In short, if you impose a U(1) symmetry on your Lagrangian, you get electromagnetism. Basically you do the action of U(1) and enforce that the equation of motion does not change. That gives you the electromagnetic force (a photon).

Putting in some of those keywords will find plenty of detailed notes online, like

http://www.physics.usu.edu/torre/Classical_Field_Theory/Lect...

or more quick overviews like

https://mugndonut.wordpress.com/2018/03/31/symmetries-yield-...