Black Hole Entropy and Finite Geometry

Understand

It is shown that the $E_{6(6)}$ symmetric entropy formula describing black holes and black strings in D=5 is intimately tied to the geometry of the generalized quadrangle GQ$(2,4)$ with automorphism group the Weyl group $W(E_6)$.

• The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ$(2,4)$.
• Different truncations with $15, 11$ and 9 charges are represented by three distinguished subconfigurations of GQ$(2,4)$, well-known to finite geometers; these are the "doily" (i.
• e.