Is there a connection between topological M theory and full M theory?

[marcus]

http://arxiv.org/abs/hep-th/0503140
A quantization of topological M theory
Lee Smolin
20 pages

"A conjecture is made as to how to quantize topological M theory. We study a Hamiltonian decomposition of Hitchin's 7-dimensional action and propose a formulation for it in terms of 13 first class constraints. The theory has 2 degrees of freedom per point, and hence is diffeomorphism invariant, but not strictly speaking topological. The result is argued to be equivalent to Hitchin's formulation. The theory is quantized using loop quantum gravity methods. An orthonormal basis for the diffeomorphism invariant states is given by diffeomorphism classes of networks of two dimensional surfaces in the six dimensional manifold. The hamiltonian constraint is polynomial and can be regulated by methods similar to those used in LQG.
To connect topological M theory to full M theory, a reduction from 11 dimensional supergravity to Hitchin's 7 dimensional theory is proposed. One important conclusion is that the complex and symplectic structures represent non-commuting degrees of freedom. This may have implications for attempts to construct phenomenologies on Calabi-Yau compactifications."

 

[Chronos]

When Smolin talks, I listen. On the surface, it looks like a challenge to M-theory, but having read it, that does not appear to be the case. Good link.

 

[marcus]

When Smolin talks, I listen. On the surface, it looks like a challenge to M-theory, but having read it, that does not appear to be the case. Good link.

I think you are right. the paper seems more like part of a conversation that he is having with string theorists Cumrun Vafa, Robbert Dijkgraaf, et al
(the authors of http://arxiv.org/hep-th/0411073 )
and with differential geometer Nigel Hitchin.
In the acknowledgments there are thanks to Dijkgraaf, Gukow, Hitchin, Neitzke, Vafa. Smolin's paper was apparently written in haste and still has lots of typos to correct. Some sections, like section 3 maybe, could be expanded. I think the paper could be seen as a response to
hep-th/0411073.

If that is the way to look at it, then one would have to go back and see what stands out in the earlier paper

 

[marcus]

just to have it handy, since it seems to go with Smolin's paper or viceversa, here is the Dijkgraaf et al abstract:

Topological M-theory as Unification of Form Theories of Gravity
Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke, Cumrun Vafa
65 pages, 2 figures


"We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G_2 holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory one can classically obtain 6-dimensional topological A and B models, the topological sector of loop quantum gravity in 4 dimensions, and Chern-Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective we find hints of an intriguing holographic link between non-supersymmetric Yang-Mills in 4 dimensions and A model topological strings on twistor space."

 

[marcus]

here is something about Nigel Hitchin
http://www.ma.utexas.edu/~hausel/hitchin/
his field is differential geometry
he is in the math department at Oxford

here's Robbert Dijkgraaf's attractive homepage
http://staff.science.uva.nl/~rhd/

 

[selfAdjoint]

Just reading a bit in these papers, an important point that Smolin makes is that the DGNV Topological M-theory isn't really topological, since he shows it has local degrees of freedom. Two of them in fact, and they don't commute with each other.

He further shows that both of these degrees of freedom are needed to specify a paticular Calabi-Yau manifold to compact your string physics on, but if they don't commute, when you come to quantize the theory, you can't specify them both at the same time because of uncertainty. Bummer! It seems amazing to me that nobody noticed this little problem before!

 

[marcus]

Just reading a bit in these papers, an important point that Smolin makes is that the DGNV Topological M-theory isn't really topological, since he shows it has local degrees of freedom. Two of them in fact, and they don't commute with each other.

He further shows that both of these degrees of freedom are needed to specify a paticular Calabi-Yau manifold to compact your string physics on, but if they don't commute, when you come to quantize the theory, you can't specify them both at the same time because of uncertainty. Bummer! It seems amazing to me that nobody noticed this little problem before!

can you explain what difference it makes? I noticed him saying that too: that there were a couple of local degrees of freedom. But I couldn't interpret it. Why should that be inconvenient, for what DGNV are up to? sorry if this is a naive question.

maybe it is merely disconcerting, that DGNV didnt notice something and had to wait for Smolin to point it out? but maybe it is not a serious impediment to what they want to do? (feeling very tentative about trying to interpret, these is unfamiliar terrain for me)

 

[Kea]

Don't know if this is helpful:

Generalized Calabi-Yau manifolds
Nigel Hitchin
http://www.arxiv.org/abs/math.dg/0209099

It wasn't referenced by Smolin. It ends up talking about gerbes. Note that Hitchin is a very important mathematician in the history of cohomological field theory and the maths of String theory in general. The above paper is expanded upon in the 2003 thesis of Hitchin's student

Generalized complex geometry
Marco Gualtieri
http://arxiv.org/abs/math.DG/0401221

Regards
Kea

 

[selfAdjoint]

can you explain what difference it makes? I noticed him saying that too: that there were a couple of local degrees of freedom. But I couldn't interpret it. Why should that be inconvenient, for what DGNV are up to? sorry if this is a naive question.

