PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model · Beyond the Standard Model
Symmetry and Flavour, Ch The relativistic string: action, light-cone quantisation, spectrum and D-branes
Problem
The split of index pairs into symmetric traceless, antisymmetric and trace pieces is pure counting, and it works for any number n of transverse directions. That the closed-string states α−1Iαˉ−1J∣p⟩ carrying such pairs are massless is a separate fact, true in the bosonic string only at D=26.
((a)) Show that the n2 index pairs split into 21n(n+1)−1 symmetric traceless, 21n(n−1) antisymmetric and 1 trace, and that with n=D−2 the first count equals 21D(D−3).
((b)) Evaluate the three counts for n=8, the number of transverse directions if spacetime had ten dimensions. Only the counting is asked for: at D=10 the bosonic states are not massless.
((c)) In the bosonic string at D=26, how many massless gauge vectors does a stack of k=4 coincident D-branes carry, and of which gauge group?
((d)) One of the four branes is now moved a proper distance d=0.50fm away from the other three, which remain coincident, with string tension T0=0.20GeV2. State the unbroken gauge group, the number of vectors that remain massless, the number that become massive, and the common mass (in GeV) of the massive vectors. Use ℏc=0.1973GeVfm.