The arrow of time and the Weyl group: all supergravity billiards are integrable

TL;DR

This paper demonstrates that all supergravity billiards derived from sigma-models on non-compact symmetric spaces in D=3 are integrable, revealing a deep connection between the arrow of time and Weyl group properties.

Contribution

It establishes the integrability of supergravity billiards via an upper triangular embedding of the solvable Lie algebra into SL(N,R) and links the arrow of time to Weyl group characteristics.

Findings

All supergravity billiards in this class are fully integrable.

The arrow of time correlates with the Weyl group ordering and entropy.

Asymptotic states correspond to Weyl group elements.

Abstract

In this paper we show that all supergravity billiards corresponding to sigma-models on any U/H non compact-symmetric space and obtained by compactifying supergravity to D=3 are fully integrable. The key point in establishing the integration algorithm is provided by an upper triangular embedding of the solvable Lie algebra associated with U/H into SL(N,R) which always exists. In this context we establish a remarkable relation between the arrow of time and the properties of the Weyl group. The asymptotic states of the developing Universe are in one-to-one correspondence with the elements of the Weyl group which is a property of the Tits Satake universality classes and not of their single representatives. Furthermore the Weyl group admits a natural ordering in terms of L(T), the number of reflections with respect to the simple roots and the direction of time flows is always towards increasing L(T), which plays the unexpected role of an entropy.