= Parallel D-brane no-force identity
{title2=$f_3^8-f_4^8-f_2^8=0$}
Identical, parallel, equally oriented type-II <D-branes> preserve the same sixteen spacetime <supercharges>. Their annulus oscillator sum has the displayed numerator, which vanishes by the <Jacobi abstruse identity>. In the closed-string channel this is cancellation of attractive NS-NS exchange and repulsive RR exchange. The force vanishes in the flat supersymmetric configuration; anti-oriented branes, relative angles or supersymmetry-breaking backgrounds need not have this cancellation.
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