A Note on q-Deformed Two-Dimensional Yang-Mills and Open Topological Strings

In this note we make a test of the open topological string version of the OSV conjecture, proposed in hep-th/0504054, in the toric Calabi-Yau manifold $X= O(-3)\to\mathbf{P}^2$ with background D4-branes wrapped on Lagrangian submanifolds. The D-brane partition function reduces to an expectation value of some inserted operators of a q-deformed Yang-Mills theory living on a chain of $\mathbf{P}^1$'s in the base $\mathbf{P}^2$ of $X$. At large $N$ this partition function can be written as a sum over squares of chiral blocks, which are related to the open topological string amplitudes in the local $\mathbf{P}^2$ geometry with branes at both the outer and inner edges of the toric diagram. This is in agreement with the conjecture.Comment: 14 pages, 3 figure

Branes, Black Holes and Topological Strings on Toric Calabi-Yau Manifolds

We develop means of computing exact degerenacies of BPS black holes on toric Calabi-Yau manifolds. We show that the gauge theory on the D4 branes wrapping ample divisors reduces to 2D q-deformed Yang-Mills theory on necklaces of P^1's. As explicit examples we consider local P^2, P^1 x P^1 and A_k type ALE space times C. At large N the D-brane partition function factorizes as a sum over squares of chiral blocks, the leading one of which is the topological closed string amplitude on the Calabi-Yau. This is in complete agreement with the recent conjecture of Ooguri, Strominger and Vafa.Comment: 50 pages, 6 figures, harvma

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

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

ABJM theory as a Fermi gas

The partition function on the three-sphere of many supersymmetric Chern-Simons-matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.Comment: 52 pages, 11 figures. v3: references, corrections and clarifications added, plus a footnote on the relation to the recent work by Hanada et a

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

Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation, previously employed for trivial bundles $\Sigma \times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $\Sigma$ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.Comment: 37 pages; v2: references adde

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

Index for Three Dimensional Superconformal Field Theories and Its Applications

We review aspects of superconformal indices in three dimension. Three dimensional superconformal indices can be exactly computed by using localization method including monopole contribution, and can be applied to provide evidences for mirror duality, AdS_4/CFT_3 correspondence and global symmetry enhancement of strongly coupled gauge theories. After reviewing, we discuss the possibility of global symmetry enhancement in a finite rank of gauge group.Comment: 14 pages, Proceedings of the Seventh International Conference Quantum Theory and Symmetries (QTS-7) in Prague, Czech Republic, August, 2011; v2: minor modifications, discussion of supersymmetry enhancement of abelian ABJM theory by using an index were 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