∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-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 $L_\infty$-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 $k$-morphisms describe the $k$-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.
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
The idea of “ghost” fields was introduced in
and expanded on in
The BRST formalism originates around
Carlo Becchi, A. Rouet, Raymond Stora, (1976). Renormalization of gauge theories. Ann. Phys. 98: 287,
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
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
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
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
With focus on the cochain cohomology:
Discussion of the BRST complex of the bosonic string/for 2d CFT includes
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]
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
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)
Last revised on January 8, 2023 at 08:45:00. See the history of this page for a list of all contributions to it.