Department of Physics, Theoretical Physics,
University of Oxford, 1 Keble Road,
Oxford, OX1 3NP United Kingdom (G. Luzón)
The relation between connections on 2-dimensional manifolds and holomorphic
bundles provides a new perspective on the role of classical gauge fields in
quantum field theory in two, three and four dimensions. In particular we show
that there is a close relation between unstable bundles and monopoles,
sphalerons and instantons. Some of these classical configurations emerge as
nodes of quantum vacuum states in nonconfining phases of
quantum field theory which suggests
a relevant role for those configurations in the mechanism of quark
confinement in QCD.
Institut für Theoretische Physik, Freie Universität
Berlin, Arnimallee 14, D-14195 Berlin, Germany
In the first part,
the sh Lie structure of brackets in field theory, described in the jet
bundle context along the lines suggested by Gel'fand, Dickey and
Dorfman, is analyzed.
In the second part, we discuss how this description allows us to
find a natural relation between the Batalin - Vilkovisky antibracket and
the Poisson bracket.
Istituto Nazionale di Fisica Nucleare, Sezione di Genova,
Via Dodecaneso, 33 I-16146 Genova, Italy (C. Imbimbo)
The BRST structure of twisted N = 2 superconformal matter coupled to
topological gravity is derived by gauging the rigid N = 2 superconformal
algebra. This construction provides BRST transformations laws for which
holomorphic factorization on the world-sheet is manifest.
International School for Advanced Studies (SISSA/ISAS)
Via Beirut 2 - 4, 34014 Trieste, Italy
Dipartimento di Matematica, Universitá di Trieste
P. le Europa 1, 34127 Trieste, Italy
We review some basic notions on anomalies in field theories and
superstring theories, with particular emphasis on the concept of locality.
The aim is to prepare the ground for a discussion on anomalies in theories
with branes. In this light we review the problem of chiral anomaly cancellation
in M-theory with a 5-brane.
The formulation of the local BRST cohomology on infinite jet bundles and its relation and reduction to gauge covariant algebras are reviewed. As an illustration, we compute the local BRST cohomology for geodesic motion in (pseudo-) Riemannian manifolds and discuss briefly the result (symmetries, constants of the motion, consistent deformations).
We discuss the generalized homology associated with a nilpotent endomorphism
d such that dN = 0. We construct such d on simplicial modules and rely
the corresponding generalized homologies to the usual simplicial ones. We
also investigate the generalization of graded differential algebras in this
Centre Automatique et Systèmes, École des Mines de Paris,
35 rue Saint-Honoré, 77305 Fontainebleau, France
E-mails: firstname.lastname@example.org, email@example.com
Centre Automatique et Systèmes, École des Mines de Paris,
60 bd. Saint-Michel, 75272 Paris Cedex 06, France
Problems of nonlinear control theory are considered in the context of
diffieties. As an application, the Dirac gauge theory is discussed.
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.
We show that any local analytic Lie pseudogroup of infinite type can be
endowed with a compatible Silva analytic manifold structure. The
compatibility condition means that the associated maximal isotropy Lie
group endowed with the induced topology becomes an analytic Lie group in
Milnor sense, i.e. an analytic manifold such that the group operations
are analytic. In that context the second fundamental theorem of Lie is
extended for the class of closed Lie subalgebras of the maximal isotropy
Lie algebra. So any closed connected subgroup of the isotropy group is a
Silva analytic Lie group. Moreover we prove that the group of local analytic
paths starting at the identity transformation in any Lie pseudogroup inherits
naturally of a Silva analytic Lie group structure in the previous sense.
Our approach treats the transitive and intransitive cases on the same footing
and our results are shown to be valid in the wider classes of quasi-analytic
transformations of Denjoy's or Gevrey's type. The Cartan - Kähler theorem is
notably shown to be valid in the quasi-analytic setting.
Our approach treats the transitive and intransitive cases on the same footing and our results are shown to be valid in the wider classes of quasi-analytic transformations of Denjoy's or Gevrey's type. The Cartan - Kähler theorem is notably shown to be valid in the quasi-analytic setting.
Using techniques of Frölicher - Nijenhuis brackets, we associate to any
formally integrable equation E a cohomology theory
HC*(E) (based on a C-complex) related to deformations
of the equation structure on
the infinite prolongation E¥. A subgroup in HC1(E) is
identified with recursion operators acting on the Lie algebra \symE
of symmetries. On the other hand, another subgroup of HC*(E)
can be understood as the algebra of supersymmetries of the ``superization''
of the equation E. This passing to superequations makes it possible to
obtain a well-defined action of recursion operators in a nonlocal setting.
Relations to Poisson structures on E¥ are briefly discussed.
The algebraic BRS method of renormalization
is applied to the supersymmetric Yang - Mills theories. The most general
invariant counterterms and the supersymmetric extension of the
Adler - Bell - Jackiw anomaly are explicitily found. In addition, masses
are added in a simple way, preserving gauge invariance to all orders
of perturbation theory while breaking supersymmetry ``softly'', in the
sense of Girardello and Grisaru.
Depto. de Matemáticas, Univ. Carlos III de Madrid, 28911
Leganés, Madrid, Spain.
Completely integrable systems are discussed in the realm of the general conjugacy
problem from the perspective offered by Lie - Scheffers theorem. Hamiltonian
and non-Hamiltonian standard completely integrable systems are briefly
reviewed as well as a natural generalization to the non-Abelian setting
suggested by the theory of double Lie groups.
Dipartimento di Metodi e Modelli Matematici, Università di Padova,
I-35131 Padova, Italy (Spera)
We give an extensive review of both the methods of approach and the available
solutions to the problem of providing a complete quantum description of 3-D
vortex dynamics. The leading technique is an appropriate form of geometric
quantization based on current algebra, implemented in the framework of the
Clebsch fluid description combined with the coadjoint orbit picture. We show
how, in the ensuing quantum field theory for the vortex gas, the dynamical
constants of motion identify with the topological invariants of the vortex
considered as an unknotted link.
After a brief history of ``cohomological physics'', the
Batalin - Vilkovisky complex is given a
revisionist presentation as homological algebra, in part classical, in part
novel. Interpretation of the higher order terms in the extended Lagrangian
is given as higher homotopy Lie algebra and via deformation theory. Examples
are given for higher spin particles and closed string field theory.
This paper is devoted to the horizontal (``characteristic'') cohomology
of systems of differential equations. Recent results on computing the
horizontal cohomology via the compatibility complex are generalized.
New results on the Vinogradov C-spectral sequenceand Krasil ¢shchik's C-cohomologyare obtained. As an application of general
theory, the examples of an evolution equation and a p-form gauge
theory are explicitly worked out.
We introduce an index associated to any birational transformation of
projective spaces. This index, which we call ``algebraic entropy'', is
conjectured to measure the obstruction to the existence invariants of
The notion of (n,k,r)-Lie algebra (n > k ³ r ³ 0),
an n-ary generalization of that of Lie algebra, is introduced and
studied. The standard Lie algebras turn out to be
(2,1,0)-Lie algebras. Two types of n-ary Lie structures
studied in recent few years in the context of the Nambu and ``non-Nambu''
generalizations of dynamics correspond to
(n,n-1,0)- and (n,1,0)- Lie algebras, respectively.