Consistent Interactions Between Gauge Fields: The Cohomological Approach

The cohomological approach to the problem of consistent interactions between fields with a gauge freedom is reviewed. The role played by the BRST symmetry is explained. Applications to massless vector fields and 2-form gauge fields are surveyed.Comment: late

Ghosts of ghosts for second class constraints

When one uses the Dirac bracket, second class constraints become first class. Hence, they are amenable to the BRST treatment characteristic of ordinary first class constraints. This observation is the starting point of a recent investigation by Batalin and Tyutin, in which all the constraints are put on the same footing. However, because second class constraints identically vanish as operators in the quantum theory, they are quantum-mechanically reducible and require therefore ghosts of ghosts. Otherwise, the BRST cohomology would not yield the correct physical spectrum. We discuss how to incorporate this feature in the formalism and show that it leads to an infinite tower of ghosts of ghosts. An alternative treatment, in which the brackets of the ghosts are modified, is also mentioned.Comment: 7 pages in LaTex, ULB-PMIF/93-0

Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket

One may introduce at least three different Lie algebras in any Lagrangian field theory : (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin-Vilkovisky antibracket, which has attracted considerable interest recently~; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket~; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number $-1$ and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number $-1$, while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from $-1$, a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin and Vilkovisky.Comment: 36 pages Latex fil

The Action for Twisted Self-Duality

One may write the Maxwell equations in terms of two gauge potentials, one electric and one magnetic, by demanding that their field strengths should be dual to each other. This requirement is the condition of twisted self-duality. It can be extended to p-forms in spacetime of D dimensions, and it survives the introduction of a variety of couplings among forms of different rank, and also to spinor and scalar fields, which emerge naturally from supergravity. In this paper we provide a systematic derivation of the action principle, whose equations of motion are the condition of twisted self-duality. The derivation starts from the standard Maxwell action, extended to include the aforementioned couplings, and proceeds via the Hamiltonian formalism through the resolution of Gauss' law. In the pure Maxwell case we recover in this way an action that had been postulated by other authors, through an ansatz based on an action given earlier by us for untwisted self-duality. Those authors also extended their ansatz to include Chern-Simons couplings. In that case, we find a different result. The derivation from the standard extended Maxwell action implies of course that the theory is Lorentz-invariant and can be locally coupled to gravity. Nevertherless we include a direct compact Hamiltonian proof of these properties, which is based on the surface-deformation algebra. The symmetry in the dependence of the action on the electric and magnetic variables is manifest, since they appear as canonical conjugates. Spacetime covariance, although present, is not manifest.Comment: Version to appear in Phys. Rev.

Asymptotic symmetries of three-dimensional higher-spin gravity: the metric approach

The asymptotic structure of three-dimensional higher-spin anti-de Sitter gravity is analyzed in the metric approach, in which the fields are described by completely symmetric tensors and the dynamics is determined by the standard Einstein-Fronsdal action improved by higher order terms that secure gauge invariance. Precise boundary conditions are given on the fields. The asymptotic symmetries are computed and shown to form a non-linear W-algebra, in complete agreement with what was found in the Chern-Simons formulation. The W-symmetry generators are two-dimensional traceless and divergenceless rank-s symmetric tensor densities of weight s (s = 2, 3, ...), while asymptotic symmetries emerge at infinity through the conformal Killing vector and conformal Killing tensor equations on the two-dimensional boundary, the solution space of which is infinite-dimensional. For definiteness, only the spin 3 and spin 4 cases are considered, but these illustrate the features of the general case: emergence of the W-extended conformal structure, importance of the improvement terms in the action that maintain gauge invariance, necessity of the higher spin gauge transformations of the metric, role of field redefinitions.Comment: 74 pages. References amended and typos corrected. Version to appear in JHE

Kac-Moody and Borcherds Symmetries of Six-Dimensional Chiral Supergravity

We investigate the conjectured infinite-dimensional hidden symmetries of six-dimensional chiral supergravity coupled to two vector multiplets and two tensor multiplets, which is known to possess the $F_{4,4}$ symmetry upon dimensional reduction to three spacetime dimensions. Two things are done. (i) First, we analyze the geodesic equations on the coset space $F_{4,4}^{++}/K(F_{4,4}^{++})$ using the level decomposition associated with the subalgebra $\mathfrak{gl}(5)\oplus \mathfrak{sl}(2)$ of $F_{4,4}^{++}$ and show their equivalence with the bosonic equations of motion of six-dimensional chiral supergravity up to the level where the dual graviton appears. In particular, the self-duality condition on the chiral $2$-form is automatically implemented in the sense that no dual potential appears for that $2$-form, in contradistinction with what occurs for the non chiral $p$-forms. (ii) Second, we describe the $p$-form hierarchy of the model in terms of its $V$-duality Borcherds superalgebra, of which we compute the Cartan matrix.Comment: 31 pages. v2: Error in section 6.3 corrected, Dynkin diagram now appears correctly, minor typo

Oscillatory Behaviour in Homogeneous String Cosmology Models

Some spatially homogeneous Bianchi type I cosmological models filled with homogeneous ``electric'' $p$-form fields are shown to mimic the never-ending oscillatory behaviour of generic string cosmologies established recently. The validity of the ``Kasner-free-flights plus collisions-on-potential-walls'' picture is also illustrated in the case of known, non-chaotic, superstring solutions.Comment: 12 pages, latex, submitted to Phys. Lett.

Comments on Unitarity in the Antifield Formalism

It is shown that the local completeness condition introduced in the analysis of the locality of the gauge fixed action in the antifield formalism plays also a key role in the proof of unitarity.Comment: 15 pages, Latex error corrected, otherwise unchange

Timelike duality, M′M'-theory and an exotic form of the Englert solution

Through timelike dualities, one can generate exotic versions of $M$-theory with different spacetime signatures. These are the $M^*$-theory with signature $(9,2,-)$, the $M'$-theory, with signature $(6,5,+)$ and the theories with reversed signatures $(1,10, -)$, $(2,9, +)$ and $(5,6, -)$. In $(s,t, \pm)$, $s$ is the number of space directions, $t$ the number of time directions, and $\pm$ refers to the sign of the kinetic term of the $3$ form. The only irreducible pseudo-riemannian manifolds admitting absolute parallelism are, besides Lie groups, the seven-sphere $S^7 \equiv SO(8)/SO(7)$ and its pseudo-riemannian version $S^{3,4} \equiv SO(4,4)/SO(3,4)$. [There is also the complexification $SO(8,\mathbb{C})/SO(7, \mathbb{C})$, but it is of dimension too high for our considerations.] The seven-sphere $S^7\equiv S^{7,0}$ has been found to play an important role in $11$-dimensional supergravity, both through the Freund-Rubin solution and the Englert solution that uses its remarkable parallelizability to turn on non trivial internal fluxes. The spacetime manifold is in both cases $AdS_4 \times S^7$. We show that $S^{3,4}$ enjoys a similar role in $M'$-theory and construct the exotic form $AdS_4 \times S^{3,4}$ of the Englert solution, with non zero internal fluxes turned on. There is no analogous solution in $M^*$-theory.Comment: 18 pages, v2: typos fixe