THE subject of quantum electrodynamics is extremely difficult, even for the case of a single electron. The usual method of solving the corresponding wave equation leads to divergent integrals. To avoid these, Prof. P. A. M. Dirac* uses the method of redundant variables. This does not abolish the difficulty, but presents it in a new form, which may be dealt with in two ways. The first of these needs only comparatively simple mathematics and is directly connected with an elegant general scheme, but unfortunately its wave functions apply only to a hypothetical world and so its physical interpretation is indirect. The second way has the advantage of a direct physical interpretation, but the mathematics is so complicated that it has not yet been solved even for what appears to be the simplest possible case. Both methods seem worth further study, failing the discovery of a third which would combine the advantages of both.

http://ieeexplore.ieee.org/iel5/3/22993/01069961.pdf

Quantum electrodynamics

We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N = 2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.

/pdf/exact-results-for-wilson-loops-in-superconformal-chern-1rdc9v6ens.pdf

Exact results for Wilson loops in superconformal Chern-Simons theories with matter

For 3-dimensional field theories with $ \mathcal{N} = 2 $
 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number ofsuch large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the super potential. In all our $ \mathcal{N} = 2 $
 superconformal examples, the local maximization of F yields answers that scale as N
 3/2 and agree with the dual M-theory backgrounds AdS
 4 × Y, where Y are 7-dimensional Sasaki-Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows. We therefore propose the “F-theorem” that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N
 5/3 at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.

Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere

The complete superconformal index for N=6 Chern–Simons theory

Quantum inverse scattering method and correlation functions

We discuss the N = 2 superspace formulation of the N = 8 superconformal Bagger-Lambert-Gustavsson theory, and of the N = 6 superconformal Aharony-Bergman-Jafferis-Maldacena U(N) x U(N) Chern-Simons theory. In particular, we prove the full SU(4)R-symmetry of the ABJM theory. We then consider orbifold projections of this theory that give non-chiral and chiral (U( N) x U(N))(n) superconformal quiver gauge theories. We argue that these theories are dual to certain AdS(4) x S-7/(Z(n) x Z (k$) over tilde) backgrounds of M-theory. We also study a SU(3) invariant mass term in the superpotential that makes the N = 8 theory flow to a N = 2 superconformal gauge theory with a sextic superpotential. We conjecture that this gauge theory is dual to the U(1)(R) x SU(3) invariant extremum of the N = 8 gauged supergravity, which was discovered by N. Warner 25 years ago and whose uplifting to 11 dimensions was found more recently.

Superconformal Chern-Simons Theories and AdS(4)/CFT(3) Correspondence

We calculate the S-matrix in the gauge-fixed sigma-model on AdS(5) x S-5 to the leading order in perturbation theory, and analyze how supersymmetry is realized on the scattering states. A mild nonlocality of the supercharges implies that their action on multi-particle states does not follow the Leibniz rule, which is replaced by a nontrivial coproduct. The plane wave symmetry algebra is thus naturally enhanced to a Hopf algebra. This structure mirrors that of the large't Hooft coupling expansion of the S-matrix for the spin chain in the dual super-Yang-Mills theory.

Worldsheet scattering in AdS(5) x S**5

We review the duality and integrability of $${\fancyscript{N}\,{=}\,6}$$
 superconformal Chern–Simons theory in three dimensions and IIA superstring theory on the background Ads4 × CP3. We introduce both of these models and describe how their degrees of freedom are mapped to excitations of a long-range integrable spin-chain. Finally, we discuss the properties of the Bethe equations, the S-matrix and the algebraic curve that are special to this correspondence and differ from the case of $${\fancyscript{N}\,{=}\,4}$$
 SYM theory and strings on Ads5 × S
 5.

Review of AdS/CFT Integrability, Chapter IV.3: $${\mathcal{N}=6}$$ Chern–Simons and Strings on AdS4 × CP3

We compute the exact S-matrix and give the Bethe ansatz solution for three sigma-models which arise as subsectors of string theory in AdS(5)xS(5): Landau-Lifshitz model (non-relativistic sigma-model on S(2)), Alday-Arutyunov-Frolov model (fermionic sigma-model with su(1|1) symmetry), and Faddeev-Reshetikhin model (string sigma-model on S(3)xR).

Bethe ansatz in stringy sigma models

We use group theoretic methods to calculate the spectrum of short multiplets around the extremum of = 8 gauged supergravity potential which possesses = 2 supersymmetry and SU(3) global symmetry. Upon uplifting to M-theory, it describes a warped product of AdS4 and a certain squashed and stretched 7-sphere. We find quantum numbers in agreement with those of the gauge invariant operators in the = 2 superconformal Chern-Simons theory recently proposed to be the dual of this M-theory background. This theory is obtained from the U(N) × U(N) theory through deforming the superpotential by a term quadratic in one of the superfields. To construct this model explicitly, one needs to employ monopole operators whose complete understanding is still lacking. However, for the U(2) × U(2) gauge theory we make a proposal for the form of the monopole operators which has a number of desired properties. In particular, this proposal implies enhanced symmetry of the U(2) × U(2) ABJM theory for k = 1,2; it makes its similarity to and subtle difference from the BLG theory quite explicit.

http://iopscience.iop.org/article/10.1088/1126-6708/2009/03/140/pdf

Thomas Klose

Papers

Superconformal Chern-Simons Theories and AdS(4)/CFT(3) Correspondence

Worldsheet scattering in AdS(5) x S**5

Review of AdS/CFT Integrability, Chapter IV.3: $${\mathcal{N}=6}$$ Chern–Simons and Strings on AdS4 × CP3

Bethe ansatz in stringy sigma models

AdS 4 /CFT 3 squashed, stretched and warped