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

• M. Asorey, F. Falceto, and G. Luzón -- Unstable bundles in quantum field theory
• G. Barnich -- Brackets in the jet-bundle approach to field theory
• C. Becchi, S. Giusto, and C. Imbimbo -- The BRST structure of twisted \(N = 2\) algebra
• L. Bonora, C. S. Chu, and M. Rinaldi -- Anomalies and locality in field theories and M-theory
• F. Brandt -- Gauge covariant algebras and local BRST cohomology
• M. Dubois-Violette -- Generalized homologies for \(d^N = 0\) and graded \(q\)-differential algebras
• M. Fliess, J. Lévine, P. Martin, and P. Rouchon -- Nonlinear control and diffieties, with an application to physics
• M. Henneaux -- Consistent interactions between gauge fields: the cohomological approach
• N. Kamran and T. Robart -- A compatible analytic manifold structure for Lie pseudogroups of infinite type
• J. Krasil'shchik -- Cohomology background in geometry of PDE
• N. Maggiore -- Algebraic renormalization of massive supersymmetric theories
• G. Marmo and A. Ibort -- A new look at completely integrable systems and double Lie groups
• V. Penna, M. Rasetti, and M. Spera -- Quantum dynamics of 3-D vortices
• J. D. Stasheff -- The (secret?) homological algebra of the Batalin-Vilkovisky approach
• A. Verbovetsky -- Notes on the horizontal cohomology
• C. M. Viallet -- Invariants of rational transformations and algebraic entropy
• A. Vinogradov -- Introdution to secondary calculus
• A. Vinogradov and M. Vinogradov -- On multiple generalizations of Lie algebras and Poisson manifolds