**Contemporary Mathematics**

Volume: 219;
1998;
287 pp;
Softcover

MSC: Primary 81; 35;
Secondary 58

Print ISBN: 978-0-8218-0828-3

Product Code: CONM/219

List Price: $76.00

AMS Member Price: $60.80

MAA member Price: $68.40

**Electronic ISBN: 978-0-8218-7810-1
Product Code: CONM/219.E**

List Price: $76.00

AMS Member Price: $60.80

MAA member Price: $68.40

# Secondary Calculus and Cohomological Physics

Share this page *Edited by *
*Marc Henneaux; Joseph Krasil′shchik; Alexandre Vinogradov*

This collection of invited lectures (at the Conference on Secondary
Calculus and Cohomological Physics, Moscow) reflects the
state-of-the-art in a new branch of mathematics and mathematical
physics arising at the intersection of geometry of nonlinear
differential equations, quantum field theory, and cohomological
algebra. This is the first comprehensive and self-contained book on
modern quantum field theory in the context of cohomological methods
and the geometry of nonlinear PDEs.

Features:

- An up-to-date and self-contained exposition of the newest results in cohomological aspects of quantum field theory and the geometry of PDEs.
- A new look at interrelations among cohomology theory, the geometry of PDEs, and field theory.
- Application to Batalin-Vilkovisky formalism, BRST formalism, anomalies, and quantum dynamics.

#### Readership

Mathematicians interested in mathematical and theoretical physics; advanced graduate students studying quantum field theory.

# Table of Contents

## Secondary Calculus and Cohomological Physics

- Contents vii8 free
- Acknowledgments ix10 free
- Foreword xi12 free
- Unstable bundles in quantum field theory 116 free
- Brackets in the jet-bundle approach to field theory 1732
- The BRST structure of twisted N = 2 algebra 2944
- Anomalies and locality in field theories and M-theory 3954
- Gauge covariant algebras and local BRST cohomology 5368
- Generalized homologies for dN = 0 and graded q-differential algebras 6984
- Nonlinear control and diffieties, with an application to physics 8196
- Consistent interactions between gauge fields: the cohomological approach 93108
- A compatible analytic manifold structure for Lie pseudogroups of infinite type 111126
- Cohomology background in geometry of PDE 121136
- Algebraic renormalization of massive supersymmetric theories 141156
- A new look at completely integrable systems and double Lie groups 159174
- Quantum dynamics of 3-D vortices 173188
- The (secret?) homological algebra of the Batalin – Vilkovisky approach 195210
- 1. The jet bundle setting for Lagrangian field theory 197212
- 2. Anti-fields as Koszul generators 199214
- 3. Noether identities and Tate generators 199214
- 4. Ghosts and the anti-bracket 201216
- 5. The Batalin- Vilkovisky complex (BV, s=) 201216
- 6. The BV version of the variational bicomplex 203218
- 7. The Master Equation and Higher Homotopy Algebra 203218
- 8. Deformation Theory and the Master Equation in Field Theory 205220
- 9. Quantization and other puzzles 206221
- References 207222

- Notes on the horizontal cohomology 211226
- Invariants of rational transformations and algebraic entropy 233248
- Introduction to Secondary Calculus 241256
- 1. Infinite jets 242257
- 2. Infinite prolongations of PDE's and diffieties 244259
- 3. Higher symmetries of PDE's 246261
- 4. The C-spectral sequence 248263
- 5. The structure of the term CE1 for 0 = ] 00 250265
- 6. The structure of the term CE1 for 0 = £00 252267
- 7. The conception of Secondary Calculus: vector fields and differential forms 254269
- 8. Secondary modules 256271
- 9. Secondary modules and flat connections 259274
- 10. ∇-C-spectral sequence 261276
- 11. Secondary multi-vector-valued differential forms: an example of secondarization 263278
- 12. Secondarization versus quantization 267282
- 13. Bibliography and comments 268283
- Acknowledgments 270285
- References 270285

- On multiple generalizations of Lie algebras and Poisson manifolds 273288