Is there any $SU(\infty)$ gauge theory in quantum field theory? The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there mentioned a $SU(\infty)$ gauge theory in physics literature?
An example of such a theory could be the following: May be $g \in SU(\infty)$ smooth and for a function $f(x,y)$ with spacetime coordinate $x$ and the new $SU(\infty)$ degree of freedom $y$ it holds $gf(x,y) = \int d^4y (g(x,y,y')f(x,y'))$. Now it is straighforward to define a gauge connection and the gauge field strength. 
In more non-theoretical words: Some quantum states have degeneracies and these degeneracies are based on a special symmetry (operator) that exists in a quantum system. If now the degeneracy symmetry operator is unitary and local symmetry one can define a gauge theory. Was this concept used in quantum mechanics or does such a concept makes sense?
Another interesting case is this: One can perform the following switch of coordinates $g(x,y,y') = g(x,x-y,x-y')$ and hence the generators $T_a(y,y')$ defined by $g(x,y,y') = \sum_a g_a(x)T_a(y,y')$ become dependent on the spacetime coordinate. Another question: Is it possible to define spacetime dependent generators of a Lie algebra?
 A: Comments to the question (v2):


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*The idea to consider the planar large $N_c\to \infty$ limit in $SU(N_c)$ QCD goes back to Ref. 1. 

*In light-cone membrane theory, pioneered in Ref. 2, the group $SU(\infty)$ is naturally identified with area-preserving diffeomorphisms ${\rm SDiff}_0(T^2)$ on the torus $T^2$ connected to the identity.

*Concretely, OP's proposal resembles a Fourier series expansion of an extra (compact) spacetime dimension. Such exercises are customary in string theory.
References:


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*G. 't Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B72 (1974) 461. 

*J. Goldstone, unpublished; J. Hoppe, MIT Ph.D. Thesis, 1982. 
A: There are apparently several thousand references to "SU(\infty)" on arxiv.org, and some of them are definitely talking about gauge fields or Yang-Mills. 
I suspect that some of the time, this will just be a way of talking about the large N limit of SU(N), i.e., not referring to a literal SU(∞) field theory, but rather the N→∞ limit of some quantity in SU(N) field theory. 
