Orientifolds and the Refined Topological String

TL;DR

This paper explores refined topological string theory with orientifolds, proposing a new integrality condition, defining SO(2N) refined Chern-Simons theory, and establishing dualities and invariants for knots and gauge groups.

Contribution

It introduces a new integrality condition for refined topological strings with orientifolds and defines the SO(2N) refined Chern-Simons theory for computing refined invariants.

Findings

New invariants for torus knots generalizing Kauffman polynomials

Duality between SO(2N) refined Chern-Simons theory and refined topological strings on orientifolds

Definition and solution of refined Chern-Simons theory for all ADE gauge groups

Abstract

We study refined topological string theory in the presence of orientifolds by counting second-quantized BPS states in M-theory. This leads us to propose a new integrality condition for both refined and unrefined topological strings when orientifolds are present. We define the SO(2N) refined Chern-Simons theory which computes refined open string amplitudes for branes wrapping Seifert three-manifolds. We use the SO(2N) refined Chern-Simons theory to compute new invariants of torus knots that generalize the Kauffman polynomials. At large N, the SO(2N) refined Chern-Simons theory on the three-sphere is dual to refined topological strings on an orientifold of the resolved conifold, generalizing the Gopakumar-Sinha-Vafa duality. Finally, we use the (2,0) theory to define and solve refined Chern-Simons theory for all ADE gauge groups.