Musings

Topological M-Theory

Topological String Theory is a rich and beautiful subject. It associates a Calabi-Yau 3-fold, $X$ with a set of invariants which have both significance in enumerative algebraic geometry and which play a physical role in the $N=2$ supergravity theory obtained by compactifying physical type-II strings on $X$.

Recently, a lot of interest has surrounded the possibility of topological M-Theory, a theory which might compute similar interesting invariants for a 7-manifold, $Y$, of $G_2$-holonomy. There’s a paper by Dijkgraaf et al on Hitchin’s theory. More recently, Nikita Nekrasov has written up his talk at Strings 2004.

Both papers are very intriguing, but neither contains what the authors (or the reader) might happily call a satisfactory formulation of “topological M-Theory.”

Since it’s the Holidays, I’m going to go out on a limb and — Scrooge-like — suggest that, perhaps, there’s a reason for this.

In a nutshell, the reason “why” there’s topological string theory is that there is a rich set of “nonrenormalization theorems” protecting various couplings in the $N=2$ supergravity theory. This protected subsector of the physical theory is what is computed by the topological theory.

On the other hand, the generic $N=1$ supergravity theory, which would result from compactifying the physical M-theory on a $G_2$-manifold, $Y$, doesn’t have (as far as we know) a set of protected coupling which might plausibly be computed by topological M-Theory on $Y$.

There are very special $N=1$ theories which do have such protected couplings. But these, typically, are related to some $N=2$ theories, and the protected subsector is computed by the corresponding topological string theory.

One of the “axioms” of topological M-Theory is that, on $Y=X\times S^1$, it should reduce to topological String Theory on $X$. The question is whether there’s anything new that might be found on manifolds, $Y$, of irreducible $G_2$ holonomy?

Even if the answer turns out to be “no,” it does not necessarily follow that the whole exercise is pointless. If nothing else, a 7-dimensional formulation might furnish a proof (or even a satisfactory statement) of the conjectured S-duality of the Topological A- and B-models on $X$.

I’m still rather optimistic about the subject. But I do have this nagging fear that the difficulties making sense of the proposals on the table are not just a case of me being dense, but are indicative of something fundamental.