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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-66/1/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 66 / 1 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

A supersymmetric type-II D-brane carries a Ramond–Ramond potential charge and saturates the corresponding generalized BPS bound in supersymmetry: its brane tension equals the charge-dependent central/tensorial-charge bound in the relevant normalization. It is therefore a BPS state, preserving sixteen of the thirty-two spacetime supercharges. Identical parallel branes with the same orientation impose the same supersymmetry projector. This statement does not apply to arbitrary unstable non-BPS branes or to the bosonic branes of the preceding parts.

One direct demonstration of the parallel D-brane no-force identity is the annulus vacuum amplitude. For separation

r

, its oscillator-dependent part is proportional to

A (r) \propto V_{p + 1} \int_{0}^{\infty} \frac{d t}{2 t} (8 π^{2} α^{'} t)^{- (p + 1) /2} e^{- r^{2} t / (2 π α^{'})} \frac{f _{3} ( q ) ^{8} - f _{4} ( q ) ^{8} - f _{2} ( q ) ^{8}}{f _{1} ( q ) ^{8}}, q = e^{- π t} .

(1)

Here

f_{1} (q) f_{3} (q) = q^{1/12} n \geq 1 \prod (1 - q^{2 n}), = q^{- 1/24} n \geq 1 \prod (1 + q^{2 n - 1}), f_{2} (q) f_{4} (q) = 2 q^{1/12} n \geq 1 \prod (1 + q^{2 n}), = q^{- 1/24} n \geq 1 \prod (1 - q^{2 n - 1}) .

(2)

The first two terms describe the projected Neveu–Schwarz sector, and the last is the oppositely signed Ramond sector contribution. The Jacobi abstruse identity gives

f_{3}^{8} - f_{4}^{8} - f_{2}^{8} = 0

for every

t

. Therefore

A (r) = 0, - \frac{d V ( r )}{d r} = 0.

(3)

In the closed-string channel, attractive graviton and dilaton exchange cancel repulsive Ramond–Ramond potential exchange. The classical probe calculation in the D-brane supergravity solution gives the same cancellation. The common supersymmetry allows separation to remain a flat modulus; relative angles, reversed RR charge or supersymmetry-breaking backgrounds can remove this protection. For a brane–antibrane pair the RR interaction reverses and does not cancel the NS-NS attraction. The charge and half-supersymmetry interpretation is established in Dirichlet-Branes and Ramond-Ramond Charges.

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