Helix Structures in Quantum Cohomology of Fano Varieties

This research monograph provides a comprehensive study of a conjecture initially proposed by the second author at the 1998 International Congress of Mathematicians (ICM). This conjecture asserts the equivalence, for a Fano variety, between the semisimplicity condition of its quantum cohomology and the existence of full exceptional collections in its derived category of coherent sheaves. Additionally, in its quantitative form, the conjecture specifies an explicit relation between the monodromy data of the quantum cohomology, characteristic classes, and exceptional collections. A refined version of the conjecture is introduced, with a particular focus on the central connection matrix, and a precise link is established between this refined conjecture and G-conjecture II, as proposed by S. Galkin, V. Golyshev, and H. Iritani. By performing explicit calculations of the monodromy data, the validity of the refined conjecture for all complex Grassmannians G(r,k) is demonstrated. Intended for students and researchers, the book serves as an introduction to quantum cohomology and its isomonodromic approach, along with its algebraic counterpart in the derived category of coherent sheaves.

- Introduction.- Gromov–Witten Theory and Quantum Cohomology.- Helix Theory in Triangulated Categories.- Non-Symmetric Orthogonal Geometry of Mukai Lattices.- The Main Conjecture.- Proof of the Main Conjecture for Projective Spaces.- Proof of the Main Conjecture for Grassmannians.

Davide Guzzetti obtained his Ph.D. in Mathematical Physics in 2000 from the Scuola Internazionale Superiore di Studi Avanzati SISSA, Trieste, Italy. After holding research positions at RIMS, Kyoto University, Japan, and KIAS, Seoul, South Korea, he became a researcher at SISSA in 2011, and an associate professor in 2018.

Giordano Cotti obtained his Ph.D. in Geometry and Mathematical Physics from SISSA in 2017.  He has held research positions at the Max Planck Institute for Mathematics in Bonn, Germany, and the University of Birmingham in the UK. Since 2020, he has been a researcher in the Group of Mathematical Physics at the University of Lisbon in Portugal.

Boris A. Dubrovin (1950–2019) obtained his Ph.D. in Geometry and Topology at Moscow State University under the supervision of S.P. Novikov. After obtaining his Habilitation in 1984, he was a full professor at Moscow State University (1988–1993) and at SISSA from 1993 to 2019.