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# Advances in Theoretical and Mathematical Physics

## Volume 9 (2005)

### Number 3

### $D$-brane dynamics in constant Ramond-Ramond potentials, $S$-duality and noncommutative geometry

Pages: 355 – 406

DOI: https://dx.doi.org/10.4310/ATMP.2005.v9.n3.a1

#### Authors

#### Abstract

We study the physics of $D$-branes in the presence of constant Ramond--Ramond potentials. In the string field theory context, we first develop a general formalism to analyze open strings in gauge trivial closed string backgrounds, and then apply it both to the RNS string and within Berkovits’ covariant formalism, where the results have the most natural interpretation. The most remarkable finding is that, in the presence of a $Dp$-brane, both a constant parallel NSNS $B$-field and RR $C^{\left(p-1\right)}$-field \textit{do not solve the open/closed equations of motion, and induce the same} \textit{non-vanishing open string tadpole}. After solving the open string equations in the presence of this tadpole, and after gauging away the closed string fields, one is left with a $U\left(1\right)$ field strength on the brane given by $F=\frac{1}{2}\left( B - \star C^{\left( p-1\right)}\right)$, where $\star$ is Hodge duality along the brane world-volume. One observes that this result differs from the usually assumed result $F=B$. Technically, this is due to the fact that supersymmetric and bosonic string world-sheet theories are different. Note, however, that the usual $F+B$ combination is \textit{still} the combination which remains gauge invariant at the $\sigma$-model level. Also, the standard result $F=B$ is, in the $D3$-brane case, *not compatible with $S$-duality.* On the other hand our result, which is derived automatically given the general formalism, offers a non-trivial check of $S$-duality, *to all orders in* $F$, and this leads to an $S$-dual invariant Moyal deformation. In an appendix, we solve the source equation describing the open superstring in a generic NSNS and RR closed string background, within the super-Poincaré covariant formalism.

Published 1 January 2005