ABJM Wilson loops in the Fermi gas approach

The matrix model of ABJM theory can be formulated in terms of a Fermi gas in an external potential. We show that, in this formalism, vevs of Wilson loops correspond to averages of operators in the statistical-mechanical problem. This makes it possible to calculate these vevs at all orders in 1/N, up to exponentially small corrections, and for arbitrary Chern-Simons coupling, by using the WKB expansion. We present explicit results for the vevs of 1/6 and the 1/2 BPS Wilson loops, at any winding number, in terms of Airy functions. Our expressions are shown to reproduce the low genus results obtained previously in the 't Hooft expansion.

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abstract: archiveprefix: arXiv primaryclass: hep-th reportnumber: HU-EP-08-20 slaccitation: %%CITATION = ARXIV:0809.2787;%%

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abstract: A bstractWe show that the ABJM theory, which is an $ \mathcal{N} = {6} $ superconformal U(N) × U(N) Chern-Simons gauge theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility of probing the quantum aspects of M-theory and testing the AdS4/CFT3 duality at the quantum level. Here we calculate the free energy, and confirm the N3/2 scaling in the M-theory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion. The method can be easily generalized to the calculations of BPS operators and to other theories that reduce to matrix models.

title: On the Moduli Space of Elliptic Maxwell-Chern-Simons Theories
abstract: We analyze the moduli space of the low-energy limit of 3-dimensional N = 3 MaxwellChern-Simons theories described by circular quiver diagrams, as for 4-dimensional elliptic models. We define the theories by using D3-NS5-(k,1)5-brane systems with an arbitrary number of fivebranes. The supersymmetry is expected to be enhanced to N =4i n the low-energy limit. We show that the Higgs branch, in which all bifundamental scalar fields develop vacuum expectation values, is an abelian orbifold of C 4 . We confirm that the same geometry is obtained as an M-theory dual of the brane system. We also consider theories realized by introducing more than two kinds of fivebranes, and obtain nontoric fourfolds as moduli spaces. Recently, there has been great interest in 3-dimensional superconformal field theories as theories for describing multiple M2-branes in various backgrounds. This was triggered by the proposal of a new class of 3-dimensional theories by Bagger and Lambert, 1)–3) and Gusstavson. 4),5) The model (BLG model) possesses N(d=3) =8 superconformal symmetry and is based on Lie 3-algebra. The action of the BLG model includes the structure constant f abc d of a Lie 3-algebra, which determines the form of the interactions, and a metric h ab , which appears in the coefficients of the kinetic terms. These tensors must satisfy certain conditions required by the supersymmetry invariance of the action. If these tensors satisfy the conditions, we can write down the action of a BLG model. The constraint imposed on the structure constant is called a fundamental identity. It was soon realized that the identity is very restrictive, 6) and it was proved that if we assume that the metric is positive definite and the algebra is finite dimensional, there is only one nontrivial Lie 3-algebra, 7),8) which is called an A4 algebra. The BLG model based on the A4 algebra is a SU(2)×SU(2) Chern-Simons theory with levels k and −k for each SU(2) factor. Analysis of this model showed that it describes a pair of M2-branes in certain orbifold backgrounds. 9)–11) As a theory for an arbitrary number of M2-branes, a model based on an algebra with a Lorenzian metric was proposed in Refs. 12)– 14). Because of the indefinite metric, the model includes unwanted ghost modes. Although the ghost modes can be removed by treating them as background fields satisfying classical equations of motion, 14),15) or by gauging certain symmetries and fixing them, 16),17) this procedure breaks the conformal invariance, and the theory becomes D2-brane theory 14),16),18) by the mechanism proposed in Ref. 19) unless the

title: A simple class of N = 3 gauge/gravity duals
abstract: We find the gravity duals to an infinite series of = 3 Chern-Simons quiver theories. They are AdS4 × M7 vacua of M-theory, with M7 in a certain class of 3-Sasaki-Einstein manifolds obtained by a quotient construction. The field theories can be engineered from a brane configuration; their geometry is summarized by a ``hyperKahler toric fan'' that can be read off easily from the relative angles of the branes. The singularity at the tip of the cone over M7 is generically not an orbifold. The simplest new manifolds we consider can be written as the biquotient U(1)\U(3)/U(1). We also comment on the relation between our theories and four-dimensional = 1 theories with the same quiver.

