The Topological Vertex

By Mina Aganagic, A. Klemm, Marcos Mariño, C. Vafa. 2003

853 citations recorded by the source corpus.

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/6a53ff2d9bb3b2e347d42c737c7279839505b1e4

How researchers see this paper

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

In Applied Mathematics

1. Marcos Mariño
2. C. Vafa
3. H. Kanno
4. A. Klemm
5. R. Pandharipande
6. V. Bouchard
7. H. Awata
8. Masato Taki

In Geometry and Topology

1. Marcos Mariño
2. C. Vafa
3. Sergei Gukov
4. R. Pandharipande
5. Tudor Dimofte
6. A. Okounkov
7. A. Klemm
8. B. Eynard

In Nuclear and High Energy Physics

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

Citations in the corpus

• Knot Homology from Refined Chern-Simons Theory (2011, background): “In the original topological vertex of [44,21] ordinary ChernSimons theory played a crucial role in solving the theory.”
• M-Strings (2013, background): “The geometry of the blow-up is a local Calabi-Yau and is given by the periodic toric diagram [10, 15] in Fig.”
• Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions (2007, background): “An important step to understand more general curves is the case of the topological vertex [67].”
• Quantum Calabi-Yau and Classical Crystals (2003, background): “In particular, the computation of topological string amplitudes when we put D-branes in this background is nontrivial and leads to the notion of a topological vertex [1, 2] (see also the recent paper [3]).”
• Random Skew Plane Partitions and the Pearcey Process (2005, result): “In [16], the partition function of the random surface model studied here was related to the topological vertex of [ 1 ] and, thus, to the Gromov-Witten theory of toric Calabi-Yau threefolds.”
• Quantum geometry of refined topological strings (2011, background): “11) The skeleton of the corresponding Riemmanian surface is illustrated in figure 1 together with the integration contours for the classical periods [70].”
• Two Dimensional Yang-Mills, Black Holes and Topological Strings (2004, background): “Moreover this theory was studied at large N beginning with [3] from a number of viewpoints [4,5,6,14,15,16].”
• Non-perturbative effects and the refined topological string (2013, background): “This can indeed be done by the topological vertex [35, 36] or by direct integration of the holomorphic anomaly equations [6, 37], for example.”
• A mathematical theory of the topological vertex (2004, background): “13) is particularly simple in the representation basis used in [1].”
• Orbifolds of M-strings (2013, methodology): “As a main tool we will use the topological vertex and its refinement [5–7, 15, 16], to compute the degeneracy of BPS states.”
• Instanton counting, Macdonald function and the moduli space of D-branes (2005, methodology): “Since the topological partition function gives the generating function of topological invariants, the equalities imply rather surprising mathematical conjectures on the relation of invariants that seem to have quite different origin at the first sight.”
• Direct integration of the topological string (2007, methodology): “to toric Fano varieties with anticanonical bundle, then the F (g) can be unambiguously calculated using the topological vertex [2].”
• Black Holes, q-Deformed 2d Yang-Mills, and Non-perturbative Topological Strings (2004, methodology): “1 Our conventions differ here from [5] by q → q.”
• Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings (2007, methodology): “The total partition function ZXp can be computed as a sum over partitions, by making use of the topological vertex formalism as described in [45].”
• Quantum Riemann Surfaces in Chern-Simons Theory (2011, background): “These ideas — leading in essence to a B-model mirror of the topological vertex formalism [28] — will not be further explored in this paper, but will rather be the topic of future work [29].”
• Topological strings and Nekrasov's formulas (2003, methodology): “ling formulas containing Chern-Simons invariants and a need for developing more powerful machinery. One obvious candidate is the operator method of two-dimensional CFT which is now being developed in [17, 33]. In a future publication we would like to study more details of topological string theory b”