The International Conference on
Secondary Calculus and Cohomological Physics,
Moscow, August 24 - August 31, 1997

Invited contributions to the Conference

Full texts published by the American Mathematical Society in the Contemporary Mathematics Series, vol. 219, 1998.

Unstable bundles in quantum field theory
Departamento de Física Teórica, Facultad de Ciencias,
Universidad de Zaragoza,
50009 Zaragoza, Spain (M. Asorey, F. Falceto)

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.

Brackets in the jet-bundle approach to field theory

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.

The BRST structure of twisted N = 2 algebra
Dipartimento di Fisica dell'Università di Genova,
Via Dodecaneso, 33 I-16146 Genova, Italy (C. Becchi, S. Giusto)

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.

Anomalies and locality in field theories and M-theory
International School for Advanced Studies (SISSA/ISAS)
Via Beirut 2 - 4, 34014 Trieste, Italy, and INFN, Sezione di Trieste

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.

Gauge covariant algebras and local BRST cohomology
Friedemann BRANDT
Departament ECM, Facultat de Física,
Universitat de Barcelona,
Diagonal 647, E-08028 Barcelona, Spain

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).

Generalized homologies for dN = 0 and
graded q-differential algebras

Laboratoire de Physique Théorique et Hautes Energies
(Laboratoire associé au Centre
National de la Recherche Scientifique - URA D0063)
Université Paris XI, Bâtiment 211
91 405 Orsay Cedex, France

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 context.

Nonlinear control and diffieties, with
an application to physics

Michel FLIESS, Jean LÉVINE, Philippe MARTIN, Pierre ROUCHON
Laboratoire des Signaux et Systèmes, CNRS-Supélec, Plateau
de Moulon, 91192 Gif-sur-Yvette, France

Centre Automatique et Systèmes, École des Mines de Paris,
35 rue Saint-Honoré, 77305 Fontainebleau, France

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.

Consistent interactions between gauge fields:
the cohomological approach

Université Libre de Bruxelles
Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium

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.

A compatible analytic manifold structure for Lie pseudogroups of infinite type
Department of Mathematics and Statistics, Mc Gill University,
Burnside Hall, 805 Sherbrooke Street West, Montreal, Qc H3A 2K6, Canada

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.

Cohomology background in geometry of PDE
Diffiety Institute and Moscow Institute for Municipal Economy, Moscow (Russia)
1st Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia

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.

Algebraic renormalization of
massive supersymmetric theories

Dipartimento di Fisica - Università di Genova,
via Dodecaneso, 33, I-16146 Genova (Italy)

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.

A new look at completely integrable systems
and double Lie groups

Dipto. di Scienze Fisiche, INFN, Universita di Napoli, Pad. 19,
Mostra d'Oltremare, 08125 Napoli, Italia.

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.

Quantum dynamics of 3-D vortices
Vittorio PENNA, Mario RASETTI, Mauro SPERA
Dipartimento di Fisica and Unità INFM, Politecnico di Torino,
I-10129 Torino, Italy (Penna, Rasetti)

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.

The (secret?) homological algebra of
the Batalin - Vilkovisky approach

Math Dept., University of North Carolina, Chapel Hill NC 27599-3250

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.

Notes on the horizontal cohomology
Moscow State Technical University and The Diffiety Institute, Russia.
Correspondence to: A. M. Verbovetsky, Profsoyuznaya 98-9-132, 117485 Moscow, Russia

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.

Invariants of rational transformations and algebraic entropy
CERN, Division Théorique, CH1211 Genève 23

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 map.

Introduction to Secondary Calculus
Dipartimento di Ingegneria dell'Informazione e Matemetica Applicata
Universitá di Salerno - Italia.
Istituto Nazionale di Fisica Nucleare, sez. Napoli - Salerno, Italia
The Diffiety Institute, Russia

First we exhibit some basic notions and constructions of modern geometry of partial differential equations that lead to the concept of diffiety, an analogue of affine algebraic varieties for partial differential equations. Then it is shown how the differential calculus on diffieties which respects the underlying infinite order contact structure is self-organized into Secondary Calculus in such a way that higher symmetries of PDE's become secondary vector fields and the first term of the C-spectral sequence becomes the algebra of secondary differential forms. Then the general secondarization problem is formulated. Its solution for modules and multi-vector-valued differential forms is proposed and the relevant homological algebra is discussed. Eventually, relations with gauge theories are briefly outlined at the end.

On multiple generalizations of Lie algebras
and Poisson manifolds

Dipartimento di Ingegneria dell'Informazione e Matemetica Applicata
Universitá di Salerno - Italia.
Istituto Nazionale di Fisica Nucleare, sez. Napoli - Salerno, Italia
The Diffiety Institute, Russia

The Diffiety Institute and Institute of Economics Forecasting, Moscow (Russia)

The notion of (n,k,r)-Lie algebra (n > k r 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.

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