Calabi-Yau manifolds appear in the expression for the energy emission rate in blackhole scattering

I received a like to a very interesting Nature article reporting the work of Driesse et al on calculation of gravitational scattering amplitudes of blackholes in a quantum field theory model. The title of the article (see this) is “Emergence of Calabi Yau manifolds in high-precision black-hole scattering”. One can also a popular article (see this) describing the findings. The motivation is that blackholes are are elementary particle-like objects characterized byt only mass, spin, and charge.

Let us look first at the abstract of the article.

When two massive objects (black holes, neutron stars or stars) in our universe fly past each other, their gravitational interactions deflect their trajectories. The gravitational waves emitted in the related bound-orbit system-the binary inspiral-are now routinely detected by gravitational-wave observatories. Theoretical physics needs to provide high-precision templates to make use of unprecedented sensitivity and precision of the data from upcoming gravitational-wave observatories. Motivated by this challenge, several analytical and numerical techniques have been developed to approximately solve this gravitational two-body problem. Although numerical relativity is accurate it is too time-consuming to rapidly produce large numbers of gravitational-wave templates. For this, approximate analytical results are also required. Here we report on a new, highest-precision analytical result for the scattering angle, radiated energy and recoil of a black hole or neutron star scattering encounter at the fifth order in Newton’s gravitational coupling G, assuming a hierarchy in the two masses. This is achieved by modifying state-of-the-art techniques for the scattering of elementary particles in colliders to this classical physics problem in our universe. Our results show that mathematical functions related to Calabi-Yau (CY) manifolds, 2n-dimensional generalizations of tori, appear in the solution to the radiated energy in these scatterings. We anticipate that our analytical results will allow the development of a new generation of gravitational-wave models, for which the transition to the bound-state problem through analytic continuation and strong-field resummation will need to be performed.

These findings look interesting from the TGD point of view. Calabi-Yau (CY) manifolds have an arbitrary complex dimension n. They generalize the notion of periodic orbit. In 1-D case orbit becomes a complex 2-D manifold, elliptic surface. But complex differential geometry allows a generalization to n-D real periodic orbits and their complex counterparts.

1. Torus is the simplest CY and 2-real-D elliptic doubly periodic surfaces appearing in complex analysis represent the basic example. I have discussed their representations at the level of space-time surfaces in the framework provided by holography= holomorphy vision. Weierstrass surfaces is one example (see this).

The periods of planetary orbits in Coulomb force expressible in terms of elliptic integrals very probably led to the notion of elliptic Riemann surfaces by making the time variable complex. Elliptic Riemann surfaces are compact but define doubly periodic structures when represented in complex plane? Could the two periods define analogs of momenta?

The K-surface (see this) is a 4-(real)-dimensional CY manifold and a purely algebraic object having a unique topology. It appears in fourth order G⁴ in the calculation. K3 surface allows a Kähler metric. It is not clear to me how unique this metric is. The existence of the Kähler metric is important from the TGD point of view since induced metric codes for the Riemannian geometric aspects of TGD.

Holography= holomorphy vision reduces TGD to algebraic geometry, which can be also regarded as Riemann geometry. Therefore an interesting question is whether a 4-real-D complex K3 surface could be represented in the TGD framework as a complex surface. Does Euclidean signature prevent this or does the K3 surface have a Minkowskian analog obtained by making the second complex coordinate hypercomplex?

K3 surface can be represented as Fermat quartic surface x⁴+y⁴+z⁴+t⁴=0 in the twistor space CP₃ assigned M⁴. Twistor space of M⁴ and CP₂ play a key role in the twistorialization of TGD (see this).

In TGD, CP₃ generalizes to its hypercomplex variant with one complex coordinate made hyperbolic and corresponds to SU(3,1)/SU(3)×U(1) (see this). This generalization allows to identify the base space of the twistor bundle as M⁴, rather than its compactified version. The hyperbolic counterpart of quartic Fermat surface might serve as a one particular space-time surface in holography= holomorphy vision.

In TGD, a generalized complex manifold is obtained from a complex manifold by making one complex coordinate hypercomplex. H=M⁴×CP₂ and space-time surfaces Xsup>4 in H are generalized complex manifolds. Suppose that the double periodicity of the 2-dimensional case generalizes so that the hyperbolic variant of K3 surface could correspond to a lattice cell of a 4-D periodic structure. Could one assign the hyperbolic counterpart of K3 surfasce a 4-D variant of a plane wave? This would conform with the view that gravitational waves are involved with the scattering of blackholes. Could kind of representation generalize to all kinds of plane waves and could K3 be one of the simplest examples?

The 3-complex-dimensional CYs were not mentioned in the article. They appear in the spontaneous compactification of the string models. Now the topology is not unique and the famous number 10⁵⁰⁰ was introduced as a rough estimate for their number. This turned out to be an untestable and fatal production.

What a 3-real-dimensional periodic “orbit” and its complex generalization could mean? By holography= holomorphy vision, space-time surfaces are representable as intersections of 2 3-D generalized complex manifolds X⁶ in H and could be seen as analogs of twistor spaces for Msup>4 and CP₂. Also the twistor space CP₃ is a CY manifold.

Could one of these 2 surfaces X⁶ be a generalized CY manifold with one hypercomplex coordinate in some cases? 6-D real periodicity would requires double periodicity in hyperbolic coordinate, which looks unrealistic with one hypercomplex coordinate could serve as a physical signature? 3-D (2-D) generalized complex homology would be non-trivial for these 6-surfaces (space-time surfaces).

For the implications of holography= holomorphy vision in TGD see for instance this , this, and this.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/05/calabi-yau-manifolds-appear-in.html