Actions and Fermionic symmetries for D-branes in bosonic backgrounds

TL;DR

The article delivers a complete construction of fully interacting D$p$-brane actions in arbitrary bosonic type II backgrounds, expanded to second order in fermions and organized around the supercovariant derivative and κ-symmetry projector. It presents a superfield-like formulation and a κ-symmetry–exposed reformulation, showing their equivalence and providing explicit SUSY and κ-symmetry transformations, including for the M2-brane, all within backgrounds with Killing spinors. A central technical achievement is the use of normal coordinate expansion and T-duality (Hassan framework) to derive and relate D2 and higher D$p$-brane actions, with higher-order fermionic couplings encoded in operators such as $ ilde{oldsymbol{ m Gamma}}_{Dp}$ and $L_p$. The D0/D4 example demonstrates a nontrivial curved-space worldvolume theory with 8 preserved supercharges in a D0-brane background, illustrating the practical impact for gauge/gravity setups and brane intersections. Overall, the work provides a robust toolkit for studying D-branes in nontrivial supergravity backgrounds and sets the stage for applications to D$p$/D$(p+4)$ systems and brane inhomogeneities in holographic contexts.

Abstract

In this article we derive the full interacting effective actions for supersymmetric D-branes in arbitrary bosonic type II supergravity backgrounds. The actions are presented in terms of component fields up to second order in fermions. As one expects, the actions are built from the supercovariant derivative operator and the $κ$-symmetry projector. The results take a compact and elegant form exhibiting $κ$-symmetry, as well as supersymmetry in a background with Killing spinors. We give the explicit transformation rules for these symmetries in all cases, including the M2-brane. As an example, we analyze the N=2 super-worldvolume field theory defined by a test D4-brane in the supergravity background produced by a large number of D0-branes. This example displays rigid supersymmetry in a curved spacetime.