Gauge-Independent Pre-Symplectic Current

• Gauge-independent pre-symplectic current is a differential form defined on field configurations that remains invariant under gauge transformations and captures only the physical data.
• It is constructed using multisymplectic and covariant phase space methods, yielding closed on-shell forms that facilitate the definition of covariant Poisson brackets and conserved charges.
• Its careful treatment of boundary terms and degenerate directions enables a systematic reduction to the physical phase space in both classical and quantum gauge field theories.

A gauge-independent pre-symplectic current is a differential form on the space of field configurations (or solutions) in a covariant field theory, constructed such that its value does not depend on arbitrary gauge choices but only on the physical (gauge-inequivalent) data. This object is a central technical input to the covariant phase space approach, the multisymplectic formalism, advanced Hamiltonian and BV-type formulations, and is foundational for covariant Poisson brackets, the classification of conserved charges, and the systematic comparison of local observables in classical and quantum gauge field theories.

1. Structural Foundations: Multisymplectic and Covariant Phase Space Approaches

The covariant phase space approach and the multisymplectic formalism both begin with a fiber bundle $\pi\colon Y\to M$ representing the configuration space of fields over an $n$-dimensional spacetime manifold $M$. For a first-order variational field theory, the Lagrangian density $L$ defines an action functional $S_U(\phi) = \int_U L(j^1\phi)$ over a compact domain $U\subset M$.

The essential differential geometric structure is provided by the infinite jet bundle $JY$ and the associated de Rham differential, decomposed as $d = d_h + d_v$, where $d_h$ differentiates along the base $M$ (spacetime directions) and $d_v$ along fibers (field directions) (Díaz-Marín et al., 2017). The key objects are:

• The Euler–Lagrange form $E(L)$, yielding the equations of motion.
• The multisymplectic potential $\Theta_L$, an $(n-1,1)$-form producing boundary contributions in variations of the action.
• The multisymplectic (pre-symplectic) current $\Omega_L = -d_v \Theta_L$, which is an $(n-1,2)$-form.

This current, $\Omega_L$, is constructed such that it is closed under $d_v$, and on-shell satisfies $d_h \Omega_L = 0$. Integration of $\Omega_L$ over a hypersurface $\Sigma$ yields a $2$-form on the space of solutions, which becomes the pre-symplectic form of the covariant phase space (Díaz-Marín et al., 2017, Ciaglia et al., 2021).

2. Gauge Invariance and Construction of the Gauge-Independent Current

Gauge invariance is central: the physical phase space is the quotient of the space of field configurations by gauge transformations. In the standard construction, the pre-symplectic current may only be invariant under gauge transformations up to a total derivative. The concept of a gauge-independent pre-symplectic current resolves this by ensuring that all physical observables and brackets are independent of the representatives within a gauge orbit.

Explicitly, for an evolutionary vector field $X$ generating an infinitesimal gauge symmetry, the defining property is

$\mathcal{L}_{jX} \Omega_L = d_h(\ldots)$

so that the current changes, at most, by a horizontal exact form. Thus, when integrated over closed or appropriately bounded hypersurfaces, the result is invariant under gauge, and $\Omega_L$ descends to the reduced phase space (Díaz-Marín et al., 2017, Paoli et al., 2018, Ramirez et al., 12 Dec 2025). In the variational tricomplex approach, the pre-symplectic current is defined as $\omega = \delta_V \theta$, and up to $d_H$-exact terms, its class in $H^{2,n-1}(d_H)$ is unique and gauge-independent (Sharapov, 2016).

Boundary terms are systematically handled by the addition of total derivatives to the presymplectic potential, ensuring that the integrated presymplectic form vanishes for pure gauge variations even near boundaries or corners, as in the improved potential $\Theta_{GI}$ for tetrad general relativity (Paoli et al., 2018).

3. Role in Hamiltonian Observables, Brackets, and Physical Charges

Hamiltonian observable currents $F$ are $(n-1,0)$-forms satisfying

$d_v F = -\iota_V \Omega_L + d_h \sigma^F$

for some boundary term $\sigma^F$, with appropriate vanishing conditions at $\partial U$. These observables, when integrated over a hypersurface, yield genuine gauge-invariant functionals on the covariant phase space (Díaz-Marín et al., 2017).

The covariant Poisson bracket of two such observables is defined via

$\{ F^V, G^W \} := i_{jW} i_{jV} \Omega_L,$

which closes (up to boundary terms) among the Hamiltonian currents, yielding a Lie or Poisson algebra structure for observables modulo gauge. The bracket descends to the space of gauge inequivalent solutions and satisfies the Jacobi identity modulo boundary terms, making the algebra structure robust under gauge (Díaz-Marín et al., 2017).

