Holst Charges in First-Order Gravity

• Holst charges are boundary charges derived from the Holst term in first-order gravity, introducing dual (parity-odd) contributions to traditional gravitational charges.
• They modify Noether-Wald, Hamiltonian, and BMS charges through presymplectic potential shifts and BF-theoretic embedding, depending on boundary conditions.
• Their behavior varies with asymptotic regimes, impacting black hole entropy and NUT charge analyses without altering bulk Einstein equations.

Holst charges are surface charges whose non-Palatini contribution is generated by the Holst term in first-order gravity,

$$S_{\rm Holst}=\frac{1}{2\kappa\gamma}\int e^I\wedge e^J\wedge F_{IJ},$$

or, equivalently in the standard bulk form,

$$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$

Although this term does not change the torsionless Einstein equations, it modifies the presymplectic potential and therefore can modify Noether-Wald charges, Hamiltonian generators, BMS charges, local Lorentz boundary charges, and Komar-type integrals. The resulting behavior is boundary-sensitive: in some sectors the Holst contribution is a total derivative or vanishes under fall-off conditions, while in others it survives and defines a distinct “dual” or parity-odd charge sector (Corichi et al., 2010, Godazgar et al., 2020, Cerdeira et al., 18 Jun 2025).

1. First-order origin and BF-theoretic embedding

In the Palatini formulation the independent fields are a co-tetrad $e^I$ and a Lorentz connection $\omega^{IJ}=-\omega^{JI}$. The Holst term is added to the usual Palatini action without changing the bulk field equations in the torsionless sector; accordingly, the equations of motion derived from the Holst action are the same as in the Palatini formulation, and in Hamiltonian language the term changes the definition of canonical variables rather than the classical vacuum solution space (Corichi et al., 2010, Mao et al., 2022).

In the constrained BF formulation of AdS gravity, used by Durka–Kowalski-Glikman and in the later SO(3,2) analysis, the relevant starting point is an $so(2,3)$ or $so(3,2)$ connection $A^{IJ}$, split into $\omega^{ab}$ and $e^a$, together with a two-form $B^{IJ}$. After solving the algebraic $$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$0-field equations, one recovers a first-order gravitational action containing the Palatini term, the negative-$$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$1 volume term, the Holst term, and the Euler, Pontryagin, and Nieh-Yan invariants. In this representation the Immirzi parameter enters through the BF couplings and accompanies the parity-odd and topological pieces from the outset (Durka et al., 2011, Durka, 2011).

This embedding is significant because it places Holst charges in a broader class of boundary observables associated not only with the Palatini sector but also with topological densities. It also clarifies why, in AdS settings, differentiability and charge regularization are closely tied to Euler, Pontryagin, and Nieh-Yan terms rather than to the Holst term in isolation (Durka, 2011).

2. Noether currents, presymplectic structure, and charge splitting

The covariant phase-space construction begins with the variation

$$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$2

with $$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$3 or $$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$4 depending on the formulation. For the Holst action one reads off a presymplectic potential of the form

$$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$5

while in BF language the symplectic potential is $$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$6. Under a diffeomorphism generated by $$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$7, the Noether current is

$$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$8

and on shell $$S_{\rm bulk}[e,\omega]=-\frac1{2\kappa}\int_M\Bigl(\Sigma_{IJ}\wedge F^{IJ}+\frac1\gamma\,\Sigma_{IJ}\wedge\star F^{IJ}\Bigr), \qquad \Sigma^{IJ}:=\tfrac12\,\epsilon^{IJ}{}_{KL}\,e^K\wedge e^L.$$9, so locally $e^I$0 (Corichi et al., 2010, Durka et al., 2011).

In the AdS BF analysis, the surface charge decomposes as

$e^I$1

with

$e^I$2

The topological contribution is an exact differential of the Euler-Pontryagin-Nieh-Yan potentials and does not produce independent local charges once boundary conditions are imposed (Durka et al., 2011).

A closely related statement appears in the asymptotically flat analysis of the Holst action: the covariant symplectic current separates into Palatini and Holst pieces, and on shell the Holst contribution is a total derivative whose integral at spatial infinity vanishes under the standard fall-off conditions. The asymptotic symplectic form then reduces to the usual Palatini one, with no $e^I$3-dependence in the ADM energy, linear momentum, or relativistic angular momentum (Corichi et al., 2010).

