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I discuss recent results obtained by our group within the Nambu-Jona Lasiniomodel with the Polyakov loop. I focus on the two flavor version of the model,discussing the phase diagram in the $\mu-T$ plane in the case in whichelectrical neutrality is required. One of the results is that pion condensationis not observed in our model when a physical mass for the light quarks isconsidered.

Phase diagram and pion modes of the neutral Nambu-Jona Lasinio model with the Polyakov loop