Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 031, 16 pages      arXiv:1710.02376      https://doi.org/10.3842/SIGMA.2020.031
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Permutation-Equivariant Quantum K-Theory X. Quantum Hirzebruch-Riemann-Roch in Genus 0

Alexander Givental
Department of Mathemetics, UC Berkeley, CA 94720, USA

Received September 28, 2019, in final form April 13, 2020; Published online April 22, 2020

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, Math. Sci. Res. Inst. Publ., 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.

Key words: Gromov-Witten invariants; K-theory; adelic characterization.

pdf (501 kb)   tex (26 kb)  

References

1. Coates T., Riemann-Roch theorems in Gromov-Witten theory, 2003, Ph.D. Thesis, University of California, Berkeley, 2003, available at https://math.berkeley.edu/~giventh/coates_thesis.pdf.
2. Coates T., Givental A., Quantum cobordisms and formal group laws, in The Unity of Mathematics, In Honor of the Ninetieth Birthday of I.M. Gelfand, Editors P. Etingof, V.S. Retakh, I.M. Singer, Progr. Math., Vol. 244, Birkhäuser Boston, Boston, MA, 2006, 155-171.
3. Givental A., Symplectic geometry of Frobenius structures, in Frobenius Manifolds, Aspects Math., Vol. E36, Friedr. Vieweg, Wiesbaden, 2004, 91-112, arXiv:math.AG/0305409.
4. Givental A., Explicit reconstruction in quantum cohomology and K-theory, Ann. Fac. Sci. Toulouse Math. 25 (2016), 419-432, arXiv:1506.06431.
5. Givental A., Permutation-equivariant quantum K-theory VII. General theory, arXiv:1510.03076.
6. Givental A., Permutation-equivariant quantum K-theory VIII. Explicit reconstruction, arXiv:1510.06116.
7. Givental A., Permutation-equivariant quantum K-theory IX. Quantum Hirzebruch-Riemann-Roch in all genera, arXiv:1709.03180.
8. Givental A., Tonita V., The Hirzebruch-Riemann-Roch theorem in true genus-0 quantum K-theory, in Symplectic, Poisson, and Noncommutative Geometry, Math. Sci. Res. Inst. Publ., Vol. 62, Editors T. Eguchi, Y. Eliashberg, Y. Mayeda, Cambridge University Press, New York, 2014, 43-91, arXiv:1106.3136.
9. Lee Y.-P., Quantum $K$-theory. I. Foundations, Duke Math. J. 121 (2004), 389-424, arXiv:math.AG/0105014.