= Vortex number
{title2=$N$}
The vortex number is an integer winding or flux charge of a finite-energy Abelian Higgs configuration. With $D=d-iA$ and a Higgs field approaching nonzero magnitude at infinity, it is
$$
N=\frac1{2\pi}\oint d\arg\phi=\frac1{2\pi}\int F.
$$
The equality follows because the covariant phase gradient vanishes asymptotically. Reversing orientation or considering an antivortex reverses the sign.
In a positive-vorticity Bogomolny sector, Higgs zeros have positive multiplicities and their total multiplicity is $N$. For a general configuration containing vortices and antivortices, winding is a signed count; the unsigned number of zeros need not equal the net vortex number. The gauge-charge normalization must also be kept when translating the flux to physical units.
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