Algorithms for representations of quiver Yangian algebras

Abstract

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 $\mathsf{Y}({\mathfrak{sl}}_2)$, $\mathsf{Y}({\widehat{\mathfrak{gl}}}_1)$ and $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$.

41 pages, 9 figures

References in corpus (13)

• Brane Tilings
• Non-commutative Donaldson-Thomas theory and the conifold
• Brane Tilings and Their Applications
• Crystal Melting and Toric Calabi-Yau Manifolds
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Shifted Quiver Yangians and Representations from BPS Crystals
• Interpolating Matrix Models for WLZZ series
• Many-body integrable systems implied by WLZZ models
• Crystal Melting and Wall Crossing Phenomena
• Gauge/Bethe correspondence from quiver BPS algebras
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• Kerov functions for composite representations and Macdonald ideal
• Multiple D4-D2-D0 on the Conifold and Wall-crossing with the Flop