Monday, November 08, 2004

Topological M-theory

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"
Of course, the authors have had discussions with some of us whose main goal was to understand what class of theories are they thinking about - what exactly "topological gravity" means. Various simple proposals have been eliminated. For example, we can't say that "topological gravity" is a theory whose fields may be written as p-forms with some action. Any generally covariant theory with a metric tensor may be rewritten using a vielbein, which is really a collection of one-forms, and the action is a function of these one-forms. Obviously, one does not want to claim that every theory of gravity is topological.


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.

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