Flowing from $AdS_5$ to $AdS_3$ with $T^{1,1}$

TL;DR

The paper constructs supersymmetric domain-wall solutions in type IIB supergravity that interpolate from $AdS_5\times T^{1,1}$ in the UV to $AdS_3\times \mathbb{R}^2\times S^2\times S^3$ in the IR, driven by two axion-like fields in the Betti multiplet. The IR fixed points exhibit enhanced $(4,2)$ superconformal symmetry and are locally related by two T-dualities to the $AdS_3\times S^3\times S^3\times S^1$ background with large $(4,4)$ supersymmetry; the flows exist within a consistent $D=5$ $N=4$ gauged supergravity derived from KK reduction on $T^{1,1}$. Flux quantisation via Page charges is carefully analyzed, yielding a central charge $c = \tfrac{3}{2}|N Q_{N5} Q_{D5}| = 3|N\bar{N}|$ for the IR CFT, and revealing a relation $2\bar{N} = -Q_{N5}Q_{D5}$, with subtleties arising from gauge choices of the $B$-field. The work also presents a broader one-parameter family of flows to generalized $AdS_3\times \mathbb{R}^2\times S^2\times S^3$ fixed points, maintaining $(0,2)$ Poincaré supersymmetry and $(4,2)$ in the IR, and discusses potential generalizations to other Sasaki–Einstein compactifications and global aspects of T-duality.

Abstract

We construct supersymmetric domain wall solutions of type IIB supergravity that interpolate between $AdS_5\times T^{1,1}$ in the UV and $AdS_3\times\mathbb{R}^2\times S^2\times S^3$ solutions in the IR. The $\mathbb{R}^2$ factor can be replaced with a two-torus and then the solution describes a supersymmetric flow across dimensions, similar to wrapped brane solutions. While the domain wall solutions preserve $(0,2)$ supersymmetry, the $AdS_3$ solutions in the IR have an enhanced $(4,2)$ superconformal supersymmetry and are related by two T-dualities to the $AdS_3\times S^3\times S^3\times S^1$ type IIB solutions which preserve a large $(4,4)$ superconformal supersymmetry. The domain wall solutions exist within the $N=4$ $D=5$ gauged supergravity theory that is obtained from a consistent Kaluza-Klein truncation of type IIB supergravity on $T^{1,1}$; a feature driving the flows is that two $D=5$ axion like fields, residing in the $N=4$ Betti multiplet, depend linearly on the two legs of the $\mathbb{R}^2$ factor.

Figures (1)

• Figure 1: Supersymmetric domain wall solutions interpolating between $AdS_5\times T^{1,1}$ and $AdS_3\times \mathbb{R}^2\times S^2\times S^3$ given by \ref{['ansatz']}. From top to bottom in the figure, from the left, we have plotted the functions $A'$ (green), $U$ (red), $B'$ (blue), and $V$ (yellow).