What is the role of homotopy super Lie-n algebras in M-theory?

[mitchell porter]

PF regular @[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] has coauthored a new paper:

http://arxiv.org/abs/1805.00233
Higher T-duality in M-theory via local supersymmetry
Hisham Sati, Urs Schreiber
(Submitted on 1 May 2018)
By analyzing super-torsion and brane super-cocycles, we derive a new duality in M-theory, which takes the form of a higher version of T-duality in string theory. This involves a new topology change mechanism abelianizing the 3-sphere associated with the C-field topology to the 517-torus associated with exceptional-generalized super-geometry. Finally we explain parity symmetry in M-theory within exceptional-generalized super-spacetime at the same level of spherical T-duality, namely as an isomorphism on 7-twisted cohomology.

There's a lot in this paper and I am looking forward to understanding it. :-)

 

[PhanthomJay]

When you do understand it, please share what it means. Insofar as our beloved Dr. Hawking felt that M Theory was the ONLY theory that could lead to a full understanding of the Universe, I hope the research continues.

 

[Fra]

I wonder if Urs could try to relate this to the thread https://www.physicsforums.com/threads/why-higher-category-theory-in-physics-comments.899167/page-2 and maybe relate how one would computationally conclude that two theories are dual to each other as per some additional computation(transformation).

We have another duality, and is there an "insight" we extract from this? What is the significance of the space of all theories, and is there a natural measure on this space so one can probabilistically "confine" the evolution by distinguishing from the mathematically possible, from the physically probable?

/Fredrik

 

[mitchell porter]

When you do understand it, please share what it means.

I will say that this paper is not about making predictions, it's more about discovering the possibilities of M-theory, which should in turn lead to new ideas about how to apply it.

In string theory, there is a transformation, T-duality, which applies to a string that extends all the way along (or around) a dimension that forms a circle. The string itself thereby forms a loop, which can circulate around that dimension, like a train on a circular track, except that the train is so long that it joins up with itself from behind.

Because of quantum mechanics, the speeds with which the string can circulate in this way are quantized. The significance of T-duality is that these states of different increasing levels of momentum, have an alternative description in a dual string theory, as states in which the string is wrapped around the circular dimension multiple times.

That's string theory. What Sati and Schreiber talk about, is a generalization of this to M-theory, where the basic objects are not strings, but rather M2-branes and M5-branes. In this "spherical T-duality" of theirs, one has an M5-brane (a 5+1 dimensional object, rather than the 1+1 dimensional string), three of whose dimensions are wrapped around a "3-sphere" (in this terminology, the spheres we are familiar with are 2-spheres, because their surface is two-dimensional).

The T-duality of string theory exchanges the circulating momentum of the string, with the number of times that it winds around the circle. So the spherical T-duality of M-theory should analogously exchange something like the number of times the M5 wraps the 3-sphere, with the circulation of the M5 "throughout" the 3-sphere (I say "throughout", because the M5 fills the 3-sphere; its rotation should define a kind of current or flow throught that volume).

However, I CANNOT CONFIRM THAT THIS IS CORRECT. I'm just reasoning through analogy. Though hopefully this simple conception of what they have found, will help me decode their equations 7 and 8, which are the formal expression of the new duality they claim.

Another thought I have... The T-duality for strings involves a "1-sphere", the circle that the string wraps around. This T-duality for M-theory involves a 3-sphere. It so happens that the unit complex numbers form a circle in the complex plane, and the unit quaternions a 3-sphere in quaternionic 4-space. The complex numbers and the quaternions are division algebras - the other two being the real numbers and the octonions - and there have long been speculations that there is some relationship between string theory and the octonions, e.g. that the 10 dimensions of string theory are the two intrinsic dimensions of the string plus the eight dimensions of the octonions.

Could it be that these T-dualities (the old and the new) derive from a division-algebra structure implicit in M-theory? I'm encouraged by the fact that Sati and Schreiber derive their new T-duality from supersymmetry, which has its own connections to the division algebras.

Another aspect of this work that intrigues me is its relationship to what I think of as the mainstream of work done to elucidate M-theory, which tends to involve geometric and algebraic ideas from quantum field theory and string theory. Urs uses a lot of category theory, and to my eye it lacks certain details that I find in the more QFT-based papers. They have equations of motion and path integrals and so on, Urs has all these functorial identities. I can't tell if Urs's theoretical language is capable of being a self-sufficient framework, or if it is inherently underdetermined and needs to be completed by concrete equations.