It sure isn't naive, I am just on the outer erdges of the math myself. I'm going to research it, including the Hitchins paper Kea linked to (Thanks Kea!) and I'll post something hopefully tomorrow.

The idea of Calabi-Yau manifolds having two structures, a metric and a symplectic, was new to me; in fact I never knew of any connection between C-Y and symplectic manifolds at all.

 

[Kea]

I'm going to research it, including the Hitchins paper Kea linked to...

Thanks selfAdjoint. I've been busy with work, mountain accidents (which I have a tendency to get mixed up in) and other adventures.

DGNV don't reference the latter Hitchin stuff either, which makes me think (perhaps erroneously) that this is why Smolin didn't look at it. It would be good if we could get some comments from an M-theory person, such as Sati, but I'm not sure if any of them read this site.

Cheers
Kea

 

[marcus]

Hi Kea, sorry to hear about the mountain accidents. Trust it is not you but someone you know, hope they are all right.

Just to clarify about the Hitchin papers (not for you or sA so much as in case anyone else is listening). The two recent papers by Hitchin that DGNV cited (besides the earlier "twistor" one from 1981) were

http://arxiv.org/math_DG/0010054
http://arxiv.org/math_DG/0107101

and these are exactly the two that Smolin referenced in his paper, which seems appropriate because his paper is in response to theirs and initiates a Loop treatment of what DGMV presented as topological M-theory.

 

[selfAdjoint]

Right, Marcus. The Hitchen "super-Calabi-Yau" paper is not in issue here. I'm still trying to get the DGNV paper into my thick head.

 

[selfAdjoint]

At long last. Here is the relevant theorem from Hitchens October 2000 paper.

Theorem 19 Let M be a closed 7-manifold with a metric with holonomy G2, with defining 3-form $$\Omega$$. Then $$\Omega$$
is a critical point of the functional $$\Phi$$ restricted to the
cohomology class $$[\Omega] \in H^3(M,R)$$.
Conversely, if $$\Omega$$ is a critical point on a cohomology class of a closed oriented 7-
manifold M such that $$\Omega$$ is everywhere positive, then $$\Omega$$ defines on M a metric with holonomy G2.

So basically just giving a 3-form with holonomy serves to define the geometry of any 7-dimensional manifold. He goes on to show that the 3-form defines the moduli space of the manifold too. Note that this is all pure math; Hitchin cross classifies it on the arxiv as Algebraic Geometry and Differentiqal Geometry.


Now on to Dijkgraaf, Gukov, Neitzke, and Vafa (DGNV) in hep-th/0411073. They are concerned with a topological theory they call the B-model:

There has been a longstanding prediction of the existence of a 7-dimensional topological theory from a very different perspective, namely the wavefunction property of the topological string partition function, which we now briefly recall in the context of the B model. The B model is a theory of variations $$\delta\Omega$$ of a holomorphic 3-form on a Calabi-Yau 3-fold X. Its partition function is written Z_B(x;0). Here x refers to the zero mode of $$\delta\Omega$$ , x ∈ H3,0(X,C) ⊕ H2,1(X,C), which is not integrated over in the B model. The other variable 0 labels a point on the moduli space of complex structures on X; it specifies the background complex structure about which one perturbs. Studying the dependence of Z_B on 0 one finds a “holomorphic anomaly equation” [7,14], which is equivalent to the statement that Z_B is a wavefunction [8], defined on the phase space H3(X, IR). Namely,
different 0 just correspond to different polarizations of this phase space, so Z_B(x;0) is related to Z_B(x;′0) by a Fourier-type transform. This wavefunction behavior is mysterious from the point of view of the 6-dimensional theory on X. On the other hand, it would be natural in a 7-dimensional theory: namely, if X is realized as the boundary of a 7-manifold Y , then path integration over Y gives a wavefunction of the boundary conditions one fixes on X.

They then define a wavefunction $$\Phi$$ as the tensor product of Z_B and its complex conjugate. This wave function then has a well defined quantum thing called a Wigner Function, which measures its density in phase space. They show an identification of this Wigner function with the partition function of Hitchin's form theory, stongly suggesting that the two theories are identical.



More to come--