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I am at the Bayrischzell workshop 2013. Yesterday I gave the following talk
on joint work, in two parts, with Domenico Fiorenza and Chris Rogers, and with Domenico Fiorenza and Hisham Sati.
This indicates how the brane spectrum of string theory/M-theory canonically flows from super-cohesion as the higher extension theory of the canonical higher super line objects $\mathbb{R}^{d;N}$ (also known as super-Minkowski spacetime) in Higher geometric prequantum theory (schreiber).
This is reminiscent of a famous cartoon of “M-theory”
Could you expand on this? Is it in terms of a single entity tying together a range of different manifestations?
Yes. That star-shaped region in the famous “cartoon” of M-theory (p. 5 here) is supposed to indicate how the five flavors of string theory together with 11-dimensional supergravity appear to be corners of one unified structure, with paths between them by “dualities”.
Now when one just classifies the iterative exceptional super $L_\infty$-algebra extensions of $\mathbb{R}^{d;N}$ (super-Minkowski spacetime regarded as the super-translation Lie algebra) then the picture of the brane content plus brane intersection laws that appears (p. 5 here) is that of these string theories plus 11d supergravity, and they appear manifestly as different corners of one unified structure (the super $L_\infty$-extension theory of $\mathbb{R}^{d;D}$) and the equivalences between them reflect the pertinent dualities.
It’s just super $L_\infty$-theory. If you had never heard of branes or string theory or M-theory, you’d be bound to eventually dicover this in higher Lie theory. It’s just there. Whatever that means.
It’s just super $L_\infty$-theory.
Does this tell us what M-theory might be like then?
So what this directly gives us first of all is a more systematic understanding of what the various $p$-brane $\sigma$-models are like.
Now, the perturbative string theories are defined to be the second quantization of the various 1-brane models that appear in the brane bouquet. By extension from this one might expect that M-theory should be something like the second quantization of the M2/M5-brane models in the brane bouuet. However, this is at least more subtle.
Among the various subtleties, the one that the brane bouquet and $\infty$-WZW theory should now shed light on is the following: the M5-brane system is not actually a traditional WZW $\sigma$-model like the other fundamental branes are, and so from the get-go it looks different. But the brane bouquet shows that it is in fact a $\sigma$-model itself, just not one whose target space is an ordinary (super)-space, but whose target space is a higher super stack (a “higher super-orbispace”, if one wants to stick to language common in $\sigma$-model theory).
So the next aim here (and the original motivation) is to make use of the fact that in higher super-geometry the single M5 brane becomes a (higher) WZW type $\sigma$-model itself, controled by the fermionic 7d term in the 11d sugra Lagrangian. This is complementary to our previous work where we discussed the bosonic 7d term controling mulitple M5-branes and found it to be a 7-cocycle on the “higher orbispace” $\mathbf{B}String^{2a}_{conn}$.
With the subtleties in both the fermionic and in the bosonic terms of the 7d parent theory of the M5 brane thus understood, the idea is then to combine them and give a good description of the 6d (2,0)-SCFT on the worldvolume of the M5 as an $\infty$-WZW $\sigma$-model. This we don’t quite have yet, but this is where this is headed.
If and when one has this or had this, then one can or could see with a description of the M5-brane as a sigma-model type theory finally in hand, if there is a sense in which maybe there is a “second quantization” of it (and the M2) that “gives M-theory”.
That’s not meant to be just around the corner, I am just saying it in reply to your question. This is a potential route from the brane bouquet of $\infty$-WZW models to an anser of “what might M-theory be like”.
But the thing is, we are not (or at least I am not) aiming to describe M-theory. I am just following the flow of cohesion. Here I am observing that the flow of super-cohesion takes one to good models of the M5-brane and all the other branes that appear in string theory. I want to see where it takes us further. If that is what elsewhere has been envisioned as M-theory, that’s fine with me. If it takes us to other places, that’s also good. The good thing about the flow of cohesion is, to my mind, that no human prejudices are intervening. I am just following what the cohesive foundations spit out.
