Gravitational instanton

Gravitational instantons are interesting, as they offer insights into the quantization of gravity. For example, positive definite asymptotically locally Euclidean metrics are needed as they obey the positive-action conjecture; actions that are unbounded below create divergence in the quantum path integral.

• A four-dimensional Ricci-flat Kähler manifold has anti-self-dual Riemann tensor with respect to the complex orientation.
• Consequently, a simply-connected anti-self-dual gravitational instanton is a four-dimensional complete hyperkähler manifold.
• Gravitational instantons are analogous to self-dual Yang–Mills instantons.

Several distinctions can be made with respect to the structure of the Riemann curvature tensor, pertaining to flatness and self-duality. These include:

• Einstein (non-zero cosmological constant)
• Ricci flatness (vanishing Ricci tensor)
• Conformal flatness (vanishing Weyl tensor)
• Self-duality
• Anti-self-duality
• Conformally self-dual
• Conformally anti-self-dual

By specifying the 'boundary conditions', i.e. the asymptotics of the metric 'at infinity' on a noncompact Riemannian manifold, gravitational instantons are divided into a few classes, such as asymptotically locally Euclidean spaces (ALE spaces), asymptotically locally flat spaces (ALF spaces).

They can be further characterized by whether the Riemann tensor is self-dual, whether the Weyl tensor is self-dual, or neither; whether or not they are Kähler manifolds; and various characteristic classes, such as Euler characteristic, the Hirzebruch signature (Pontryagin class), the Rarita–Schwinger index (spin-3/2 index), or generally the Chern class. The ability to support a spin structure (i.e. to allow consistent Dirac spinors) is another appealing feature.

Eguchi et al. list a number of examples of gravitational instantons.^[1] These include, among others:

• Flat space ${\displaystyle \mathbb {R} ^{4}}$ , the torus ${\displaystyle \mathbb {T} ^{4}}$  and the Euclidean de Sitter space ${\displaystyle \mathbb {S} ^{4}}$ , i.e. the standard metric on the 4-sphere.
• The product of spheres ${\displaystyle S^{2}\times S^{2}}$ .
• The Schwarzschild metric ${\displaystyle \mathbb {R} ^{2}\times S^{2}}$  and the Kerr metric ${\displaystyle \mathbb {R} ^{2}\times S^{2}}$ .
• The Eguchi–Hanson instanton ${\displaystyle T^{*}\mathbb {CP} (1)}$ , given below.
• The Taub–NUT solution, given below.
• The Fubini–Study metric on the complex projective plane ${\displaystyle \mathbb {CP} (2).}$ ^[2] Note that the complex projective plane does not support well-defined Dirac spinors. That is, it is not a spin structure. It can be given a spinc structure, however.
• The Page space, which exhibits an explicit Einstein metric on the connected sum of two oppositely oriented complex projective planes ${\displaystyle \mathbb {CP} (2)\#{\overline {\mathbb {CP} }}(2)}$ .
• The Gibbons–Hawking multi-center metrics, given below.
• The Taub-bolt metric ${\displaystyle \mathbb {CP} (2)\setminus \{0\}}$  and the rotating Taub-bolt metric. The "bolt" metrics have a cylindrical-type coordinate singularity at the origin, as compared to the "nut" metrics, which have a sphere coordinate singularity. In both cases, the coordinate singularity can be removed by switching to Euclidean coordinates at the origin.
• The K3 surfaces.
• The ALE (asymptotically locally Euclidean) anti-self-dual manifolds. Among these, the simply connected ones are all hyper-Kähler, and each one is asymptotic to a flat cone over ${\displaystyle S^{3}=SU(2)}$  modulo a finite subgroup. Each finite sub-group of ${\displaystyle SU(2)}$  actually occurs. The complete list of possibilities consists of the cyclic groups ${\displaystyle \mathbb {Z} _{k+1}}$  together with the inverse images of the dihedral groups, the tetrahedral group, the octahedral group, and the icosahedral group under the double cover ${\displaystyle SU(2)\to SO(3)}$ . Note that ${\displaystyle \mathbb {Z} _{2}}$  corresponds to the Eguchi–Hanson instanton, while for higher k, the cyclic group ${\displaystyle \mathbb {Z} _{k+1}}$  corresponds to the Gibbons–Hawking multi-center metrics, each of which diffeomorphic to the space obtained from the disjoint union of k copies of ${\displaystyle T^{*}S^{2}}$  by using the Dynkin diagram ${\displaystyle A_{k}}$  as a plumbing diagram.

This is a very incomplete list; there are many other possibilities, not all of which have been classified.

