Equivariant K-theory of Hilbert schemes via shuffle algebra

Abstract

In this paper we construct actions of Ding-Iohara and shuffle algebras on the sum of localized equivariant K-groups of Hilbert schemes of points on C 2 . We show that commutative elements K i of shuffle algebra act through vertex operators over the positive part {h i } i>0 of the Heisenberg algebra in these K-groups. This provides an action of the Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space C[h 1 , h 2 , . . .].

Explicit examples of DIM constraints for network matrix models

“…As was mentioned earlier, the vertical representation of the DIM algebra has a combinatorial description in terms of Young diagrams, [60,63]. We have…”

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

“…(1.1), the job of the R-matrix is to permute the components in the tensor product of representations of the algebra G. This is the property we will use in refined topological strings. The representations in question are going to be Fock modules [37][38][39] and their permutation exchanges the legs of the toric diagram corresponding to a DIM intertwiner [40,41]. The permutation of the legs performed by the R-matrix has a simple interpretation in terms of the corresponding conformal blocks of the q-Virasoro or qW N -algebras.…”

“…a bosonization of the DIM generators, which are expressed through exponentials of the free bosons (for concrete expressions see [37][38][39][40]). The second central charge of this representation is trivial, C 2 = 1, while the first one is given by C 1 = (t/q) 1/2 .…”

“…horizontal one, but with a different action of the DIM generators [37][38][39][40]. In the basis of Macdonald symmetric polynomials, M (q,t) Y (a −n )|u , the generators x + n add a box to the Young diagram Y , while x − n delete one box, and ψ ± n act diagonally.…”

“…In the basis of Macdonald symmetric polynomials, M (q,t) Y (a −n )|u , the generators x + n add a box to the Young diagram Y , while x − n delete one box, and ψ ± n act diagonally. The central charges of this representation are (1, (t/q) 1/2 ) (we refer the reader to [37][38][39][40] for the complete construction).…”