Invariant Theory

Taken from Shamil Shakirov Nov 4 GRASP talk

Consider a path integral, it is invariant under the special linear group acting on the vector space of functions. So based on this goal let us consider the simpler case of integral over a finite n-dimensional space.

We want invariants of polynomials like



where i and j runs from 1 to n. This would be a degree 2 polynomial in n variables. The only invariants for this system are functions of the determinant. This is as it should be because in free field theories the path integral gives the square root of the determinant of the appropriate differential operator (the analog for A in the functional case). This means that the invariant ring which describes all possible polynomial invariants is generated by one element, the determinant. All invariants of the above polynomial will be powers of the determinant.

For interacting field theories we need higher degree polynomials. In these cases the invariant ring is more complicated than just being simply generated by one element and having no relations to quotient out by.

Consider a cubic in two variables (u,v) call it A



The ring of algebraic invariants is generated by the discriminant which is some function of a,b,c,d. Call this
D(A) This means that the integral



must be some function of the discriminant at least formally. Obviously like path integrals it doesn't actually converge. We can pin down what exactly this function is with scaling arguments. If you scale all of the coefficients by some power, then do a substitution (x,y)=C(u,v) to compensate, you find



this implies the function has to be a power law with power -1/6 with some constant in front. This constant is infinite of course, but it has no dependence on the cubic A. We could expect no better from a badly behaved divergent integral.

We continue this process for higher and higher number of variables with cubic and quartic polynomials, and hopefully the dependence of the ring of invariants on n (variables) and r (polynomial degree) gives an idea for how it should behave in an infinite dimensional vector space which we want. It would be nice to explain the answers we get in path integrals more so by invariance properties that have to hold, but I have to admit this sounds very hopeful considering we don't know a lot of the invariant rings for different values of n and r.