Abstract:
The geometric interpretation of the antibracket formalism given by Witten is extended to cover the anti-BRST symmetry. This enables one to formulate the quantum master equation for the BRST--anti-BRST formalism in terms of integration theory over a supermanifold. A proof of the equivalence of the standard antibracket formalism with the antibracket formalism for the BRST--anti-BRST symmetry is also given.

Abstract:
The antifield-BRST formalism and the various cohomologies associated with it are surveyed and illustrated in the context of Yang-Mills gauge theory. In particular, the central role played by the Koszul-Tate resolution and its relation to the characteristic cohomology are stressed.

Abstract:
Let $B_{\mu \nu }^a$ ($a=1,...,N$) be a system of $N$ free two-form gauge fields, with field strengths $H_{\mu \nu \rho }^a = 3 \partial _{[\mu }B_{\nu \rho ]}^a$ and free action $S_0$ equal to $(-1/12)\int d^nx\ g_{ab}H_{\mu \nu \rho }^aH^{b\mu \nu \rho }$ ($n\geq 4$). It is shown that in $n>4$ dimensions, the only consistent local interactions that can be added to the free action are given by functions of the field strength components and their derivatives (and the Chern-Simons forms in $5$ mod $3$ dimensions). These interactions do not modify the gauge invariance $B_{\mu \nu }^a\rightarrow B_{\mu \nu }^a+\partial _{[\mu }\Lambda _{\nu ]}$ of the free theory. By contrast, there exist in $n=4$ dimensions consistent interactions that deform the gauge symmetry of the free theory in a non trivial way. These consistent interactions are uniquely given by the well-known Freedman-Townsend vertex. The method of proof uses the cohomological techniques developed recently in the Yang-Mills context to establish theorems on the structure of renormalized gauge-invariant operators.

Abstract:
A crucial property of the standard antifield-BRST cohomology at non negative ghost number is that any cohomological class is completely determined by its antifield independent part. In particular, a BRST cocycle that vanishes when the antifields are set equal to zero is necessarily exact.\ \ This property, which follows from the standard theorems of homological perturbation theory, holds not only in the algebra of local functions, but also in the space of local functionals. The present paper stresses how important it is that the antifields in question be the usual antifields associated with the gauge invariant description. By means of explicit counterexamples drawn from the free Maxwell-Klein-Gordon system, we show that the property does not hold, in the case of local functionals, if one replaces the antifields of the gauge invariant description by new antifields adapted to the gauge fixation. In terms of these new antifields, it is not true that a local functional weakly annihilated by the gauge-fixed BRST generator determines a BRST cocycle; nor that a BRST cocycle which vanishes when the antifields are set equal to zero is necessarily exact.

Abstract:
The requirement that the action be stationary for solutions of the Dirac equations in anti-de Sitter space with a definite asymptotic behaviour is shown to fix the boundary term (with its coefficient) that must be added to the standard Dirac action in the AdS/CFT correspondence for spinor fields.

Abstract:
The (perturbative) renormalization properties of the BF formulation of Yang-Mills gauge models are shown to be identical to those of the usual, second order formulation. This result holds in any number of spacetime dimensions and is a direct consequence of cohomological theorems established by G. Barnich, F. Brandt and the author (Commun.Math.Phys., 174 (1995) 57).

Abstract:
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.

Abstract:
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.

Abstract:
We analyze carefully the problem of gauge symmetries for Bianchi models, from both the geometrical and dynamical points of view. Some of the geometrical definitions of gauge symmetries (=``homogeneity preserving diffeomorphisms'') given in the literature do not incorporate the crucial feature that local gauge transformations should be independent at each point of the manifold of the independent variables ( = time for Bianchi models), i.e, should be arbitrarily localizable ( in time). We give a geometrical definition of homogeneity preserving diffeomorphisms that does not possess this shortcoming. The proposed definition has the futher advantage of coinciding with the dynamical definition based on the invariance of the action ( in Lagrangian or Hamiltonian form). We explicitly verify the equivalence of the Lagrangian covariant phase space with the Hamiltonian reduced phase space. Remarks on the use of the Ashtekar variables in Bianchi models are also given.

Abstract:
We compute the billiards that emerge in the Belinskii-Khalatnikov-Lifshitz (BKL) limit for all pure supergravities in D=4 spacetime dimensions, as well as for D=4, N=4 supergravities coupled to k (N=4) Maxwell supermultiplets. We find that just as for the cases N=0 and N=8 investigated previously, these billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. Hence, the dynamics is chaotic in the BKL limit. A new feature arises, however, which is that the relevant Kac-Moody algebra can be the Lorentzian extension of a twisted affine Kac-Moody algebra, while the N=0 and N=8 cases are untwisted. This occurs for N=5, N=3 and N=2. An understanding of this property is provided by showing that the data relevant for determining the billiards are the restricted root system and the maximal split subalgebra of the finite-dimensional real symmetry algebra characterizing the toroidal reduction to D=3 spacetime dimensions. To summarize: split symmetry controls chaos.