Ka Tiritiri O Te Moana
In the Rosebrugh-Wood characterisation of Set there is a string of adjunctions between Set and [Setop,Set],
namely U-|V-|W-|X-|Y where Y is the Yoneda embedding. There is both a monad and a comonad hidden in here. To deal with annihilation and creation operators we are going to need both structures.
M-theorists would like to have a similar characterisation for the quantum topos of M-theory. By its higher dimensional nature, this is not simply a 2-category or 3-category. The dimension raising aspects of Gray type products are crucial to explaining the physical combination of systems. If the basic object is modelled on projective geometry, such as in twistor space, then one expects the combination of two particles to yield two copies of twistor space, more or less as noted long ago by Hughston and Hurd, who used the Kunneth formula to study mass generation in the twistor setting.
Witten's study of N=4 SYM showed that certain amplitudes were localised on curves. Clearly something interesting is happening here. If one steps back and considers supersymmetry from a Machian perspective, then one finds it is related to the Real-Symplectic duality of Mulase et al in matrix models.
This manifests T-duality, but what about S-duality? Could this arise from including the octonion case? One shouldn't be bothered by nonassociativity. After all, it comes up in the K-theory studies of T-duality, and in a simple monoidal categorical way. Moreover, the association of Jordan algebras and projective geometry means that one can make the connection to twistor string theory very explicit. In doing so, one ends up studying moduli spaces as orbifolds, with cell decompositions. It is well known that both these decompositions, and Deligne-Mumford type compactifications, can be described by operad like polytopes and their duals. In this higher categorical language, the web of dualities is a kind of Poincare duality in higher cohomology.
namely U-|V-|W-|X-|Y where Y is the Yoneda embedding. There is both a monad and a comonad hidden in here. To deal with annihilation and creation operators we are going to need both structures.
M-theorists would like to have a similar characterisation for the quantum topos of M-theory. By its higher dimensional nature, this is not simply a 2-category or 3-category. The dimension raising aspects of Gray type products are crucial to explaining the physical combination of systems. If the basic object is modelled on projective geometry, such as in twistor space, then one expects the combination of two particles to yield two copies of twistor space, more or less as noted long ago by Hughston and Hurd, who used the Kunneth formula to study mass generation in the twistor setting.
Witten's study of N=4 SYM showed that certain amplitudes were localised on curves. Clearly something interesting is happening here. If one steps back and considers supersymmetry from a Machian perspective, then one finds it is related to the Real-Symplectic duality of Mulase et al in matrix models.
This manifests T-duality, but what about S-duality? Could this arise from including the octonion case? One shouldn't be bothered by nonassociativity. After all, it comes up in the K-theory studies of T-duality, and in a simple monoidal categorical way. Moreover, the association of Jordan algebras and projective geometry means that one can make the connection to twistor string theory very explicit. In doing so, one ends up studying moduli spaces as orbifolds, with cell decompositions. It is well known that both these decompositions, and Deligne-Mumford type compactifications, can be described by operad like polytopes and their duals. In this higher categorical language, the web of dualities is a kind of Poincare duality in higher cohomology.
posted by Kea | 1:11 PM
11 Comments:
Hi Kea
How did your talk on ribbon graphs go? I hope it sparked the interest of a few category theorists out there. :)
It's nice that you brought up T-duality and S-duality in the context of M-theory. One can study the union of both (U-duality), when investigating the symmetries of toroidal compactifications of 11D supergravity. The U-duality groups of these toroidal compactifications are non-compact symmetry groups, expected to be exact symmetries of the full quantum M-theory.
Recent interest in U-duality groups stems from the observation that U-duality groups in 3 and 4 dimensions act as spectrum generating symmetry groups in the charge space of certain BPS black hole solutions (see hep-th/0502235 and hep-th/0512296). It was also noted that such spectrum generating symmetry groups correspond to symmetry groups of Jordan algebras of degree three and their corresponding Freudenthal triple systems (FTS) (see hep-th/0606209).
In the case of 4D N=2 supergravity, the relevant symmetric space is M=E_6(-26)/F_4. In Jordan algebraic form, M=Str(J)/Aut(J), where Str(J) and Aut(J) are the reduced structure group and automorphism group of the exceptional Jordan algebra, respectively. The exceptional Jordan algebra is the algebra of 3x3 Hermitian matrices over the octonions. Str(J)=E_6(-26) is just the group of determinant preserving transformations of these matrices. Geometrically, E_6(-26) and F_4 are the collineation and isometry groups of the projective space OP^2 (the space of primitive idempotents of the exceptional Jordan algebra).
Now recalling Witten's twistor amplitudes, which localize on lines in projective space, it can be conjectured that in the octonionic case E_6(-26) will be the relevant collineation group for amplitudes over OP^2. This has yet to be studied and is still an open problem.
