## December 31, 2004

### 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

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/489

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

Posted by: WL on December 31, 2004 3:54 AM | Permalink | Reply to this

### Re: Topological M-Theory

There are, it’s true, lots of holomorphic objects in $N=1$ theories. I was thinking that “topological” meant something stronger than “holomorphic” — that it meant independence of (some of) the moduli.

Clearly that’s too much to hope for in a generic $N=1$ theory.

Holomorphy, supplemented by some further constraints can be very powerful. On a case-by-case basis, we can often find these extra constraints and solve for the F-terms of an $N=1$ theory. I’m asking whether there’s some generic underlying characterization (as would follow from topological M-Theory on $Y$) of those extra constraints?

Posted by: Jacques Distler on December 31, 2004 8:14 AM | Permalink | PGP Sig | Reply to this

### Re: Topological M-Theory

At tree level, the situation for holomorphic objects in N=1 is almost as good as for N=2. There are decoupling theorems concering the moduli-dependence (Kahler and complex structure) of F- and D- terms, allowing to use methods of mirror symmetry much in the same way as for N=2. Some further constraints follow from a N=1 generalization of N=2 special geometry, and that’s more than a simple deformation (or spontaneous breaking) of N=2.
At higher order, though, it seems that such decoupling theorems are violated.

All this refers to N=1 D-brane configurations on CY’s, and I am not sure to what extent this carries over to M-theory compactrifications (or a topological version of it). So far, M-theory was not very useful for doing actual computations, like eg instanton corrected Yukawa couplings, and I share your doubts that a topological version would do much better.

Posted by: WL on December 31, 2004 9:38 AM | Permalink | Reply to this

### No MathML?

Looks like the itex2MML filter is not applied to content.

Posted by: Henri Sivonen on December 31, 2004 5:07 AM | Permalink | Reply to this

### Re: No MathML?

The “default” text filter does not apply any itex2MML filtering. The commenter actively needs to select one of the itex2MML-enabled filters (of the four currently available).

I took the liberty of changing the filter on Wolfgang’s comment from the default one to “Markdown with itex2MML” and — presto changeo — his comment appears as intended (including the Markdownism of _emphasis_ for emphasis).

Posted by: Jacques Distler on December 31, 2004 8:34 AM | Permalink | PGP Sig | Reply to this

### Re: No MathML?

When I made the comment, even your main post came through unfiltered, so it appeared the filter was not working at all.

Posted by: Henri Sivonen on January 1, 2005 4:08 AM | Permalink | Reply to this

Post a New Comment