# Relations between diffeomorphism symmetry theories and invariant $SU(N), N \rightarrow \infty$ theories

Is it possible to have, an exhaustive panorama (as much as possible), about the relations between theories having a diffeomorphism symmetry, and theories having a $SU(N), N\rightarrow\infty$ invariance ?

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I cannot quite vouch for exhaustive panoramas, but the crucial point is that GL(N), SU(N) matrices are representable in a nonhermitean basis discovered by Sylvester in 1882, the clock and shift matrices which he called nonions for N=3 (long before the Gell-Mann basis!), sedenions, etc. Their braiding relations, and maximal grading, and hence commutators, structure constants, and Casimirs!, are thus analytic in N and hence handily amenable to the N⟶∞ limit.

They undergird a discrete truncation of the Heisenberg group explored by Weyl in 1927, but that is almost besides the point, except for the fact that, in a toroidal phase space, they can be organized to SU(N) generators with two integer indices, cf. a quarter-century old talk of mine which also does the SU(∞) gauge theory.

That is to say, the Moyal algebra on a toroidal phase space amounts to SU(N), through Fourier transformation. And the classical, ħ⟶0, limit of that algebra, which is the Poisson-Bracket algebra is thus a Fourier-transform description of SU(∞), an observation first made on the sphere by Hoppe, but made manifest on the torus here.

The above summary talk bird-eye-views an expansive panoramas implicit in here ; and here ;and ; available at ; and ; [and finally] (https://www.academia.edu/6688013/Infinite-Dimensional_Algebras_and_a_Trigonometric_Basis_for_the_Classical_Lie_Algebras) with apologies for the massive document dump.

Now the Poisson-bracket algebra describes area-preserving diffeomorphisms on a notional phase space, and as such it lends itself to connecting this SU(∞) to the null string of Schild, basically with the Nambu action squared. Intriguingly, it has found applications in 2d hydrodynamics and the systematic study of Casimirs, also in use in large-N models in QFT, and the predictable supersymmetrization of such.

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