Higher-rank tensor non-Abelian field theory: Higher-moment or subdimensional polynomial global symmetry, algebraic variety, Noether's theorem, and gauging

With a view toward fracton theory in condensed matter, we introduce higher-moment polynomial degree-p global symmetry, acting on complex scalar/vector/tensor fields (e.g., ordinary or vector symmetry for p$=0$ and p$=1$ respectively). We relate this of $n$-dimensional space, to lower degree (either higher-moment, e.g., degree-(p-$\ell$)) subdimensional subsystem layers $(n-\ell)$-submanifolds. These submanifolds are algebraic affine varieties (i.e., solutions polynomials). The structure as subvarieties can be studied via mathematical tools embedding, foliation, geometry. also generalize Noether's theorem symmetry. promote the local derive new family higher-rank-m symmetric tensor gauge by gauging, with m = p$+1$. By further gauging discrete $\mathbb{Z}_2^C$ charge conjugation (particle-hole) general class rank-m non-abelian field (the is non-commutative thus but not an group): hybrid (symmetric non-symmetric) anti-symmetric topological theory, generalizing [arXiv:1909.13879], interplaying between gapless gapped sectors.