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In this work we introduce the concept of Cartan-Polyakov loops, a specialsubset of Polyakov loops in the fundamental representation of the$\mathrm{SU}(N_c)$ group, with charges $k=1,\ldots,(N_c-1)/2$. It constitutes asufficient set of independent degrees of freedom and it is used to parametrizethe thermal Wilson line. Polyakov loops not contained in this set areclassified as non-Cartan-Polyakov loops. Using properties of the characteristicpolynomial of the thermal Wilson line, we write a non-Cartan-Polyakov loopcharge decomposition formula. This formalism allows one to readily buildeffective models of quarks and gluons with an arbitrary number of colors. Weapply it to the Polyakov$-$Nambu$-$Jona-Lasinio model and to an effective gluemodel, in the mean field approximation, showing how to directly extend thesemodels to higher values of $N_c$.

Building models of quarks and gluons with an arbitrary number of colors using Cartan-Polyakov loops