 Bogomolny bound for the inverse-density Abelian Higgs model

ID: bogomolny-bound-for-the-inverse-density-abelian-higgs-model

Let

j_{i} = Im (\overset{ˉ}{Φ} D_{i} Φ)

. The identity

∣ D Φ ∣^{2} = ∣ (D_{1} + i D_{2}) Φ ∣^{2} + B h + \partial_{1} j_{2} - \partial_{2} j_{1}

and completing the square give, when the current boundary integral vanishes,

V = \frac{1}{2} \int [∣ (D_{1} + i D_{2}) Φ ∣^{2} + \frac{( B - \frac{1}{2} h ( 1 - h ) ) ^{2}}{h}] + \frac{1}{2} \int B .

2 π N > 0

, the lower bound is

π N

, saturated by the two first-order Bogomolny equations. Division by

h

is interpreted in the finite-energy sense at its zeros.

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