It will be convenient to write the gravitational instanton solutions below using left-invariant 1-forms on the three-sphere S³ (viewed as the group Sp(1) or SU(2)). These can be defined in terms of Euler angles by

${\displaystyle {\begin{aligned}\sigma _{1}&=\sin \psi \,d\theta -\cos \psi \sin \theta \,d\phi \\\sigma _{2}&=\cos \psi \,d\theta +\sin \psi \sin \theta \,d\phi \\\sigma _{3}&=d\psi +\cos \theta \,d\phi .\\\end{aligned}}}$ 

Note that ${\displaystyle d\sigma _{i}+\sigma _{j}\wedge \sigma _{k}=0}$  for ${\displaystyle i,j,k=1,2,3}$  cyclic.

${\displaystyle ds^{2}={\frac {1}{4}}{\frac {r+n}{r-n}}dr^{2}+{\frac {r-n}{r+n}}n^{2}{\sigma _{3}}^{2}+{\frac {1}{4}}(r^{2}-n^{2})({\sigma _{1}}^{2}+{\sigma _{2}}^{2})}$ 

The Eguchi–Hanson space is defined by a metric the cotangent bundle of the 2-sphere ${\displaystyle T^{*}\mathbb {CP} (1)=T^{*}S^{2}}$ . This metric is

${\displaystyle ds^{2}=\left(1-{\frac {a}{r^{4}}}\right)^{-1}dr^{2}+{\frac {r^{2}}{4}}\left(1-{\frac {a}{r^{4}}}\right){\sigma _{3}}^{2}+{\frac {r^{2}}{4}}(\sigma _{1}^{2}+\sigma _{2}^{2}).}$ 

where ${\displaystyle r\geq a^{1/4}}$ . This metric is smooth everywhere if it has no conical singularity at ${\displaystyle r\rightarrow a^{1/4}}$ , ${\displaystyle \theta =0,\pi }$ . For ${\displaystyle a=0}$  this happens if ${\displaystyle \psi }$  has a period of ${\displaystyle 4\pi }$ , which gives a flat metric on R⁴; However, for ${\displaystyle a\neq 0}$  this happens if ${\displaystyle \psi }$  has a period of ${\displaystyle 2\pi }$ .

Asymptotically (i.e., in the limit ${\displaystyle r\rightarrow \infty }$ ) the metric looks like

${\displaystyle ds^{2}=dr^{2}+{\frac {r^{2}}{4}}\sigma _{3}^{2}+{\frac {r^{2}}{4}}(\sigma _{1}^{2}+\sigma _{2}^{2})}$ 

which naively seems as the flat metric on R⁴. However, for ${\displaystyle a\neq 0}$ , ${\displaystyle \psi }$  has only half the usual periodicity, as we have seen. Thus the metric is asymptotically R⁴ with the identification ${\displaystyle \psi \,{\sim }\,\psi +2\pi }$ , which is a Z₂ subgroup of SO(4), the rotation group of R⁴. Therefore, the metric is said to be asymptotically R⁴/Z₂.

There is a transformation to another coordinate system, in which the metric looks like

${\displaystyle ds^{2}={\frac {1}{V(\mathbf {x} )}}(d\psi +{\boldsymbol {\omega }}\cdot d\mathbf {x} )^{2}+V(\mathbf {x} )d\mathbf {x} \cdot d\mathbf {x} ,}$ 

where ${\displaystyle \nabla V=\pm \nabla \times {\boldsymbol {\omega }},\quad V=\sum _{i=1}^{2}{\frac {1}{|\mathbf {x} -\mathbf {x} _{i}|}}.}$ 

(For a = 0, ${\displaystyle V={\frac {1}{|\mathbf {x} |}}}$ , and the new coordinates are defined as follows: one first defines ${\displaystyle \rho =r^{2}/4}$  and then parametrizes ${\displaystyle \rho }$ , ${\displaystyle \theta }$  and ${\displaystyle \phi }$  by the R³ coordinates ${\displaystyle \mathbf {x} }$ , i.e. ${\displaystyle \mathbf {x} =(\rho \sin \theta \cos \phi ,\rho \sin \theta \sin \phi ,\rho \cos \theta )}$ ).