3. Asymptotic regimes and null-boundary sectors

At asymptotic infinity the status of Holst charges depends on the boundary conditions and on the class of asymptotic symmetries considered.

Regime	Holst contribution	Outcome
Asymptotically flat spatial infinity with standard fall-off	Total derivative with vanishing integral	ADM/Poincaré charges unchanged
Future null infinity	Survives as dual or magnetic charge variation	Dual BMS sector appears
Spatial infinity with relaxed BMS boundary conditions	Naive Lorentz piece diverges linearly; compensator regularizes it	Finite $e^I$4-shift of Lorentz charges, none for supertranslations
3D null boundary with local Lorentz symmetry	Purely Holst charge for complex $e^I$5-$e^I$6 rotation	New boundary degree of freedom

In the Godazgar–Godazgar–Perry Hamiltonian analysis, the Holst term yields the dual BMS variation

$e^I$7

and, after a Wald-Zoupas split into integrable and flux parts, one obtains a finite leading-order dual charge and a charge algebra with a field-dependent central term. The same works classify the standard BMS sector as “electric” and the Holst-induced sector as “magnetic,” with the dual supermomentum proportional to $e^I$8 (Godazgar et al., 2020, Godazgar et al., 2020).

Oliveri and Speziale emphasized that these dual charges originate from an exact 3-form in the tetrad symplectic potential. Using the Kosmann variation, they showed that the dual contribution vanishes for exact isometries and for asymptotic symmetries at spatial infinity under standard parity conditions, but persists at future null infinity, where it reproduces the dual supertranslation charge (Oliveri et al., 2020).

A newer spatial-infinity analysis with relaxed boundary conditions admitting the full BMS group, including non-trivial supertranslations, reaches a different conclusion for the Lorentz sector. There the naive Holst surface integrals are linearly divergent because large Lorentz diffeomorphisms drag the background tetrad, but a compensating internal Lorentz transformation removes the divergence. The resulting charges are finite and integrable; the Holst term shifts boosts and rotations, while supertranslation charges remain identically invariant (Bakhoda et al., 2 Apr 2026).

At null boundaries, local Lorentz symmetries produce an additional effect. In Newman-Penrose language there is one more charge derived from the local Lorentz transformation, and this new charge is purely from the Holst term. Specifically, the complex rotation in the $e^I$9-$\omega^{IJ}=-\omega^{JI}$0 plane carries a charge

$\omega^{IJ}=-\omega^{JI}$1

yielding five independent charges on a 3D null boundary when combined with the Palatini-sector charges (Mao et al., 2022).

4. Horizons, entropy, and black-hole thermodynamics

For bifurcate Killing horizons, the Holst contribution to the Wald-type entropy formula often vanishes. In the constrained BF analysis, if $\omega^{IJ}=-\omega^{JI}$2 is the timelike Killing field vanishing on the bifurcation two-surface $\omega^{IJ}=-\omega^{JI}$3, then $\omega^{IJ}=-\omega^{JI}$4 on $\omega^{IJ}=-\omega^{JI}$5. Every term proportional to $\omega^{IJ}=-\omega^{JI}$6 therefore vanishes there, including the Holst piece. The Palatini term reproduces the Bekenstein-Hawking area law, while the Euler term contributes only an additive constant proportional to $\omega^{IJ}=-\omega^{JI}$7 (Durka et al., 2011).

The broader AdS BF analysis reaches the same conclusion for several ordinary AdS black holes. For AdS-Schwarzschild, topological black holes, and AdS-Kerr, the $\omega^{IJ}=-\omega^{JI}$8-dependent piece cancels or vanishes under the relevant integrals, so mass, angular momentum, and the area-scaling part of the entropy are independent of the Immirzi parameter. The explicit AdS-Kerr charges are

$\omega^{IJ}=-\omega^{JI}$9

with no surviving $so(2,3)$0-dependence (Durka, 2011).

In the asymptotically flat Euclidean approach, the on-shell Holst action for static solutions containing horizons also reproduces the standard thermodynamical relations. Since the bulk terms vanish on shell, only boundary terms contribute, and the extra Holst boundary integral vanishes for the static solutions considered because one can choose a diagonal tetrad so that $so(2,3)$1 on each boundary. The result is the familiar

$so(2,3)$2

with no $so(2,3)$3-dependence at the semiclassical level (Corichi et al., 2010).

A separate horizon analysis concerns asymptotic symmetry charges rather than entropy. On the future Schwarzschild horizon, standard and dual supertranslation charges can be constructed from the Holst action. For smooth supertranslation parameters $so(2,3)$4, the integrable parts are

$so(2,3)$5

$so(2,3)$6

Neither charge depends on $so(2,3)$7 at leading order, but the dual-electric split does. For singular supertranslations the Dirac bracket acquires a central term, and Akhoury–Choi–Perry show that an $so(2,3)$8 Chern-Simons theory on the horizon cancels the anomaly and restores closure of the horizon BMS algebra (Akhoury et al., 2022).

5. NUT charge, Komar modifications, and parity-odd sectors

The clearest classical situations in which a Holst charge survives are those with nonzero NUT charge. In the AdS-Taub-NUT geometry, the connection and curvature components entering the $so(2,3)$9-weighted term do not vanish: $so(3,2)$0 is nonzero, and the Noether charge acquires a genuine Immirzi-dependent contribution. The resulting total mass and entropy are shifted by terms proportional to the NUT charge $so(3,2)$1 (Durka, 2011).

A related development is the Komar analysis with the parity-breaking Barbero parameter $so(3,2)$2. In first-order Palatini gravity modified by the Holst term, the Noether-Wald charge becomes

$so(3,2)$3

so the generalized Komar 2-form is

$so(3,2)$4

For the time-translation Killing field one defines

$so(3,2)$5

and the total mass becomes

$so(3,2)$6

The paper describes this as the gravitational analogue of the Witten effect: a non-vanishing NUT charge induces a non-vanishing mass $so(3,2)$7 (Cerdeira et al., 18 Jun 2025).

The null-infinity literature expresses the same structure in asymptotic language. The Holst contribution supplies a tower of dual or magnetic supermomenta,

$so(3,2)$8

where $so(3,2)$9 is the NUT or dual mass aspect. This places ordinary ADM/BMS charges and Holst-induced dual charges into parallel electric and magnetic sectors (Godazgar et al., 2020).

6. Ambiguities, on-shell equivalence, and interpretive disputes

A central issue in the literature is whether Holst charges are invariant observables or artifacts of a particular choice of symplectic potential. Oliveri–Speziale showed that the dual charges derive from an exact 3-form $A^{IJ}$0 in the tetrad symplectic potential. Since Iyer-Wald methods allow one to shift $A^{IJ}$1, one may choose a dressed tetrad potential equal to the metric potential, in which case the Holst-induced dual piece disappears. In that sense the dual contribution is a cohomological ambiguity of type II in Iyer-Wald (Oliveri et al., 2020).

The generalized Holst analysis with torsion, nonmetricity, and timelike boundaries reaches a related but stronger on-shell statement. Barbero G., Margalef-Bentabol, Varo, and Villaseñor prove that the generalized Holst theory has the same solution space as the Palatini action and hence the same metric sector as Einstein-Hilbert gravity. They also show that the Lagrangians are not cohomologically equal, so the presymplectic structure and charges provided by the covariant phase-space method might differ; however, within the relative bicomplex framework the covariant phase spaces and charges are equivalent on shell. In particular, the dual and internal-Lorentz charges differ off shell but vanish on shell once the algebraic field equations setting the Holst torsion/non-metricity pieces to zero are imposed (G. et al., 2022).

A common misconception is that the Holst term either always produces new classical charges or never does. The literature supports a narrower statement: the answer depends on formulation, boundary conditions, the handling of $A^{IJ}$2-ambiguities, and whether one studies exact isometries, asymptotic symmetries, null boundaries, or NUT sectors. This suggests that “Holst charges” are best understood not as a single universal observable, but as a family of boundary contributions whose physical status is controlled by the covariant phase-space prescription and by the geometry of the boundary itself (Oliveri et al., 2020, G. et al., 2022).