Yangian associated with 2D𝒩= 1 SCFT

Abstract

Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at each site is identical to the Hilbert space of free boson in two dimensions. We give a brief review of their construction and explain the relation with W n algebra and Calogero-Sutherland model. As a generalization, we examine the Yangian associated with N = 1 superconformal algebra which describes a supersymmetric extension of Calogero-Sutherland model and compare it with the literature.

Instanton R-matrix and $$ \mathcal{W} $$-symmetry

“…In [64] an expansion of (3.30) at infinite value of the central element a − 0(cyl) was studied. In the rest of this section, we will write simply a m instead of a − m(cyl) .…”

Higher spins and Yangian symmetries

“…We note that the affine Yangian (as well as W 1+∞ [λ]) also possess larger representations; for example, for λ = N there are also representations that are labelled by N independent Young diagrams. (This follows from the fact that the algebra SH c , which is isomorphic to the affine Yangian of gl 1 , see [28] and section 5 below, has such representations, see e.g. [29].…”

“…The affine Yangian is believed to be isomorphic [11,28] to the spherical degenerate double affine Hecke algebra, the so-called SH c algebra of [21], although the detailed dictionary has, to our knowledge, not been written down before. In this section we exhibit this isomorphism in detail; we also explain how this fits together with the equivalence to the universal enveloping algebra of W 1+∞ [λ].…”

“…(The alert reader will notice that this is a generalization of (2.29), to which it reduces if a i = h 3 (i − 1).) With these conventions, the action of D ±1 (z) on these states is defined as 28) JHEP04 (2017)152 where Res is the residue, while D 0 (z) acts diagonally as…”

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

“…By exchanging the spectral parameters, the R-matrix inverts the momenta, and therefore acts exactly as the Liouville reflection matrix introduced in [42]. This connection (first noted in [43], see also [44]) is quite interesting, since, as we will see in the following, the R-matrix can be evaluated explicitly by solving for the eigenfunctions of the generalized Macdonald Hamiltonian with known eigenvalues.…”