Refined knot invariants and Hilbert schemes

Moduli of flags of sheaves and their K-theory

“…It is easy to see that these formulas are actually imposed by iterating (4.12) k times to compute the action of z n1 * ... * z n k . 6 4.9. The shuffle algebra formalism allows us write down explicitly many elements of A, and then the above formulas tell us how they act on K. In particular, an important class of elements of A that were defined in [9] are:…”

“…Theorem 4.10. For any Laurent polynomial m(z 1 , ..., z k ), the geometric correspondence x ± m : K −→ K of (3.16) is given by the shuffle element: 6 The reason the order of the contours differs in the cases + and − is that the creation shuffle elements P + satisfy the opposite algebra relations from the annihilation shuffle elements P − Remark 4.11. Theorem 4.10 shows us why the geometric correspondences x m are not simply compositions of simple Nakajima correspondences.…”

“…Our formulas allow us to explicitly compute the matrix coefficients of P k,d in the basis of torus fixed points. Certain special cases of these coefficients are interpreted as refined knot invariants in [6], where many combinatorial and representation theoretic consequences are explored.…”

“…On a deeper level, the moduli spaces Z k of flags of sheaves (1.2) appear in a conjecture of Bezrukavnikov and Okounkov concerning filtrations on the category of coherent sheaves on the Hilbert scheme. As a special case of this conjecture, they are expected to be related to certain modules of the rational Cherednik algebra (see [6]). We intend to develop the structure of Z k in more detail in subsequent papers.…”

Refined Chern-Simons theory and (q, t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit

“…Refined Chern-Simons theory has been of recent interest because of the rich structure of the new knot and three-manifold invariants that it computes, and also because of its connection to refined topological string theory and the refined topological vertex [1,2,3,4,5,6,7]. In this paper we explore its relation to refined topological string theory on the local Calabi-Yau threefold X = O(−p) ⊕ O(p − 2) −→ P 1 .…”