Changing the preferred direction of the refined topological vertex

TL;DR

Awata and Kanno investigate the slice invariance of the refined topological vertex by testing two cases: a five-dimensional ${U(1)}$ theory with matter and homological Hopf-link invariants from refined open strings. They develop a Macdonald-operator framework to perform the partition-sum in the antisymmetric sector, obtaining a closed, polynomial-form expression for the antisymmetric Hopf-link superpolynomials and proving invariance in that sector. They show that slice invariance holds for totally antisymmetric representations but can fail for other representations, providing explicit counterexamples and analyzing the structure via symmetric-function techniques and Nekrasov-Whitney-type factorization. The work connects refined vertex calculus to Macdonald theory and Nekrasov partition functions, clarifying the scope of slice invariance for refined BPS counting and homological link invariants.

Abstract

We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first example is a geometric engineering of five-dimensional U(1) gauge theory with a matter. The slice invariance follows from a highly non-trivial combinatorial identity which equates two known ways of computing the chi_y genus of the Hilbert scheme of points on C^2. The second example is concerned with the proposal that the superpolynomials of the colored Hopf link are obtained from a refinement of topological open string amplitudes. We provide a closed formula for the superpolynomial, which confirms the slice invariance when the Hopf link is colored with totally anti-symmetric representations. However, we observe a breakdown of the slice invariance for other representations.

Figures (3)

• Figure 1: Change of the preferred direction, which is indicated by the white arrow.
• Figure 2: Slicing and Flop; the preferred direction is indicated by the bold line .
• Figure 3: Choice of the preferred direction for $U(1)^N$ theory