• B ′ is the induced Kalb-Ramond field on the brane’s surface. The precise co-ordinate definition can be found in [9], but all it represents is the two-form field which has been induced on the surface of the D-brane as a result of the Kalb-Ramond field in the bulk. • F is the usual electromagnetic field potential for a single D-brane, and is attributable to the open string sector. In fact the B ′ fields and F fields interact, and it can be shown [50] that only the combination F + B ′ is gauge invariant. In many publications, it has been conventional to re-define F as 2 πα ′ F , and for clarity we follow this convention in this thesis. Furthermore, the Kalb-Ramond field will be switched off for all the examples in this thesis. Hence the DBI action for the rest of this thesis is taken to mean There will be cases though when we turn off the electromagnetic field tensor F in (1.53) too. For completeness, we add that it is possible to add a Chern-Simons term to (1.52) [51,52]: However this is a topological term that is only relevant if we are considering vacuums with non-trivial topologies. In all that follows, we implicitly as- sume that the vacuum is topologically trivial, and hence we will not include a Chern-Simons term in any subsequent action. 33 S ∼ ∫ M e Φ / 2 √ det( G + 2 πα ′ F ) (1.53) S ∼ ∫ M e Φ / 2 √ det( G + B ′ + F ) + i ∫ δM Tr [ exp(2 $_{πα}$′$_{F}$$_{2}$ + B$_{2}$ ) ∧ ∑ q C$_{q}$ ] (1.54)