# Effective actions for finite temperature Lattice Gauge Theories

## TL;DR

This paper develops an exact effective action for Polyakov loops in finite temperature lattice gauge theories, using character expansion and Schwinger-Dyson equations, and applies it to determine the deconfinement temperature in SU(2).

## Contribution

It introduces a systematic method to derive an effective Polyakov loop action to all orders in spatial coupling, improving understanding of finite temperature phase transitions.

## Key findings

- Constructed the first non-trivial order effective action for SU(2) in (3+1) dimensions.
- Extracted the deconfinement temperature from the effective action.
- Demonstrated the method's exactness to all orders in temporal coupling.

## Abstract

We consider a lattice gauge theory at finite temperature in ($d$+1) dimensions with the Wilson action and different couplings $\beta_t$ and $\beta_s$ for timelike and spacelike plaquettes. By using the character expansion and Schwinger-Dyson type equations we construct, order by order in $\beta_s$, an effective action for the Polyakov loops which is exact to all orders in $\beta_t$. As an example we construct the first non-trivial order in $\beta_s$ for the (3+1) dimensional SU(2) model and use this effective action to extract the deconfinement temperature of the model.