Black Hole Entropy and Finite Geometry

Abstract

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 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 $E_{7(7)}$ 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 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$(2,4)$.

17 pages, 3 figures, v2 a new paragraph added, typos corrected

References in corpus (26)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Microscopic Origin of the Bekenstein-Hawking Entropy
• Hidden Variables and the Two Theorems of John Bell
• Universality of Sypersymmetric Attractors
• Orbits of Exceptional Groups, Duality and BPS States in String Theory
• E(7) Symmetric Area of the Black Hole Horizon
• Branes, central charges and U-duality invariant BPS conditions
• Black Holes and U-Duality
• String triality, black hole entropy and Cayley's hyperdeterminant
• Black Holes, Qubits and Octonions
• Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
• Strings, Black Holes, and Quantum Information
• Magic Supergravities, N= 8 and Black Hole Composites
• Stringy Black Holes and the Geometry of Entanglement
• E_7 and the tripartite entanglement of seven qubits
• E_{7(7)} Duality, BPS Black-Hole Evolution and Fixed Scalars
• Branes at Angles and Black Holes
• Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes
• Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane
• Freudenthal triple classification of three-qubit entanglement
• E_6 and the bipartite entanglement of three qutrits
• Wrapped branes as qubits
• Regular BPS black holes: macroscopic and microscopic description of the generating solution
• A three-qubit interpretation of BPS and non-BPS STU black holes
• Projective Ring Line Encompassing Two-Qubits
• Five Dimensional U-Duality, Black-Hole Entropy and Topological Invariants

Cited by in corpus (34)

• The black-hole/qubit correspondence: an up-to-date review
• On the Black-Hole/Qubit Correspondence
• STU Black Holes as Four Qubit Systems
• Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Small Orbits
• Observations on Integral and Continuous U-duality Orbits in N=8 Supergravity
• Grassmannian Connection Between Three- and Four-Qubit Observables, Mermin's Contextuality and Black Holes
• The Veldkamp space of multiple qubits
• Qubits from extra dimensions
• Mermin's Pentagram as an Ovoid of PG(3,2)
• Hitchin functionals are related to measures of entanglement
• Quantum contextual finite geometries from dessins d'enfants
• Mermin pentagrams arising from Veldkamp lines for three qubits
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• The attractor mechanism as a distillation procedure
• Contextuality degree of quadrics in multi-qubit symplectic polar spaces
• Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
• STU attractors from vanishing concurrence
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• Aspects of the Segre variety S_{1,1,1}(2)
• The Veldkamp Space of GQ(2,4)
• Geometric Hyperplanes of the Near Hexagon L_3 times GQ(2, 2)
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2)
• Magic Three-Qubit Veldkamp Line and Veldkamp Space of the Doily
• On the Veldkamp Space of GQ(4, 2)
• Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4)
• Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive
• A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line
• Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra $sl(3,C)\oplus u(1)$
• Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits
• Moebius Pairs of Simplices and Commuting Pauli Operators
• Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups