Gravity and holography

Let us assume that we have a gravity theory in the bulk manifold with the boundary to which we approach taking the limiting value of one "holographic" component. For the theory to be consistent on the boundary we must impose certain "boundary conditions" which together with symmetries and gauge conditions, define the asymptotic symmetry algebra on the boundary and subsequently the corresponding field theory. The correspondence says that quantities computed in d dimensions on the boundary of the bulk theory are equal to the quantities computed in d-1 dimensions using the field theory on the boundary. The popular name of the correspondence is AdS/CFT correspondence, in which the quantities computed on the boundary of the d dimensional anti--de Sitter(AdS) space in which gravity theory lives, correspond to quantities computed in d-1 dimensional conformal field theory (CFT) that lives on the boundary. The two theories have not been proven to be equal yet, however, it was shown on the number of examples that once we compute a physical quantity in one theory, it gives equal result as the connecting or corresponding theory. Since then, scientists investigate this correspondence to prove it, and one of the excellent starting points is consideration of highly symmetric theories. In this research we want to study these highly symmetric theories, precisely, conformal higher spin (CHS) theory, its consistent covariant description and the physical quantities such as partition function and correlation functions. CHS theory includes models that generalise conformally invariant theories which are known as Maxwell and Weyl theories. It is using gauge/gravity correspondence related to more general higher spin theories and string theory, into which it can also give us insights. We also want to learn about that field theory when the boundary is flat space, which is natural to research since we live in such space. The methods we use for that are method for formulation of covariant CHS action in curved background, method for evaluation of partition function called "heat kernel" combined with "group theoretic" approach, which relies on the background on which the theory is considered. For analysis of the gravity theory with desired boundary we use "holographic renormalization" method which makes sure the theory is well defined. The novelty is definition of the CHS action on curved background beyond the linear term in curvature, search for relations among the partition functions on given background in d dimensions, as well as consideration of the four dimensional theory of given symmetry on flat boundary.

Everything that we see around us is built from small fundamental constituents of nature. One of the ways to classify these constituents, beside via their mass and charge, is according to their third property, intrinsic angular momentum, called spin. The additional property that they can have is that they can be conformal. The property of being conformal is related to conformal symmetry. Conformal symmetry is a special type of symmetry which dictates that laws and behaviour of the particle do not change with change of the scale. In other words, one can say that laws on the surface of the balloon before and after it has been inflated remain the same. This also implies that the angles are conserved but distances are not. Most important research finding in this project was theoretical prediction of the new consistent family of particles whose property is that they have higher spin, and conformal symmetry. They are called conformal higher spin particles and in this moment they are purely theoretical particles which exist in three dimensions. Beside finding these particles we have also studied holographic consequence of their existence. In holography we have two theories. One is a theory in the space-time called bulk, and then we choose one of the coordinates and use it to approach the boundary. We call this coordinate holographic coordinate. The boundary of the bulk can be represented by different things, it can be a point of no return of the black hole, which is called horizon, or it can be some distant point which we refer to as infinity. The other theory lives at that boundary. The quantities calculated in the theory that lives at the boundary, are equal to the quantities calculated in the theory in the bulk when holographic coordinate is taken to its limiting boundary value. The equality of these theories, the theory in the bulk in d dimensions with holographic coordinate at limiting value, and one at the boundary in d-1 dimensions, is called gauge/gravity correspondence. In this research, our current indications show that these objects that we find at the boundary of three dimensional conformal higher spin theory, can describe certain types of black holes.