Definition of the Wilson action on a lattice plaquette?

[James1238765]

TL;DR

What are the terms to be calculated in the "Wilson action" definition on a Yang-Mills plaquette?

The definition of the Wilson action relating to discrete Yang-Mills model is:

$$ S_{plaq} (\sigma) := \frac{1}{2}\sum_{plaq}\|I_N - \sigma_p\|^2 $$

(from [here] at 5:55)

It is mentioned that $\sigma_p$ is some kind of a matrix. Could anyone give an explicit example of what a $\sigma_p$ matrix look like, please?

Does the multiplication of sigmas

$$ \sigma_p = \sigma_{e1} \sigma_{e2} \sigma_{e3} \sigma_{e4} $$

mean consecutive matrix multiplication of the four square $\sigma_e$ matrices?

 

[James1238765]

This seems to be a question of how to do algebra on a toric code grid: https://www.physicsforums.com/threads/how-to-do-algebra-on-the-toric-code-grid.1048715/#post-6838379

 

[jedishrfu]

TL;DR Summary: What are the terms to be calculated in the "Wilson action" definition on a Yang-Mills plaquette?

Does the multiplication of sigmas

$$ \sigma_p = \sigma_{e1} \sigma_{e2} \sigma_{e3} \sigma_{e4} $$

mean consecutive matrix multiplication of the four square $\sigma_e$ matrices?

In general, matrix multiplications are associative so you can multiply in whatever order works for you. However, matrix multiplcations are not commutative so you can't switch the order around at all.