Composite Representation Invariants and Unoriented Topological String Amplitudes

TL;DR

This paper advances the understanding of topological string amplitudes on orientifolds by explicitly computing composite invariants in Chern-Simons theory, verifying key conjectures, and tabulating BPS invariants for unoriented amplitudes.

Contribution

It provides explicit calculations of composite invariants for knots and links, verifies generalized Rudolph's theorem, and confirms Marino's conjectures on integrality of topological string amplitudes.

Findings

Verified generalized Rudolph's theorem relating composite invariants to SO(N) invariants.

Confirmed Marino's conjectures on integrality of topological string amplitudes.

Tabulated BPS invariants for cross-cap c=0 and c=2 amplitudes.

Abstract

Sinha and Vafa \cite {sinha} had conjectured that the $SO$ Chern-Simons gauge theory on $S^3$ must be dual to the closed $A$-model topological string on the orientifold of a resolved conifold. Though the Chern-Simons free energy could be rewritten in terms of the topological string amplitudes providing evidence for the conjecture, we needed a novel idea in the context of Wilson loop observables to extract cross-cap $c=0,1,2$ topological amplitudes. Recent paper of Marino \cite{mar9} based on the work of Morton and Ryder\cite{mor} has clearly shown that the composite representation placed on the knots and links plays a crucial role to rewrite the topological string cross-cap $c=0$ amplitude. This enables extracting the unoriented cross-cap $c=2$ topological amplitude. In this paper, we have explicitly worked out the composite invariants for some framed knots and links carrying composite representations in $U(N)$ Chern-Simons theory. We have verified generalised Rudolph's theorem, which relates composite invariants to the invariants in $SO(N)$ Chern-Simons theory, and also verified Marino's conjectures on the integrality properties of the topological string amplitudes. For some framed knots and links, we have tabulated the BPS integer invariants for cross-cap $c=0$ and $c=2$ giving the open-string topological amplitude on the orientifold of the resolved conifold.