### 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.

Posted by distler at December 31, 2004 1:12 AM
## Re: Topological M-Theory

Hi Jaques,

you write

“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.”

What’s wrong with superpotentials and gauge couplings (plus other holomophic objects) in

genericN=1 theories ? Those can be computed eg with topological open strings, and those are in general not related to deformations of N=2 theories.