What is Non-Abelian Z-theory and how does it relate to gluon amplitudes?

In summary: Iranians... in the amplitude business.)In summary, there is ongoing research into the connections between E-theory, Z-theory, and their relation to M-theory, F-theory, and the double copy construction. These theories provide a deeper understanding of the symmetries and structures underlying quantum field theory and string theory, with potential implications for the unification of fundamental forces and the development of new mathematical tools for studying these theories. The study of these theories is an active and exciting area of research, with many promising avenues for future exploration.
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mitchell porter
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Move over, M- theory and F-theory!

A brief review of E theory
Peter West
(Submitted on 22 Sep 2016)
I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of non-linear realisations and Kac-Moody algebras, I explain how to construct the non-linear realisation based on the Kac-Moody algebra E11 and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space time, lead to precisely the equations of motion of eleven dimensional supergravity theory. By taking different group decompositions of E11 we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the non-linear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the E11 conjecture given many years ago.

Non-abelian Z-theory: Berends-Giele recursion for the α′-expansion of disk integrals
Carlos R. Mafra, Oliver Schlotterer
(Submitted on 22 Sep 2016)
We present a recursive method to calculate the α′-expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as Z-theory, we pinpoint the equation of motion of Z-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order α′7 is made available on the website this http URL
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  • #2
Are the E and Z related?

But E is older than F, isn't it?
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The connection is the SU(N) nonlinear sigma model, which for N=2 describes the pions, and for N=3 also includes kaons and eta meson. The classic understanding of this theory is that it results from chiral symmetry breaking in QCD with N flavors. The chiral symmetry SU(N)L x SU(N)R breaks to SU(N)diagonal, the pions etc are the Goldstone bosons, and they realize the broken symmetries nonlinearly. Peter West proposes that the fundamental symmetry of M theory is E11 and that it too is realized nonlinearly, in a way that acts on spacetime and fields together.

Meanwhile, today people are studying the double copy construction of graviton amplitudes from gauge boson amplitudes, that derives from the analogous relation between closed and open strings. The double copy construction requires that the gauge theory amplitudes satisfy special properties (color-kinematic duality, BCJ relations). The string theory origin of this is now being uncovered. Specifically, tree-level open string amplitudes can be written as a product of gauge amplitudes and amplitudes of a scalar field theory, dubbed Z-theory, with a BCJ property. The simplest case of Z-theory turns out to be the U(N) nonlinear sigma model, which is just the SU(N) theory with an extra U(1) factor.
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atyy said:
But E is older than F, isn't it?

Not sure. F-theory originates around 1996 with
while "E-theory" (if it is to be called this way now) originates only around 2001 with
  • Peter West, "##E_{11}## and M theory", Class. Quant. Grav., 18:4443–4460, 2001. (arXiv:hep-th/0104081)
The latter has a precursor in
but that would still be 2 years after F-theory, it seems.
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  • #5
mitchell porter said:
Move over, M- theory and F-theory!

A brief review of E theory
Peter West

The derivation of the equations of motion of supergravities in the article is a review of
  • Alexander G. Tumanov, Peter West,
    "##E_{11}## must be a symmetry of strings and branes",
    Physics Letters B Volume 759, 10 August 2016, Pages 663–671
  • Alexander G. Tumanov, Peter West,
    "##E_{11} in ##11d##",
    Physics Letters B Volume 758, 10 July 2016, Pages 278–285
I have trouble extracting the exact algorithm that is being used. It seems that an ##E_{11} \ltimes l_{1}## valued connection on ##l_1## is considered (in analogy to an ##SO(d-1,1) \ltimes \mathbb{R}^{d-1,1}##-valued Poincare connection on ##\mathbb{R}^{d-1,1}##) and then they look for any equations on this that are invariant under global ##E_{11}##-symmetry and local ##I_c(E_{11})## symmetry. The equations of motion for the various maximal supergravity theories are claimed to be shown to be such (to low order in levels of ##E_{11}##). Is the claim that these are the unique such equations?

It is noteworthy that there are no fermions in this picture. The sugra equations are purely bosonic.
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Nima Arkani-Hamed's latest amplitude paper (with three Chinese coauthors) presents a different interpretation of topics like NLSMs, field theory of a "bi-adjoint scalar" (a limit of "Z-theory"), and color-kinematic duality, in terms of the amplituhedron. A taste of the paper: "the gauge-invariant object for gluon scattering in YM theory completely dictates the infinite series of higher-dimensional corrections from superstrings".

They don't cite some of the papers by Cheung et al already mentioned in this thread, so the perspectives may be complementary. (By the way, I'm impressed by the number of Chinese physicists now working on these topics, and making major contributions. It's a historical milestone and I suspect a sign of things to come.)

Related to What is Non-Abelian Z-theory and how does it relate to gluon amplitudes?

1. What is string theory?

String theory is a theoretical framework that attempts to explain the fundamental nature of the universe by describing all particles as tiny vibrating strings. It combines elements of quantum mechanics and general relativity to provide a unified understanding of the forces and particles in the universe.

2. How is string theory different from other theories of the universe?

String theory is different from other theories because it proposes that the building blocks of the universe are not point-like particles, but rather tiny strings. These strings vibrate at different frequencies, which give rise to the different particles in the universe. This is in contrast to other theories that describe particles as point-like objects.

3. Is string theory proven?

No, string theory is still a theoretical framework and has not been proven. However, it has been extensively studied and has shown promise in addressing some of the unanswered questions in physics, such as unifying quantum mechanics and general relativity.

4. What are the potential implications of string theory?

If string theory is proven to be correct, it could have profound implications for our understanding of the universe. It could provide a unified theory of all forces and particles, and potentially even reconcile the discrepancies between quantum mechanics and general relativity. It could also help us better understand the nature of space and time.

5. Can string theory be tested?

Yes, there are ongoing efforts to test string theory through experiments and observations. However, due to the extremely high energies required to observe the effects of string theory, it has not yet been possible to definitively test it. As technology advances, researchers hope to find ways to test and potentially validate the predictions of string theory.