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Graded Hilbert spaces, quantum distillation and connecting SQCD to QCD

The dimension of the Hilbert space of QFT scales exponentially with the volume of the space in which the theory lives, yet in supersymmetric theories, one can define a graded dimension (such as the supersymmetric index) that counts just the number of bosonic minus fermionic ground states. Can we make this observation useful in non-supersymmetric QFTs in four dimensions? In this work, we construct symmetry graded state sums for a variety of non-supersymmetric theories. Among the theories we consider is one that is remarkably close to QCD: Yang--Mills theory with $N_f = N_c$ fundamental Dirac fermions and one adjoint Weyl fermion, QCD(F/adj). This theory can be obtained from SQCD by decoupling scalars and carry exactly the same anomalies. Despite the existence of fundamental fermions, the theory possess an exact 0-form color-flavor center (CFC) symmetry for a particular grading/twist under which Polyakov loop is a genuine order parameters. By a two-loop analysis, we prove that CFC-symmetry remains unbroken at small $β$ due to grading. Chiral symmetry is spontaneously broken within the domain of validity of semi-classics on $\mathbb R^3 \times S^1$ in a pattern identical to $N_f=N_c$ SQCD on $\mathbb R^4$ and the two regimes are adiabatically connected. The vacuum structures of the theory on $\mathbb R^4$ and $\mathbb R^3 \times S^1$ are controlled by the same mixed 't Hooft anomaly condition, implying a remarkable persistent order.