Constrained Gravitational Instantons

• Constrained gravitational instantons are non-perturbative Euclidean solutions with additional constraint functionals beyond the Einstein–Hilbert action.
• They are constructed using advanced gluing, collapse, and hyperKähler perturbations to produce metrics with fixed topology and boundary conditions.
• These instantons contribute to moduli stabilization, influence path integral corrections, and affect axion inflation through precise non-perturbative effects.

Constrained gravitational instantons are non-perturbative Euclidean solutions in quantum gravity and supergravity that satisfy additional (often non-covariant) constraints beyond the standard Einstein–Hilbert variational principle. These configurations contribute to path integrals, scalar potentials, and supersymmetric superpotentials in sectors with fixed topology or boundary data. Recent constructions range from wormhole metrics dominating AdS path integrals (Cotler et al., 2020), to non-linear superpositions and codimension–1 gluing of Atiyah–Hitchin and Taub–NUT geometries (Schroers et al., 2020, Salm, 24 Jun 2024), to gauge-invariant contributions in flux compactifications (Grimm et al., 2011), and precision bounds in axion inflation (Hebecker et al., 2016). The subject intricately connects to moduli stabilization, topological classification of instanton families, and the extension of Weak Gravity Conjecture principles to non-perturbative settings.

1. Formal Definition and Variational Structure

The general framework for constrained gravitational instantons arises from the Einstein–Hilbert action in $d+1$ Euclidean dimensions,

$$S_{\rm EH}[g] = -\frac{1}{16\pi G} \int_{\mathcal{M}} d^{d+1}x\,\sqrt{g}\,(R - 2\Lambda)\,, \qquad \Lambda = -\frac{d(d-1)}{2L^2}\,,$$

augmented with one or more constraint functionals,

$$C[g] = \frac{1}{8\pi G} \int d^{d+1}x\,\Lambda\,\sqrt{g_{\rho\rho}}\,F(x)\,,$$

imposed via Lagrange multipliers $\lambda$ so that

$$S_{\rm cons}[g,\lambda] = S_{\rm EH}[g] + \lambda\,C[g]\,.$$

Variation with respect to $g_{\mu\nu}$ and $\lambda$ at fixed constraint yields modified field equations with correction terms proportional to the constraints. Physical solutions with $\lambda \neq 0$ are referred to as constrained instantons (Cotler et al., 2020).

These instantons are not always classical saddle points of $S_{\rm EH}$ but can dominate contributions in path integrals with fixed topology, boundary conformal structure, or topological charge.

2. Classification by Cosmological Constant and Boundary Conditions

Constrained instantons exhibit a rich zoo of solutions depending on the sign and value of the cosmological constant $\Lambda$:

• AdS wormholes ($\Lambda<0$): Metrics solve modified equations connecting two asymptotic AdS regions, e.g.

$$ds^2 = d\rho^2 + b^2 [2\cosh(\tfrac{d\rho}{2L})]^{4/d} \delta_{ij} dy^i dy^j\,,$$

where $b$ and twist parameters $\tau^i$ label instanton moduli (Cotler et al., 2020). These provide leading semiclassical contributions in the two-boundary AdS path integral and encode coarse-grained energy-level statistics of black hole microstates.

• Flat-space and de Sitter instantons ($\Lambda=0$ and $\Lambda>0$): Solutions include big bang/crunch cosmologies and quantum bounces, with half-line or toroidal boundary topologies.
• Non-maximal volume growth instantons: Types ALG, ALG*, ALH, ALH* arise in codimension-1 collapse constructions, where the metric collapses away from exceptional points modeled on Taub–NUT or Atiyah–Hitchin metrics (Salm, 24 Jun 2024). At the ends, metrics glue into precise local models depending on the fixed-point structure of $\mathbb{Z}_2$ actions on the base space.

Boundary conditions are fixed by conformal structure (e.g., $b$, $\beta_{i}$ for thermal circles) and topology (e.g., $\mathbb{T}^d\times I$). The constraint parameter $\zeta$ (wormhole length or energy) labels continuous families of constrained solutions.

3. Construction Methodologies: Gluing, Collapse, and HyperKähler Triples

Modern instanton construction employs advanced gluing and collapse techniques:

• Non-linear superpositions: $D_k$ ALF gravitational instantons are built via 5-manifold fibrations, with constituent Atiyah–Hitchin (AH) and Taub–NUT (TN) geometries appearing as boundary components in the adiabatic parameter limit $\varepsilon \to 0$ (Schroers et al., 2020).
• Codimension-1 collapse: Instantons of types ALG, ALH, etc., are constructed by removing small balls around exceptional points (fixed points $q_j$ for AH ends, nuts $p_i$ for TN ends) in a flat 3-manifold base, then gluing in local models using balanced harmonic functions and $S^1$-bundle connections (Salm, 24 Jun 2024). A global closed triple of hyperKähler forms is assembled, matching exactly to local models outside small gluing regions and satisfying desired decay.

$$g_\epsilon^{GH} = h_\epsilon\,g_B + \frac{\epsilon^2}{h_\epsilon}\,\eta^2\,,\quad \omega_i^{GH} = \epsilon\,dx_i\wedge\eta + h_\epsilon\,{}^B{*}\,dx_i$$

• HyperKähler perturbations: The glued approximate triple is corrected to an exact hyperKähler triple using inverse-function–theorem techniques in weighted Sobolev spaces. This yields complete, smooth instanton metrics classifiable by moduli count, intersection forms, and asymptotic rates (Salm, 24 Jun 2024, Schroers et al., 2020).

