String triality, black hole entropy and Cayley's hyperdeterminant

TL;DR

The paper analyzes the STU model as a four-dimensional $N=2$ supergravity theory with a triple $[SL(2,Z)]$-duality and an on-shell $S$-$T$-$U$ triality, focusing on extremal black holes carrying eight charges. It shows these charges arrange into a $2\times2\times2$ hypermatrix $a_{ijk}$ and that the black hole entropy is given by $S=\pi\sqrt{-\mathrm{Det}~a_3}$, with $\mathrm{Det}~a_3$ a quartic invariant under the dualities and triality. A key result is the correspondence between black hole entropy and Cayley’s hyperdeterminant, which also appears as the 3-tangle in three-qubit entanglement, where $\tau_{ABC}=4|\mathrm{Det}~a_3|$. This work highlights deep mathematical connections between string-theoretic black hole microphysics and quantum information theory, linking moduli-independent entropy to a tripartite entanglement measure via Cayley’s hyperdeterminant.

Abstract

The four-dimensional N=2 STU model of string compactification is invariant under an SL(2,Z)_S x SL(2,Z)_T x SL(2,Z)_U duality acting on the dilaton/axion S, complex Kahler form T and the complex structure fields U, and also under a string/string/string triality S-T-U. The model admits an extremal black hole solution with four electric and four magnetic charges whose entropy must respect these symmetries. It is given by the square root of the hyperdeterminant introduced by Cayley in 1845. This also features in three-qubit quantum entanglement.