Frobenius manifolds, quantum cohomology, and moduli spaces.

*(English)*Zbl 0952.14032
Colloquium Publications. American Mathematical Society (AMS). 47. Providence, RI: American Mathematical Society (AMS). xiii, 303 p. (1999).

This monograph grew out of several lecture courses given by the author at the Max-Planck-Institut für Mathematik in Bonn, Germany, between 1994 and 1998, and many shorter lecture series delivered at various summer schools and conferences. Large parts of the material presented here have been circulating, for quite a while, in preprint form under the title “Frobenius manifolds, quantum cohomology, and moduli spaces. I, II, III” [cf.: MPI preprint 96-113, Max-Planck-Institut für Mathematik (Bonn 1996)]. The present book is the final and complete version of the author’s lectures on these subjects. Its main goal is to summarize, in a coherent and systematic way, some of the pioneering mathematical developments that took place in the past ten years and that aimed at establishing a mathematical version of quantum cohomology.

The author, being one of the great pioneers, ultimate experts, leading inspirators and most active researchers in the field, has contributed a great deal to these developments, whether by his own work or by the joint work with his collaborators K. Behrend, C. Hertling, R. Kaufmann, M. Kontsevich, and others. In view of this fact, it is a matter of course that the approach to quantum cohomology described in this book is closely related to the author’s original work and/or his research colleagues: The text consists of six chapters, each of which is divided into several sections, and a foregoing introduction dedicated to the general motivation for mathematical quantum cohomology. The contents of the single chapters are as follows:

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. The author introduces formal Frobenius manifolds, \(\text{Comm}_\infty\)-algebras, abstract (polynomial) correlation functions, and the Euler operator in the formal case. Then he discusses prestable pointed algebraic curves, together with their associated graphs and their moduli spaces, stratifications of moduli spaces of stable pointed curves, the particular structure of these moduli spaces in the case of genus zero and, in particular, the recent work of S. Keel [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)]. The fourth section of this chapter establishes the link between formal Frobenius manifolds and their abstract correlation functions, on the one hand, and cohomological field theories (with so-called tree level structure) and their natural correlation functions, on the other hand. Then, in section 5, Gromov-Witten invariants and the quantum cohomology ring of a projective manifold are described via the axiomatic approach by the author and M. Kontsevich [cf. M. Kontsevich and Yu. Manin, Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)]. This framework is then tied up with the structural theory of formal Frobenius supermanifolds introduced at the beginning of this chapter. In this context, the author also discusses cohomological field theories and formal Frobenius supermanifolds of rank one in connection with Weil-Petersson volume forms on moduli spaces of stable \(n\)-pointed rational curves and Mumford classes. – The last four sections of this chapter are devoted to R. Kaufmann’s study of tensor products in the categories of local and global Frobenius manifolds [cf. R. M. Kaufmann, “The geometry of moduli spaces of pointed curves, the tensor product in the theory of Frobenius manifolds and the explicit Künneth formula in quantum cohomology” (Bonn 1998; Zbl 0918.14011)], K. Saito’s framework families over Frobenius manifolds, Gepner’s Frobenius manifolds, Gerstenhaber-Batalin-Vilkovyski supercommutative algebras and their Maurer-Cartan equations, relations to symplectic manifolds, and special Frobenius manifolds associated to Calabi-Yau manifolds and symplectic Lefschetz manifolds.

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. The author reviews the classical linear operads and their occurrence in the homology theory of moduli spaces of stable \(n\)-pointed rational curves, sketches then the general functorial interpretation of operads, proves subsequently some formal identities for certain infinite sums taken over graphs of various topological types, and concludes this chapter by calculating several types of generating functions related to moduli spaces of curves and quantum cohomology rings. The latter topic is discussed along the lines of the work of W. Fulton and R. MacPherson [Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)] and the material of the whole chapter is closely related to the work of E. Getzler and M. Kapranov on cyclic and modular operads (1995-1998).

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 approach described here was formulated by the author himself and K. Behrend in their 1996 paper [K. Behrend and Yu. Manin, Duke Math. J. 85, No. 1, 1-60 (1996; Zbl 0872.14019)]. Later on this programme has been rigorously worked out by K. Behrend [Invent. Math. 127, No. 3, 601-617 (1997; Zbl 0909.14007) and by K. Behrend and B. Fantechi, Invent. Math. 128, No. 1, 45-88 (1997; Zbl 0909.14006)]. The Manin-Behrend-Fantechi theory of algebro-geometric Gromov-Witten invariants is heavily based on the technical framework reviewed in chapter V. The author discusses this construction procedure in all of its crucial steps, turns then to the resulting quantum cohomology structure via gravitational descendants and Virasoro constraints, and shows how to calculate the occurring correlator functions on the rational cohomology of the underlying projective manifold (phase space) via the established Gromov-Witten correspondences and the intersection theory of the involved moduli spaces. Moreover, it is explained how the quantum cohomology constructed here can be interpreted as a Frobenius manifold, and why this viewpoint might be useful with regard to further generalizations of the Martin-Behrend-Fantechi approach to more general Frobenius manifolds.

In the course of the text, the author frequently points to other available approaches to Gromov-Witten theories and quantum cohomology models. However, although the celebrated “mirror conjecture” for Calabi-Yau manifolds initially provided the main stimulus for the development of mathematical quantum cohomology theories, it is not treated in this book. Fortunately, there are at least two recent (complementary) monographs which stress this particular aspect, together with the alternate approaches to Gromov-Witten invariants and quantum cohomology. The booklet “Mirror, symmetry” by C. Voisin [cf. SMF/AMS Texts Monogr. 1 (1999; Zbl 0945.14021)] gives a nice panoramic overview of the recent developments concerning the mirror conjecture and the related quantum cohomology theory, whereas the voluminous book “Mirror symmetry and algebraic geometry” by D. Cox and S. Katz [cf. Math. Surveys Monogr. 68 (1999; 951.14026)] provides a nearly encyclopaedic exposition of these topics. Together with these two books, the monograph under review represents the current standard literature in textbook form on these subjects, without any doubt.

