What is the effect of including additional representations in the action of a lattice gauge theory?

Question

I'm reading Introduction to Quantum Fields on a Lattice by Jan Smit. When introducing the lattice gauge-field action as a sum over plaquettes, Smit says that in general the action should include a sum over representations. He states that the 'Wilson action' includes only the fundamental representation, which takes the form

$$S(U) = \frac{1}{g^2\rho}\sum_{p} \text{Re}\, \left[\text{Tr} \ \{U_p\}\right],$$

whereas in general, for representations denoted by $r$, we could have a lattice action

$$S(U) = \sum_p \sum_r \beta_r \frac{\text{Re}\left[ \chi_r(U_p)\right]}{\chi_r(1)}, $$

where

$$\chi_r(U) = \text{Tr}\{{D^r(U)}\},$$

$$\frac{1}{g^2} = \sum_r \frac{\beta_r \rho_r}{d_r},\qquad d_r = \chi_r(1)$$

Unfortunately, Smit doesn't explain why one would generalize the action like this. Why all this focus on the representation? What does it mean physically for the model?

Answer 1

The idea is that, instead of $Tr[U_p]$, you could use any function $f: G \to \mathbb{C}$ which is invariant under conjugation $f(U) \mapsto f(gUg^{-1})$, as this condition is enough to guarantee invariance under gauge transformation. Such invariant functions are called "class functions".

One variation on the Peter-Weyl Theorem states that

The characters of the irreducible representations of G form an orthonormal basis for the space of square-integrable class functions on G.

In other words, any class function can be written as a sum of characters, so it's enough to consider the actions you've written.