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My question is about web diagram in paper of Aganagic et al. How to get the $\alpha-\beta$ line rather than $-\alpha+\beta$ line of rightmost diagram of fig.3? More precisely, how to show the charge conservation at each vertex?

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Something similar happens in the earlier paper Fig. 13.

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The symbols $\alpha,\beta$ in these toric geometry diagrams refer to the two 1-dimensional cycles of a torus, $T^2$, and the labels which are combinations of $\alpha,\beta$ correspond to the 1-cycles of the torus that degenerate at the given line of the diagram.

If $c$ degenerates, so does $-c$, so all these labels are pretty much unoriented and the overall sign doesn't matter. The vectors are indeed sort of "conserved" in the vertices but you shouldn't imagine that it's completely the same diagram as a Feynman diagram with momenta.

However, the relative signs between the coefficients of $\alpha,\beta$ (and the whole ratios) are important, linked to some $(p,q)$ five-branes, and correlated with the orientation of the lines on which the labels are attached.

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