Interacting fermions and N=2 Chern-Simons-matter theories

The partition function on the three-sphere of N=3 Chern-Simons-matter theories can be formulated in terms of an ideal Fermi gas. In this paper we show that, in theories with N=2 supersymmetry, the partition function corresponds to a gas of interacting fermions in one dimension. The large N limit is the thermodynamic limit of the gas and it can be analyzed with the Hartree and Thomas-Fermi approximations, which lead to the known large N solutions of these models. We use this interacting fermion picture to analyze in detail N=2 theories with one single node. In the case of theories with no long-range forces we incorporate exchange effects and argue that the partition function is given by an Airy function, as in N=3 theories. For the theory with g adjoint superfields and long-range forces, the Thomas-Fermi approximation leads to an integral equation which determines the large N, strongly coupled R-charge.Comment: 29 pages, 4 figure

Comments on F-maximization and R-symmetry in 3D SCFTs

We report preliminary results on the recently proposed F-maximization principle in 3D SCFTs. We compute numerically in the large-N limit the free energy on the three-sphere of an N=2 Chern-Simons-Matter theory with a single adjoint chiral superfield which is known to exhibit a pattern of accidental symmetries associated to chiral superfields that hit the unitarity bound and become free. We observe that the F-maximization principle produces a U(1) R-symmetry consistent with previously obtained bounds but inconsistent with a postulated Seiberg-like duality. Potential modifications of the principle associated to the decoupling fields do not appear to be sufficient to account for the observed violations.Comment: 17 pages, 3 figures; v2 a reference has been added, a missing factor of 2 has been corrected in eq (3.3) and the numerical results have been accordingly updated. The new results do not show any obvious signs of violation of previously obtained bounds. A potential disagreement with a postulated Seiberg-like duality is note

Matrix Models for Supersymmetric Chern-Simons Theories with an ADE Classification

We consider N=3 supersymmetric Chern-Simons (CS) theories that contain product U(N) gauge groups and bifundamental matter fields. Using the matrix model of Kapustin, Willett and Yaakov, we examine the Euclidean partition function of these theories on an S^3 in the large N limit. We show that the only such CS theories for which the long range forces between the eigenvalues cancel have quivers which are in one-to-one correspondence with the simply laced affine Dynkin diagrams. As the A_n series was studied in detail before, in this paper we compute the partition function for the D_4 quiver. The D_4 example gives further evidence for a conjecture that the saddle point eigenvalue distribution is determined by the distribution of gauge invariant chiral operators. We also see that the partition function is invariant under a generalized Seiberg duality for CS theories.Comment: 20 pages, 3 figures; v2 refs added; v3 conventions in figure 3 altered, version to appear in JHE

Instantons on ALE spaces and orbifold partitions

We consider N=4 theories on ALE spaces of $A_{k-1}$ type. As is well known, their partition functions coincide with $A_{k-1}$ affine characters. We show that these partition functions are equal to the generating functions of some peculiar classes of partitions which we introduce under the name 'orbifold partitions'. These orbifold partitions turn out to be related to the generalized Frobenius partitions introduced by G. E. Andrews some years ago. We relate the orbifold partitions to the blended partitions and interpret explicitly in terms of a free fermion system.Comment: 28 pages, 10 figures; reference adde

Large N Free Energy of 3d N=4 SCFTs and AdS/CFT

We provide a non-trivial check of the AdS_4/CFT_3 correspondence recently proposed in arXiv:1106.4253 by verifying the GKPW relation in the large N limit. The CFT free energy is obtained from the previous works (arXiv:1105.2551, arXiv:1105.4390) on the S^3 partition function for 3-dimensional N=4 SCFT T[SU(N)]. This is matched with the computation of the type IIB action on the corresponding gravity background. We unexpectedly find that the leading behavior of the free energy at large N is 1/2 N^2 ln N. We also extend our results to richer theories and argue that 1/2 N^2 ln N is the maximal free energy at large N in this class of gauge theories.Comment: 20 pages, 3 figure

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

The ABCDEF's of Matrix Models for Supersymmetric Chern-Simons Theories

We consider N = 3 supersymmetric Chern-Simons gauge theories with product unitary and orthosymplectic groups and bifundamental and fundamental fields. We study the partition functions on an S^3 by using the Kapustin-Willett-Yaakov matrix model. The saddlepoint equations in a large N limit lead to a constraint that the long range forces between the eigenvalues must cancel; the resulting quiver theories are of affine Dynkin type. We introduce a folding/unfolding trick which lets us, at the level of the large N matrix model, (i) map quivers with orthosymplectic groups to those with unitary groups, and (ii) obtain non-simply laced quivers from the corresponding simply laced quivers using a Z_2 outer automorphism. The brane configurations of the quivers are described in string theory and the folding/unfolding is interpreted as the addition/subtraction of orientifold and orbifold planes. We also relate the U(N) quiver theories to the affine ADE quiver matrix models with a Stieltjes-Wigert type potential, and derive the generalized Seiberg duality in 2 + 1 dimensions from Seiberg duality in 3 + 1 dimensions.Comment: 30 pages, 5 figure

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

Exact Results in D=2 Supersymmetric Gauge Theories

We compute exactly the partition function of two dimensional N=(2,2) gauge theories on S^2 and show that it admits two dual descriptions: either as an integral over the Coulomb branch or as a sum over vortex and anti-vortex excitations on the Higgs branches of the theory. We further demonstrate that correlation functions in two dimensional Liouville/Toda CFT compute the S^2 partition function for a class of N=(2,2) gauge theories, thereby uncovering novel modular properties in two dimensional gauge theories. Some of these gauge theories flow in the infrared to Calabi-Yau sigma models - such as the conifold - and the topology changing flop transition is realized as crossing symmetry in Liouville/Toda CFT. Evidence for Seiberg duality in two dimensions is exhibited by demonstrating that the partition function of conjectured Seiberg dual pairs are the same.Comment: 78 pages, LaTeX; v2: small corrections and references added; v3: JHEP version, discussing factorization further in new appendix F; v4: sign corrected for non simply-connected gauge grou

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