Lars Brink

N=4 Yang-Mills and N=8 Supergravity as Limits of String Theories

The formulation of supersymmetric string theories in ten dimensions is generalized to incorporate compactified dimensions. Expressions for the one-loop four-particle S -matrix elements of N = 4 Yang-Mills and N = 8 supergravity in four dimensions are obtained by studying the string-theory loop amplitudes in the limit that the radii of the compactified dimensions and the Regge slope parameter simultaneously approach zero. If certain patterns that emerge should persist in the higher orders of perturbation theory, then N = 4 Yang-Mills in four dimensions would be ultraviolet finite to all orders, whereas N = 8 supergravity in four dimensions would have ultraviolet divergences starting at three loops.

N=4 Yang-Mills Theory on the Light Cone

Abstract The 10-dimensional supersymmetric Yang-Mills theory is constructed in the light-cone gauge. When the theory is dimensionally reduced to four dimensions it is shown that the corresponding N = 4 theory is conveniently described in terms of a scalar superfield. This formalism avoids the problem of auxiliary fields but is Lorentz invariant only on the mass shell. Similar formalisms in terms of scalar superfields are also sketched for the other supersymmetric Yang-Mills as well as for N = 8 supergravity.

Cubic interaction terms for arbitrary spin

Abstract Using the light-cone formulation of dynamics, we construct cubic self-interaction terms for massless fields of arbitrary helicity λ. The coupling constants have dimension [length]λ−1 and there is a gauge group for any odd λ.

Cubic interaction terms for arbitrarily extended supermultiplets

Abstract Using a light-cone superspace formalism we construct cubic interaction terms for all maximally extended supersymmetry multiplets where the maximum helicity λ is an integer. The coupling constants have dimension [length]λ−1 and there is a gauge group for any odd λ.

Superfield theory of type (II) superstrings

Abstract Type (II) superstrings are described by light-cone superfields that are functionals of string superspace coordinates x i ( σ ) and ϑ a ( σ ) or the conjugate momenta p i ( σ ) and λ a ( σ ). The interaction hamiltonian and supersymmetry charges are determined to first order in the coupling constant by requiring invariance under the super-Poincare algebra, some of whose generators are implemented locally in σ. In a suitable low-energy limit the formalism goes smoothly into that of extended supergravity in ten dimensions.

How massless are massless fields in AdS d

Massless fields of generic Young symmetry type in AdSd space are analyzed. It is demonstrated that in contrast to massless fields in Minkowski space whose physical degrees of freedom transform in irreps of o(d−2) algebra, AdS massless mixed symmetry fields reduce to a number of irreps of o(d −2) algebra. From the field theory perspective this means that not every massless field in flat space admits a deformation to AdSd with the same number of degrees of freedom, because it is impossible to keep all of the flat space gauge symmetries unbroken in the AdS space. An equivalent statement is that, generic irreducible AdS massless fields reduce to certain reducible sets of massless fields in the flat limit. A conjecture on the general pattern of the flat space limit of a general AdSd massless field is made. The example of the three-cell “hook” Young diagram is discussed in detail. In particular, it is shown that only a combination of the three-cell flat-space field with a graviton-like field admits a smooth deformation to AdSd.

Explicit solution to the N-body Calogero problem

Abstract We solve the N-body Calogero problem, i.e., N particles in one dimension subject to a two-body interaction of the form 1 2 Σ i,j [(x i −x j ) 2 + g (x i −x j ) 2 ] , by constructing annihilation and creation operators of the form ⊣ 1 = ( 1 √2 )(x i ± i p i ) , where p i is a modified momentum operator obeying Heisenberg-type commutation relations with xi, involving explicitly permutation operators. On the other hand, D j = i p j can be interpreted as a covariant derivative corresponding to a flat connection. The relation to fractional statistics in 1+1 dimensions and anyons in a strong magnetic field is briefly discussed.

Eleven-Dimensional Supergravity on the Mass-Shell in Superspace

Abstract The eleven-dimensional supergravity theory is presented in superspace on the mass shell.

The Calogero model — anyonic representation, fermionic extension and supersymmetry

Abstract We discuss several applications and extensions of our previous operator solution of the N-body quantum-mechanical Calogero problem, i.e. N particles in one dimension subject to a two-body interaction of the form 1 2 Σ i,j [(x i − x j ) 2 + g/(x i − x j ) 2 ] . Using a complex representation of the deformed Heisenberg algebra underlying the Calogero model, we explicitly establish the equivalence between this system and anyons in the lowest Landau level. A construction based on supersymmetry is used to extend our operator method to include fermions, and we obtain an explicit solution of the supersymmetric Calogero model constructed by Freedman and Mende. We also show how the dynamical OSp(2; 2) supersymmetry is realized by bilinears of modified creation and annihilation operators, and how to construct a supersymmetic extension of the deformed Heisenberg algebra.

The N = 8 supergravity in superspace

Abstract The N =8 supergravity is presented in superspace. The theory is on-shell and has the expected local SU(8) and global E 7 invariances.