title: Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
abstract: We study the vacuum structure and don spectrum of N =2 supersymmetric gauge theory in four dimensions, with gauge group SU(2). The theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and dyon masses and the metric on the moduli space of vacua. The description involves a version of Olive-Montonen electric-magnetic duality. The “strongly coupled” vacuum turns out to be a weakly coupled theory of monopoles, and with a suitable perturbation confinement is described by monopole condensation.

title: An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications
abstract: We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) series Z=log(eXeY) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y) generated by X and Y. It is based on the close relationship of L(X,Y) with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree of 20 [111 013 independent elements in L(X,Y)] takes less than 15min on a personal computer and requires 1.5Gbytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when X and Y are real or complex matrices.

title: Asymptotic ~-expansions of stationary quantum states
abstract: We discuss the mathematical framework for uniform asymptotic expansions in the parameter h of non-relativistic quantum mechanics. Quantum observables of interest are taken in an algebra of pseudo-differential operators defined with the help of the Wigner transformation. We define a class of asymptotic quantum states as certain linear functionals on those operators. These functionals are microlocal, and we study their supports in phase space in analogy with wave front set theory; they are also shown to be covariant under the « metaplectic representation » of the affine symplectic transformations in phase space. In this asymptotic framework we can formulate the eigenstate problem for the most general observable. This problem is formally solved by quadratures for onedimensional systems : a Bohr-Sommerfeld formula correct to all orders in h is the obtained for the discrete spectrum.  Nous formulons un cadre mathématique pour développer la mécanique quantique non-relativiste en series asymptotiques uniformes dans le paramètre h. II faut se restreindre a des observables quantiques appartenant a une certaine algèbre d’opérateurs pseudo-différentiels, définie a l’aide de la transformation de Wigner. Nous définissons alors une classe d’états quantiques asymptotiques comme fonctionnelles linéaires sur ces observables. Ces fonctionnelles sont microlocales, et nous étudions leurs supports dans l’espace de phase en analogie avec la théorie des fronts d’onde ; nous montrons aussi qu’elles sont covariantes par la « représen(*) Detache du C. N. R. S. Annales de l’Institut Henri Poincaré Section A Vol. XXVI, n° 4 1977.

title: Supergravity on AdS 5 / 4 × Hopf Fibrations and Conformal Field Theories
abstract: We obtain three and four dimensional conformal field theories with less than maximal supersymmetry by using their supergravity duals. These supergravity

title: Knot Invariants and Topological Strings
abstract: We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation) agrees to all orders in N with the corresponding quantity on the topological string side. For a general knot, we find a reformulation of the knot invariant in terms of new integral invariants, which capture the spectrum (and spin) of M2 branes ending on M5 branes embedded in the resolved conifold geometry. We also find an intriguing link between knot invariants and superpotential terms generated by worldsheet instantons in N=1 supersymmetric theories in 4 dimensions.

title: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
abstract: In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to a well-defined algebro-geometric computation very similar to well-known localization techniques in Gromov–Witten theory. In particular, our computation of the invariants for multiple covers of a generic disc bounding a special Lagrangian submanifold in a Calabi–Yau threefold agrees completely with the original predictions of Ooguri and Vafa based on string duality. Our proposed invariants depend more generally on a discrete parameter which came to light in the work of Aganagic, Klemm, and Vafa which was also based on duality, and our more general calculations agree with theirs up to sign.

title: Interacting fermions and $ \mathcal{N} $ = 2 Chern-Simons-matter theories
abstract: A bstractThe partition function on the three-sphere of $ \mathcal{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 $ \mathcal{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 $ \mathcal{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 $ \mathcal{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.

title: Topological Strings and (Almost) Modular Forms
abstract: The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP2 and IP1  ×  IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold $${{\mathbb {C}^3} / {\mathbb {Z}_3}}$$.