Gauge-Fixed Version

• Gauge-fixed version is a reformulated gauge theory that eliminates redundant field configurations by imposing explicit gauge conditions, ensuring each physical state is uniquely integrated.
• The method introduces auxiliary fields such as ghosts, antighosts, and Lagrange multipliers to compensate for the gauge orbit volume, making the action suitable for perturbative and nonperturbative analyses.
• Applications range from standard QFT and lattice computations to effective field theories in scenarios like abelian projection and D-brane constructions, with full and partial gauge-fixing strategies maintaining precise BRST invariance.

A gauge-fixed version of a gauge theory is any reformulation in which one or more components of the original gauge redundancy have been eliminated via explicit gauge conditions. In such a formulation, the gauge symmetry is fixed (fully or partially), auxiliary degrees of freedom such as ghost, antighost, and Lagrange multiplier fields are introduced to correctly account for the gauge orbit volume in the path integral, and the resulting action becomes suitable for perturbative expansion, quantization, or nonperturbative numerical methods. Gauge-fixed versions appear universally in quantum field theory (QFT), quantization of constrained systems, lattice gauge theory, string theory, and quantum gravity.

1. Concept and Rationale of Gauge Fixing

Gauge theories possess local symmetries generated by first-class constraints, resulting in physically equivalent field configurations (gauge orbits). Naively integrating over all field configurations in the path integral grossly overcounts physically equivalent points. The gauge-fixing procedure introduces explicit constraints—gauge conditions—and modifies the action and measure to guarantee each physical configuration is integrated exactly once, up to global/topological ambiguities.

Classically, gauge fixing amounts to imposing a set of nondegenerate conditions $G_j[\phi]=0$ that intersect each gauge orbit transversely; quantum mechanically, it is implemented via the insertion of a $\delta$-functional and a Faddeev–Popov determinant, generically exponentiated using ghost fields. In BRST formalism, these structures are elegantly encoded by extending the field space and introducing nilpotent differentials acting on both physical and ghost sectors (Varshovi, 2016, Maas, 2013).

A gauge-fixed version need not refer to full gauge fixing: partial gauge fixing restricts the gauge group from $G$ to a subgroup $H\subset G$, preserving partial symmetry while eliminating unwanted redundancies (Ferrari, 2013).

2. Algebraic and Cohomological Structure

The algebraic characterization of gauge-fixed versions leverages differential graded algebras generated by field/ghost content and BRST operators.

For full gauge fixing, the BRST differential $s$ encodes both the infinitesimal gauge transformation and the gauge-fixing redundancy. The cohomology of $s$ restricted to ghost number zero classifies gauge-invariant physical observables. In the case of partial gauge-fixing ($G\to H$), the BRST structure splits into $s = s_H + \delta$, where $s_H$ is the $H$-BRST operator and $\delta$ encodes the residual $G/H$ transformations, giving rise to equivariant cohomology (Ferrari, 2013). The relevant mathematical models are the Weil and Cartan models for equivariant cohomology, connected via the Kalkman automorphism. The partial gauge-fixed BRST algebra satisfies $[s_H, s_H]=0, \; [s_H, L_a]=0, \; [s_H, i_a]=L_a$, and the equivariant (Cartan) differential $D = s_H - p^a i_a$ satisfies $D^2=0$ on $H$-invariants.

In general, physical-state conditions and Ward identities in the gauge-fixed theory are ensured by the nilpotency of $s$ and anti-BRST operator $\bar s$ (Varshovi, 2016).

3. Ghost Lagrangian and Gauge-Fixing Fermions

The gauge-fixing sector introduces auxiliary fields:

• Ghosts $c^A$ (Grassmann, gh +1), antighosts $\bar c^A$ (gh –1), and Nakanishi–Lautrup fields $b^A$ (gh 0) in the adjoint representation,
• Gauge-fixing fermion $\Psi$ of ghost number $-1$,
• For partial gauge fixing, quartic ghost couplings, essential even at tree level for consistency of the equivariant BRST symmetry.

The ghost Lagrangian is constructed as the BRST (or equivariant) variation of $\Psi$. For partial gauge fixing, the generic ghost sector includes (Ferrari, 2013):

$$\mathcal{L}_{\text{ghost}} = b^a F^a(\Phi) + \bar c^a s_H F^a(\Phi) + \tfrac{\xi}{2}(b^a)^2 + \tfrac{1}{4} f^{aij}f^{a}{}_{kl} \bar c^i \bar c^k c^j c^l,$$

where $F^a(\Phi)$ are $H$-covariant gauge-fixing functions. The quartic term is dictated by the requirement for equivariant cohomology and to maintain invariance under $s_H$ (Ferrari, 2013).

