100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

316 T. BanksS int = µ p∫d σ a1 ∧ . . . ∧ d σ ap+1 A a1...a p+1(x(σ)) = µ p∫W pA p+1 .The potential A p+1 is a p + 1 form defined in all of space-time, but theintegral is only over the brane world volume. This coupling is invariantunder the gauge transformationA p+1 → A p+1 + dω p ,where ω p is a p form, if the brane world volume is closed or the gaugetransformation vanishes on its boundary. The gauge invariant field strength∫F p+2 = dA p+1 . If we add a Maxwell term − 1 4 Fp+2 ∧∗F f p+2 to the action,F p+2 satisfiesd ∗ F p+2 = µ p J p+1 ,where J p+1 is the de Rham current concentrated on the brane world volume.The p form charge Z a1...a pobtained by integrating the time component ofthe current over a space like surface is conserved. Notice that only thespatial components of the brane charge are non-zero. Non-vanishing branecharge indicates the existence of an infinite brane whose asymptotic worldvolume picks out an asymptotic Lorentz frame where the brane is at rest.We can also introduce magnetic sources as the world volumes of d−p−4branes where the Bianchi identity dF p+2 = 0 breaks down. Nepomechieand Teitelboim 8 showed that the electric and magnetic charges satisfied ageneralization of the Dirac quantization condition.The relation between p brane charges and SUSY is a consequence of thefact that the product of two spinor representations is a direct sum of antisymmetrictensor representations. Let us elaborate in eleven dimensions,the maximal dimension in which interacting low energy supersymmetriceffective field theories can exist. The spinor super-charge, Q a , has 32 realcomponents. The most general right hand side for the supercharge anticommutationrelation is[Q a , Q b ] + = (γ 0 [γ µ P µ + γ µν Z µν + γ µ1...µ5 Z µ1...µ 5]) ab .This suggests the existence of 2-branes and 5-branes in any supersymmetricquantum theory in eleven dimensions. It is indeed true that the uniqueinteracting supersymmetric low energy field theory in eleven dimensions,11D SUGRA, has solutions corresponding to these branes. They are calledthe M2 and M5 branes.f ∗ represents the Hodge duality operator.