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Composite representation invariants and unoriented topological string amplitudes. (English) Zbl 1207.81077
Summary: Sinha and Vafa 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 based on the work of Morton and Ryder 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.

MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)
58J28 Eta-invariants, Chern-Simons invariants
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