Christiane Schomblond

Characteristic cohomology of p-form gauge theories

The characteristic cohomologyHkchar(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.

Hyperbolic Kac Moody algebras and Einstein billiards

We identify the hyperbolic Kac Moody algebras for which there exists a Lagrangian of gravity, dilatons, and p-forms which produces a billiard that can be identified with their fundamental Weyl chamber. Because of the invariance of the billiard upon toroidal dimensional reduction, the list of admissible algebras is determined by the existence of a Lagrangian in three space–time dimensions, where a systematic analysis can be carried out since only zero-forms are involved. We provide all highest dimensional parent Lagrangians with their full spectrum of p-forms and dilaton couplings. We confirm, in particular, that for the rank 10 hyperbolic algebra, CE10=A15(2)∧, also known as the dual of B8∧∧, the maximally oxidized Lagrangian is nine-dimensional and involves besides gravity, 2 dilatons, a 2-form, a 1-form, and a 0-form.We identify the hyperbolic Kac Moody algebras for which there exists a Lagrangian of gravity, dilatons, and p-forms which produces a billiard that can be identified with their fundamental Weyl chamber. Because of the invariance of the billiard upon toroidal dimensional reduction, the list of admissible algebras is determined by the existence of a Lagrangian in three space–time dimensions, where a systematic analysis can be carried out since only zero-forms are involved. We provide all highest dimensional parent Lagrangians with their full spectrum of p-forms and dilaton couplings. We confirm, in particular, that for the rank 10 hyperbolic algebra, CE10=A15(2)∧, also known as the dual of B8∧∧, the maximally oxidized Lagrangian is nine-dimensional and involves besides gravity, 2 dilatons, a 2-form, a 1-form, and a 0-form.

Einstein billiards and spatially homogeneous cosmological models

In this paper, we analyse the Einstein and Einstein–Maxwell billiards for all spatially homogeneous cosmological models corresponding to three- and four-dimensional real unimodular Lie algebras and provide a list of those models which are chaotic in the Belinskii, Khalatnikov and Lifschitz (BKL) limit. Through the billiard picture, we confirm that, in D = 5 spacetime dimensions, chaos is present if off-diagonal metric elements are kept: the finite volume billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac–Moody algebras. The most generic cases bring in the same algebras as in the inhomogeneous case, but other algebras appear through special initial conditions.

Consistent elimination of redundant second-class constraints

Abstract An elementary criterium for the consistent elimination of redundant second-class constraints is given. Failure to fulfil this criterium leads to inconsistencies which are pointed out. A naive covariant separation of first-class and second-class constraints for the superstring does not meet the criterium and hence is not yet acceptable. Similar considerations apply also to the supermenbrame.

Strong coupling limit of gφ4 theory: General formalism and applications

Abstract A method, independently proposed by Kaiser and by us, to study the strong coupling limit of the Green functions is described and discussed. Although its quantum mechanical version exhibits unusual features, an application to the anharmonic oscillator indicates that the method is able to reproduce correctly known numerical results. In spite of difficulties in the setting up of a renormalization program for the theory, a preliminary study of the CS β function for the gφ4 interaction shows that β(g r is asymptotically linear in the renormalization coupling constant. Evidence is given for the compatibility of this behaviour with the information that can be drawn from the known perturbative expansion.

Strong coupling limit of the gφ4 theory

Abstract The strong coupling limit of the gφ4 theory in the framework of the path integral formalism. An expansion of the Greens functions in negative powers of the coupling constant is obtained; at each order the dependence on the external momenta is of polynomial type. A renormalization procedure is proposed; the asymptotic behaviour of the Callan-Symanzik β function is studied and the existence of a stable ultraviolet fixed point is established.

Results on the Wess-Zumino consistency condition for arbitrary Lie algebras

The so-called covariant Poincare lemma on the induced cohomology of the space–time exterior derivative in the cohomology of the gauge part of the BRST differential is extended to cover the case of arbitrary, nonreductive Lie algebras. As a consequence, the general solution of the Wess–Zumino consistency condition with a nontrivial descent can, for arbitrary (super) Lie algebras, be computed in the small algebra of the one-form potentials, the ghosts and their exterior derivatives. For particular Lie algebras that are the semidirect sum of a semisimple Lie subalgebra with an ideal, a theorem by Hochschild and Serre is used to characterize more precisely the cohomology of the gauge part of the BRST differential in the small algebra. In the case of an Abelian ideal, this leads to a complete solution of the Wess–Zumino consistency condition in this space. As an application, the consistent deformations of 2+1 dimensional Chern–Simons theory based on iso(2,1) are rediscussed.

BRST Cohomology of the Chapline–Manton Model

We completely compute the local BRST cohomology H(s|d) of thecombined Yang–Mills 2-form system coupled through theYang–Mills Chern–Simons term (Chapline–Manton model). Weconsider the case of a simple gauge group and explicitly include in the analysis the sources for theBRST variations of the fields (antifields).We show that there is an antifield independentrepresentative in each cohomological class of H(s|d) at ghost number 0 or 1. Accordingly, any counterterm may beassumed to preserve the gauge symmetries. Similarly,there is no new candidate anomaly beside those alreadyconsidered in the literature, evenwhen one takes the antifields into account.We then characterize explicitly all the nontrivialsolutions of the Wess–Zumino consistencyconditions. In particular, we providea cohomological interpretation of the Green–Schwarzanomaly cancellation mechanism.

Couplings of gravity to antisymmetric gauge fields

Abstract We classify all the first-order vertices of gravity consistently coupled to a system of 2-form gauge fields by computing the local BRST cohomology H ( s | d ) in ghost number 0 and form degree n . The consistent deformations are at most linear in the undifferentiated two-form, confirming the previous results of Damour et al. [Phys. Rev. D 45 (1992) 3289; D 47 (1993) 1541] that geometrical theories constructed from a nonsymmetric gravity theory are physically inconsistent or trivial. No assumption is made here on the degree of homogeneity in the derivatives nor on the form of the gravity action.