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The quantum numbers of monopoles in $R^3$in the presence of massless fermions have been analyzed using a uniform flux background in $S^2\times R$coupled to fermions. An analogous study in $T^2\times R$is performed by studying the discrete symmetries of the Dirac Hamiltonian in the presence of a static uniform field on $T^2$with a total flux of $Q$in the continuum. The degenerate ground states are classified based on their transformation properties under $\frac{\pi }{2}$rotations of $T^2$that leave the background field invariant. We find that the lattice analysis with overlap fermions exactly reproduces the transformation properties of the single-particle zero modes in the continuum. Whereas the transformation properties of the single-particle negative energy states can be studied in the continuum and the lattice, we are also able to study the transformation properties and the particle number (charge) of the many-body ground state on a finite lattice, and we show that the contributions from the fully filled single-particle states cannot be ignored. Published by the American Physical Society2025