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We explore 4-dimensional SU(N) gauge theory with a Weyl fermion in anirreducible self-conjugate representation. This theory, in general, has adiscrete chiral symmetry. We use 't Hooft anomaly matching condition of thecenter symmetry and the chiral symmetry, and find constraints on thespontaneous chiral symmetry breaking in the confining phase. The domain-wallsconnecting different vacua are discussed from the point of view of the 't Hooftanomaly. We consider the SU(6) gauge theory with a Weyl fermion in the rank 3anti-symmetric representation as an example. It is argued that this theory islikely to be in the confining phase. The chiral symmetry $\mathbb{Z}_6$ shouldbe spontaneously broken to $\mathbb{Z}_2$ under the assumption of theconfinement, although there cannot be any fermion bilinear condensate in thistheory.

't Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate