The Holst Term and Immirzi Parameter

The Holst action adds an Immirzi-parameter term to first-order Palatini gravity. For an invertible tetrad, real nonzero parameter, and no connection-sourcing matter, the connection equation still gives zero torsion and the extra coframe equation vanishes by the Bianchi identity. That is a conditional statement about classical bulk equations—not a claim that the term is topological off shell or that the parameter has no boundary, matter, or quantum significance.

Required background. Frames, Coframes, and Anholonomy supplies the internal Lorentz algebra, The Gibbons-Hawking-York Boundary Term supplies the metric comparison problem, and Presymplectic Potentials and Currents supplies the distinction between equal field equations and equal symplectic data.

Helpful background. The Einstein Equation supplies the recovered bulk dynamics, Hypersurface Normals and Tangential Projectors supplies the later canonical split, and Formulations of General Relativity places the Holst family among classical formulations.

Internal dual and the Holst action

Keep $\eta_{IJ}=\operatorname{diag}(-1,1,1,1)$ and $\epsilon_{0123}=+1$. The internal dual of a Lorentz bivector is

$$(\star X)^{IJ}=\frac12\epsilon^{IJ}{}_{KL}X^{KL}, \qquad \star^2=-1.$$

Define

$$\Sigma_{IJ}=\star(e_I\wedge e_J) =\frac12\epsilon_{IJ}{}^{KL}e_K\wedge e_L.$$

With $\kappa=8\pi G$, one convention for the Palatini–Holst action is

$$\begin{aligned} S_\gamma[e,\omega] ={}&\frac1{2\kappa}\int_{\mathcal M} \left(\Sigma_{IJ}-\frac1\gamma e_I\wedge e_J\right) \wedge F^{IJ}\\ &-\frac{\Lambda}{24\kappa}\int_{\mathcal M} \epsilon_{IJKL}e^I\wedge e^J\wedge e^K\wedge e^L +S_{\partial\mathcal M}. \end{aligned}$$

Here $\gamma$ is a dimensionless, finite, nonzero Immirzi parameter; the Palatini theory is recovered as $\gamma\to\infty$. Some sources put a plus sign or the reciprocal parameter in the second term. All later formulas on this page refer to the displayed convention. The sign is tied to the canonical connection: with $\Gamma^a=-\epsilon^a{}_{bc}\omega^{bc}/2$ and the chapter’s extrinsic-curvature convention, this action gives $A_i{}^a=\Gamma_i{}^a-\gamma K_{ij}e^{ja}$ and a positive $\kappa\gamma$ in its bracket with the densitized triad. Changing the Holst sign while keeping those definitions requires $\gamma\mapsto-\gamma$ in the canonical formulas as well.

The original derivation of the generalized action and its real canonical variables is Holst 1996, equations (1)–(14), pp. 1–5, PDF. A boundary-conscious comparison with Palatini and Nieh–Yan terms is given in Corichi, Rubalcava-García, and Vukašinac 2016, §IV.B, pp. 24–27, PDF.

Why the vacuum equations agree

Since $\delta F^{IJ}=D\delta\omega^{IJ}$, the bulk connection equation is

$$D\!\left(\Sigma_{IJ}-\frac1\gamma e_I\wedge e_J\right)=0.$$

On internal bivectors it is controlled by the map

$$P_\gamma=\star-\frac1\gamma.$$

For real $\gamma\ne0$, this map is invertible. With an invertible coframe, the connection equation therefore implies

$$T^I=De^I=0.$$

The Holst contribution to the coframe equation is proportional to $e_J\wedge F^{IJ}$. On the torsion-free branch, the first Bianchi identity

$$DT^I=F^I{}_J\wedge e^J$$

sets this term to zero. The remaining equation is the Palatini coframe equation and hence Einstein’s equation. The classical vacuum solution space is therefore independent of real $\gamma$ within the invertible, torsion-free sector and for matched boundary conditions.

At $\gamma=\pm i$, $P_\gamma$ is proportional to a projector onto a complex self-dual or anti-self-dual sector instead of being invertible. The resulting variables are complex and recovering real Lorentzian geometry requires reality conditions. They are not obtained by simply inserting $\gamma=\pm i$ into every conclusion for real variables.

Holst versus Nieh–Yan

The Nieh–Yan identity in these conventions is

$$d(e_I\wedge T^I) =T_I\wedge T^I-e_I\wedge e_J\wedge F^{IJ}.$$

It shows that the Holst density is not itself a total derivative off shell. It differs from a torsion-squared term by the exact Nieh–Yan form. When $T^I=0$, the Holst bulk density vanishes, consistently with the Bianchi argument. When torsion is nonzero, these statements separate: the Holst term, the Nieh–Yan invariant, and a torsion-square term have different variations unless coefficients and boundary completions are chosen together.

Fermions provide the standard warning. A Dirac spin current sources contorsion, and eliminating that contorsion can make the effective classical interaction depend on $\gamma$. Certain nonminimal couplings reorganize this dependence into a Nieh–Yan term, but that is an additional matter assumption, not a property of the pure gravitational action.

Classical equivalence and quantum questions

For real $\gamma$, the canonical split leads to a family of classically related real connection variables. This chapter stops at that classical canonical transformation. Claims about holonomies, flux operators, spectra, black-hole state counting, or whether distinct $\gamma$ values define unitarily equivalent quantum theories belong to quantum gravity and do not follow from classical equivalence of the bulk equations.

Exercise: invert the Holst map

Find $P_\gamma^{-1}$ using $\star^2=-1$, and identify the values of $\gamma$ for which the inverse fails to exist.

Solution

Let $a=1/\gamma$. Then

$$(\star-a)(-\star-a)=-\star^2+a^2=1+a^2.$$

Therefore

$$P_\gamma^{-1} =\frac{-\star-\gamma^{-1}}{1+\gamma^{-2}}.$$

The denominator vanishes when

$$1+\gamma^{-2}=0, \qquad \gamma=\pm i.$$

It never vanishes for real nonzero $\gamma$. Thus the real connection equation is equivalent to the Palatini torsion equation, whereas the complex self-dual values require a separate analysis.

References

• Corichi, Alejandro, Irais Rubalcava-García, and Tatjana Vukašinac. “Actions, Topological Terms and Boundaries in First Order Gravity: A Review.” International Journal of Modern Physics D 25, no. 4 (2016): 1630011. DOI. Open PDF.
• Holst, Sören. “Barbero’s Hamiltonian Derived from a Generalized Hilbert–Palatini Action.” Physical Review D 53, no. 10 (1996): 5966–5969. DOI. Open PDF.