Changing the preferred direction of the refined topological vertex

TL;DR

This paper investigates the slice invariance of refined topological string amplitudes through examples involving gauge theory and link invariants, confirming invariance in some cases and observing breakdowns in others.

Contribution

It demonstrates slice invariance in refined topological string amplitudes for specific cases and identifies conditions where invariance fails, providing new insights into topological string theory.

Findings

Slice invariance holds for certain gauge theory examples.

Closed formula for superpolynomial of the Hopf link confirms invariance in some cases.

Breakdown of slice invariance observed for non-antisymmetric representations.

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.