...it can be conjectured that in the octonionic case E_6(-26) will be the relevant collineation group for amplitudes over OP^2.
That would be nice. I'm afraid I tried to rush through too many things in the talk (and it was only mathematicians), but they seemed to enjoy it, although they were much more interested in news of the Fields medallists than in the discovery of Dark Matter.
I think for now I will stick with the generalised-knot-invariants approach to working with labelled ribbons, which is very category theoretic. I'm afraid I struggle quite a bit with the String theory terminology.
The orthogonal and symplectic analysis of Mulase and Waldron seems to involve integrals over real and quaternionic NxN Jordan algebra matrices. It appears that their approach can be generalized to the octonionic case, as their integral merely involves exponentials of the norm of Jordan algebra elements, e.g., tr X^2 = tr(XoX)= ||X||^2. In the octonion case, however, 3x3 is the largest our matrices can be, with the N>3 cases failing to be Jordan algebras.
The SO(N) and Sp(N) gauge groups act as isometry groups on the projective spaces RP^N-1 and HP^N-1, respectively. In the 3x3 octonionic generalization, the relevant gauge group would be the exceptional group F_4, acting as the isometry group on OP^2.
As Waldron was one of the authors of the BPS black hole paper (hep-th/0512296) I mentioned in my last post, it's possible that he has considered a graphical expansion technique for a 3x3 octonionic matrix integral. If not, maybe we can. ;)
As Waldron was one of the authors of the BPS black hole paper...
Oh! I didn't spot that.
...possible that he has considered a graphical expansion technique for a 3x3 octonionic matrix integral.
Can't we just ask him? If he hasn't, well then, yes, it would clearly be a fun thing to do.
Can't we just ask him? If he hasn't, well then, yes, it would clearly be a fun thing to do.
Yeah, the direct approach would be best. Are you going to shoot him an email?
Will do.
Hi kneemo
Apparently he did play around with octonions, but didn't find anything significant. He said, "Octonions are dangerous"!
I guess that means that we can think about it now.
Initial thought (may be wrong):
Use the Fano plane lines to label triple ribbons. When only colinear labels are allowed we effectively have quaternionic labels. When a non-associativity appears in the triple, there is a swap.
Kea,
I wanted to thank you for kindly mentioning my insane physics theory on your blog. It's not every insane theory that gets mentioned by the pros, even if the purpose is to piss off the ozzies.
The insanity started with my losing my faith in special relativity. One night I realized that I could get the kinematics of SR by switching proper time and coordinate time. Promoting proper time from to be a part of the geometry in the form of a rolled up dimension, and demoting coordinate time.
In this version of SR, the usual QFT is a mathematical artifact while the Wick rotated QFT is the true one. But making the change is not so simple as just that one flip. Instead, it modifies the interpretation of everything else in physics. I made a list of all the points where my version differed from what was acceptable and I found about 18.
It turns out that violating Lorentz invariance isn't a good way to get physicists interested. So I hid all the rabbit hole stuff by restricting the theory to finite dimensional QM (i.e. analyze the masses of point particles). That way I can sort of dance around the Coleman-Mandula theorems (hep-th/9605147) which assume perfect Lorentz invariance.
Carl Brannen
When only colinear labels are allowed we effectively have quaternionic labels.
I think that's the right idea: look at quaternionic subalgebras! When we try to build 3x3 primitive idempotents over the octonions, it turns out this isn't possible unless the off-diagonal octonionic entries lie in a quaternionic subalgebra (i.e., pick 3 imaginary units from the Fano plane). So there aren't any *pure* octonionic primitive idempotents, because the construction requires associativity.
I've been wondering how ribbon diagrams are depicted in twistor space. From Mulase and Waldron's pictures, it seems that the diagrams are drawn in Minkowski space. Looking at Fig. 5 (pg. 28) in Lectures on Twistor Strings and Perturbative Yang-Mills Theory we can see how to translate a Minkowski tree diagram into a twistor diagram.
Representing Mulase's Fig. 2.8 ribbon diagram on pg. 22 of Lectures on the Combinatorial Structure of the Moduli Spaces of Riemann Surfaces in twistor space, I'm thinking it would be two intersecting "ribbon lines" with 4 points identified on one and 2 points identified on the other. In other words, a vertex maps to a line in projective space, while half-edges meeting at a vertex become points on that line. This allows us to assign a primitive idempotent to each half edge where collinearity is measured by a vanishing Jordan cubic form (see end of Baez's node 12 for more on collinearity and the cubic form).
HI Kea: Your mountain-climbing adventures sound very exciting! Have you seen TOUCHING THE VOID? Your movie could be next!
Since we are all interested in maths, this week I am experimenting with equations on my blog. If successful, we can all use the technology. Ours will be the blogs of real math and science.
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