The Topological Vertex

TLDR: In this paper, a cubic field theory was constructed for all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefold.

Abstract: 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 Kahler 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.

Figures

Fig. 1 The degenerate locus of the T 2 × R fibration of C3 in the base R3 = (rα, rβ , rγ). This locus is a graph Γ. The labels (−pi, qi) correspond to the cycles of T 2 which vanish over the corresponding edge.

Fig. 9 The figure on the left corresponds to Chern-Simons theory on S3 with three source D-branes. Q1,2,3, Q denote the bifundamental strings. In the large N limit, the S3 undergoes a geometric transition. The figure on the right depicts the large N dual geometry, with Lagrangians L1,2,3, after the transition. The local patch where the D-branes are is a C3.

Fig. 7 Three stacks of D-branes on C3. We have introduced graphs F1,2,3 to help us keep track of framing. Fi are straight lines in the base, corresponding to vectors fi in the text. Different choices of fi give different amplitudes. The choice in this figure is defined to be canonical framing.

Fig. 14 This shows O(−1) +O(−1) → P1 with D-branes on external legs

Fig. 16 There are three Kähler parameters in the problem. The size of leg corresponding to R0 is s, and the sizes of R3 and R2 correspond to t1, t2, respectively.

Fig. 3 The graph Γ̂ of O(−3) → P2. The black points correspond to vectors ~wi. Its dual is the graph of degenerate fibers Γ.