We study the framing dependence of the Wilson loop observable of U(N) Chern-Simons gauge theory at large N. Using proposed geometrical large N dual, this leads to a direct computation of certain topological string amplitudes in a closed form. This yields new formulae for intersection numbers of cohomology classes on moduli of Riemann surfaces with punctures (including all the amplitudes of pure topological gravity in two dimensions). The reinterpretation of these computations in terms of BPS degeneracies of domain walls leads to novel integrality predictions for these amplitudes. Moreover we find evidence that large N dualities are more naturally formulated in the context of U(N) gauge theories rather than SU(N).

Framed knots at large N

Two different conformal field theories can be joined together along a defect line. We study such defects for the case where the conformal field theories on either side are single free bosons compactified on a circle. We concentrate on topological defects for which the left- and right-moving Virasoro algebras are separately preserved, but not necessarily any additional symmetries. For the case where both radii are rational multiples of the self-dual radius we classify these topological defects. We also show that the isomorphism between two T-dual free boson conformal field theories can be described by the action of a topological defect, and hence that T-duality can be understood as a special type of order–disorder duality.

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Topological defects for the free boson cft

We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and $\mathcal{R}$-matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and $\mathcal{R}$-matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental $\mathcal{R}$-matrices and clarifying some conjectures formulated in previous papers.

Cabling procedure for the colored HOMFLY polynomials

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.

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Chern–Simons Invariants of Torus Links

We elaborate the Chern-Simons eld theoretic method to obtain colored HOM- FLY invariants of knots and links. Using multiplicity-free quantum 6j-symbols for Uq(slN ), we present explicit evaluations of the HOMFLY invariants colored by symmetric represen- tations for a variety of knots, two-component links and three-component links.

Colored homfly polynomials from chern–simons theory

U(N) framed links, three-manifold invariants, and topological strings

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 $c=0$, $c=1$ and $c=2$ giving the open-string topological amplitude on the orientifold of the resolved conifold.

Composite Invariants and Unoriented Topological String Amplitudes

Composite Representation Invariants and Unoriented Topological String Amplitudes

SO(N) reformulated link invariants from topological strings

Motivated by the duality conjecture of Dijkgraaf and Vafa between supersymmetric gauge theories and matrix models, we derive the effective superpotential of N=1 supersymmetric gauge theory with gauge group SO(N_c) and arbitrary tree-level polynomial superpotential of one chiral superfield in the adjoint representation and $N_f$ fundamental matter multiplets.

Pravina Borhade

Papers

U(N) framed links, three-manifold invariants, and topological strings

Composite Invariants and Unoriented Topological String Amplitudes

Composite Representation Invariants and Unoriented Topological String Amplitudes

SO(N) reformulated link invariants from topological strings

Exact Effective Superpotential for SO(N_c) Gauge Theory with $N_f$ Flavors