Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}(\mathfrak{sl}(2|1))$

Abstract

This paper is a continuation of study by J. Ding and B.Feigin [1]. We find a bosonization of the deformed W -superalgebras Wqt(sl(2|1)) that commutes up-to total difference with deformed screening currents. Using our bosonization, we derive a set of quadratic relations of generators of the deformed W -superalgebra Wqt(sl(2|1)). The deformed W -superalgebra is independent of the choice of a Dynkindiagram for the superalgebra sl(2|1), though the deformed screening currents depend on it.

Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

“…• Studying horizontal representations (C = 1) [10,[16][17][18][19][20][21][22][23][24]69] of shifted QQTA is also one of the studies that must be done. Studying generalized intertwiners [29,30,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] with shift parameters as Φ r,r : (vertical) r ⊗ (horizontal) r → (horizontal) r+r is interesting.…”

“…On the other hand, representations with C = 1 are called horizontal representations. We will not discuss about these representations, so see [10,[16][17][18][19][20][21][22][23][24] for details. See also [69] for a good review.…”

“…Infinite-dimensional alebra has been one of the most powerful tools to study supersymmetric gauge theories [1][2][3][4][5]. In particular, quantum toroidal algebras [6][7][8][9][10][11][12][13][14][15] and their truncations [10,[16][17][18][19][20][21][22][23][24] have played significant roles in the context of 5d AGT correspondence [25][26][27][28][29][30][31][32] and topological vertex [29,30,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]…”

Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}\bigl(A(M,N)\bigr)$

“…For that matter it is worthwhile to concretely construct W q,t (g) in each case. This paper is a continuation of the paper [11] for W q,t A(1, 0) . The purpose of this paper is to generalize the result of case A(1, 0) to A(M, N ).…”

“…Shiraishi et al [1] obtained a free field construction of the deformed Virasoro algebra W q,t sl (2) , which is a one-parameter deformation of the Virasoro algebra, to construct a deformation of the correspondence between conformal field theory and the Calogero-Sutherland model. The theory of the deformed Walgebras W q,t (g) has been developed in papers [2][3][4][5][6][7][8][9][10][11]. However, in comparison with the conformal case, the theory of the deformed W -algebra is still not fully developed and understood.…”