Jim Stasheff

Introduction to SH Lie algebras for physicists

UNC-MATH-92/2originally April 27, 1990, revised September 24, 1992INTRODUCTION TO SH LIE ALGEBRAS FOR PHYSICISTSTom LadaJim StasheffMuch of point particle physics can be described in terms of Lie algebras andtheir representations. Closed string field theory, on the other hand, leads to ageneralization of Lie algebra which arose naturally within mathematics in the studyof deformations of algebraic structures [SS]. It also appeared in work on higherspin particles [BBvD]. Representation theoretic analogs arose in the mathematicalanalysis of the Batalin-Fradkin-Vilkovisky approach to constrained Hamiltonians[S6].The sh Lie algebra of closed string field theory [SZ], [KKS], [K], [Wies], [WZ],[Z] is defined on the full Fock complex of the theory, with the BRST differential Q.Following Zwiebach [Z], we stipulate that the string fields B

Obstructions to homotopy equivalences

Abstract An obstruction theory is developed to decide when an isomorphism of rational cohomology can be realized by a rational homotopy equivalence (either between rationally nilpotent spaces, or between commutative graded differential algebras). This is used to show that a cohomology isomorphism can be so realized whenever it can be realized over some field extension (a result obtained independently by Sullivan). In particular an algorithmic method is given to decide when a c.g.d.a. has the same homotopy type as its cohomology (the c.g.d.a. is called formal in this case). The chief technique is the construction of a canonically filtered model for a commutative graded differential algebra (over a field of characteristic zero) by perturbing the minimal model for the cohomology algebra. This filtered model is also used to give a simple construction of the Eilenberg-Moore spectral sequence arising from the bar construction. An example is given of a c.g.d.a. whose Eilenberg-Moore sequence collapses, yet which is not formal.

The Lie algebra structure of tangent cohomology and deformation theory

Abstract Tangent cohomology of a commutative algebra is known to have the structure of a graded Lie algebra; we account for this by exhibiting a differential graded Lie algebra (in fact, two of them) equivalent as cochain complex to Harrisons yielding the tangent cohomology. This d.g. Lie algebra, called the tangent Lie algebra, also provides an interpretation of the cohomology in terms of perturbations of multiplicative resolutions and hence clarifies the relation to deformation theory. In particular, the higher order obstructions to deformations appear as Massey-Lie brackets. Moreover, we obtain homological constructions for the base and total spaces of a versal deformation.

Existence, uniqueness and cohomology of the classical BRST charge with ghosts of ghosts

A complete canonical formulation of the BRST theory of systems with redundant gauge symmetries is presented. These systems includep-form gauge fields, the superparticle, and the superstring. We first define the Koszul-Tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic. The global existence of the BRST generator is then demonstrated, and the BRST charge is proved to be unique up to canonical transformations in the extended phase space, which includes the ghosts. Finally, the BRST cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits, as in the irreducible case. This is done by re-expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree, which are identified with the ghosts of ghosts. The quantum cohomology is not dealt with.

On operad structures of moduli spaces and string theory

We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory.

The intrinsic bracket on the deformation complex of an associative algebra

In his pioneering work on deformation theory of associative algebras, Gerstenhaber created a bracket on the Hochschild cohomology Hoch(A,A), but this bracket seemed to be rather a tour de force since it was not induced from a differential graded Lie algebra structure on the underlying complex. Schlessinger and Stasheff constructed a differential graded Lie algebra structure on a complex giving the Harrison cohomology Harr(A,A) of a commutative algebra A in characteristic 0. Here we present a differential graded Lie algebra structure on a complex giving the Hochschild cohomology Hoch(A,A) and inducing the Gerstenhaber bracket for any associative algebra in any characteristic. Although the principal is the same as in the commutative case, the details as well as the essential idea will hopefully be revealed more transparently.

Solutions to {Yang-Mills} Field Equations in Eight-dimensions and the Last Hopf Map

AbstractWe will show that the Hopf map

Invitation to composition

S^{15} \xrightarrow{{S^7 }}S^8