Is N = 4 gauge theory on S4 an effective description of topological gravity?

[atyy]

It's often been hoped that gravity is topological, eg. Witten, Xu, Gu & Wen, Rovelli.

Heckman & Verlinde make a new suggestion:
http://arxiv.org/abs/1112.5210
Instantons, Twistors, and Emergent Gravity
"The basic idea is to view N = 4 gauge theory on S4 as an effective low energy description of an underlying topological large N gauge theory, without any local propagating degrees of freedom."

Interestingly, they say their model has some similarities to an earlier condensed matter model by Zhang and Hu, which Gu and Wen also refer to:
http://arxiv.org/abs/cond-mat/0110572
A Four Dimensional Generalization of the Quantum Hall Effect
"On the other hand, the main problem with the current model seems to be the “embarrassment of riches”. ... As a result, there are not only photons and gravitons in the collective modes spectrum, there are also other massless relativistic particles with higher spins."

 

[mitchell porter]

Jonathan Heckman just gave a talk about this at Perimeter. Herman Verlinde also gave a talk about it last year (my comments).

It looks important but unrelated to the "topological theories of gravity" you list. They get their twistor matrix model from topological string theory on supertwistor space, but they also get it from bound states of branes in full string theory (see section 8 of their paper), and Heckman says he thinks that is the true UV completion.

If you go to 54-56 minutes into Heckman's talk, you will find how this twistor matrix model differs from the twistor string, and why it contains Einstein supergravity rather than conformal supergravity. The expression mentioned here (in equation 3.4) is employed, instead of the D-instantons employed in sections 4.6 and 4.7 of Witten's twistor string paper.

How the matrix model works is described at 40-45 minutes (action at 45 minutes). We started with a "holomorphic Chern-Simons theory" on supertwistor space, and then added flux in a "Yang monopole" configuration. This makes the supertwistor space noncommutative, which is why we are now using matrices rather than objects located at points. But enough of the commutative space's structure is retained that we can talk about "bulk modes" which propagate throughout the CP3, and "defect modes" which propagate along CP1 fibers (CP3 can be regarded as CP1 fibered over S^4). The "bulk modes" are analogous to gauge fields, the "defect modes" to quarks. (Since we are in N=4 supersymmetry, presumably these "modes" resemble large supermultiplets each containing both fermionic and bosonic degrees of freedom.)

An S^4 space has emerged, and in fact the low-energy theory is N=4 Yang-Mills coupled to Einstein gravity, on S^4 - so it's the Euclidean theory. We can build certain spin-1 and spin-2 currents out of the "defect modes", which in the flat-space limit (zoom into a small region of the S^4) behave like the MHV sector of gluon and graviton scattering, respectively. (MHV = maximum helicity violating; MHV amplitudes are basic to the twistor revolution in QFT - you get the extreme simplifications of Feynman sums by reorganizing them so that you have MHV vertices; look up "BCFW" for more.)

An S^4 is also a space with positive curvature and so one might hope that this is relevant to defining quantum gravity in de Sitter space. In Heckman's talk I notice a few similarities to Tom Banks's ideas - the remark at 27 minutes about a cutoff on representations sounds like his fuzzy spinor geometry, and (35 minutes) Banks talks about space-time pixels as well - and a recent Banks paper does get cited right at the end of Heckman & Verlinde's latest.

So we have a lot of things happening at once here: emergent gravity; Einstein gravity rather than conformal gravity in a twistor model; gravity in a space of positive curvature; as well as a new take on the twistor derivation of N=4 theory.

 

[member 11137]

Are not the recent works concerning the modified GTR models incorporating the Gauss-Bonnet invariant a kind of positive answer to your question?

Personaly (but it does not matter here), I think that yes: the topology certainly plays a crucial role in gravity.

What is your own opinion?