The orthogonal ensemble of random matrices and QCD in three dimensions

We consider the parity-invariant Dirac operator with a mass term in three-dimensional QCD for $N_c=2$ and quarks in the fundamental representation. We show that there exists a basis in which the matrix elements of the Euclidean Dirac operator are real. Assuming there is spontaneous breaking of flavor and/or parity, we read off from the fermionic action the flavor symmetry-breaking pattern $Sp(4N_f) \to Sp(2N_f) \times Sp(2N_f)$ that might occur in such a theory. We then construct a random matrix theory with the same global symmetries as two-color QCD$_3$ with fundamental fermions and derive from here the finite-volume partition function for the latter in the static limit. The expected symmetry breaking pattern is confirmed by the explicit calculation in random matrix theory. We also derive the first Leutwyler-Smilga-like sum rule for the eigenvalues of the Dirac operator.