Triality of Majorana-Weyl Spacetimes With Different Signatures

TL;DR

This paper explores the triality automorphisms of Spin(8) and their implications for dualities among Majorana-Weyl spacetimes with various signatures, revealing deep algebraic structures and symmetries in high-dimensional theories.

Contribution

It provides a detailed analysis of triality automorphisms acting on Majorana-Weyl spacetimes, establishing exact translations and dualities among different signatures and formulating invariant theories.

Findings

Identifies dualities among (1+9), (5+5), and (9+1) spacetimes.

Establishes a six-element permutation group S_3 for superstring dualities.

Constructs invariants for manifest space-time symmetry.

Abstract

Majorana-Weyl spacetimes offer a rich algebraic setup and new types of space-time dualities besides those discussed by Hull. The triality automorphisms of Spin(8) act non-trivially on Majorana-Weyl representations and Majorana-Weyl spacetimes with different signatures. In particular relations exist among the (1+9)-(5+5)-(9+1) spacetimes, as well as their transverse coordinates spacetimes (0+8)-(4+4)-(8+0). Larger dimensional spacetimes such as (2+10)-(6+6)-(10+2) also show dualities induced by triality. A precise three-languages dictionary is here given. It furnishes the exact translations among, e.g., the three different versions (one in each signature) of the ten-dimensional N=1 superstring and superYang-Mills theories. Their dualities close the six-element permutation group S_3. Bilinear and trilinear invariants allowing to formulate theories with a manifest space-time symmetry are constructed.