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On isomorphisms between quiver Yangians

Vishnu Jejjala, Dumisani Nxumalo, Konstantinos Zoubos

TL;DR

This work provides an explicit isomorphism between the quiver Yangians of the two Seiberg-dual phases of the $\\mathbb{F}_0$ quiver, a setting that lies beyond previously treated generalised conifolds. Central to the construction is a novel square-root of a bilinear fermionic operator on the dualised node, which, when combined with carefully chosen bosonic bilinears, reproduces the Phase II bond factors and OPEs from Phase I data (and vice versa). The authors give detailed maps for the dualised node and adjacent nodes, verify all relevant current relations, and present an inverse map, while discussing limitations and the potential need for mode-level treatment. The results demonstrate Seiberg-duality isomorphisms at the level of currents for a chiral quiver with compact four-cycles and point toward broader applications to other quivers, including dP quivers and toroidal/elliptic generalisations, with implications for representations and crystal structures across chambers.

Abstract

Quiver Yangians are infinite-dimensional algebras capturing the BPS structure of a large class of supersymmetric models. Quiver theories related by Seiberg duality are expected to have isomorphic quiver Yangians, and this isomorphism has previously been shown for quivers corresponding to generalised conifold geometries. In this work, we present an explicit isomorphism for the two Seiberg dual phases of the F0 quiver theory, which falls outside of the above class. Some aspects of our construction are similar to the known cases, while others appear to be specific to the F0 quiver. In particular, the map involves square roots of operators bilinear in the fermionic fields of the mode being dualised.

On isomorphisms between quiver Yangians

TL;DR

This work provides an explicit isomorphism between the quiver Yangians of the two Seiberg-dual phases of the quiver, a setting that lies beyond previously treated generalised conifolds. Central to the construction is a novel square-root of a bilinear fermionic operator on the dualised node, which, when combined with carefully chosen bosonic bilinears, reproduces the Phase II bond factors and OPEs from Phase I data (and vice versa). The authors give detailed maps for the dualised node and adjacent nodes, verify all relevant current relations, and present an inverse map, while discussing limitations and the potential need for mode-level treatment. The results demonstrate Seiberg-duality isomorphisms at the level of currents for a chiral quiver with compact four-cycles and point toward broader applications to other quivers, including dP quivers and toroidal/elliptic generalisations, with implications for representations and crystal structures across chambers.

Abstract

Quiver Yangians are infinite-dimensional algebras capturing the BPS structure of a large class of supersymmetric models. Quiver theories related by Seiberg duality are expected to have isomorphic quiver Yangians, and this isomorphism has previously been shown for quivers corresponding to generalised conifold geometries. In this work, we present an explicit isomorphism for the two Seiberg dual phases of the F0 quiver theory, which falls outside of the above class. Some aspects of our construction are similar to the known cases, while others appear to be specific to the F0 quiver. In particular, the map involves square roots of operators bilinear in the fermionic fields of the mode being dualised.

Paper Structure

This paper contains 17 sections, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quiver Yangians
  3. The two phases of $\mathbb{F}_0$
  4. Quiver Yangian for phase I
  5. Quiver Yangian for phase II
  6. Mapping the fields
  7. Dualised node
  8. Adjacent nodes
  9. Checking the map
  10. The $\psi$ fields
  11. The inverse map
  12. A remark on simpler maps
  13. Discussion
  14. Quadratic relations for the $\mathbb{F}_0$ quiver Yangian
  15. Phase I
  16. ...and 2 more sections

Figures (3)

  • Figure 1: The square move/urban renewal transformation implementing Seiberg duality at the level of the bipartite graph.
  • Figure 2: The quiver and bipartite diagram for phase I of $\mathbb{F}_0$. We have indicated the charges satisfying the loop and vertex constraints @(\ref{['LoopVertex']}). To better visualise the direction of the arrows in the bipartite graph, we have placed the charges on the side of each edge where the arrow is pointing.
  • Figure 3: The quiver and bipartite diagram for phase II of $\mathbb{F}_0$. Note the exchange of white and black nodes for face 4 (the one being dualised), corresponding to the reversal of all the arrows involving that node in the quiver. We again indicate the charges satisfying @(\ref{['LoopVertex']}), with the charge assignments for node/face 2 having been kept the same as for phase I.