Complexified sigma model and duality

TL;DR

This paper introduces 'complexoids', a new mathematical concept related to complex numbers and matroids, used to explore duality in complexified sigma-models with potential implications for quantum mechanics.

Contribution

It presents the concept of complexoids and demonstrates their application in formulating a complexified sigma-model with (1+1)-worldsheet metric, challenging existing symmetry assumptions.

Findings

Complexoids relate to complex numbers and matroids.

They enable a complexified sigma-model with Lorentzian worldsheet.

The work suggests potential for advancing complexified quantum mechanics.

Abstract

We show that the equations of motion associated with a complexified sigma-model action do not admit manifest dual SO(n,n) symmetry. In the process we discover new type of numbers which we called `complexoids' in order to emphasize their close relation with both complex numbers and matroids. It turns out that the complexoids allow to consider the analogue of the complexified sigma-model action but with (1+1)-worldsheet metric, instead of Euclidean-worldsheet metric. Our observations can be useful for further developments of complexified quantum mechanics.