Permutation-equivariant quantum K-theory X. Quantum Hirzebruch-Riemann-Roch in genus 0

TL;DR

This paper derives genus 0 consequences from the all-genera quantum Hirzebruch-Riemann-Roch formula, generalizing previous characterizations and analyzing invariance properties of the quantum K-theory big J-function.

Contribution

It extends the adelic characterization of genus 0 quantum K-theory and explores invariance under finite difference operators, providing explicit descriptions for point targets.

Findings

Re-proved and generalized the adelic characterization of genus 0 quantum K-theory.

Proved invariance of the big J-function under finite difference operators.

Provided explicit description of the big J-function for point targets.

Abstract

We extract genus $0$ consequences of the all genera quantum HRR formula proved in Part IX. This includes re-proving and generalizing the adelic characterization of genus $0$ quantum K-theory found in [Givental A., Tonita V., in Symplectic, Poisson, and Noncommutative Geometry, <I>Math. Sci. Res. Inst. Publ.</I>, Vol. 62, Cambridge University Press, New York, 2014, 43-91]. Extending some results of Part VIII, we derive the invariance of a certain variety (the "big J-function"), constructed from the genus $0$ descendant potential of permutation-equivariant quantum K-theory, under the action of certain finite difference operators in Novikov's variables, apply this to reconstructing the whole variety from one point on it, and give an explicit description of it in the case of the point target space.