Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces (American Mathematical Society Colloquium Publications, Volume 47)

This is the first monograph dedicated to the systematic exposition of the whole variety of topics related to quantum cohomology. The subject first originated in theoretical physics (quantum string theory) and has continued to develop extensively over the last decade. The author's approach to quantum cohomology is based on the notion of the Frobenius manifold. The first part of the book is devoted to this notion and its extensive interconnections with algebraic formalism of operads, differential equations, perturbations, and geometry. In the second part of the book, the author describes the construction of quantum cohomology and reviews the algebraic geometry mechanisms involved in this construction (intersection and deformation theory of Deligne-Artin and Mumford stacks). Yuri Manin is currently the director of the Max-Planck-Institut f?r Mathematik in Bonn, Germany. He has authored and coauthored 10 monographs and almost 200 research articles in algebraic geometry, number theory, mathematical physics, history of culture, and psycholinguistics. Manin's books, such as Cubic Forms: Algebra, Geometry, and Arithmetic (1974), A Course in Mathematical Logic (1977), Gauge Field Theory and Complex Geometry (1988), Elementary Particles: Mathematics, Physics and Philosophy (1989, with I. Yu. Kobzarev), Topics in Non-commutative Geometry (1991), and Methods of Homological Algebra (1996, with S. I. Gelfand), secured for him solid recognition as an excellent expositor. Undoubtedly the present book will serve mathematicians for many years to come. Contents......Page page000008.djvu Preface......Page page000010.djvu 0. Introduction: What Is Quantum Cohomology?......Page page000014.djvu 1. Definition of Frobenius manifolds and the structure connection......Page page000030.djvu 2. Identity, Euler field, and the extended structure connection......Page page000035.djvu 3. Semisimple Frobenius manifolds......Page page000040.djvu 4. Examples......Page page000048.djvu 5. Weak Frobenius manifolds......Page page000055.djvu 1. The second structure connection......Page page000062.djvu 2. Isomonodromic deformations......Page page000068.djvu 3. Semisimple Frobenius manifolds as special solutions to the Schlesinger equations......Page page000073.djvu 4. Quantum cohomology of projective spaces......Page page000079.djvu 5. Dimension three and Painlevé VI......Page page000084.djvu 1. Formal Frobenius manifolds and Comm_∞-algebras......Page page000096.djvu 2. Pointed curves and their graphs......Page page000101.djvu 3. Moduli spaces of genus 0......Page page000104.djvu 4. Formal Frobenius manifolds and Cohomological Field Theories......Page page000112.djvu 5. Gromov-Witten invariants and quantum cohomology: Axiomatic theory......Page page000128.djvu 6. Formal Frobenius manifolds of rank one and Weil-Petersson volumes of moduli spaces......Page page000134.djvu 7. Tensor product of analytic Frobenius manifolds......Page page000147.djvu 8. K. Saito's frameworks and singularities......Page page000159.djvu 9. Maurer-Gartan equations and Gerstenhaber-Batalin-Vilkovyski algebras......Page page000170.djvu 10. From dGBV-algebras to Frobenius manifolds......Page page000181.djvu 1. Classical linear operads......Page page000188.djvu 2. Operads and graphs......Page page000197.djvu 3. Sums over graphs......Page page000200.djvu 4. Generating functions......Page page000206.djvu 1. Prestable curves and prest able maps......Page page000214.djvu 2. Flat families of curves and maps......Page page000221.djvu 3. Groupoids and moduli groupoids......Page page000225.djvu 4. Morphisms of groupoids and moduli groupoids......Page page000228.djvu 5. Stacks......Page page000234.djvu 6. Homological Chow groups of schemes......Page page000241.djvu 7. Homological Chow groups of DM-stacks......Page page000247.djvu 8. Operational Chow groups of schemes and DM-stacks......Page page000251.djvu 1. Virtual fundamental classes......Page page000260.djvu 2. Gravitational descendants and Virasoro constraints......Page page000265.djvu 3. Correlators and forgetful maps......Page page000271.djvu 4. Correlators and boundary maps......Page page000281.djvu 5. The simplest Virasoro constraints......Page page000285.djvu 6. Generalized correlators......Page page000287.djvu 7. Generating functions on the large phase space......Page page000293.djvu Bibliography......Page page000300.djvu Subject Index......Page page000314.djvu This is the first monograph dedicated to the systematic exposition of the whole variety of topics related to quantum cohomology. The subject first originated in theoretical physics (quantum string theory) and has continued to develop extensively over the last decade. The author's approach to quantum cohomology is based on the notion of the Frobenius manifold. The first part of the book is devoted to this notion and its extensive interconnections with algebraic formalism of operads, differential equations, perturbations, and geometry. In the second part of the book, the author describes the construction of quantum cohomology and reviews the algebraic geometry mechanisms involved in this construction (intersection and deformation theory of Deligne-Artin and Mumford stacks). Yuri Manin was once the director of the Max-Planck-Institut für Mathematik in Bonn, Germany, one of the most prestigious mathematics institutions in the world. He has authored and coauthored 10 monographs and almost 200 research articles in algebraic geometry, number theory, mathematical physics, history of culture, and psycholinguistics. Manin's books, such as Cubic Forms: Algebra, Geometry, and Arithmetic (1974), A Course in Mathematical Logic (1977), Gauge Field Theory and Complex Geometry (1988), Elementary Particles: Mathematics, Physics and Philosophy (1989, with I. Yu. Kobzarev), Topics in Non-commutative Geometry (1991), and Methods of Homological Algebra (1996, with S. I. Gelfand), secured for him solid recognition as an excellent expositor. Undoubtedly the present book will serve mathematicians for many years to come. This monograph summarizes some of the developments that have taken place in quantum cohomology in the last decade, but does not explain the history or physical motivations. Manin begins by developing the local and global geometric and analytic theory of Frobenius manifolds, then introduces the more algebraic aspects formal Frobenius manifolds, moduli spaces and their homology operads. The last two chapters focus on the algebraic geometric constructions of the Gromov-Witten invariants. Annotation c. Book News, Inc., Portland, OR (booknews.com) Presents an exposition of the whole variety of topics related to quantum cohomology. In this monograph, the author's approach to quantum cohomology is based on the notion of the Frobenius manifold. It deals with this notion and its extensive interconnections with algebraic formalism of operads, differential equations, perturbations, and geometry.