4. Path Integral and Gauge-Fixed Measure

The gauge-fixed version produces the modified functional integral:

$$Z = \int D\Phi\, Dc\, D\bar c\, Db\, e^{i S_{GF}[\Phi, c, \bar c, b]},$$

with $S_{GF}$ the fully gauge-fixed action. For partial gauge fixing, the path integral is first constrained to the $H$-invariant subspace; then, after (optional) full fixing of $H$, one matches the original $G$-theory partition function (Ferrari, 2013). The measure is invariant under the (equivariant) BRST differential and all dependence on the gauge-fixing fermion cancels in the computation of gauge-invariant observables.

In practice, the choice of gauge condition (e.g., Landau gauge, $R_\xi$ gauge, Maximal Abelian gauge) determines the explicit content of $\Psi$ and the corresponding modifications to the ghost and Nakanishi–Lautrup sectors.

5. Applications and Physical Significance

Gauge-fixed versions are essential in:

• Perturbative quantization—fixing the gauge allows computation of propagators and Feynman rules,
• Nonperturbative computations, notably lattice gauge theory, where gauge-fixing (e.g., Landau gauge via minimization or maximization) enables calculation of gauge-dependent correlation functions,
• The systematic construction of effective field theories by partial gauge fixing, as in abelian projection for dual superconductivity in QCD, or the construction of effective symmetry-broken descriptions in grand unified models (Ferrari, 2013).

Partial gauge fixing is central in scenarios such as the 't Hooft Abelian projection, D-brane models of emergent space, and the effective theory of monopoles and vortices.

In Abelian models, the gauge-fixed version is related to non-gauge-invariant (massive) models by integrating out the Stueckelberg or Wess–Zumino scalar introduced to restore gauge invariance (or vice versa, "gauge enhancing" a massive action to a gauge theory by the Harada–Tsutsui trick) (Lima, 2014).

6. Comparison: Full vs. Partial Gauge Fixing

Feature	Full Gauge Fixing	Partial Gauge Fixing
BRST Complex	Standard (cohomology of $s$ on all fields)	Equivariant (cohomology of $s_H$ or $D$ on $H$-invariants)
Ghost Structure	Quadratic ghost terms, standard Faddeev–Popov	Includes essential quartic ghost terms in coset directions (Ferrari, 2013)
Residual Symmetries	None (up to global or discrete symmetries)	$H$-gauge symmetry persists, may be further fixed
Path Integral	Over fields modulo all gauge redundancy	Over fields modulo $G/H$, with $H$ unfixed
Applications	Standard QFT, lattice gauge theory	Abelian projections, emergent gauge theories, D-brane constructions, effective field theory below symmetry-breaking scales

The essential algebraic and physical difference is that partial gauge fixing requires a consistent treatment of the remaining $H$-redundancy at the cohomological level. Equivariant BRST cohomology and the appearance of nontrivial higher-order ghost couplings (e.g., quartic terms) are generic.

7. Equivariant Cohomology: Cartan vs. Weil Models

Equivariant cohomology provides the mathematical structure underlying partial gauge-fixed versions:

• Weil model: starts with the $H$-BRST algebra and tensor with independent "Weil" auxiliary variables $(w^a, p^a)$, with differential $Q_W = s_H + d_W$ and computes the cohomology on basic elements (annihilated by $i_a$ and $L_a$).
• Cartan model: eliminates $w^a$ via $i_a=0$, leaving the algebra with differential $D_C = s_H - p^a i_a$, manifestly nilpotent on $H$-invariants.
• The Kalkman automorphism provides the canonical change of variables relating the two.

This duality ensures equivalence of various explicit constructions of the partially gauge-fixed action; both formalisms recover the same physical content (Ferrari, 2013).

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References:

• (Ferrari, 2013) Partial Gauge Fixing and Equivariant Cohomology
• (Varshovi, 2016) Gauge Fixing Invariance and Anti-BRST Symmetry
• (Maas, 2013) Local and global gauge-fixing
• (Lima, 2014) The Relation Between Gauge and Non-Gauge Abelian Models