 

[Fra]

Another aspect of this work that intrigues me is its relationship to what I think of as the mainstream of work done to elucidate M-theory, which tends to involve geometric and algebraic ideas from quantum field theory and string theory. Urs uses a lot of category theory, and to my eye it lacks certain details that I find in the more QFT-based papers. They have equations of motion and path integrals and so on, Urs has all these functorial identities. I can't tell if Urs's theoretical language is capable of being a self-sufficient framework, or if it is inherently underdetermined and needs to be completed by concrete equations.

I can't speak for Urs, but still my understanding of this - as per the association i made to the higher category theory as mentioned in Why Higher Category Theory in Physics? - Comments would suggest something like this:

I see the abstractions themselves (lacking the concrete equations of motions etc) might be thought of as defining relations between different theories, and all you need is a starting point. And the exploit in envision is that there is a unique starting point corresponding to the complexity -> 0 limit. And from that minimal seed, framework defining relations between theories should "generate" the full physical theoryspace. The complexity -> 0 should then in the "observer problem" association be understood as the observers information and computation capacity -> 0, ie. smaller and smaller inside observers, ie. we are talking about the ultimate TOE unificatation scale.

This is how i understand this CONCEPTUALLY, but indeed Urs technical stuff might first be seen in another way as well i presume, and also it is very easy to loose the holistic concepts due to the extremely dense technical details?

/Fredrik

 

[Urs Schreiber]

Another aspect of this work that intrigues me is its relationship to what I think of as the mainstream of work done to elucidate M-theory, which tends to involve geometric and algebraic ideas from quantum field theory and string theory. Urs uses a lot of category theory, and to my eye it lacks certain details that I find in the more QFT-based papers. They have equations of motion and path integrals and so on, Urs has all these functorial identities. I can't tell if Urs's theoretical language is capable of being a self-sufficient framework, or if it is inherently underdetermined and needs to be completed by concrete equations.

Indeed, there is a separation into the kinematical/cohomological aspect and the dynamical aspect. So far we have been focusing on the former, but just for lack of time. We comment on how the dynamical aspect will be brought about in the Introduction of Real ADE-equivariant (co)homotopy and Super M-branes .

A quick way to see how this comes about is to consider the equations of motion for the supergravity C-field

d G4 = 0
d star G4 = 1/2 G4 /\ G4

One may separate this in two steps, first a purely cohomological system of equations

1)
d G4 = 0
d star G7 = 1/2 G4 /\ G4

and then a dynamical self-duality condition

2)
G7 = star G4.

This is indeed a self-duality if one regards the pair (G4,G7) jointly as the actual M-flux field, just as the tuples (F2, F4, F6, ...) of RR-flux forms are jointly the "D-brane flux field". And indeed, for these the analogous logic holds: The equations of motion for the RR-flux forms may be separated into 1) a purely cohomological condition, saying that they are (the Chern character of) cocycles in K-theory, and then 2) in a second step imposing geometric self-duality.

Since M-theory won't be solved in a single day, we proceed step by step, but that does not mean that the later steps are to be ignored. In any case, so far we have been analysing the cohomological consequences that result from (1) and are relegating analysis of (2) to later. Not for lack of ideas, but for lack of hours in a day.

 

[Urs Schreiber]

By the way, there takes place a conference later this year, focused to one half on this approach:

http://ims.nus.edu.sg/events/2018/wstring/index.php
(10 - 14 Dec 2018)
National University of Singapore

Sub-theme 2: The role of homotopy super Lie-n algebras in M-theory

• • Homotopy super Lie-n algebras and higher WZW models of branes.
  • Homotopy super Lie-n algebras and string dualities.

http://ims.nus.edu.sg/events/2018/wstring/index.php#collapseOrganizerhttp://ims.nus.edu.sg/events/2018/wstring/index.php#collapseOrganizer

Chair

• 
  
  • Meng-Chwan Tan (National University of Singapore)
    
    Members
  • Fei Han (National University of Singapore)
  • Neil Lambert (King's College London)
  • Hirosi Ooguri (California Institute of Technology)
  • Christian Saemann (Heriot-Watt University)
  • Hisham Sati (New York University)
  • Mathai Varghese (University of Adelaide)
  • Edward Witten (Institute for Advanced Study)
  • Shing-Tung Yau (Harvard University)