Massless fermions in uniform flux background on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:mi>R</mml:mi></mm

The quantum numbers of monopoles in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msup><a:mrow><a:mi>R</a:mi></a:mrow><a:mrow><a:mn>3</a:mn></a:mrow></a:msup></a:mrow></a:math> in the presence of massless fermions have been analyzed using a uniform flux background in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>S</c:mi><c:mn>2</c:mn></c:msup><c:mo>×</c:mo><c:mi>R</c:mi></c:math> coupled to fermions. An analogous study in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>T</e:mi><e:mn>2</e:mn></e:msup><e:mo>×</e:mo><e:mi>R</e:mi></e:math> is performed by studying the discrete symmetries of the Dirac Hamiltonian in the presence of a static uniform field on <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>T</g:mi><g:mn>2</g:mn></g:msup></g:math> with a total flux of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>Q</i:mi></i:math> in the continuum. The degenerate ground states are classified based on their transformation properties under <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mfrac><k:mi>π</k:mi><k:mn>2</k:mn></k:mfrac></k:math> rotations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mi>T</m:mi><m:mn>2</m:mn></m:msup></m:math> 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.