The key to quantum gravity theory may lie within harmonic superspace.

A series of articles based on their research has been published in recent years in the Journal of High Energy Physics. It is well known that the classical Einsteinian general theory of relativity excellently explains macroscopic gravitational phenomena, but its description using quantum field theory leads to significant difficulties. This may imply that general relativity is an effective theory and likely requires modification at high energies.

It is not possible to construct a quantum theory of gravity within the framework of general relativity essentially due to the so-called renormalization problem. In quantum field theory, each particle is viewed as a quantum of the corresponding field. Particles interact with one another by emitting and absorbing virtual particles, which is vividly described using Feynman diagrams.

The interaction of a particle with virtual particles leads to changes in its observable characteristics depending on the energy scale, such as electric charge. This phenomenon is known as charge screening. Within the framework of quantum field theory, such behavior of electric charge is formulated in terms of the renormalization procedure, which is structured differently in various field theories. Specifically, attempting to carry out renormalization in quantum gravity leads to the necessity of introducing an infinite number of new interaction constants in addition to the gravitational one.

Thus, the theory of gravity at the quantum level loses its predictive power.
To study effects at very high energies, new models need to be developed that incorporate additional degrees of freedom and can provide consistent predictions at any energy levels. Current research suggests that there may need to be infinitely many such degrees of freedom.

A lack of understanding of how gravity behaves at extremely high energies, where quantum laws begin to apply, is one of the major gaps in our understanding of phenomena such as the initial stage of the universe's evolution or the final stage of black hole evolution. The influence of quantum effects on gravity becomes noticeable at Planck scales (approximately 10^19 GeV), which far exceed the capabilities of current accelerators, typically around 10^4 GeV. Therefore, the only approach to studying quantum gravity remains theoretical research based on hypotheses that at Planck scales, significant new degrees of freedom, suppressed at low energies, must play a crucial role, and that all local and global symmetries of nature are unbroken.

In the 1970s, attempts were made to construct a theory of strong interactions using the idea of relativistic strings. It was later realized that the string concept could potentially provide a solution to the problem of quantum gravity, leading to the understanding of the importance of higher spin fields as new degrees of freedom necessary for modifying the theory of gravity at extremely high energies.

In string theory, the states of strings correspond to particles with different spins. States with higher spins (greater than two) have large masses and do not contribute to physical processes at low energies. In the language of field theory, such states correspond to higher spin fields. This indicates that string theory might be one of the phases of a theory of massless higher spin fields, where these fields acquire mass as a consequence of an analogue of the Higgs mechanism. At very high energies, on the contrary, all particles can be considered massless, and the symmetry of the theory significantly expands. This gives rise to natural tasks of studying the structure of massless higher spin field theories, higher spin symmetries, mechanisms of their breaking, interaction structures, and so forth.

Currently, the theory of higher spin fields is an actively developing area of research. Several fundamental results in this field have been achieved by the scientific group of Mikhail Vasiliev at the Department of Theoretical Physics of the Physics Institute of the Russian Academy of Sciences, the most notable of which are the discovery of a system of consistent equations describing interacting higher spin fields and a new type of symmetry linking an infinite tower of such fields.

Another potentially important new symmetry of nature at extremely high energies could be supersymmetry, which extends the symmetry of special relativity and provides a unification of particles with integer spins (bosons) and half-integer spins (fermions). Supersymmetry is essential in string theory to eliminate physically unacceptable states with negative mass squared, known as tachyons, from the string spectrum.

For the explicit realization of supersymmetry, special mathematical spaces called superspaces are used, which include, in addition to Minkowski coordinates, additional anticommuting coordinates taking values in Grassmann algebra. The introduction of anticommuting coordinates leads to bosons and fermions being described as components of a single superfield, transforming into each other under supersymmetry transformations.

Thus, the supersymmetric theory of higher spin fields can serve as a natural development of quantum field theory in two fundamental directions simultaneously: in the introduction of higher spin fields as new degrees of freedom and their symmetries, which are important at extremely high energies, and in the introduction of supersymmetry as a principle of unification of bosons and fermions and their interactions. As a result, a new approach to the study of gravity at the quantum level is formed.

Scientists from MIPT and JINR in their work investigated how to construct such a theory using N=2 harmonic superspace, which is a special type of superspace containing a specific set of anticommuting coordinates and auxiliary variables called harmonics, associated with the internal symmetries of the corresponding field theory.

These auxiliary variables are necessary for the explicit realization of a special supersymmetry known as N=2 extended supersymmetry. The harmonic superspace was introduced by a group of scientists from Dubna, including Alexander Galperin, Evgeny Ivanov, Viktor Ogievetsky, and Emery Sokachev in 1984 and became a breakthrough in the field of supersymmetry. For this work, they received the first prize from JINR in 1987. A series of works by Iosif Bukhbinder, Evgeny Ivanov, and Nikita Zaigraev extends these studies to describe higher spin fields and their interactions.

The introduction of harmonics leads to harmonic superfields containing an infinite number of bosonic and fermionic fields. Remarkably, this turns out to be a key feature of this approach, which led to the discovery of the geometric structure of all N=2 supersymmetric theories. The fact is that the introduction of harmonics leads to the concept of analytic superspace, in which many properties are radically simplified. This approach to supersymmetric theories was named the principle of Grassmann analyticity by the pioneers of harmonic superspace. It restricts various structures that can appear in N=2 supersymmetric theories.

Surprisingly, this principle of Grassmann analyticity also works in the theory of supersymmetric higher spin fields! Thus, many remarkable properties and features of harmonic superfields can be applied to describe supersymmetric higher spin fields and their interactions. In particular, based on it, the authors obtained a set of superfields and their super gauge transformations corresponding to N=2 supermultiplets of higher spins. It has been shown that the requirements of supersymmetry, analyticity, and super gauge invariance fully fix the form of cubic interactions of supersymmetric higher spin fields.

“We found that the principle of harmonic analyticity plays a key role in the formulation of N = 2 non-conformal multiplets of higher spins as well as N = 2 superconformal higher spins,” said Nikita Zaigraev, a research associate at the Laboratory of Mathematical and Theoretical Physics at MIPT and a member of the supersymmetry sector of the Bogoliubov Laboratory of Theoretical Physics at JINR. “All fundamental gauge potentials are analytic superfields.

This, considering the requirements of N = 2 supersymmetry and conformal supersymmetry, significantly restricts the possible form of interaction Lagrangians. In turn, this opens up a wide range of new tasks for constructing consistent theories of higher spin fields. In particular, in this way, we managed to construct a consistent theory of the interaction of the entire infinite tower of higher spins with particles forming a special supermultiplet of particles called a hypermultiplet! Such a theory has no analogues among known field models, as it relies on the properties of analytic superfields of higher spins and the superfield of the hypermultiplet.”