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 do not solve the open/closed equations of motion, and induce the samenon-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 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.

In the context of $2+1$-dimensional quantum gravity with negative cosmological constant and topology $\mathbb{R} \times T^2$, constant matrix-valued connections generate a $q$-deformed representation of the fundamental group, and signed area phases relate the quantum matrices assigned to homotopic loops. Some features of the resulting quantum geometry are explored, and as a consequence a quantum version of the Goldman bracket is obtained.

The problem of finding a holographic dual to string theory on $AdS_3 \times \mathbf{S}^3 \times \mathbf{S}^3 \times \mathbf{S}^1$ is examined in depth. This background supports a large $\mathcal{N}=4$ superconformal symmetry. While in some respects similar to the familiar small $\mathcal{N}=4$ systems on $AdS_3\times \mathbf{S}^3\times K3$ and $AdS_3\times \mathbf{S}^3\times T^4$, there are important qualitative differences. Using an analogue of the elliptic genus for large $\mathcal{N}=4$ theories we rule out all extant proposals--in their simplest form--for a holographic duality to supergravity at generic values of the background fluxes. Modifications of these extant proposals and other possible duals are discussed.