We've spent some time by waiting for - and looking forward to - the new paper by Dijkgraaf, Gukov, Neitzke, and Vafa, and now it's finally here:
Their goal is to unify various theories of gravity based on p-forms: topological theories of gravity, to use a simpler language, as manifestations or compactifications of the same seven-dimensional theory ("topological M-theory").
The conjectured seven-dimensional "master" theory is supposed to be a UV completion of the so-called Hitchin theory. It has relations to the G2 holonomy metrics, and stable p-forms, and by assuming the solution (background) to have various fibered forms, they "dimensionally" reduce the Hitchin theory (and its completion) to various lower-dimensional theories, namely those:
- in six dimensions: the topological A-model (classically it's the Kähler gravity) and the topological B-model (classically it's the Kodaira-Spencer gravity)
- in four dimensions: the SU(2) BF-theory, or the "topological sector of loop quantum gravity", as they call it
- in three dimensions: the Chern-Simons theory with the gauge groups like SL(2,C) that is (and is not) equivalent to three-dimensional gravity
- in two dimensions: a two-dimensional form gravity - by which they mean two-dimensional (non-critical) string theory, dual to the old matrix models; they relate these two-dimensional theories to their favorite theories via the "effective theory of a string on the Calabi-Yau generated by the Fermi surface of the corresponding fermions"
What do they say about this question in their paper? Well, the answer, as far as I see after a very fast first search, is not too clear, but their focus is clearly on theories that have something to do with calibrated geometries.
Is there some evidence that the seven-dimensional theory is well-defined at the quantum level? In fact, what does it exactly mean that such a theory is well-defined? Is it about the existence of some partition sum which is a function of some parameters (moduli)? Which conditions must be satisfied by such a partition function? Obviously, some function always exists. ;-) In the case of physical string theory, we require the theory to predict a whole unitary S-matrix, which is highly non-trivial. But what exactly do we require from these topological theories that would make them non-trivial?
I know what we require for the topological A-model and B-model perturbatively - they are well-defined worldsheet theories obtained by twisting the physical superstring.
Obviously, they mean something rather specific by the "quantum completion" of the Hitchin theory - it's supposed to be a theory that includes M2-brane instantons wrapped on the calibrated 3-cycles. But does this theory make sense? In the case of type IIA - M-theory duality, we know that there should be a non-perturbative completion of type IIA superstring theory, and many of its objects and properties are determined by supersymmetry, which naturally leads to M-theory in 11 dimensions.
But can we do the same thing here? Is there some really good reason to believe in a unique non-perturbative completion of topological string theory? Are there some "objects" analogous to D0-branes that become light at strong coupling, and that show that the physics (or mathematics, in this case) becomes seven-dimensional?
Is not there an eight-dimensional theory, a topological F-theory? Such a theory would require to be compactified on a torus before it would make sense.
Cannot all these theories be simply defined as some truncations of the full M-theory and its compactifications to its BPS subsector(s)? Well, I feel a bit uneasy to imagine that these topological theories generate another type of string/M-theory. If one imagines that topological string theories (A-model and B-model) have a non-perturbative completion, as especially Cumrun likes to say, is not this completion the whole string/M-theory? Cannot one derive the relations between the different "topological theories of gravity" directly from the stringy dualities, even if the topological theories are still just a truncation?
For example, is not the partition function of topological M-theory defined simply by a superpotential in the real M-theory?
Well, these are not the only questions that I may only have a chance to understand after I read the whole article. Other mysterious points involve the interpretation of the partition sum as a "wave function"; and the new relations to holography and twistors. So far I don't quite see the meat, but it is very likely that the meat is there and it is very exciting.