Helix structures in quantum cohomology of Fano varieties / Giordano Cotti, Boris A. Dubrovin, Davide Guzzetti.

Format:

Book

Author/Creator:

Cotti, Giordano, author.

Dubrovin, B. A. (Boris Anatolʹevich), author.

Guzzetti, Davide, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 0075-8434 2356.

Lecture notes in mathematics, 0075-8434 ; Volume 2356

Language:

English

Subjects (All):

Helices (Algebraic topology).

Homology theory.

Coherent analytic sheaves.

Geometry, Algebraic.

Mathematical physics.

Differential equations.

Geometry, Differential.

Algebra, Homological.

Physical Description:

xiii, 234 pages : illustrations (chiefly color) ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2024]

Summary:

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 Γ-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.

Contents:

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.

Notes:

Includes bibliographical references (pages 223-231) and index.

ISBN:

9783031690662

3031690664

OCLC:

1470938757