4. Instanton Actions, Induced Potentials, and Trustworthiness

Gravitational instantons induce non-perturbative corrections to scalar potentials via dilute-gas summation: $$V(\theta) \simeq A\,e^{-S_{\rm inst}} \cos(n\,\theta/f_{\rm ax})$$ where all coefficients and actions depend on instanton family. For axion systems (Hebecker et al., 2016):

• Extremal instantons (C=0): $S_{\rm inst} = (2/\alpha)\,(n/f_{\rm ax})$
• Cored instantons (C>0): Actions exceed extremal, with discrete moduli.
• Wormholes (C<0): $$S_{\rm inst} = (2/\alpha)\,(n/f_{\rm ax})\,\sin[(\alpha\,\pi/4)\sqrt{3/2}]$$

Determinant prefactor $A$ is model dependent; string compactifications yield $A \lesssim \mathcal{V}^{-5/3}$ for Calabi–Yau volume $\mathcal{V}$.

UV completion imposes trustworthiness conditions: the 4D effective action breaks down below KK or moduli mass scales, requiring $S_{\rm inst} \gtrsim c \cdot r_c^2$ for cutoff $r_c$. For all plausible cutoffs, instanton-induced $\delta V$ is far below inflationary potential scales $V_{\rm inf}$.

5. Gauge and Flux Constraints: Anomaly Cancellation and M/F-Theory Embedding

In circle compactifications with flux-induced gaugings, gravitational instantons (notably Taub–NUT) acquire constraints to maintain gauge invariance (Grimm et al., 2011):

• Without fluxes: Taub–NUT instantons generate $W_{TN} \sim A\,e^{-2\pi T_0}$, where $T_0$ is the radion multiplet.
• With fluxes: The naive $e^{-2\pi T_0}$ breaks gauge invariance unless bound gauge instantons supply compensating charge $n^i$: $W \sim e^{-2\pi T_0 - 2\pi n^i T_i}\,.$ Gauge invariance requires anomaly cancellation $M_\alpha - i n^i X_{i\alpha} = 0$ (pullback vanishing), realized in M-theory as a Freed–Witten-type constraint $G_4|_D = 0$ for the 4-form flux $G_4$ on M5-instanton worldvolume.

Such constrained instantons contribute to superpotentials, stabilize moduli, and can give AdS$_3$ vacua. The resulting Kähler and superpotential structure fits into the broader landscape of supersymmetric lower-dimensional compactifications and non-perturbative effects.

6. Stability, Physical Interpretation, and Connections to Black Hole Microstates

Constrained gravitational instantons are generically stable under quadratic fluctuations:

• The symmetric AdS torus wormhole exhibits no negative eigenmodes for $d\ge 2$, and scalar, vector, and tensor fluctuations possess positive-definite kinetic and mass terms (Cotler et al., 2020).
• Minimally coupled fields obey stability bounds (e.g., $m^2 \ge m^2_{\rm BF} = -d^2/4$).

From a quantum gravity perspective, constrained wormholes dominate sectors with fixed topology and encode ensemble-averaged statistics of energy levels, especially in AdS black hole microstates. Their on-shell, holographically renormalized action reduces to pure boundary contributions,

$$S_{\rm ren} = \frac{(d-1)\,\mathrm{Vol}(\mathbb{T}^d)}{2\pi G}\,b^d$$

and more generally, combinations of boundary temperatures. This suggests deep links between semiclassical gravity, random matrix theory, and quantum black hole statistics.

7. Classification and Moduli Spaces of Instantons with Constraints

Gravitational instantons are classified by their asymptotic geometry, volume growth, and moduli:

• ALE ($r^4$), ALF ($r^3$), ALG ($r^2$), ALH ($r$), ALG* ($r^2$), ALH* ($r^{4/3}$) correspond to manifold ends and singularity structures built via gluing and collapse (Salm, 24 Jun 2024).
• Exceptional points (fixed points $q_j$, nuts $p_i$) are locally modeled on AH or TN metrics. The number of moduli and intersection forms match the extended Dynkin diagram $D_n$ or its degenerations.
• The continuous gluing parameters and collapse scale are in direct correspondence with the dimension of moduli spaces for each instanton family.

A plausible implication is that the structure and classification of constrained instantons provide a road map for enumerating all possible semiclassical non-perturbative sectors in quantum gravity, informing both phenomenological model building and the ongoing development of black hole microstate models.