Algorithms for representations of quiver Yangian algebras

• Published in: The journal of high energy physics Vol. 2024; no. 8; pp. 209 - 47
• Main Authors: Galakhov, Dmitry, Gavshin, Alexei, Morozov, Alexei, Tselousov, Nikita
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 27.08.2024; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Classical and Quantum Gravitation; Crystals; D-Branes; Elementary Particles; Linear algebra; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Groups; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Representations; String Theory; Superstring Vacua; Supersymmetric Effective Theories; D-Branes; Superstring Vacua; Quantum Groups; Supersymmetric Effective Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP08(2024)209

A bstract In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm’s implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of Y sl 2 , Y gl ̂ 1 and Y gl ̂ 1 1 .