Quantum Groups as a Generalization of String Theory

[billtodd]

TL;DR

https://doi.org/10.1016/0370-2693(89)91375-0

Every once in a while I use my ancient trick of searching something in google with keywords, and found the above article. I don't think there's a free copy of it, because it's from 1989.

I guess I need to read the pink book on foundations of Q-Groups by Majid.
You know who also has written a book on Quantum Riemmannian Geometry, I wonder if it's worth the effort of reading it in the future towards a reliable Quantum Gravity theory, i.e. with actual predictions.

 

[renormalize]

TL;DR Summary: https://doi.org/10.1016/0370-2693(89)91375-0

This paper certainly hasn't been very influential. According to Google Scholar, it's only been cited 5 times in the past 35 years, most recently in 1998.
(But amusingly, the online abstract includes a typo that mentions "the Heidelberg commutation rules", which sounds like a reference to local German traffic laws.)

 

[billtodd]

This paper certainly hasn't been very influential. According to Google Scholar, it's only been cited 5 times in the past 35 years, most recently in 1998.
(But amusingly, the online abstract includes a typo that mentions "the Heidelberg commutation rules", which sounds like a reference to local German traffic laws.)

Let's say it's April's Fool.



Did your read the paper it's only 5-6 pages, but I guess you like me don't know a lot about Q-Groups..
And reading from Wikipedia is never a substitute from the book.

 

[mitchell porter]

That particular paper may have gone nowhere, but the general theme of quantum deformations of string theory has had a revival in recent years. String theory had its start in some particle scattering formulas (Veneziano amplitude, Virasoro-Shapiro amplitude) which began life purely as algebraic formulas, but which were later derived from the scattering of quantized strings.

Quantum deformations of these formulas (the first such may have been the "Coon amplitude"), which complicate them through the introduction of an extra parameter called "q", do exist. The question is whether these deformations can consistently be extended into a full theory. Recent work suggests No. But quantum groups do show up elsewhere in quantum field theory, e.g. here.

 

[OlderWannabeNewton]

q-deformed groups are probably important in some respects for understanding quantum gravity, though I don't know enough about the string theory aspect in particular. But there has recently been some interest in q-deformations in the context of double scaled SYK (which is a proposed boundary dual of a 2d quantum gravity Hilbert space with a notion of length operator that has a discrete spectrum). q-deformations seem to play a role in low-dimensional discrete quantum gravity models more generally. See here and here.

 

[mitchell porter]

@OlderWannabeNewton: in the first paper you mention ("Holography on the Quantum Disk"), the 2d space is a noncommutative version of the hyperbolic disk, described by Leonid Vaksman (died 2007). The quantum groups here are q-deformations of SU(1,1), the symmetry group of the usual hyperbolic disk.

It turns out that Vaksman was also interested in q-deformations of the Penrose transform, which is far more physical, being the basis of 4d twistor theory. Meanwhile, quantum groups are also showing up in twistorial versions of 4d "celestial holography" (e.g. see Kevin Costello). Incidentally, Costello also has a paper on how a version of Yang-Mills theory "is controlled by the Yangian, in the same way that Chern-Simons theory is controlled by the quantum group".