And here we observe: the cohesive foundations applied to super cohesion spit out the brane spectrum of string/M-theory as well-defined $\infty$-WZW models. Whatever that implies…
Thanks! I only wish I had time to follow a little closer in the trail of this work, instead of having to wade through an enormous stack of essays on topics in the philosophy of medicine.
I wonder if other kinds of cohesion will allow other good flows, e.g., if we’d discovered what a Berkovich analytic cohesion might be.
I wonder if other kinds of cohesion will allow other good flows, e.g., if we’d discovered what a Berkovich analytic cohesion might be.
Yes, true. Maybe one day…
To see what one would need to check: At the bottom of it, cohesion produces prequantum field theories from cocycles. So the question is: where do special cocycles show up?
For instance the basic higher line objects in smooth cohesion are $\mathbb{R}^d$ and happen to have no interesting cohomology. But the basic line objects $\mathbb{R}^{d;N}$ in super-cohesion, regarded as equipped with their group structure/Lie algebra structure turn out to have interesting cohomology, and in fact interesting super-Whitehead towers. This is where the M-theory-like structure all comes from in the brane bouquet. One $\sigma$-model per non-trivial step in the Whitehead tower of “the fundamental cohesive object”.
Now maybe there are some other types of cohesion with canonical higher line objcts $\mathbb{A}^n$ that similarly have non-trivial interesting higher Whitehead towers. That would be interesting. But I don’t know.
Wild conjecture for the future then: if there’s a reason for Weil’s Rosetta Stone it will be down to cohesion, accounting for the knot-number field analogy, flavours of Langlands,…
As another desperate distraction from marking: given the puns people enjoy with brane/brain, have you considered
brane frieze/brain freeze?
Wild conjecture for the future then
I’ll think about it :-)
given the puns people enjoy
Sometimes the apparent puns are accidental though. It’s easy to get used to a technical term that leads to strange connotations when read with an every-day mind. For instance I well remember back in the 90s people would make fun of articles on “$p$-branes”, due to its sound when read out of context. Surely not an intentional pun…
Urs 3 > and the equivalences between them reflect the pertinent dualities
How do you show equivalences in your approach ? It seems that in your approach these are different cases a priori.
For instance S-duality:
the 3-cocycles that defines the fundamental string and the D-string are
$\Psi \Gamma^a \otimes \sigma^1 \psi \wedge E_a$
and
$\Psi \Gamma^a \otimes \sigma^3 \psi \wedge E_a$
respectively. So $E_a \mapsto E_a$ and $\Psi \mapsto \exp(i \sigma_1 \sigma_3) \psi$ is an equivalence of super $L_\infty$-algebras. This mixes fundamental strings with d-strings. This is S-duality.
Nice. On the other hand, this is just the cocycle input data, not the equivalence of physical theories, say as SCFT-s.
It gives an equivalence of the corresponding sigma-models, since these are entirely controled by these cocycles. However, these Green-Schwarz type sigma-models are not worldsheet supersymmetric, a priori. They are instead manifestly spacetime supersymmetric. The quantization of these Green-Schwarz type string sigma-models is much, much less understood than the quantization of the worldsheet-supersymmetric Ramond-Neveu-Schwarz type superstring, which gives the 2d SCFTs that you mention.
Any interest in An octonionic formulation of the M-theory algebra?
Rather than thinking of the M-theory algebra as an eleven-dimensional real algebra, it may be fruitful to think of it as a four-dimensional octonionic one.
This group has several papers, including one on the Freudenthal magic square being extended to a magic pyramid.
Thanks for the pointer. I haven’t looked at this latest article yet. Of course after all we know it’s not at all surprising that the M-theory Lie algebra is usefully studied in terms of octonionic structures. That’s due to the well known relation between supersymmetry and division algebras which goes back to
and has since been further discussed by the above authors.
But I haven’t yet looked at what specifically they achieve in the new article. Need to do some other things first now…
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