"AdS/CFT from F-theory?"

At the lunch group meeting James told us about a fairly recent paper by Polchinski and Silverstein, entitled Dual Purpose Landscaping Tools: Small Extra Dimensions in AdS/CFT. Their goal is to construct vacua of string or M-theory with minimal supersymmetry of the form $AdS_d \times \text{small}$, where small means that the internal space is small compared to the AdS radius. The problem is that in the case of the standard Freund-Rubin backgrounds (whose dual CFTs are rather well understood in terms of the near-horizon limit of D3- or M2-branes at the tip of a Calabi-Yau cone), the radius of the internal space is of the same size as the AdS radius.

The new idea is to add some D7-branes. These contribute to the potential energy with an opposite sign to the curvature of the internal manifold, and so could be chosen such that the resulting cosmological constant is small.

This is where F-theory comes into play. The construction there is to glue IIB solutions using the $SL(2,\mathbb{Z})$ duality symmetry, and to allow for D7-branes. A D7-brane, which is codimension two and thus can be surrounded by a circle, is a unit magnetic source for the axion $C_0$ (a periodic RR scalar field): \[\int_{S^1} d C_0 =1~.\] The axion combines with the dilaton $\phi$in a complex field in the upper half-plane \[\tau = C_0 + i e^{-\phi} = C_0 +\frac{i}{g_s}~.\] $C_0$ has monodromy one around the D7-brane, meaning that going around the circle transforms it as $C_0 \to C_0 +1$.

Now splitting the 10d metric as a 4d Minkowski space-time times a 6d manifold $B$ and requiring $\mathcal{N}=1$ susy in 4d implies that $B$ is a Kaehler manifold and that $\tau$ is (anti-)holomorphic: $\bar\partial \tau =0$.

Mathematically, specifying the modular parameter $\tau$ in the fundamental region is equivalent to specifying an elliptic curve, and so this whole construction can be seen as an elliptic fibration $\pi: X\to B$, with $X$ a four complex dimensional manifold. The fibre over a point $p\in B$ on the base is an elliptic curve $E_{\tau}$ that is "biholomorphically" equivalent to a torus determined by a lattice $(1,\tau)$:
\[\pi^{-1} (p) \cong E_{\tau} \cong \mathbb{C}/(1,\tau)~.\]
What is very nice is that the D7-branes are located where the elliptic fibration degenerates, i.e. where the torus gets pinched off and becomes singular.
James made the point that this is not just a mathematical construction since one can T-dualise to IIA and the lift to M-theory, where the resulting solution is the product of a 3d Minkowski space-time and a CY four-fold. Polchinski and Silverstein's idea is to get $AdS_5 \times \text{small}$ solutions by looking at CY four-fold cones that are elliptically fibered...