DiracBornInfeld

Dirac-Born-Infeld (DBI) theory refers to the low-energy effective action describing the dynamics of a single Dp-brane in type II string theory. It encapsulates the worldvolume degrees of freedom, including the brane’s transverse embedding coordinates and a U(1) gauge field living on the brane. The DBI action provides a non-linear generalization of Maxwell electrodynamics and, in the appropriate limits, reduces to familiar gauge dynamics. It is central to describing how D-branes interact with background fields and with each other in string theory.

The standard form of the DBI action in the string frame is S_DBI = -T_p ∫ d^{p+1}ξ e^{-φ} sqrt(-det(P[g]_{ab} + 2π α' F_{ab} - P[B]_{ab})). Here P[g]_{ab} is the pullback of the spacetime metric to the brane, F_{ab} is the worldvolume field strength, B is the NS-NS two-form, and φ is the dilaton. The gauge-invariant combination is F_{ab} = ∂_a A_b - ∂_b A_a + P[B]_{ab}, and the brane tension is T_p = (2π)^{-p} α'^{-(p+1)/2} g_s^{-1} (string coupling g_s). The action is often accompanied by a Chern-Simons term that couples the brane to Ramond-Ramond fields.

Extensions and limitations: For multiple coincident branes, a non-Abelian generalization exists but is technically intricate, often addressed with a symmetrized-trace prescription at leading order. The DBI action is an effective, α'-corrected description valid for slowly varying fields and weak curvature; higher-derivative and loop corrections may be important in other regimes. The DBI framework is widely used in brane cosmology, holography, and the study of non-linear electrodynamics in string theory.

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