Secondary Calculus and Cohomological Physics,

Moscow, August 24 - August 31, 1997

SECONDARY CALCULUS

AND COHOMOLOGICAL PHYSICS

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

M. ASOREY, F. FALCETO, G. LUZÓN

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.

Glenn BARNICH

Institut für Theoretische Physik, Freie Universität

Berlin, Arnimallee 14, D-14195 Berlin, Germany

E-mail: barnich@physik.fu-berlin.de

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.

C. BECCHI, S. GIUSTO, C. IMBIMBO

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.

L. BONORA, C.S. CHU, M. RINALDI

International School for Advanced Studies (SISSA/ISAS)

Via Beirut 2 - 4, 34014 Trieste, Italy, and INFN, Sezione di Trieste

E-mail: bonora@sissa.it

International School for Advanced Studies (SISSA/ISAS)

Via Beirut 2 - 4, 34014 Trieste, Italy

E-mail: cschu@sissa.it

Dipartimento di Matematica, Universitá di Trieste

P. le Europa 1, 34127 Trieste, Italy

E-mail: rinaldi@uts.univ.trieste.it

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.

Friedemann BRANDT

Departament ECM, Facultat de Física,

Universitat de Barcelona,

Diagonal 647, E-08028 Barcelona, Spain

E-mail: brandt@itp.uni-hannover.de

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).

graded q-differential algebras

Michel DUBOIS-VIOLETTE

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

E-mail: flad@qcd.th.u-psud.fr

We discuss the generalized homology associated with a nilpotent endomorphism
d such that d^{N} = 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.

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

E-mail: fliess@lss.supelec.fr

Centre Automatique et Systèmes, École des Mines de Paris,

35 rue Saint-Honoré, 77305 Fontainebleau, France

E-mails: levine@cas.ensmp.fr, martin@cas.ensmp.fr

Centre Automatique et Systèmes, École des Mines de Paris,

60 bd. Saint-Michel, 75272 Paris Cedex 06, France

E-mail: rouchon@cas.ensmp.fr

Problems of nonlinear control theory are considered in the context of diffieties. As an application, the Dirac gauge theory is discussed.

the cohomological approach

Marc HENNEAUX

Université Libre de Bruxelles

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

E-mail: henneaux@ulb.ac.be

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.

Niky KAMRAN

Department of Mathematics and Statistics, Mc Gill University,

Burnside Hall, 805 Sherbrooke Street West, Montreal, Qc H3A 2K6, Canada

E-mail: nkamran@scylla.math.mcgill.ca

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.

Joseph KRASIL'SHCHIK

Diffiety Institute and Moscow Institute for Municipal Economy, Moscow (Russia)

1st Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia

E-mail: josephk@glasnet.ru

Using techniques of Frölicher - Nijenhuis brackets, we associate to any
formally integrable equation *E* a cohomology theory
H_{C}^{*}(*E*) (based on a *C*-complex) related to deformations
of the equation structure on
the infinite prolongation *E*^{¥}. A subgroup in H_{C}^{1}(*E*) is
identified with recursion operators acting on the Lie algebra \sym*E*
of symmetries. On the other hand, another subgroup of H_{C}^{*}(*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.

massive supersymmetric theories

Nicola MAGGIORE

Dipartimento di Fisica - Università di Genova,

via Dodecaneso, 33, I-16146 Genova (Italy)

E-mail: maggiore@ge.infn.it

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.

and double Lie groups

G. MARMO, A. IBORT

Dipto. di Scienze Fisiche, INFN, Universita di Napoli, Pad. 19,

Mostra d'Oltremare, 08125 Napoli, Italia.

E-mail: gimarmo@na.infn.it

Depto. de Matemáticas, Univ. Carlos III de Madrid, 28911

Leganés, Madrid, Spain.

E-mail: alberto@ciruelo.fis.ucm.es

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.

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 Batalin - Vilkovisky approach

Jim STASHEFF

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

E-mail: jds@math.unc.edu

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.

Alexander VERBOVETSKY

Moscow State Technical University and The Diffiety Institute, Russia.

Correspondence to: A. M. Verbovetsky, Profsoyuznaya 98-9-132, 117485 Moscow, Russia

E-mail: verbovet@mail.ecfor.rssi.ru

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.

Claude M. VIALLET

CERN, Division Théorique, CH1211 Genève 23

E-mail: viallet@lpthe.jussieu.fr

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.

Alexandre VINOGRADOV

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

E-mail: vinograd@ponza.dia.unisa.it

and Poisson manifolds

Alexandre VINOGRADOV, Michael VINOGRADOV

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

E-mail: vinograd@ponza.dia.unisa.it

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

E-mail: vin@mail.ecfor.rssi.ru

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.

File translated from T