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

[mitchell porter]

Move over, M- theory and F-theory!

http://arxiv.org/abs/1609.06863
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.

http://arxiv.org/abs/1609.07078
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

 

[atyy]

Are the E and Z related?

But E is older than F, isn't it?

 

[mitchell porter]

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.

 

[Urs Schreiber]

But E is older than F, isn't it?

Not sure. F-theory originates around 1996 with

• Cumrun Vafa, "Evidence for F-theory", Nucl.Phys.B469:403-418,1996, (arXiv:hep-th/9602022)

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

• Bernard Julia, "Dualities in the classical supergravity limits" (arXiv:hep-th/9805083)

but that would still be 2 years after F-theory, it seems.

 

[Urs Schreiber]

Move over, M- theory and F-theory!

http://arxiv.org/abs/1609.06863
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
  (arXiv:1512.01644)
• Alexander G. Tumanov, Peter West,
  "$E_{11} in$11d##",
  Physics Letters B Volume 758, 10 July 2016, Pages 278–285
  (arXiv:1601.03974)

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.

 

[mitchell porter]

Last week, Renata Kallosh had a paper on amplitude identities for nonlinear supersymmetry (also see the companion paper). It's actually connected both to "Z-theory" (e.g. the use of recursion relations for amplitudes) and to "E-theory" (e.g. the $E_7$ symmetry of d=4 maximal supergravity). Add fermions for $E_{11}$, and even standard model fermions for $E_{10}$, and there's a lot to think about.

 

[mitchell porter]

On the "Z" side, I want to report three papers in which new basics for quantum field theory, are applied to the nonlinear sigma model:

Flavor-kinematics duality for NLSM.

NLSM belongs to a class of exceptional scalar EFTs controlled by their S-matrix properties.

NLSM locality and unitarity from hypotheses about singularities and soft limits.

I'll reiterate that NLSM describes pions, so this is all actual physics, and not just mathematics.

 

[mitchell porter]

M-branes from E-theory (starts with a review of other attempts at M-brane theory).

NLSM pions from higher-dimensional gluons (the color-kinematic algebra comes from higher-dimensional Poincare symmetry).

 

[pat777b]

It's also worth mentioning an older paper by Sugawara related to the M-branes from E-theory link above.
https://arxiv.org/abs/1701.06894

 

[mitchell porter]

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.)