Symmetric deformed 2D/3D Hurwitz–Kontsevich model and affine Yangian of {mathfrak {gl}}(1)

Abstract

Since the (beta -deformed) Hurwitz Kontsevich model corresponds to the special case of affine Yangian of {mathfrak {gl}}(1). In this paper, we construct two general cases of the beta -deformed Hurwitz Kontsevich model. We find that the W-operators of these two models can be represented by the generators e_k, f_k,psi _k of the affine Yangian of {mathfrak {gl}}(1), and the eigenstates (the symmetric functions Y_lambda and 3-Jack polynomials) can be obtained from the 3D Young diagram representation of affine Yangian of {mathfrak {gl}}(1). Then we can see that the W-operators and eigenstates are symmetric about the permutations of coordinate axes.