Open crepant resolution transformation formula for disk functions

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Let X→X←Y\mathcal{X}\rightarrow X\leftarrow Y be toric Calabi--Yau threefolds satisfying the closed crepant resolution conjecture, let O:HX→HY\mathbb{O}:\mathcal{H}_\mathcal{X}\to\mathcal{H}_Y be the transformation relating the corresponding open theories, and let W⊂K(Z)\mathfrak{W}\subset K(Z) be a grade-restriction window descending to bases for K(X)⊗CK(\mathcal{X})\otimes\mathbb{C} and K(Y)⊗CK(Y)\otimes\mathbb{C}. Write CH^∙=z−12deg⁡CH∙\widehat{CH}_\bullet=z^{-\frac12\deg}CH_\bullet for the homogenized Chern-character matrix in these bases, and let Θ∙\Theta_\bullet be the operator defined by the displayed Gamma-class and disk-function formula. Open crepant resolution transformation formula. The transformation should satisfy

O=ΘY∘CH^Y∘CH^X−1∘ΘX−1.\mathbb{O}=\Theta_Y\circ\widehat{CH}_Y\circ\widehat{CH}_\mathcal{X}^{-1}\circ\Theta_\mathcal{X}^{-1}.

This prediction expresses the open transformation through the grade-restriction-window correspondence and homogenized Chern characters; it is presented as a prediction, with no resolution status supplied.

References

Primary source

Andrea Brini, Renzo Cavalieri and Dustin Ross, “The open string McKay correspondence for type A singularities”, arXiv:1303.0723 (2014).

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