# Geometric Interpretation of BRST Symmetry

BRST quantization (and BRST symmetry in general), at least in this point in my understanding of them, seem rather arbitrary and slightly miraculous. However, the cohomological nature of the BRST charge $Q$ and the fact that a BRST transformation takes the form of an "extended" gauge transformation (which is purely geometric in nature) seems highly suggestive that there is a simple geometric interpretation of this symmetry.

So I am lead to ask my question: What is happening geometrically in a BRST transformation. What geometric roles do the ghost fields play? Any insight would be helpful.

[Note: I am mostly asking in the context of Yang-Mills gauge theories, but answers in the context of string theory are welcome.]

• Are you asking for the "geometric" meaning in the Hamiltonian or the Lagrangian formalism? There's a lot one can say about BRST in both formalisms, depending on what you already know, but I am uncertain what sort of answer you're looking for when you ask for the "geometric" meaning, so it would help to clarify the geometry of what space you're thinking of here. – ACuriousMind Jun 20 '17 at 10:11
• I am mostly referring to the Lagrangian formalism. The space I'm thinking of is a manifold $\mathcal{M}$ equipped with a principle bundle of gauge group $G$ that has been augmented with ghost fields and a Nakanishi–Lautrup field for gauge fixing. – Bob Knighton Jun 20 '17 at 12:04
• – Qmechanic Jun 20 '17 at 12:33
• Ah, I meant not the spacetime manifold $\mathcal{M}$ - if there is a "geometric" interpretation of BRST, it will be in field space, which in the Hamiltonian point of view would have been the phase space, and in the Lagrangian formalism a geometric interpretation will probably only arise if you can "geometrically" understand the anti-bracket in the BRST anti-field formalism, which is not evident to me. – ACuriousMind Jun 20 '17 at 12:36
• Without the Fadeev-Popov ghosts, a gauge symmetry is still a very geometric object that can be geometrically understood without reference to field space (namely, a gauge transformation corresponds to changing the section chosen over the principle bundle, something which is purely geometric in nature). It seems entirely feasible that the BRST transformations could have a similar interpretation. – Bob Knighton Jun 20 '17 at 15:38

1. In superfield formalism, there is a long tradition in the literature to consider constructions that interpret geometrically BRST (& anti-BRST) transformations as translations of Grassmann-odd $\theta$ and $\bar{\theta}$ coordinates in various physical systems, see e.g. Ref. 3 and references therein. The earliest articles seem to be Refs. 1 & 2. (We caution that BRST supersymmetry should not be conflated with Poincare supersymmetry.)

2. If we are not allowed to introduce Grassmann-odd $\theta$ and $\bar{\theta}$ coordinates, then it seems that OP's quest for a "geometric interpretation" becomes just a matter of providing explicit, manifest coordinate-independent, differential-geometric bundle constructions for the BRST formulation of various gauge theories. This will depend on the gauge theory. E.g. Yang-Mills theory, BF-theory, string theory, etc.

References:

1. S. Ferrara, O. Piquet & M. Schweda, Nucl. Phys. B119 (1977) 493.

2. K. Fujikawa, Prog. Theor. Phys. 59 (1978) 2045.

3. C.M. Hull. B. Spence. & J.L. Vázquez-Bello, Nucl. Phys. B348 (1991) 108.

Construct a principal bundle with base M and structure group G. Define Projection map and trivialization of the bundle in usual way. Denote this bundle as $P_1$. Construct another trivial principle fiber bundle $P_2= P_1\times G$ with base $P_1$ and Structure group G with trivialization of $P_2$ consisting of $P_1$ and an identity map from $P_2$ to $P_1\times G$. Construct $P_3$ as $P_2\times G$ as in the second step with local trivialization consisting of $P_2$ and an identity map from $P_3$ to $P_2\times G$.

BRS transformations are identified with infinitesimal gauge transformation on $P_3$ with parameters related to ghost fields, where these ghost fields are identified with part of certain one-forms on base space $P_2$. For details consult ref. 1 and 2.

There is another approach of group manifold in which you can gauge the algebra of $G+Q$ to obtain BRS transformation of Gauge fields where $G+Q$ has the structure of a group manifold. In short, BRST transformation are a sort of diffeomorphic invariance of this group manifold. Consult ref. 3 for details.

1- Geometric structure of Faddeev- Popov fields and invariance properties of gauge theories: Quiros, Urries, Hoyos, Mazon and Rodriguez.

2- Geometrical gauge theory of ghost and Goldstone fields and of ghost symmetries: Ne'eman and Thierry-Miec.

3- Supergravity and superstrings (a geometric perspetive): Castellani, Auria, Fre (3 vol. set with first vol. containing the necessary Group manifold machinery).