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]
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.
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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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