Wednesday, February 22, 2017

Gravity as the dimensional reduction of a theory of forms in six or seven dimensions

Tuesday, February 21st
Kirill Krasnov, University of Nottingham
Title: 3D/4D gravity as the dimensional reduction of a theory of differential forms in 6D/7D 
PDF of the talk (5M)
Audio+Slides [.mp4 16MB]

by Jorge Pullin, Louisiana State University

Ordinary field theories, like Maxwell’s electromagnetism, are physical systems with infinitely many degrees of freedom. Essentially the values of the fields at all the points of space are the degrees of freedom. There exist a class of field theories that are formulated as ordinary ones in terms of fields that take different values at different points in space,  but that whose equations of motion imply that the number of degrees of freedom are finite. This makes some of them particularly easy to quantize. A good example of this is general relativity in two space and one time dimensions (known as 2+1 dimensions). Unlike general relativity in four-dimensional space-time, it only has a finite number of degrees of freedom that depend on the topology of the space-time considered. This type of behavior tends to be generic for these types of theories and as a consequence they are labeled Topological Field Theories (TFT). These types of theories have encountered application in mathematics to explore geometry and topology issues, like the construction of knot invariants, using quantum field theory techniques. These theories have the property of not requiring any background geometric structure to define them unlike, for instance, Maxwell theory, that requires a given metric of space-time in order to formulate it.

Remarkably, it was shown some time ago by Plebanski, in 1977 and later further studied by Capovilla-Dell-Jacobson and Mason in 1991 that certain four dimensional TFTs, if supplemented by additional constraints among their variables, were equivalent to general relativity. The additional constraints had the counterintuitive effect of adding degrees of freedom to the theory because they modify the fields in terms of which the theory is formulated. Formulating general relativity in this fashion leads to new perspectives on the theory. In particular it suggests certain generalizations of general relativity, which the talk refers to as deformations of GR.

The talk considered a series of field theories in six and seven dimensions. The theories do not require background structures for their definition but unlike the topological theories we mentioned before, they do have infinitely many degrees of freedom. Then the dimensional reduction to four dimensional of these theories was considered. Dimensional reduction is a procedure in which one “takes a lower dimensional slice” of a higher dimensional theory, usually by imposing some symmetry (for instance assuming that the fields do not depend on certain coordinates). One of the first such proposals was considered in 1919 by Kaluza and further considered later by Klein so it is known as Kaluza-Klein theory. They considered general relativity in five dimensions and by assuming the metric does not depend on the fifth coordinate, were able to show that the theory behaved like four-dimensional general relativity coupled to Maxwell’s electromagnetism and a scalar field. In the talk it was shown that the seven dimensional theory considered, when reduced to four dimensions, was equivalent to general relativity coupled to a scalar field. The talk also showed that certain topological theories in four dimensions known as BF theories (because the two variables of the theory are fields named B and F) can be viewed as dimensional reductions from topological theories in seven dimensions and finally that general relativity in 2+1 dimensions can be viewed as a reduction of a six dimensional topological theory.

At the moment is not clear whether these theories can be considered as describing nature, because it is not clear whether the additional scalar field that is predicted is compatible with the known constraints on scalar-tensor theories. However, these theories are useful in illuminating the structures and dynamics of general relativity and connections to other theories.

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