Confluence for the K-theoretic J-function

I am planning to talk about my recent paper (1) written in collaboration with Alexis Roquefeuil. In the first part of the talk I would like to explain the background of our project: quantum differential equations and K-theoretic quantum q-difference equations in genus-0 Gromov--Witten theory. In the second part of the talk, I would like to explain our main result with an interesting application. Namely, under the assumption that the first Chern class of the tangent bundle is positive, we proved that the small J-function in quantum cohomology can be obtained as a limit q -->1 of the small J-function in quantum K-theory. In the case of a Fano toric manifold of Picard rank 2, we proved the K-theoretic version of an identity due to Iritani that relates the I-function of the toric manifold and the oscillatory integral of the toric mirror. In particular, our confluence result yields a new proof of Iritani's identity in the case of a Fano toric manifold of Picard rank 2.

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