Two-Center Black Holes, Qubits and Elliptic Curves

TL;DR

This paper explores the deep connections between black hole solutions in supergravity, quantum entanglement of four-qubits, and elliptic curves, revealing new invariants and mathematical structures relevant to physics.

Contribution

It establishes a novel link between U-duality invariants of two-center black holes, four-qubit entanglement invariants, and elliptic curves, introducing new mathematical tools for understanding black hole physics.

Findings

U-duality invariants relate to four-qubit entanglement classes.

Elliptic curves can be associated with charge configurations.

The hyperdeterminant's j invariant plays a role in two-center solutions.

Abstract

We relate the U-duality invariants characterizing two-center extremal black hole solutions in the stu, st^2 and t^3 models of N=2, d=4 supergravity to the basic invariants used to characterize entanglement classes of four-qubit systems. For the elementary example of a D0D4-D2D6 composite in the t^3 model we illustrate how these entanglement invariants are related to some of the physical properties of the two-center solution. Next we show that it is possible to associate elliptic curves to charge configurations of two-center composites. The hyperdeterminant of the hypercube, a four-qubit polynomial invariant of order 24 with 2894276 terms, is featuring the j invariant of the elliptic curve. We present some evidence that this quantity and its straightforward generalization should play an important role in the physics of two-center solutions.