Shlomo Sternberg

Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras

Abstract This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite-dimensional Clifford algebras and their spin representations. We find that in the infinite-dimensional case, the analog of the finite-dimensional construction of Lie algebra cohomology breaks down, the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes.

Coadjoint structures, solitions, and integrability

In this paper we explain some recent results relating the geometry of the dual space of a Lie algebra to the complete integrability of certain non-linear partial differential equations. Our main purpose is to give a more or less self contained exposition of the recent results of Adler I and Lebedev-Manin 25 on the Poisson structures associated with non linear evolution equations. We briefly recall the background to this problem. By an evolution equation we mean a partial differential equation of the type

The interaction of spin and torsion. II: The principle of general covariance

Abstract The principle of general covariance, as formulated by V. Guillemin and S. Sternberg, is used to derive the passive equations of motion for a spinning matter field in the presence of an external gravitational field with torsion. In the case where the matter field becomes concentrated along a curve, the equations of motion for a spinning particle studied by D. Rappaport and Sternberg are recovered. The results of A. Einstein, L. Infeld, and B. Hoffmann and of J. M. Souriau are thus generalized to include spin and torsion.

Duality, crossing and MacLane’s coherence

It is shown that MacLane’ rectangle, pentagon and hexagon identities in category theory, when applied in particle physics to duality diagrams or to rational conformal field theories in two dimensions, yield the necessary physical algebraic constraints.

Magnetic moments and general covariance

The principle of general covariance for deriving the passive equations is applied to the group of automorphisms of compact support of a principal bundle acting on the space of Cartan connections. Under an orbital constraint this yields the equations of motion postulated in [11] which have a symplectic character. When specialized to an affine connection these equations become the equations of interaction of spin and torsion studied in [5]. With the incorporation of a Higgs field, a forcing term is added to the equations which, in the case of an electromagnetic field, incorporate the effects of the magnetic moment. The Higgs field is of the character obtained by dimensional reduction.

Sequential internal supersymmetry

The supergroups SU(2/1), SU(5/1) and SU(5 = k/1) provide fitting classifications, unifying weak‐electromagnetic, color, and sequential flavor respectively.