Higher spin BRS cohomology of supersymmetric chiral matter inD=4

We examine the BRS cohomology of chiral matter inN=1,D=4 supersymmetry to determine a general form of composite superfield operators which can suffer from supersymmetry anomalies. Composite superfield operators _{(a, b)} are products of the elementary chiral superfieldsS and and the derivative operatorsD , and . Such superfields _{(a, b)} can be chosen to have a symmetrized undotted indices _i and b symmetrized dotted indices . The result derived here is that each composite superfield _(a,b) is subject to potential supersymmetry anomalies ifa–b is an odd number, which means that _(a,b) is a fermionic superfield.     

BRS cohomology and topological anomalies

The occurrence of non-abelian anomalies in gauge theories and gravitation, first discovered via perturbative techniques, is now completely explained from the mathematical point of view by means of the family index theorem of Atiyah and Singer. Here we make contact between this approach and BRS cohomology, by showing that they yield the same non-abelian anomalies, provided a certain restriction to local functionals is not introduced from the very beginning. In particular, this solves the unicity problem for this kind of anomalies. Local BRS cohomology is still relevant for the abelian case.Work partially supported by Gruppo Nazionale di Fisica Matematica del CNR and Progetto Nazionale Geometria e Fisica del MPI     

BRS cohomology and Casimir operators

Casimir operators play a central role in the study of cohomology problems for semisimple Lie algebras. An attempt is made to generalize this to strings. This generalization may be useful for studying small oscillations around nontrivial backgrounds.     

BRS cohomology of the supertranslations inD=4

Supersymmetry transformations are a kind of square root of spacetime translations. The corresponding Lie superalgebra always contains the supertranslation operator . We find that the cohomology of this operator depends on a spin-orbit coupling in anSU(2) group and has a quite complicated structure. This spin-orbit type coupling will turn out to be basic in the cohomology of supersymmetric field theories in general.     

Calculation of BRS cohomology with spectral sequences

A method for finding the general form of the BRS cohomology spaceH for the various gauge and supersymmetry theories is presented. The method is adapted for use in the space of integrated local polynomials of the gauge fields and ghosts with arbitrary numbers of fields and dervivatives. The technique uses the Hodge decomposition in a Fock space with a Euclidean inner product, and combines this with spectral sequences to generate simple and soluble equations whose solutions span a simple spaceE isomorphic to the complicated spaceH. The technique is illustrated for pedagogic purposes by the detailed calculation of the ghost charge zero and one sectors ofH for Yang-Mills theory with gauge groupSO (32) in ten dimensions. The method is appropriate for supersymmetric theories, gravity, supergravity and superstrings where higher order terms with many derivatives occur naturally in the effective action.Research supported in part by the Robert A. Welch Foundation and NSF Grants PHY 77-18762 and PHY 9009850     

BRS cohomology in string theory and the no-ghost theorem

It is shown that the operator counting the number of non-transverse modes of the bosonic string in 26 dimensions can be expressed as the anticommutator of the BRS charge Q with another operator. As a result it is easy to exhibit the cohomology of Q and express the transverse state projection operator of Brink and Olive in terms of Q.     

Remarks on the cohomology of the BRS operator in string theory

The cohomology of the open bosonic string BRS operator is computed in a physical fashion. The method can be applied to the closed and to the fermionic models.     

Superfield effective action within the general chiral superfield model

The chiral superfield model associated with the low-energy limit of superstring theory and characterized by the Kähler and the chiral potential [K(Φ, $\overline \Phi$ ) and W(Φ), respectively] is analyzed. An approach to solving a general problem is developed, and quantum loop corrections at arbitrary K(Φ, $\overline \Phi$ ) and W(Φ) are found. Various aspects of a supergraph technique that are associated with calculating perturbative contributions to the superfield effective action are analyzed. Explicit expressions for the one-and two-loop corrections to the Kähler potential are calculated. The leading two-loop correction to the chiral potential is obtained, and it is shown that, irrespective of the form of K(Φ, $\overline \Phi$ ) and W(Φ), counterterms are not needed for deducing this correction.     

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

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.     

One-loop effective action in three-dimensional general chiral superfield model

Using the heat kernel approach we calculate one-loop Kähler effective potential for the general chiral superfield model in three dimensions.     

Some remarks on BRS transformations,anomalies and the cohomology of the Lie algebra of the group of gauge transformations

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ? of gauge transformations. We examine the cohomology of the Lie algebra of ? and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.     

Variant superfield representations

We discuss some variant superfield representations which can arise by the replacement of some of the usual fields in a multiplet with p-form gauge fields.     

Representations of BRS algebra

In the canonical formulation of gauge theories the BRS transformation plays a fundamental rôle. The generator of this transformation along with the ghost number forms an algebra called the BRS algebra. Certain properties of this algebra are essential to the proof of unitarity of the S matrix in the physical sector and also to the discussion of color confinement.In the present paper we present all the possible representations of the BRS algebra in the light of indefinite metric.