A Duality between Strings and Loops in M Theory

[atyy]

"Historically speaking, string theory and the loop representation of quantum gravity are twins separated at birth. (Baez, Class. Quantum Grav. 15 (1998) 1827–1858)"

Now with AdS/CFT for strings, and group field theory for loops, both can reach their true destiny for describing condensed matter

 

[atyy]

I was kidding, but apparently there is some analogy!

"The advantage of this formulation of quantum gravity is that it precisely fixes the sum over spin foams, and that it allows a number of theoretical tools from standard quantum field theory to be imported directly into the background independent formalism. In this sense, this approach has similarities with the philosophy of the Maldacena duality in string theory: a nonperturbative theory is dual to a more quantum field theory. But here there is no conjecture involved: the duality between certain spin-foam models and certain group field theories is a theorem. (Rovelli, http://relativity.livingreviews.org/Articles/lrr-2008-5/ )"

 

[marcus]

I was kidding, but apparently there is some analogy!

"...But here there is no conjecture involved: the duality between certain spin-foam models and certain group field theories is a theorem." (Rovelli, http://relativity.livingreviews.org/Articles/lrr-2008-5/)"[/QUOTE]

That's impressive. In the sense of not starting with any fixed spacetime geometry, I would say that GFT is background independent. Group elements are viewed as labels. So the cartesian product of N copies of the group G is just the space of all possible labelings of some structure. Spacetime can be any shape you want, depending on how the N cells or edges of the structure are labeled. So working on G^N doesn't precommit you to any particular spacetime geometry.
You do field theory on the cartesian product G^N. (Which can be thought of kind of like a "space of all geometries".)

And I do not think of spinfoam as background dependent either. So it doesn't seem to me as if the duality bridges any chasm between choosing a prior spacetime geometry and not choosing one.

But the duality itself is still impressive. I didn't know it was a proved theorem. I hope it has been proven in 4D and not in some toy model case. Maybe I will see if Rovelli cites some references. Thanks for spotlighting that.

=======================
EDIT TO REPLY TO NEXT
Atyy, I am getting drowsy and have to turn in, but before doing so, I want to acknowledge how helpful your guidance to the literature is. It's great to have those links to the seminal GFT papers from 1999 and 2000, as well as the three more recent ones---not having to grope, given my limited memory, energy, and willpower. I hope other people are getting the good out of it too. As a general rule you seem quicker and better than I at searching for interesting papers. If it weren't completely off topic I'd ask for some tips on how to do it.

 

[atyy]

That's impressive. In the sense of not starting with any fixed spacetime geometry, I would say that GFT is background independent. Group elements are viewed as labels. So the cartesian product of N copies of the group G is just the space of all possible labelings of some structure. Spacetime can be any shape you want, depending on how the N cells of the structure are labeled. So you don't precommit to any particular spacetime geometry.
You do field theory on the cartesian product G^N.

And I do not think of spinfoam as background dependent either. So it doesn't seem to me as if the duality bridges any chasm between choosing a prior spacetime geometry and not choosing one.

But the duality itself is still impressive. I didn't know it was a proved theorem. I hope it has been proven in 4D and not in some toy model case. Maybe I will see if Rovelli cites some references. Thanks for spotlighting that.

The proof is supposed to be here:
Spacetime as a Feynman diagram: the connection formulation
Michael P. Reisenberger, Carlo Rovelli
http://arxiv.org/abs/gr-qc/0002095

Group field is to spin foam as Boulatov-Ooguri is to Pozano-Regge:
Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space
R. De Pietri, L. Freidel, K. Krasnov, C. Rovelli
http://arxiv.org/abs/hep-th/9907154

OK, it's not my policy to cite Rovelli so much since I detest background independence , so let me cite some other interesting group field theory stuff to balance it out:

3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory
Winston Fairbairn, Etera R. Livine
http://arxiv.org/abs/gr-qc/0702125

Group field theory renormalization - the 3d case: power counting of divergences
Laurent Freidel, Razvan Gurau, Daniele Oriti
http://arxiv.org/abs/0905.3772

Scaling behaviour of three-dimensional group field theory
Jacques Magnen, Karim Noui, Vincent Rivasseau, Matteo Smerlak
http://arxiv.org/abs/0906.5477

And to get back to twins separated at birth, let's compare:

Oriti http://arxiv.org/abs/gr-qc/0607032 "Group field theories were developed at first as a generalisation of matrix models for 2d quantum gravity to 3 and 4 spacetime dimensions to produce a lattice formulation of topological theories. More recently, they have been developed further in the context of spin foam models for quantum gravity"

McGreevy http://arxiv.org/abs/0909.0518 "Now we can see some similarities between this expansion and perturbative string expansions ... This story is very general in the sense that all matrix models define something like a theory of two-dimensional fluctuating surfaces via these random triangulations."

 

[atyy]

A short review which gives examples of non-commutative geometry in spin foams (section 2.3), condensed matter (2.5), and string theory (2.6):

Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime
Richard J. Szabo
http://arxiv.org/abs/0906.2913

 

[MTd2]

http://arxiv.org/abs/hep-th/0006137

The cubic matrix model and a duality between strings and loops

Lee Smolin
(Submitted on 19 Jun 2000)
We find evidence for a duality between the standard matrix formulations of M theory and a background independent theory which extends loop quantum gravity by replacing SU(2) with a supersymmetric and quantum group extension of SU(16). This is deduced from the recently proposed cubic matrix model for M theory which has been argued to have compactifications which reduce to the IKKT and dWHN-BFSS matrix models. Here we find new compactifications of this theory whose Hilbert spaces consist of SU(16) conformal blocks on compact two-surfaces. These compactifications break the SU(N) symmetry of the standard M theory compactifications, while preserving SU(16), while the BFSS model preserve the SU(N) but break SU(16) to the SO(9) symmetry of the 11 dimensional light cone coordinates. These results suggest that the supersymmetric and quantum deformed SU(16) extension of loop quantum gravity provides a dual, background independent description of the degrees of freedom and dynamics of the M theory matrix models.

*****

This paper was never published in a peer review magazine, but it has got 33 citations, mostly from string theorists.