Duality Cascades and Parallelotopes

TL;DR

This paper explores the geometric structure of duality cascades in field theories, proposing that the fundamental domain of brane configurations forms a parallelotope that tiles the parameter space, based on physical and mathematical reasoning.

Contribution

It introduces the novel idea that the fundamental domain of supersymmetric brane configurations in duality cascades is a parallelotope, providing a geometric perspective on the structure of these cascades.

Findings

The fundamental domain can tile the entire parameter space.

Duality cascades always end, with the final state depending only on initial conditions.

The proposed parallelotope structure is supported by theoretical arguments.

Abstract

Duality cascades are a series of duality transformations in field theories, which can be realized as the Hanany-Witten transitions in brane configurations on a circle. In the setup of the ABJM theory and its generalizations, from the physical requirement that duality cascades always end and the final destination depends only on the initial brane configuration, we propose that the fundamental domain of supersymmetric brane configurations in duality cascades can tile the whole parameter space of relative ranks by translations, hence is a parallelotope. We provide our arguments for the proposal.