Non-commutative tachyon action and D-brane geometry

TL;DR

This paper derives a non-commutative tachyon action from open string correlators in non-constant backgrounds, revealing a unique differential structure on D-branes where the connection is unaffected by the gauge field.

Contribution

It provides a new derivation of the tachyon action considering non-constant backgrounds and uncovers a distinctive geometric property of D-branes with a connection independent of the gauge field.

Findings

Derived a tachyonic on-shell condition and kinetic term.

Obtained a non-commutative tachyon potential from 3-point correlator.

Discovered the connection on the D-brane is the same as in closed string theory, independent of the gauge field.

Abstract

We analyse open string correlators in non-constant background fields, including the metric $g$, the antisymmetric $B$-field, and the gauge field $A$. Working with a derivative expansion for the background fields, but exact in their constant parts, we obtain a tachyonic on-shell condition for the inserted functions and extract the kinetic term for the tachyon action. The 3-point correlator yields a non-commutative tachyon potential. We also find a remarkable feature of the differential structure on the D-brane: Although the boundary metric $G$ plays an essential role in the action, the natural connection on the D-brane is the same as in closed string theory, i.e. it is compatible with the bulk metric and has torsion $H=dB$. This means, in particular, that the parallel transport on the brane is independent of the gauge field $A$.