BRST complex


\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



What is called the “BRST complex” in the physics literature is the qDGCA which is the Chevalley-Eilenberg algebra of the L L_\infty-algebroid which is the differential version in Lie theory of the \infty-groupoid

  • whose space of objects is the space of configurations/histories of a given physical system;

  • whose morphisms describe the gauge transformations between these configurations/histories;

  • whose kk-morphisms describe the kk-fold gauge-of-gauge transformations.

The generators of the BRST complex are called

The cochain cohomology of the BRST complex is called, of course, BRST cohomology.


For details see at

The BRST complex described a homotopical quotient of a space by an infinitesimal action. Combined with a homotopical intersection, it is part of the BRST-BV complex.

gauge field: models and components

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map



The idea of “ghost” fields was introduced in

  • Richard Feynman, Quantum theory of gravitation In: Acta physica polonica. vol 24, 1963, S. 697

and expanded on in

The BRST formalism originates around

see also the references at BRST.

A classical standard references for the Lagrangian formalism is

  • Marc Henneaux, Lectures on the Antifield-BRST formalism for gauge theories, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 (pdf)

Similarly the bulk of the textbook

considers the Hamiltonian formulation. Chapters 17 and 18 are about the Lagrangian (“antifield”) formulation, with section 18.4 devoted to the relation between the two.

The L-infinity algebroid-structure of the local BRST complex on the jet bundle is made manifest in

  • Glenn Barnich, A note on gauge systems from the point of view of Lie algebroids, in P. Kielanowski, V. Buchstaber, A. Odzijewicz, M.. Schlichenmaier, T Voronov, (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) (arXiv:1010.0899, doi:/10.1063/1.3527427)

Discussion with more emphasis on the applications to quantum field theory of interest is in lecture 3 of

The perturbative quantization of gauge theories (Yang-Mills theory) in causal perturbation theory/perturbative AQFT is discussed (for trivial principal bundles and restricted to gauge invariant observables) via BRST-complex/BV-formalism in

and surveyed in

  • Kasia Rejzner, section 7 of Perturbative algebraic quantum field theory Springer 2016

Discussion of the BRST complex of the bosonic string/for 2d CFT includes

  • Graeme Segal, p.114 and following of The definition of conformal field theory , preprint, 1988; also in Ulrike Tillmann (ed.) Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. (pdf)

Discussion of the BRST complex for the superstring (hence with the corresponding Lie algebroid being actually a super Lie algebroid) is for instance in

The perspective on the BRST complex as a formal dual to a space in dg-geometry is relatively clearly stated in section 2 of

For more along these lines see BV-BRST formalism.


Revised on February 5, 2018 12:30:21 by Urs Schreiber (