Knots, links and branes at large N

By J. Labastida, Marcos Mariño, C. Vafa. 2000

248 citations recorded by the source corpus.

We consider Wilson loop observables for Chern-Simons theory at large N and its topological string dual and extend the previous checks for this duality to the case of links. We find an interesting structure involving representation/spin degeneracy of branes ending on branes which features in the large-N dual description of Chern-Simons theory. This leads to a refinement of the integer invariants for links and knots. We illustrate our results with explicit computations on the Chern-Simons side.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/ea3b08f6de84799105dde55d36185ee678e66425

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. C. Vafa
2. Marcos Mariño
3. Sergei Gukov
4. Mina Aganagic
5. E. Witten
6. Pavel Putrov
7. A. Klemm
8. Tudor Dimofte

In Mathematical Physics

1. Sergei Gukov
2. C. Vafa
3. Marcos Mariño
4. E. Witten
5. Mina Aganagic
6. Pavel Putrov
7. A. Klemm
8. B. Eynard

In Nuclear and High Energy Physics

1. Marcos Mariño
2. C. Vafa
3. Sergei Gukov
4. E. Witten
5. Mina Aganagic
6. Pavel Putrov
7. A. Klemm
8. D. Gaiotto

Citations in the corpus

• Disk Instantons, Mirror Symmetry and the Duality Web (2001, methodology): “To see how this works it is simplest to consider the case where the D-brane topological amplitudes were computed using the observables of Chern-Simons theory [4][27][26].”
• All Loop Topological String Amplitudes from Chern-Simons Theory (2002, methodology): “Various aspects of the duality were studied in [3,4,5,6,7,8,9,10] from different points of view.”
• Knot Invariants from Four-Dimensional Gauge Theory (2011, background): “See [5]-[13] for a sampling of additional developments.”
• Khovanov-Rozansky Homology and Topological Strings (2004, methodology): “reformulated in terms of integers N ,Q,s which capture the spectrum of BPS states in the string Hilbert space(1) [7,8]: PN (q) = 1 q − q−1 ∑”
• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial (2003, background): “11 A similar situation occurs in a different, though possibly not unrelated context in string theory [58].”
• The Superpolynomial for Knot Homologies (2005, background): “∗(Mg,Q). On the other hand, the invariants χ(Mg,Q), which in the physics literature are called “integer BPS invariants”, contain the in formation about all-genus open GromovWitten invariants of (X,LK)[25, 18]. The relation between the open Gromov-Witten invariants and the integer BPS invariants is v”
• R-Twisting and 4d/2d Correspondences (2010, background): “However, it has been argued [23, 24] that these contributions actually do admit a simple structure.”
• Torus Knots and Mirror Symmetry (2011, background): “As explained in [33], this can be evaluated by using the fact that the torus knot operators provide a representation of the fusion rules of the affine Lie algebra [31]; therefore, we can write W ) R1,.”
• Worldsheet derivation of a large N duality (2002, background): “Many checks have been performed for this conjectured duality including highly nontrivial tests involving Wilson loop operators to all order in the gauge coupling constant [8,9,10,11].”
• BPS spectra and 3-manifold invariants (2017, background): “to M5-branes in (2.13). Therefore, the sl(N) homology of M 3 = L K gets related to the bottom row of -colored HOMFLY-PT homology of K, where is a partition with Nrows (or Ncolumns). Moreover, as in [15,57], we need to relate the \partition basis" (natural on the LHS of (2.73)) to the \holonomy ”
• Mirror symmetry, D-branes and counting holomorphic discs (2000, methodology): “Using the large N duality conjecture [12] there have been some cases where one can compute certain corrections involving holomorphic maps from Riemann surfaces with boundaries to target space geometry [6][13][14].”
• Partition functions of holographic minimal models (2011, methodology): “2) is similar to the Ooguri-Vafa operator [33] (see also [34, 35, 36]) that arises in the topological string computation of knot invariants.”
• The Topological Vertex (2003, background): “Notice that the BPS degeneracies N(R1,···,RL),g,Q in the representation basis are more fundamental than the degeneracies in the winding number basis, as emphasized in [34].”