Twisted K-theoretic Gromov–Witten invariants

Abstract

We introduce twisted K-theoretic Gromov–Witten (GW) invariants in the frameworks of both “ordinary” and permutation-equivariant K-theoretic GW theory defined recently by Givental. We focus on the case when the twisting is given by the Euler class of an index bundle which allows one (under a convexity assumption on the bundle) to relate K-theoretic GW invariants of hypersurfaces with those of the ambient space. Using the methods developed in Givental and Tonita (Math Sci Res Inst Publ 62:43–92. Cambridge Univ. Press, Cambridge, 2014) we characterize the range of the J-function of the twisted theory in terms of the untwisted theory. As applications we use the $$\mathcal {D}_q$$ module structure in the permutation-equivariant case to generalize results of Givental (Permutation-equivariant quantum K-theory I–VIII. https://math.berkeley.edu/~giventh/perm/perm.html , 2015): we prove a general “quantum Lefschetz” type theorem for complete intersections given by zero sections of convex vector bundles and we relate points on the cones of the total space with those of the base of a toric fibration.