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• 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 

<img src='content_image/321375.jpg'>

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]: 

<img src='content_image/321376.jpg'>

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. 

