BRST complex (changes) in nLab

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Contents

Idea

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

• in degree 0: fields;

• in degree 1: ghost field;

• in degree 2: ghost-of-ghost fields;

• etc.

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

Details

For details see at

• A first idea of quantum field theory the chapter Gauge symmetries.

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 transformation, higher gauge transformation

• ghost field, ghost-of-ghost field

• antifield, antighost field

• local BRST complex

• variational BRST-bicomplex

• equivariant de Rham cohomology

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map

References

General

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

• Richard Feynman in John Wheeler, Klauder (eds.), Magic without Magic, Wheeler-Festschrift (1972)

The BRST formalism originates around

• Carlo Becchi, A. Rouet, Raymond Stora: Renormalization of gauge theories, Ann. Phys. 98 287 (1976)

• I. V. Tyutin, (1975), Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism, arXiv:0812.0580.

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 (doi pdf)

Similarly the bulk of the textbook

• Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems

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.

Review includes:

• Shuhan Jiang, Shuhan Jiang, BRST cohomology, Section 3 in: Mathematical Structures of Cohomological Field Theories, PhD thesis (1995) [[urn:nbn:de:bsz:15-qucosa2-869402](https://nbn-resolving.org//urn:nbn:de:bsz:15-qucosa2-869402), spire:2701230, pdf]

The $L_\infty$-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: XXIX Workshop on Geometric Methods in Physics, AIP Conference Proceedings, 1307 7 (2010) [[arXiv:1010.0899](https://arxiv.org/abs/1010.0899), doi:/10.1063/1.3527427]

Lecture notes from this perspective:

• Urs Schreiber: around Ex. 10.28 in geometry of physics -- perturbative quantum field theory (2018)

Discussion with more emphasis on the applications to quantum field theory of interest:

• Edward Witten, in lecture 3 of: Dynamics of Quantum Field Theory in vol II, starting page 1119, of Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999.

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

• Stefan Hollands, Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys.20:1033-1172, 2008 (arXiv:0705.3340)

• Klaus Fredenhagen, Kasia Rejzner, Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun. Math. Phys. 314(1), 93–127 (2012) (arXiv:1101.5112)

• Klaus Fredenhagen, Kasia Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys. 317(3), 697–725 (2012) (arXiv:1110.5232)

• Katarzyna Rejzner, Remarks on local symmetry invariance in perturbative algebraic quantum field theory (arXiv:1301.7037)

• Katarzyna Rejzner, Remarks on local symmetry invariance in perturbative algebraic quantum field theory (arXiv:1301.7037)

• Mojtaba Taslimi Tehrani, Quantum BRST charge in gauge theories in curved space-time (arXiv:1703.04148)

and surveyed in

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

With focus on the cochain cohomology:

• José Figueroa-O'Farrill, Takashi Kimura, The cohomology of BRST complexes (1988) [[pdf](https://www.maths.ed.ac.uk/~jmf/Research/PVBLICATIONS/homol.pdf), pdf]

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

• José Figueroa-O'Farrill, Takashi Kimura, The BRST cohomology of the NSR string: vanishing and “-ghost’‘ theorems, Comm. Math. Phys. 124 1 (1989) 105-132. [[euclid:cmp/1104179078](http://projecteuclid.org/euclid.cmp/1104179078)]

• Alexander Belopolsky, De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology, Phys.Lett. B403 (1997) 47-50 (arXiv:hep-th/9609220)

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

• Kevin Costello, Renormalisation and the Batalin-Vilkovisky formalism (arXiv).

For more along these lines see BV-BRST formalism.

In relation to equivariant de Rham cohomology:

• Jaap Kalkman, BRST model applied to symplectic geometry, Ph.D. Thesis, Utrecht, 1993 (arXiv:hep-th/9308132 (broken), cds:9308132, pdf)

• Jaap Kalkman, BRST Model for Equivariant Cohomology and Representatives for the Equivariant Thom Class, Comm. Math. Phys. Volume 153, Number 3 (1993), 447-463. (euclid:1104252784)

Making explicit that general observables constitute the functions on the BRST complex regarded as a dg-manifold:

• Anton Kapustin, Yi Li: Open String BRST Cohomology for Generalized Complex Branes, Adv. Theor. Math. Phys. 9 (2005) 559-574 [[arXiv:hep-th/0501071](https://arxiv.org/abs/hep-th/0501071), euclid]

• Shuhan Jiang, Def. 4.3 in: Mathematical Structures of Cohomological Field Theories, Journal of Geometry and Physics 185 (2023) 104744 [[doi:10.1016/j.geomphys.2022.104744](https://doi.org/10.1016/j.geomphys.2022.104744), arXiv:2202.12425]

History

• Carlo Becchi, BRS “Symmetry”, prehistory and history (arXiv:1107.1070)