Thursday, December 19, 2019

Stringy canonical forms

There are two new hep-th papers co-authored by Nima Arkani-Hamed today, a long one and a short one:
Stringy Canonical Forms by Nima+He+Lam, 59 pages,

Non-perturbative geometries for planar \(\NNN=4\) SYM amplitudes by Nima+Lam+Spradlin, 8 pages
They continue to elaborate upon deep links between the post-twistor ways to rewrite the scattering amplitudes in the maximally supersymmetric gauge theory in \(d=4\) using the amplituhedrons and similar polytopes and similar geometric notions; and the deep mathematics, especially one found in string theory.

The short paper is an application of the long one that addresses the \(G(4,n)\) cluster algebra whose links to \(n\)-particle scattering amplitudes to the gauge theory have looked clear to exist for some time but the link was never articulated too comprehensively.



OK, the stringy canonical forms are integrals of some polynomials exponentiated to some powers. The polynomials are polynomials in \({\bf X}\), some variables, and the exponents are \(-\alpha' c_I\) where \(\alpha'\) is a new parameter introduced to emulate the usual \(\alpha' = 1/2\pi T\) in perturbative string theory.



That was a boring complex paragraph, perhaps, although I am sure that some readers will have found it more useful than anything else. The qualitative point is to realize – as many of us have for years – that the integrals that appear in the amplituhedron papers are rather similar to the integrals in string theory.

The simplest nontrivial amplitude in perturbative string theory is the Euler Beta function\[

B(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt

\] where you can see an integral over \(t\) – Nima et al. call their integration variables \(x_i\) – and the exponents are objects such as \(x,y\) which are \(\alpha' s\) in string theory, some Mandelstam variables or products of momenta, and \(c_I\) in the new Nima et al. papers.

This particular 4-point function may be evaluated in terms of a quasi-elementary function, the Euler Gamma function (generalized factorial) as\[

B(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}

\] but you may add additional variables aside from \(x,y\) and the same number of integration variables. In perturbative open string theory, the integrals are over the locations of the vertex operators on the boundary of the disk. Here the locations within the disk (parameters of some moduli space) are replaced with some geometric properties of the polytopes although it's hard for me to describe them by words.

So the integrals that define the amplitudes are similar as in string theory and so are some properties, including

* meromorphicity in the variables \({\bf X},c_I\), analogous to the stringy meromorphicity in the momenta
* exponentially decreasing amplitudes in the "high energy" limit
* Regge behavior, i.e. poles as functions of \(s\) are precisely \(t\) and similar statements
* channel duality, \(s\leftrightarrow t\), although it has to be generalized in some way

In other words, the whole beef of the amplituhedron business is being reinterpreted as some generalized string theory with a newly invented string scale \(\ell_s = \sqrt{\alpha'}\). The usual amplitudes from the amplituhedron papers are obtained as the \(\alpha'\to 0\) limit while, if I understand well, the whole behavior for \(\alpha'\gt 0\) is unequivalent.

If they have deformed the gauge theory amplitudes in a stringy-like way, I have an obvious question: is the deformation an actual string theory that may be calculated by some other, "totally stringy" tools, or is it just an analogy? Do the new amplitudes for \(\alpha'\gt 0\) correspond to some unitary processes in a spacetime? And are these deformations unique in a similar way in which string theory seems to contain all the consistent UV-completions of low-energy effective field theories?

These are my basic questions – well, my question is basically "are these constructions as 'physically real' as string theory, or just mathematical auxiliary constructs to calculate real things in the limit?" Note that an obvious point is that a consistent, finite \(\alpha'\), finite \(g\) generalization of gauge theory contains gravitons, not just gluons!

Their amplituhedron analogy to string theory shares some properties with string theory but it differs in some others. It is more "number-theoretical". In particular, the short paper is full of sequences of abruptly growing integers and combinatorial methods to fill numbers into a \(4\times 4\) square. The expansion in \(\alpha'\) has rational coefficients at the leading order, transcendental at higher orders, like in string theory. Lots of their figures involve grids and polytopes connecting vertices with integral coordinates.

Some people, including other Persian ones, could immediately shout "toric geometry" or "quantum foam". Or at least, I do shout it! ;-)

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