Toroidal and elliptic quiver BPS algebras and beyond

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 $$ \mathcal{N} $$ 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.

Crystal melting, BPS quivers and plethystics

“…For example, it was conjectured in [51] that the positive part of the Maulik-Okounkov Yangian [39] of a quiver is isomorphic to the COHA associated to its tripled quiver. Moreover, recent progress on related quantum algebras and crystals has been made including shifted Yangians, toroidal and (hyper)elliptic BPS algebras [90][91][92][93]. In particular, it would be interesting to compare the crystals for different chambers in this paper with those studied in [91].…”

Quantum loop groups for arbitrary quivers

“…for each i ∈ I separately 4 . 4 Although the ζ functions might seem to contribute simple poles at z ia − z ib for a = b to the right-hand side of (2.5), these poles disappear when taking the symmetrization (the poles in question can only have even order in any symmetric rational function).…”

“…After establishing Theorem 1.5 in Section 2, the main purpose of Section 3 is to discuss the "thorny" question of Remark 1.3 in the general setting of Subsection 1. 4. In other words, we would like to explicitly describe the two-sided ideal…”

“…Our construction also applies to quiver quantum toroidal algebras. These are trigonometric versions (introduced in [4,15]) of the quiver Yangians (introduced in [7], see also [20] for a related mathematical construction) that act on BPS states for non-compact toric Calabi-Yau threefolds X. More specifically, one associates to such an X a quiver Q endowed with arrow parameters {q ij e } i,j∈I,e∈{1,...,#ij} ∈ Rep C * ×C * See [15, Section B] for the precise construction; the two-dimensional torus C * × C * should be interpreted as the kernel of the Calabi-Yau form.…”

5d AGT correspondence of supergroup gauge theories from quantum toroidal $\mathfrak{gl}_{1}$

“…Then, we can construct representations using simultaneous eigenstates of these Cartan operators. The eigenstates have a crystal like interpretation and actually they are related to BPS crystals [85][86][87][88][89][90][91][92][93], so we call them crystal representations 5 . The basis of the representations we consider is labeled by Young diagrams (see Appendix A for the notations).…”