However, as has been stated before, Yu. I. Manin’s book is not entirely self-contained. 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.

The author, being one of the great pioneers, ultimate experts, leading inspirators and most active researchers in the field, has contributed a great deal to these developments, whether by his own work or by the joint work with his collaborators K. Behrend, C. Hertling, R. Kaufmann, M. Kontsevich, and others. In view of this fact, it is a matter of course that the approach to quantum cohomology described in this book is closely related to the author’s original work and/or his research colleagues: The text consists of six chapters, each of which is divided into several sections, and a foregoing introduction dedicated to the general motivation for mathematical quantum cohomology. The contents of the single chapters are as follows:

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. The author introduces formal Frobenius manifolds, \(\text{Comm}_\infty\)-algebras, abstract (polynomial) correlation functions, and the Euler operator in the formal case. Then he discusses prestable pointed algebraic curves, together with their associated graphs and their moduli spaces, stratifications of moduli spaces of stable pointed curves, the particular structure of these moduli spaces in the case of genus zero and, in particular, the recent work of S. Keel [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)]. The fourth section of this chapter establishes the link between formal Frobenius manifolds and their abstract correlation functions, on the one hand, and cohomological field theories (with so-called tree level structure) and their natural correlation functions, on the other hand. Then, in section 5, Gromov-Witten invariants and the quantum cohomology ring of a projective manifold are described via the axiomatic approach by the author and M. Kontsevich [cf. M. Kontsevich and Yu. Manin, Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)]. This framework is then tied up with the structural theory of formal Frobenius supermanifolds introduced at the beginning of this chapter. In this context, the author also discusses cohomological field theories and formal Frobenius supermanifolds of rank one in connection with Weil-Petersson volume forms on moduli spaces of stable \(n\)-pointed rational curves and Mumford classes. – The last four sections of this chapter are devoted to R. Kaufmann’s study of tensor products in the categories of local and global Frobenius manifolds [cf. R. M. Kaufmann, “The geometry of moduli spaces of pointed curves, the tensor product in the theory of Frobenius manifolds and the explicit Künneth formula in quantum cohomology” (Bonn 1998; Zbl 0918.14011)], K. Saito’s framework families over Frobenius manifolds, Gepner’s Frobenius manifolds, Gerstenhaber-Batalin-Vilkovyski supercommutative algebras and their Maurer-Cartan equations, relations to symplectic manifolds, and special Frobenius manifolds associated to Calabi-Yau manifolds and symplectic Lefschetz manifolds.

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. The author reviews the classical linear operads and their occurrence in the homology theory of moduli spaces of stable \(n\)-pointed rational curves, sketches then the general functorial interpretation of operads, proves subsequently some formal identities for certain infinite sums taken over graphs of various topological types, and concludes this chapter by calculating several types of generating functions related to moduli spaces of curves and quantum cohomology rings. The latter topic is discussed along the lines of the work of W. Fulton and R. MacPherson [Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)] and the material of the whole chapter is closely related to the work of E. Getzler and M. Kapranov on cyclic and modular operads (1995-1998).

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 approach described here was formulated by the author himself and K. Behrend in their 1996 paper [K. Behrend and Yu. Manin, Duke Math. J. 85, No. 1, 1-60 (1996; Zbl 0872.14019)]. Later on this programme has been rigorously worked out by K. Behrend [Invent. Math. 127, No. 3, 601-617 (1997; Zbl 0909.14007) and by K. Behrend and B. Fantechi, Invent. Math. 128, No. 1, 45-88 (1997; Zbl 0909.14006)]. The Manin-Behrend-Fantechi theory of algebro-geometric Gromov-Witten invariants is heavily based on the technical framework reviewed in chapter V. The author discusses this construction procedure in all of its crucial steps, turns then to the resulting quantum cohomology structure via gravitational descendants and Virasoro constraints, and shows how to calculate the occurring correlator functions on the rational cohomology of the underlying projective manifold (phase space) via the established Gromov-Witten correspondences and the intersection theory of the involved moduli spaces. Moreover, it is explained how the quantum cohomology constructed here can be interpreted as a Frobenius manifold, and why this viewpoint might be useful with regard to further generalizations of the Martin-Behrend-Fantechi approach to more general Frobenius manifolds.

In the course of the text, the author frequently points to other available approaches to Gromov-Witten theories and quantum cohomology models. However, although the celebrated “mirror conjecture” for Calabi-Yau manifolds initially provided the main stimulus for the development of mathematical quantum cohomology theories, it is not treated in this book. Fortunately, there are at least two recent (complementary) monographs which stress this particular aspect, together with the alternate approaches to Gromov-Witten invariants and quantum cohomology. The booklet “Mirror, symmetry” by C. Voisin [cf. SMF/AMS Texts Monogr. 1 (1999; Zbl 0945.14021)] gives a nice panoramic overview of the recent developments concerning the mirror conjecture and the related quantum cohomology theory, whereas the voluminous book “Mirror symmetry and algebraic geometry” by D. Cox and S. Katz [cf. Math. Surveys Monogr. 68 (1999; 951.14026)] provides a nearly encyclopaedic exposition of these topics. Together with these two books, the monograph under review represents the current standard literature in textbook form on these subjects, without any doubt.

However, as has been stated before, Yu. I. Manin’s book is not entirely self-contained. 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.

##### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |