**Abstract.**In contrast with QFT, classical ﬁeld theory can be formulated in a strict mathematical way if one deﬁnes even classical ﬁelds as sections of smooth ﬁber bundles. Formalism of jet manifolds provides the conventional language of dynamic systems (nonlinear diﬀerential equations and operators) on ﬁber bundles. Lagrangian theory on ﬁber bundles is algebraically formulated in terms of the variational bicomplex of exterior forms on jet manifolds where the Euler–Lagrange operator is present as a coboundary operator. This formulation is generalized to Lagrangian theory of even and odd ﬁelds on graded manifolds. Cohomology of the variational bicomplex provides a solution of the global inverse problem of the calculus of variations, states the ﬁrst variational formula and Noether’s ﬁrst theorem in a very general setting of supersymmetries depending on higher-order derivatives of ﬁelds. A theorem on the Koszul–Tate complex of reducible Noether identities and Noether’s inverse second theorem extend an original ﬁeld theory to prequantum ﬁeld-antiﬁeld BRST theory. Particular ﬁeld models, jet techniques and some quantum outcomes are discussed.

**Contents**

I. Introduction

II. ACFT. The general framework

1. The main postulate, 2. Jet manifolds, 3. Jets and connections, 4.
Lagrangian theory

of even ﬁelds, 5. Odd ﬁelds, 6. The algebra of even and odd ﬁelds, 7. Lagrangian theory

of even and odd ﬁelds, 8. Noether’s
ﬁrst theorem in a general
setting, 9. The Koszul–Tate complex of Noether identities, 10. Noether’s
inverse second theorem, 11. BRST extended ﬁeld theory, 12.
Local BRST cohomology.

III. Particular models

13. Gauge theory of principal connections, 14. Topological Chern–Simons
theory, 15.

Topological BF theory, 16. SUSY gauge theory, 17. Field theory on
composite bundles,

18. Symmetry breaking and Higgs ﬁelds, 19. Dirac
spinor ﬁelds, 20. Natural and gauge natural
bundles. 21. Gauge gravitation theory, 22. Covariant Hamiltonian ﬁeld theory,

23. Time-dependent mechanics, 24. Jets of submanifolds, 25. Relativistic
mechanics, 26. String theory.

IV. Quantum outcomes

27. Quantum master equation, 28. Gauge ﬁxing procedure, 28. Green
function identities

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