Euclidean On-Shell Action in Quantum Gravity

• Euclidean on-shell actions are defined by evaluating the gravitational action on solutions to the equations of motion, enabling the semiclassical saddle-point approximation of the path integral.
• The incorporation of first-order formalisms and boundary terms, such as the Gibbons-Hawking-York term, yields finite partition functions that correctly reproduce black hole energy and entropy values.
• Additional contributions from topological terms, supersymmetric localization, and foliation effects reveal quantum corrections and ambiguities that are central to contemporary gravitational thermodynamics.

Euclidean spacetime actions evaluated on-shell play a pivotal role in both semiclassical quantum gravity and gravitational thermodynamics. The on-shell action—meaning the action evaluated when the fields satisfy the equations of motion—governs the semiclassical saddle-point approximation of the gravitational path integral, encodes thermodynamic quantities of black holes and other gravitational instantons, and generates the partition function that interfaces with boundary or dual quantum field theories. The precise definition and evaluation of the Euclidean on-shell action is highly sensitive to choice of variational principle, treatment of boundaries, topological characteristics, and analytic continuations, and can be approached in multiple distinct frameworks, some of which have become central to contemporary gravitational research.

1. First-Order (Palatini) Formulation and Asymptotically Flat Euclidean Actions

The first-order or Palatini formalism recasts the gravitational action in terms of an orthonormal coframe $e^I$ and an independent Lorentz connection $A^{IJ}$. The metric is reconstructed as $g = \delta_{IJ} e^I \otimes e^J$. The first-order action without added counterterms is

$$I[e, A] = \frac{1}{2\kappa} \int_{M} \Sigma_{IJ} \wedge \Omega^{IJ} - \frac{1}{2\kappa} \oint_{\partial M} \Sigma_{IJ} \wedge A^{IJ}$$

where $\kappa = 8\pi$, $\Sigma^{IJ}$ is a $(D-2)$-form constructed from the coframe, and $\Omega^{IJ}$ is the curvature two-form.

Crucially, for asymptotically flat Euclidean spacetimes, this action is finite and stationary on-shell, provided one imposes the correct asymptotic expansions for the coframe and the Lorentz connection (e.g., $e$ and $A$ expanded in inverse powers of the radial coordinate $r$) (0810.0297). The resulting partition function in the stationary-phase approximation is $Z = \exp(-I[e_0, A_0])$, evaluated on the classical solution.

For the four-dimensional Euclidean Schwarzschild solution, this formalism yields the on-shell action $I = \beta^2 / (16\pi)$ and partition function $Z=\exp(-\beta^2/16\pi)$, with regularity at the horizon fixing $\beta=8\pi M$. The thermodynamic energy and entropy follow as $E=M$, $S=4\pi M^2 = A/4$, matching the Bekenstein-Hawking area law. NUT-charged solutions such as Taub-NUT exhibit analogous results, with $I=4\pi MN$ and entropy proportional to $N$ (0810.0297).

2. Boundary Terms, Membrane Paradigm, and Thermodynamic Interpretation

The proper definition of the Euclidean action and its thermodynamic interpretation involves careful attention to boundary terms. In the standard (metric) formulation, the action comprises a bulk Einstein-Hilbert term and a boundary term (typically the Gibbons-Hawking-York (GHY) term),

$$I_g = -\frac{1}{16\pi} \int d^4x \sqrt{g} R - \frac{1}{8\pi} \int d^3x \sqrt{\gamma} (K - K_0)$$

where $K$ is the extrinsic curvature on the boundary and $K_0$ is the reference value in flat spacetime (Lemos et al., 2011).

The membrane paradigm introduces a stretched horizon (timelike membrane just outside the event horizon), permitting the application of standard variational techniques. Properly accounting for the contributions from both external and membrane boundaries is essential: in the limit where the membrane approaches the horizon, the additional boundary entropy $S_{mb}$ recovers $S = A/4$. Analysis limited to gravitational and matter fields (excluding gauge/electric fields) guarantees that entropy is purely geometric and associated with the horizon area (Lemos et al., 2011).

These structural features yield a universal thermodynamic relation: $I = \beta E - S$ with $Z \approx e^{-I}$, $E = -\partial_\beta \ln Z$, $S = \beta E - I$, seamlessly connecting the on-shell action to black hole thermodynamics.

3. Topological Contributions, Holst Term, and the Barbero-Immirzi Parameter

Topologically nontrivial terms—especially those associated with the Holst action and the Barbero-Immirzi parameter $\gamma$—can affect the value and interpretation of the Euclidean on-shell action. Nonvanishing contributions from the Holst term

$$I_{Holst} \propto \frac{1}{\gamma} \int_M \Omega \wedge \Omega$$

arise only if the coframe is non-diagonalizable (i.e., $e \wedge de \neq 0$) and the Pontryagin number of the manifold is nonzero (Liko, 2011). The Taub-NUT-ADS geometry provides a concrete example: the Holst term produces finite shifts in the energy and entropy associated with the NUT charge, with the partition function acquiring a $\gamma$-dependent phase. This demonstrates that topological sectors in the Euclidean path integral can probe quantization ambiguities and have physical semiclassical consequences, notably shifting the on-shell partition function and thus all derived thermodynamic observables (Liko, 2011).

