Universal algebraic varieties and ideals in physics: Field theory on algebraic varieties

A class of universal algebraic varieties in physics is discussed herein using the concepts of determinant ideals in algebraic geometry. It is shown that these algebraic varieties arise with very different physical contexts in many branches of physics and mathematics from high energy physics theory to chaos theory. In these physical systems the models are constructed by using the fields on usual manifolds such as vector fields in a Euclidean space and a Minkowskian space. But there is a universal mathematical aspect of linear algebra for linear vector spaces, where the linear independency and dependency are described using the Gramians of the vectors. These Gramians form a class of hypersurfaces in a higher-dimensional mathematical space: If there exist g vectors v(i) in an n-dimensional Euclidean space, the Gramian G(g) is given as a g x g determinant G(g) = Det[x(ij)] with the inner products x(ij) (v(i),v(j)), and exists in a g(g-1)/2-[g(g + 1)/2-] dimensional space if the vectors are (not) normalized, x(ii) = 1 (x(ii) not equal 1). It is also shown that the Gramians are invariant under automorphisms of the vectors. The mathematical structure of the Gramians is revealed to be equivalent to the concepts of determinant ideals I-g(v), each element of which is a g x g determinant constructed from components of an arbitrary N x N matrix with N > n and which have inclusion relation: R = I-0(v) superset of I-1(v) superset of ... superset of I-g(v) superset of ..., and I-g(v) = 0 if g > n. In the various physical systems the ideals naturally emerge to give us dynamical flows on the hypersurfaces, and therefore, it is called the field theory on algebraic varieties. This viewpoint provides us a grand viewpoint in physics and mathematics.