymmetry and U(N)^d gauge groups in the limit of large N. In its simplest application, the matrix model computes the free energy of the gauge theory on S^3. The conjectured F-theorem states that this quantity should decrease under renormalization group flow. We show that for a simple class of such flows, the F-theorem holds for our necklace theories. We also provide a relationship between matrix model eigenvalue distributions and numbers of chiral operators that we conjecture holds more generally. Through the AdS/CFT correspondence, there is therefore a natural dual geometric interpretation of the matrix model saddle point in terms of volumes of 7-d tri-Sasaki Einstein spaces and some of their 5-d submanifolds. As a final bonus, our analysis gives us the partition function of the T(U(N)) theory on S^3.Comment: 3 figures, 41 pages; v2 minor improvements, refs adde

Baryonic symmetries and M5 branes in the AdS_4/CFT_3 correspondence

We study U(1) symmetries dual to Betti multiplets in the AdS_4/CFT_3 correspondence for M2 branes at Calabi-Yau four-fold singularities. Analysis of the boundary conditions for vector fields in AdS_4 allows for a choice where wrapped M5 brane states carrying non-zero charge under such symmetries can be considered. We begin by focusing on isolated toric singularities without vanishing six-cycles, and study in detail the cone over Q^{111}. The boundary conditions considered are dual to a CFT where the gauge group is U(1)^2 x SU(N)^4. We find agreement between the spectrum of gauge-invariant baryonic-type operators in this theory and wrapped M5 brane states. Moreover, the physics of vacua in which these symmetries are spontaneously broken precisely matches a dual gravity analysis involving resolutions of the singularity, where we are able to match condensates of the baryonic operators, Goldstone bosons and global strings. We also argue more generally that theories where the resolutions have six-cycles are expected to receive non-perturbative corrections from M5 brane instantons. We give a general formula relating the instanton action to normalizable harmonic two-forms, and compute it explicitly for the Q^{222} example. The holographic interpretation of such instantons is currently unclear.Comment: 92 pages, 10 figure