Frobenius manifolds, quantum cohomology, and moduli spaces / Yuri I. Manin

Yuri Manin received the Bolyai Prize of the Hungarian Academy of Sciences for this title. (source : AMS)

Bibliogr. p. 285-297. Index

Chapter 0: “Introduction: What is quantum cohomology?”: This introduction gives a rather detailed overview of the two central themes of the book: quantum cohomology and Frobenius manifolds. The author explains the (preliminary) definitions underlying these concepts, gives some illustrations by important examples, and derives from this motivating discussion the strategic plan of the book. Typically for the author’s well-known style of writing, already the introduction is pointed, concise, directing and highly enlightening.

Chapter I: “Introduction to Frobenius manifolds”: This chapter is based on B. Dubrovin’s innovating work on Frobenius (super-)manifolds [in: Integrable systems and quantum groups, Montecatini 1993, Lect. Notes Math. 1620, 120-348 (1996; Zbl 0841.58065)] and provides, together with some important enhancements by the author himself, a systematic exposition of the fundaments of this theory. This includes the definition of Frobenius manifolds, Dubrovin’s structure connection, Euler fields, the extended structure connection, semi-simple Frobenius manifolds, examples of Frobenius manifolds and a first encounter with quantum cohomology in this context, weak Frobenius manifolds, and relations to Poisson structures.

Chapter II: “Frobenius manifolds and isomonodromic deformations”: In this chapter, the author continues the study of Frobenius (super-)manifolds from the deformation-theoretic viewpoint. The main topics treated here are the so-called second structure connection on Frobenius manifolds, the formal Laplace transform, isomonodromic deformations of connections, versal deformations, Schlesinger equations and their Hamiltonian structure, semisimple Frobenius manifolds as special solutions to the Schlesinger equations, and applications to the quantum cohomology ring of a projective space. The concluding section of this chapter discusses, in greater detail, the three-dimensional semisimple case of Frobenius manifolds and its connection with a special family of nonlinear ordinary differential equations, the so-called family “Painlevé VI”. Again, much of the material presented here originates from Dubrovin’s fundamental work cited above.

Chapter III: “Frobenius manifolds and moduli spaces of curves”: This chapter turns to the more algebraic aspects of Frobenius manifolds in their supergeometric setting. ... Chapter IV: “Operads, graphs, and perturbation series”: This chapter serves as a concise introduction to the more technical framework of operads and generating functions for moduli spaces of curves and quantum cohomology rings. ... Chapter V: “Stable maps, stacks, and Chow groups”: Although quantum cohomology, the main subject of the book, has been invoked in several places in the first four chapters, whether in the form of illustrating examples in chapter II or as an axiomatic framework in chapter III, its proof of existence as well as its systematic treatment had to be postponed until the final chapter VI. This is due to the fact that either construction of a mathematical quantum cohomology structure on the cohomology ring of a projective manifold requires a tremendous amount of advanced algebro-geometric techniques. Chapter V provides an overview of these methods and results needed, in addition, for the author’s construction of quantum cohomology: prestable curves and prestable maps, flat families of these objects, groupoids and moduli groupoids, algebraic stacks à la Artin and Deligne-Mumford, homological Chow groups of schemes, homological Chow groups of stacks, operational Chow groups of schemes and stacks, and the related intersection and deformation theory of schemes and stacks.

Whereas chapters I–IV are reasonably self-contained and offer complete proofs of the main results, this chapter V is comparatively sketchy and survey-like. As the author points out in the preface of the book, this chapter and the following chapter VI are meant as an introduction to the wealth of original papers on the subjects discussed here and cannot replace the study of those.

Chapter VI: “Algebraic geometric introduction to the gravitational quantum cohomology”: This concluding chapter focuses on the algebro-geometric construction of explicit Gromov-Witten-type invariants. ... The exposition of the material is rather concise and condensed, nevertheless coherent, comprehensible and educating. The reader is required to have quite a bit of expertise in algebraic geometry, complex differential geometry, category theory, non-commutative algebra, Hamiltonian systems, and modern quantum physics. On the other hand, the wealth of both mathematical information and inspiration provided by the text is absolutely immense, and in this vein, the book is an excellent source for experts and beginning researchers in the field. (Zentralblatt)