### Abstract

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 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 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 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).

Original language | English |
---|---|

Article number | 084036 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 79 |

Issue number | 8 |

DOIs | |

Publication status | Published - Apr 1 2009 |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*79*(8), [084036]. https://doi.org/10.1103/PhysRevD.79.084036

**Black hole entropy and finite geometry.** / Lévay, P.; Saniga, Metod; Vrana, Péter; Pracna, Petr.

Research output: Contribution to journal › Article

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 79, no. 8, 084036. https://doi.org/10.1103/PhysRevD.79.084036

}

TY - JOUR

T1 - Black hole entropy and finite geometry

AU - Lévay, P.

AU - Saniga, Metod

AU - Vrana, Péter

AU - Pracna, Petr

PY - 2009/4/1

Y1 - 2009/4/1

N2 - 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 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 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 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).

AB - 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 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 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 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).

UR - http://www.scopus.com/inward/record.url?scp=66849087527&partnerID=8YFLogxK

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U2 - 10.1103/PhysRevD.79.084036

DO - 10.1103/PhysRevD.79.084036

M3 - Article

AN - SCOPUS:66849087527

VL - 79

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 8

M1 - 084036

ER -