Black Hole Entropy and Finite Geometry

It is shown that the E6(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 (E6). 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. GQ(2, 2)) with 15, the “perp-set” of a point with 11, and the “grid” (i. e. GQ(2, 1)) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non-commutative labelling for the points of GQ(2, 4). For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares — objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the E7(7) symmetric entropy formula in D = 4 by observing that the structure of GQ(2, 4) is linked to a particular kind of geometric hyperplane of the split Cayley hexagon of order two, featuring 27 points located on 9 pairwise disjoint lines (a distance-3-spread). We conjecture that the different possibilities of describing the D = 5 entropy formula using Jordan algebras, qubits and/or qutrits correspond to employing different coordinates for an underlying non-commutative geometric structure based on GQ(2, 4).