In the new coordinates, ${\displaystyle \psi }$  has the usual periodicity ${\displaystyle \psi \ {\sim }\ \psi +4\pi .}$ 

One may replace V by

${\displaystyle \quad V=\sum _{i=1}^{n}{\frac {1}{|\mathbf {x} -\mathbf {x} _{i}|}}.}$ 

For some n points ${\displaystyle \mathbf {x} _{i}}$ , i = 1, 2..., n. This gives a multi-center Eguchi–Hanson gravitational instanton, which is again smooth everywhere if the angular coordinates have the usual periodicities (to avoid conical singularities). The asymptotic limit (${\displaystyle r\rightarrow \infty }$ ) is equivalent to taking all ${\displaystyle \mathbf {x} _{i}}$  to zero, and by changing coordinates back to r, ${\displaystyle \theta }$  and ${\displaystyle \phi }$ , and redefining ${\displaystyle r\rightarrow r/{\sqrt {n}}}$ , we get the asymptotic metric

${\displaystyle ds^{2}=dr^{2}+{\frac {r^{2}}{4}}\left({d\psi \over n}+\cos \theta \,d\phi \right)^{2}+{\frac {r^{2}}{4}}[(\sigma _{1}^{L})^{2}+(\sigma _{2}^{L})^{2}].}$ 

This is R⁴/Z_n = C²/Z_n, because it is R⁴ with the angular coordinate ${\displaystyle \psi }$  replaced by ${\displaystyle \psi /n}$ , which has the wrong periodicity (${\displaystyle 4\pi /n}$  instead of ${\displaystyle 4\pi }$ ). In other words, it is R⁴ identified under ${\displaystyle \psi \ {\sim }\ \psi +4\pi k/n}$ , or, equivalently, C² identified under z_i ~ ${\displaystyle e^{2\pi ik/n}}$  z_i for i = 1, 2.

To conclude, the multi-center Eguchi–Hanson geometry is a Kähler Ricci flat geometry which is asymptotically C²/Z_n. According to Yau's theorem this is the only geometry satisfying these properties. Therefore, this is also the geometry of a C²/Z_n orbifold in string theory after its conical singularity has been smoothed away by its "blow up" (i.e., deformation).^[3]

The Gibbons–Hawking multi-center metrics are given by^[4][5]

${\displaystyle ds^{2}={\frac {1}{V(\mathbf {x} )}}(d\tau +{\boldsymbol {\omega }}\cdot d\mathbf {x} )^{2}+V(\mathbf {x} )d\mathbf {x} \cdot d\mathbf {x} ,}$ 

where

${\displaystyle \nabla V=\pm \nabla \times {\boldsymbol {\omega }},\quad V=\varepsilon +2M\sum _{i=1}^{k}{\frac {1}{|\mathbf {x} -\mathbf {x} _{i}|}}.}$ 

Here, ${\displaystyle \epsilon =1}$  corresponds to multi-Taub–NUT, ${\displaystyle \epsilon =0}$  and ${\displaystyle k=1}$  is flat space, and ${\displaystyle \epsilon =0}$  and ${\displaystyle k=2}$  is the Eguchi–Hanson solution (in different coordinates).

In 2021 it was found^[6] that if one views the curvature parameter of a foliated maximally symmetric space as a continuous function, the gravitational action, as a sum of the Einstein–Hilbert action and the Gibbons–Hawking–York boundary term, becomes that of a point particle. Then the trajectory is the scale factor and the curvature parameter is viewed as the potential. For the solutions restricted like this, general relativity takes the form of a topological Yang–Mills theory.

• Gravitational anomaly
• Hyperkähler manifold

• Gibbons, G.W.; Hawking, S.W. (October 1978). "Gravitational multi-instantons". Physics Letters B. 78 (4): 430–432. Bibcode:1978PhLB...78..430G. doi:10.1016/0370-2693(78)90478-1.
• Gibbons, G. W.; Hawking, S. W. (October 1979). "Classification of Gravitational Instanton symmetries". Communications in Mathematical Physics. 66 (3): 291–310. Bibcode:1979CMaPh..66..291G. doi:10.1007/BF01197189. S2CID 123183399.
• Eguchi, Tohru; Hanson, Andrew J. (April 1978). "Asymptotically flat self-dual solutions to euclidean gravity". Physics Letters B. 74 (3): 249–251. Bibcode:1978PhLB...74..249E. doi:10.1016/0370-2693(78)90566-X. OSTI 1446816. S2CID 16380482.
• Eguchi, Tohru; Hanson, Andrew J (July 1979). "Self-dual solutions to euclidean gravity". Annals of Physics. 120 (1): 82–106. Bibcode:1979AnPhy.120...82E. doi:10.1016/0003-4916(79)90282-3. OSTI 1447072. S2CID 48866858.
• Eguchi, Tohru; Hanson, Andrew J. (December 1979). "Gravitational instantons". General Relativity and Gravitation. 11 (5): 315–320. Bibcode:1979GReGr..11..315E. doi:10.1007/BF00759271. S2CID 123806150.
• Kronheimer, P. B. (1989). "The construction of ALE spaces as hyper-Kähler quotients". Journal of Differential Geometry. 29 (3): 665–683. doi:10.4310/jdg/1214443066.