A geometric dual of $c$-extremization

Understand

We consider supersymmetric AdS$_3 \times Y_7$ and AdS$_2 \times Y_9$ solutions of type IIB and $D=11$ supergravity, respectively, that are holographically dual to SCFTs with $(0,2)$ supersymmetry in two dimensions and $\mathcal{N}=2$ supersymmetry in one dimension.

• The geometry of $Y_{2n+1}$, which can be defined for $n\ge 3$, shares many similarities with Sasaki-Einstein geometry, including the existence of a canonical R-symmetry Killing vector, but there are also some crucial differences.
• We show that the R-symmetry Killing vector may be determined by extremizing a function that depends only on certain global, topological data.
• In particular, assuming it exists, for $n=3$ one can compute the central charge of an AdS$_3 \times Y_7$ solution without knowing its explicit form.