4. Generalizations: Observer Dependence, Signature Interpolation, and Foliation Effects

The explicit construction of the Euclidean metric from Lorentzian data, based on foliation by a timelike vector field $u^a$, yields important corrections to the standard continuation $t \rightarrow -i\tau$ (Kothawala, 2017). Defining a parameter $\Theta(\lambda)$ which interpolates the signature between Lorentzian ($\Theta=0$) and Euclidean ($\Theta=-2$), the Ricci scalar associated with the interpolated metric contains extra foliation-dependent terms: $$\widehat{R} = (1+\Theta)R - \Theta\, {}^3R + (d\Theta/d\lambda)K$$ where ${}^3R$ is the curvature scalar of the spatial slice and $K$ is its extrinsic curvature. In the Lorentzian limit, this reproduces the standard Einstein-Hilbert Lagrangian with the GHY boundary term, but in the Euclidean regime an additional $2\,{}^3R$ term appears, encoding observer/foliation dependence (Kothawala, 2017). This approach suggests an intrinsic relation between the choice of foliation and the action weighting in the gravitational path integral, impacting both quantum cosmology and the interpretation of no-boundary proposals.

5. Extensions: Modular Hamiltonians, Replica Trick, and Boundary Fluctuations

In spacetimes with bifurcate Killing horizons, such as causal diamonds in flat space, the on-shell Euclidean action on the $n$-fold replica space directly encodes the moments of the modular Hamiltonian $K$. Explicit path integral constructions show that

$$\langle K \rangle = \langle (\Delta K)^2 \rangle = \frac{A_{\mathcal{B}}}{4 G_N}$$

where $A_{\mathcal{B}}$ is the area of the bifurcate horizon (Fransen et al., 30 Jul 2025). This framework provides a concrete realization of the relation between Rényi entropies (and modular free energy) and the on-shell action, generalizing entanglement entropy calculations to gravitational theories and linking geometric entropy bounds to quantum statistical mechanics.

Moreover, in the context of soft graviton dynamics and memory/shockwave boundary conditions at null infinity, the on-shell infrared action reduces to a gauge-invariant "corner" integral,

$$S_{OS} = -\frac{1}{16\pi G_N} \int_S d^2z\, {}_z^2 C\, {}^2 N$$

where $C$ and $N$ parameterize the soft Goldstone and memory modes, respectively (He et al., 2 Aug 2024). The resulting action is equivalent to soft supertranslation charges and to the modular Hamiltonian, with quantum fluctuations exhibiting an area law, further highlighting the deep interplay between boundary data, horizon symmetries, and thermodynamics.

6. Supersymmetry, Localization, and Higher-Dimensional Generalizations

Supersymmetric Euclidean actions, constructed via off-shell reduction from higher dimensions and using extended superconformal calculus, feature multiplets (Weyl, vector, tensor, hyper) with characteristic transformation properties and reality conditions unique to Euclidean signature (Wit et al., 2017). Off-shell approaches enable the construction of invariant densities and facilitate supersymmetric localization techniques, crucial for computing black hole entropy and exact partition functions. Notably, chiral and anti-chiral multiplets become independent and real, while the R-symmetry structure is modified to $SU(2)\times SO(1,1)$. Supersymmetric actions obtained in this formalism are not necessarily bounded from below, a marked difference from their Minkowski analogues.

Equivariant localization extends these ideas, allowing the on-shell supergravity action to be precisely evaluated in odd and even dimensions by localizing the integrand to fixed-point sets of symmetry generators (R-symmetry Killing vectors) (Couzens et al., 9 Apr 2025, Cassani et al., 2 Sep 2024). In Romans supergravity, for instance, the on-shell action is rendered in terms of equivariant data and topological invariants of fixed loci (zero-, two-, and four-dimensional fixed sets), obviating the need to solve the full supergravity field equations.

7. Physical and Conceptual Implications

The Euclidean on-shell spacetime action, particularly in semiclassical and supersymmetric gravity, is the linchpin connecting gravitational path integrals, black hole thermodynamics, topological invariants, and quantum entropy bounds. It encodes the semiclassical weighting of saddle points, determines the statistical mechanics of both geometric and topological features of spacetime, enables precise Witten index calculations in supersymmetric theories, and is an essential element in holographic dualities (e.g., relating the on-shell SUGRA action to conformal anomalies and free energies in dual field theories as in AdS/CFT (Chakrabarti et al., 2022)). The action is sensitive to topological sectors (via the Holst/Pontryagin terms), foliation and observer structure, and boundary data, and must be carefully regulated by additional boundary terms to ensure finiteness and a well-posed variational problem, particularly outside of the AdS context (Dei et al., 14 Aug 2025).

Recent developments extend these ideas to more general spacetimes: inclusion of memory and soft gravitational effects, near-equilibrium steady-states in non-equilibrium settings (with Onsager reciprocity and entropic force relations (An, 14 Jul 2025)), and matching between bulk on-shell actions and relevant deformed boundary field theory quantities as in $T\overline{T}$-deformed holographic correspondences (Dei et al., 14 Aug 2025). In even-dimensional de Sitter spacetimes, the imaginary part of the effective Euclidean action encodes non-perturbative pair production (analogous to the QED Schwinger effect), with the caveat that perturbation theory fails to capture these essential non-analytic contributions (Zhou et al., 28 Oct 2024).

In all these approaches, the Euclidean on-shell action is the universal generator of gravitational thermodynamics, quantum entropy, and interface with boundary field theory; its precise evaluation and interpretation constitute a central problem and tool in contemporary gravitational research.