For theories with nontrivial gauge structure, such as general relativity or Yang–Mills, conserved charges (momentum maps) associated to residual global symmetries (e.g., Poincaré, BMS) are constructed as gauge-independent functionals using the presymplectic current, and their algebra is determined by the structure of $\Omega_L$ (Ciaglia et al., 2021, Bac et al., 2023).

4. Generalizations and Systematic Reductions: Kernel and Cohomological Methods

If the presymplectic current $\omega$ is degenerate, its kernel defines the gauge distribution $K$ (the space of gauge directions) (Gaset, 2022). The pullback of $\omega$ along two variations $\delta_1\phi, \delta_2\phi$ is

$$J(\delta_1\phi, \delta_2\phi) = \phi^*[i_{\xi_2} i_{\xi_1} \omega],$$

which vanishes whenever one variation is in $K$, ensuring the current is defined on gauge-inequivalent data. The presymplectic current hence uniquely characterizes the geometry of the quotient by gauge orbits and allows identification of the reduced multisymplectic structure on $Y/K$ (Gaset, 2022).

Gotay's coisotropic embedding theorem is employed to symplectically regularize the presymplectic manifold (the space of solutions), promoting the degenerate form to a genuine symplectic form on an extended space, facilitating systematic quantization and charge definition (Ciaglia et al., 2021).

Cohomological (variational tricomplex) methods, in both Lagrangian and BV–BRST formulations, identify the presymplectic current as the descendant (n–1,2) form, and the gauge-independence as the invariance of its cohomology class under BRST or homological vector fields, modulo $d_h$-exact forms (Sharapov, 2016, Grigoriev, 2022, Dneprov et al., 5 Feb 2024).

5. Applications: Gravity, Gauge Theories, Higher Spin, and Minimal Models

Gauge-independent pre-symplectic currents have been explicitly constructed and utilized in multiple contexts:

• General Relativity: In tetrad and connection formulations, the improved symplectic potential (including surface/corner terms) yields a pre-symplectic current that is strictly invariant under Lorentz gauge transformations, reduces to the Einstein–Hilbert current in the torsion-free case, and controls the proper definition of black hole entropy and the first law independently of gauge (Paoli et al., 2018, Ramirez et al., 12 Dec 2025).
• Yang–Mills Theories and Conformal Gravity: Normal conformal Cartan connections yield conformally invariant presymplectic currents, with explicit decomposition into topological and physical pieces and correct behavior at holographic or null boundaries (Bac et al., 2023).
• Electromagnetism and Metric-Affine Gravity: The degenerate kernel is spanned by the generators of $U(1)$ or projective connection shifts, so the presymplectic current naturally descends to the reduced (physical) data (Gaset, 2022).
• Higher-Spin Fields: For unfolded non-Lagrangian systems, gauge-independent presymplectic currents are constructed explicitly (e.g., in AdS backgrounds), and remain closed and gauge-invariant modulo equations of motion (Sharapov, 2016).
• Minimal BV Models: In the BV and AKSZ-type formulations, the presymplectic current is defined on the minimal graded manifold after reduction, and its cohomology class uniquely encodes the physical phase space (Grigoriev, 2022, Dneprov et al., 5 Feb 2024).

6. Boundaries, Local Structure, and Gluing

For theories defined on manifolds with boundaries or corners, correct handling of boundary terms in the symplectic potential is essential. The addition of surface/corner terms to the potential is necessary to ensure complete gauge invariance of the presymplectic current, particularly in the presence of edge degrees of freedom or in entropy computations for black holes (Paoli et al., 2018, Ramirez et al., 12 Dec 2025).

Algebras of observable currents associated to bounded domains exhibit nontrivial gluing properties, enabling consistent local-to-global constructions and ensuring the matching of physical observables across adjoining regions (Díaz-Marín et al., 2017).

Boundary analysis reveals that the presymplectic current can vanish for specific gauge modes at asymptotic boundaries (e.g., for BMS variations at null infinity), aligning with expectations from holography or topological decoupling (Bac et al., 2023).

7. Minimal Uniqueness, Redundancy, and Physical Significance

The gauge-independent presymplectic current is unique up to the addition of $d_h$-exact forms, reflecting the freedom to add trivial, locally exact currents without physical effect (Sharapov, 2016, Grigoriev, 2022). This minimal cohomological ambiguity corresponds precisely to the ambiguity in defining surface terms, and does not affect the bracket or Poisson structure on the reduced, physical phase space.

The resulting structure underpins the consistent classification of local and global observables, the definition of conserved charges, and the implementation of quantization procedures compatible with gauge symmetry. These properties ensure that physical predictions obtained from the covariant phase space, Peierls brackets, or BV quantization are robust and independent of redundant gauge choices.