We consider branes $N$ in a Schwarzschild-$\text{AdS}_{(n+2)}$ bulk, where the stress-energy tensor is dominated by the energy density of a scalar fields map $\varphi{:}\ N\to \mathcal{S}$ with potential $V$, where $\mathcal{S}$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field equation that is smooth across the singularity $r=0$, and which has smooth and uniquely determined solutions which exist across the singularity in an interval $(-\epsilon,\epsilon)$. Restricting a solution to $(-\epsilon,0)$ \resp $(0,\epsilon)$, and assuming $n$ odd, we obtain branes $N$ resp. $\hat{N}$ which together form a smooth hypersurface. Thus a smooth transition from big crunch to big bang is possible both geometrically as well as physically.

We demonstrate that for a broad class of local Calabi--Yau geometries built around a string of $\I\mathbb{P}^1$'s — those whose toric diagrams are given by triangulations of a strip — we can derive simple rules, based on the topological vertex, for obtaining expressions for the topological string partition function in which the sums over Young tableaux have been performed. By allowing non-trivial tableaux on the external legs of the corresponding web diagrams, these strips can be used as building blocks for more general geometries. As applications of our result, we study the behavior of topological string amplitudes under flops, as well as check Nekrasov's conjecture in its most general form.

We derive a formula for D3-brane charge on a compact spacetime, which includes torsion corrections to the tadpole cancellation condition. We use this to classify D-branes and Ramond-Ramond fluxes in type II string theory on $\mathbb{R}P^3\times\mathbb{R}P^{2k+1}\times S^{6-2k}$ with torsion $H$-flux and to demonstrate the conjectured $T$-duality to $S^3\times S^{2k+1}\times S^{6-2k}$ with no flux. When $k=1$, $H\neq 0$ and so the $K$-theory that classifies fluxes is twisted. When $k=2$, the square of the $H$-flux yields an $S$-dual Freed-Witten anomaly, which is canceled by a D3-brane insertion that ruins the dual $K$-theory flux classification. When $k=3$, the cube of $H$ is nontrivial and so the D3 insertion may itself be inconsistent and the compactification unphysical. Along the way we provide a physical interpretation for the Atiyah-Hirzebruch spectral sequence in terms of the boundaries of branes within branes.

The triangle anomalies in conformal field theory, which can be used to determine the central charge $a$, correspond to the Chern-Simons (CS) couplings of gauge fields in $\mathrm{AdS}_5$ under the gauge$/$gravity correspondence. We present a simple geometrical formula for the CS couplings in the case of type IIB supergravity compactified on a five-dimensional Einstein manifold $X$. When $X$ is a circle bundle over del Pezzo surfaces or a toric Sasaki-Einstein (SE) manifold, we show that the gravity result is in perfect agreement with the corresponding quiver gauge theory. Our analysis reveals an interesting connection with the condensation of giant gravitons or dibaryon operators which effectively induces a rolling among SE vacua.