Deformation of $D_p$-Brane Worldvolume in Weakly Curved Background

We study a $D_p$-brane in a parallelizable NS-NS background. The article starts with a brief review of the non-associative deformation of $D$-brane worldvolume in presence of torsion \cite{cornalba}. We suggest an alternative form and heuristic derivation of the open string metric for weakly curved backgrounds, by promoting the constant two-form in the flat space formula to a dynamical two-form and then Taylor expanding the bulk fields in Riemann Normal Coordinates at the origin. For weakly curved backgrounds, terms only upto the leading order in the NS-NS field strength or torsion contribute. This formalism differs from the author's earlier works in a collaboration \cite{richa}. We use the open string metric proposed in this paper to determine the deformation of $D_5$-brane for a particular NS-NS background. It turns out that a spherical $D_5$-brane with torsion acts like an extremal black 5-brane in the limit of radius $r \rightarrow 0$. This is an interesting case of gauge/gravity duality with an intractable non-commutative and non-associative gauge theory on a $D_p$-brane with torsion, which is equivalent to a simpler and ordinary gravity theory on the extremal black brane.