Black hole entropy and finite geometry

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. GQ(2, 2)] with 15, the “perp set” of a point with 11, and the “grid” [i.e. GQ(2, 1)] with nine points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a noncommutative labeling for the points of GQ(2, 4). For the 40 different possible truncations with nine charges this labeling 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 $E_{7(7)}$ 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 2, featuring 27 points located on nine 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 noncommutative geometric structure based on GQ(2, 4).

Figure 1

A diagrammatic illustration of the structure of the generalized quadrangle GQ(2, 4) after Polster [31]. In both the figures, each picture depicts all 27 points (circles). The top picture shows only 19 lines (line segments and arcs of circles) of GQ(2, 4), with the two points located in the middle of the doily being regarded as lying one above and the other below the plane the doily is drawn in. Sixteen out of the missing 26 lines can be obtained by successive rotations of the figure through 72 degrees around the center of the pentagon. The bottom picture shows a couple of lines which go off the doily’s plane; the remaining 8 lines of this kind are again gotten by rotating the figure through 72 degrees around the center of the pentagon.

Figure 2

A qutrit labeling of the points of GQ(2, 4). Three different colors (online only) are used to illustrate a triple of grids partitioning the point set.

Figure 3

An illustration of the noncommutative labeling of the points of GQ(2, 4). For better readability of the figure, the sign of an operator is placed above the latter.