arXiv subjects
Publications and source records attributed to Lars Brink.
In this article I describe my own stumblings in the first string era. This was a time when most of the active people were very young, not very knowledgeable and the field was completely new. Many of us had little training for what we came to work on, and it took quite some time to accomplish the new conceptual discoveries.
We conjecture that the light-cone Hamiltonian of N=8 Supergravity can be expressed as a quadratic form. We explain why this rewriting is unique to maximally supersymmetric theories. The N=8 quartic interaction vertex is constructed and used to verify that this conjecture holds to second order in the coupling constant.
The symmetries of the N=4 SuperYang-Mills theory on the light-cone are discussed, solely in terms of its physical degrees of freedom. We derive explicit expressions for the generators of the PSU(2,2|4) superalgebra, both in the free theory, and to all orders in the gauge coupling of the classical theory. We use these symmetries to construct its Hamiltonian, and show that it can be written as a quadratic form of a fermionic superfield.
Eugene Wigner showed already in 1939 that the elementary particles are related to the irreducible representations of the Poincare algebra. In the light-cone frame formulation of quantum field theory one can extend these representations to depend also on a coupling constant. The representations then become non-linear and contain the interaction terms which are shown to have strong uniqueness. Extending the algebra to supersymmetry it is shown that two field theories stick out, N=4 Yang-Mills and N=8 Supergravity and their higher dimensional analogues. I also discuss string theory from this starting point.
We show that Supergravity in eleven dimensions can be described in terms of a constrained superfield on the light-cone, without the use of auxiliary fields. We build its action to first order in the gravitational coupling constant κ, by "oxidizing" (N=8,d=4) Supergravity. This is simply achieved, as for N=4 Yang-Mills, by extending the transverse derivatives into superspace. The eleven-dimensional SuperPoincare algebra is constructed and a fourth order interaction is conjectured.
We introduce superspace generalizations of the transverse derivatives to rewrite the four-dimensional N=4 Yang-Mills theory into the fully ten-dimensional N=1 Yang-Mills in light-cone form. The explicit SuperPoincare algebra is constructed and invariance of the ten-dimensional action is proved.
The history of anticommuting coordinates is decribed.
We construct Wigner's continuous spin representations of the Poincaré algebra for massless particles in higher dimensions. The states are labeled both by the length of a space-like translation vector and the Dynkin indices of the {\it short little group} $SO(d-3)$, where $d$ is the space-time dimension. Continuous spin representations are in one-to-one correspondence with representations of the short little group. We also demonstrate how combinations of the bosonic and fermionic representations form supermultiplets of the super-Poincaré algebra. If the light-cone translations are nilpotent, these representations become finite dimensional, but contain zero or negative norm states, and their supersymmetry algebra contains a central charge in four dimensions.
In a preceding report (hep-th/0112261), Pierre Ramond outlined a programme we have followed recently. An important ingredient in a future unique theory for basic physics must be a unique mathematical structure. It is then interesting to investigate the underlying mathematical structure of 11-dimensional space-time. A search for exceptional group structures specific to eleven dimensions leads us then to a study of the coset F_4/SO(9),which may provide a generalization of eleven-dimensional supergravity. This scheme is more general though and in this report we will show the corresponding structure related to the coset SU(3)/SU(2) x U(1), where we interpret the U(1) as the helicity group in four dimensions.
After a brief historical survey that emphasizes the role of the algebra obeyed by the Dirac operator, we examine an algebraic Dirac operator associated with Lie algebras and Lie algebra cosets. For symmetric cosets, its ``massless'' solutions display non-relativistic supersymmetry, and can be identified with the massless degrees of freedom of some supersymmetric theories: N=1 supergravity in eleven dimensions (M-theory), type IIB string theory in ten and four dimensions, and in four dimensions, N=8 supergravity, N=4 super-Yang-Mills, and the N=1 Wess-Zumino multiplet. By generalizing this Dirac operator to the affine case, we generate superconformal algebras associated with cosets ${\bf g}/\bf h$, where $\bf h$ contains the {\it space} little group. Only for eleven dimensional supergravity is $\bf h$ simple. This suggests, albeit in a non-relativistic setting, that these may be the limit of theories with underlying two-dimensional superconformal structure.
We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the A_N, BC_N, B_N, C_N and D_N Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed -- for arbitrary values of the coupling constants -- as quadratic polynomials in the generators of a Borel subalgebra of the Lie algebra gl(N+1) or the Lie superalgebra gl(N+1|N) for the supersymmetric case. These algebras are realized by first order differential operators. This fact establishes the exact solvability of the models according to the general definition given by one of the authors in 1994, and implies that the Calogero and Jack-Sutherland polynomials, as well as their supersymmetric generalizations, are related to finite-dimensional irreducible representations of the Lie algebra gl(N+1) and the Lie superalgebra gl(N+1|N).
In this paper we introduce a unified approach to Toda field theories which allows us to formulate the classes of $A_n$, $B_n$ and $C_n$ models as unique models involving an arbitrary continuous parameter $ν$. For certain values of $ν$, the model describes the standard Toda theories. For other values of $ν$ it defines a class of models that involve infinitely many fields. These models interpolate between the various standard Toda field theories. They are conformally invariant and possess infinitely many conserved higher-spin currents thus making them candidates for a new set of integrable systems. A general construction is performed, which can effectively be used for the derivation of explicit forms of particular higher-spin currents. We also study the currents in a different representation in which they are linear in the dynamical variables having, however, a non-linear Poisson bracket algebra. An explicit formula for this Poisson structure is found.
We discuss lowest-weight representations of the $S_N$-Extended Heisenberg Algebras underlying the $N$-body quantum-mechanical Calogero model. Our construction leads to flat derivatives interpolating between Knizhnik-Zamolodchikov and Dunkl derivatives. It is argued that based on these results one can establish new links between solutions of the Knizhnik-Zamolodchikov equations and wave functions of the Calogero model.
The superstring in $D\!=\!3,4$ and 6 is invariant under an $N\!=\!D\!-\!2$ superconformal algebra based on $\widehat{S^{D-3}}$. There is a direct relationship between this (world-sheet) symmetry and the super-Poincaré (target space) symmetry. We establish this relationship using the light-cone gauge, show how the statement generalizes to $D\!=\!10$ and examine the properties of the $N\!=\!8$ superconformal algebra and the possible implications of its existence.