Abstract
The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional mathcal{N} = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.
Highlights
Introduction and summaryFor centuries physics and mathematics are tied together and their mutual influence goes in both directions
While physicists have often used known mathematical results to derive new results in physics, physicists can use their insights to create new mathematics, which can be of independent interest to mathematicians
The quiver Yangian in our opinion is an excellent example for such a new mathematical output arising from the physics of supersymmetric gauge theories and string theory
Summary
For centuries physics and mathematics are tied together and their mutual influence goes in both directions. We hope that the study of our general elliptic and toroidal quiver algebras and their representation theories will grow into a new exciting research area In physics language, these algebras are realized by 3d/2d/1d quantum field theories with four supercharges, in which the algebras are related by dimensional reductions. While we leave many questions for future work, we argue in this paper that we can define quiver BPS algebras for more general cases, associated with moduli spaces of supersymmetric field theories and/or generalized cohomology theories (and “formal group laws”). For this reason the possible family of quiver BPS algebras will go well beyond the traditional elliptictrigonometric-rational hierarchy of quiver BPS algebras (and the word ‘beyond’ in the title of this paper)..
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