Perturbative Approach to Integrability in Three-Dimensional Chern-Simons Theories (original) (raw)
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N=2 Superstring Theory Generates Supersymmetric Chern-Simons Theories
Modern Physics Letters A, 1994
We show that the action of self-dual supersymmetric Yang-Mills theory in four dimensions, which describes the consistent massless background fields for N=2 superstring, generates the actions for N=1 and N=2 supersymmetric non-Abelian Chern-Simons theories in three dimensions after some dimensional reductions. Since the latters play important roles for supersymmetric integrable models, this result indicates the fundamental significance of the N=2 superstring theory controlling (possibly all) supersymmetric integrable models in lower dimensions.
đť’© = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
Journal of High Energy Physics, 2008
We construct three dimensional Chern-Simons-matter theories with gauge groups U (N ) Ă—U (N ) and SU (N ) Ă—SU (N ) which have explicit N = 6 superconformal symmetry. Using brane constructions we argue that the U (N ) Ă— U (N ) theory at level k describes the low energy limit of N M2-branes probing a C 4 /Z k singularity. At large N the theory is then dual to M-theory on AdS 4 Ă— S 7 /Z k . The theory also has a 't Hooft limit (of large N with a fixed ratio N/k) which is dual to type IIA string theory on AdS 4 Ă— CP 3 . For k = 1 the theory is conjectured to describe N M2-branes in flat space, although our construction realizes explicitly only six of the eight supersymmetries. We give some evidence for this conjecture, which is similar to the evidence for mirror symmetry in d = 3 gauge theories. When the gauge group is SU (2) Ă— SU (2) our theory has extra symmetries and becomes identical to the Bagger-Lambert theory.
The Integral Form of D = 3 Chern-Simons Theories Probing Cn/Γ Singularities
Fortschritte der Physik, 2017
We consider D=3 supersymmetric Chern Simons gauge theories both from the point of view of their formal structure and of their applications to the AdS 4 /CFT 3 correspondence. From the structural viewpoint , we use the new formalism of integral forms in superspace that utilizes the rheonomic Lagrangians and the Picture Changing Operators, as an algorithmic tool providing the connection between different approaches to supersymmetric theories. We provide here the generalization to an arbitrary Kähler manifold with arbitrary gauge group and arbitrary superpotential of the rheonomic lagrangian of D=3 matter coupled gauge theories constructed years ago. From the point of view of the AdS 4 /CFT 3 correspondence and more generally of M2-branes we emphasize the role of the Kähler quotient data in determining the field content and the interactions of the Cherns Simons gauge theory when the transverse space to the brane is a non-compact Kähler quotient K 4 of some flat variety with respect to a suitable group. The crepant resolutions of C n /Γ singularities fall in this category. In the present paper we anticipate the general scheme how the geometrical data are to be utilized in the construction of the D=3 Chern-Simons Theory supposedly dual to the corresponding M2-brane solution.
Integrability of N ÂĽ 6 Chern-Simons theory at six loops and beyond
We study issues concerning perturbative integrability of N ÂĽ 6 Chern-Simons theory at planar and weak 't Hooft coupling regime. By Feynman diagrammatics, we derive so-called maximal-ranged interactions in the quantum dilatation generator, originating from homogeneous and inhomogeneous diagrams. The dilatation operator requires proper regularization of ultraviolet and infrared divergences and also bears scheme dependence depending on operator-mixing or two-point function methods adopted. We first consider the standard operator-mixing method. We show that homogeneous diagrams are obtainable by recursive method to all orders. The method, however, is not easily extendable to inhomogeneous diagrams. We thus consider two-point function method and study both operator contents and the spectrum of the quantum dilatation generator up to six-loop orders. Within this scheme, we show that, of two possible classes of operators, only one linear combination actually contributes. Curiously, this is exactly the same combination as in N ÂĽ 4 super Yang-Mills theory. From these operators, we extract a spectrum of anomalous dimension up to six loops. We find that the spectrum agrees perfectly with the prediction based on quantum integrability. In evaluating the six-loop diagrams, we utilized a remarkable integer-relation algorithm developed
Integrability of N=6 Chern-Simons theory at six loops and beyond
Physical Review D, 2010
We study issues concerning perturbative integrability of N = 6 Chern-Simons theory at planar and weak 't Hooft coupling regime. By Feynman diagrammatics, we derive so called maximalranged interactions in the quantum dilatation generator, originating from homogeneous and inhomogeneous diagrams. These diagrams require proper regularization of not only ultraviolet but also infrared divergences. We first consider standard operator mixing method. We show that homogeneous diagrams are obtainable by recursive method to all orders. The method, however, is not easily extendable to inhomogeneous diagrams. We thus consider two-point function method and study both operator contents and spectrum of the quantum dilatation generator up to six loop orders. We show that, of two possible classes of operators, only one linear combination actually contributes. Curiously, this is exactly the same combination as in N = 4 super Yang-Mills theory. We then study spectrum of anomalous dimension up to six loops. We find that the spectrum agrees perfectly with the prediction based on quantum integrability. In evaluating the six loop diagrams, we utilized remarkable integer-relation algorithm (PSLQ) developed by Ferguson, Baily and Arno.
We revisit unitary representation of centrally extended psu(2|2) excitation superalgebra. We find most generally that 'pseudo-momentum', not lattice momentum, diagonalizes spin chain Hamiltonian and leads to generalized dynamic spin chain. All known results point to lattice momentum diagonalization for N = 4 super-Yang–Mills theory. Having different interacting structure, we ask if N = 6 superconformal Chern– Simons theory provides an example of pseudo-momentum diagonalization. For SO(6) sector, we study maximal shuffling and next-to-maximal shuffling terms in the dilatation operator and compare them with results expected from psu(2|2) superalgebra and integrability. At two loops, we rederive maximal shuffling term (3-site) and find perfect agreement with known results. At four loops, we first find absence of next-to-maximal shuffling term (4-site), in agreement with prediction based on integrability. We next extract maximal shuffling term (5-site), the most relevant term for checking the possibility of pseudo-momentum diagonalization. Curiously, we find that result agrees with integrability prediction based on lattice momentum , as in N = 4 super-Yang–Mills theory. Consistency of our results is fully ensured by checks of renormalizability up to six loops.
Quantum Spectral Curve of the N = 6 Supersymmetric Chern-Simons Theory
Physical Review Letters, 2014
Recently, it was shown that the spectrum of anomalous dimensions and other important observables in planar N ¼ 4 supersymmetric Yang-Mills theory are encoded into a simple nonlinear Riemann-Hilbert problem: the Pμ system or quantum spectral curve. In this Letter, we extend this formulation to the N ¼ 6 supersymmetric Chern-Simons theory introduced by Aharony, Bergman, Jafferis, and Maldacena. This may be an important step towards the exact determination of the interpolating function hðλÞ characterizing the integrability of this model. We also discuss a surprising relation between the quantum spectral curves for the N ¼ 4 supersymmetric Yang-Mills theory and the N ¼ 6 supersymmetric Chern-Simons theory considered here.
ℵ0-extended supergravity and Chern-Simons theories
Nuclear Physics B, 1996
We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in threedimensions by gauging its intrinsic global SO(N) symmetry. We call these ℵ 0 (Aleph-Null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ 0 supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three-dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ 0 globally and locally supersymmetric theories in three-dimensions. As an interesting application, we present ℵ 0 supergravity Liouville theory in two-dimensions after appropriate dimensional reduction from threedimensions.