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Manin Yu.I. — Frobenius manifolds, quantum cohomology, and moduli spaces
Manin Yu.I. — Frobenius manifolds, quantum cohomology, and moduli spaces

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Название: Frobenius manifolds, quantum cohomology, and moduli spaces

Автор: Manin Yu.I.


Book News, Inc.
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. — Copyright © 1999 Book News, Inc., Portland, OR All rights reserved

Product Description:
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 für 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.

Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 303

Добавлена в каталог: 15.03.2005

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Предметный указатель
$S_n$-covariance axiom for GW-invariants      III.5.2 VI.1.2
A-model      0.1
Absolute stabilization      V.1.7 V.4.6
Abstract correlation functions      III.1.3
Admissible metric      1.5.4
Affine flat structure on supermanifold      1.1.2
Algebraic space      V.5.5.3
Algebraic stack      V.5.5
Artin stacks      V.5.5
Associative pre-Frobenius manifold      1.1.3
Associativity (WDW) equations      0.2
Atlas      V.5.5
B-model      0.1
Boundary morphism of moduli stacks      V.4.7
Cartesian square of groupoids      V.4.3.1 '
Central operations      V.8.7
Chern classes      V.8.6
Class of prestable map      V.l.4.1
Classical linear operad      IV.1.1.1
Classifying groupoid      V.3.2.5
Cohomological correlators      VI.2.2
Cohomological Field Theory (CohFT)      0.3.1 III.4.1
Combinatorial type of prestable curve      III.2.5
Combinatorial type of prestable map      V.1.5
Complete Cohomological Field Theory      III.4.5
Configuration space      IV.4.3
Correlation functions of CohFT      III.4.1
Cycles on DM-stacks      V.7.1.1
Cycles on schemes      V.6.1
Cyclic $Comm_{\infty}$-algebra      III.1.2
Cyclic operad      IV.2.6
d-spectrum of a Frobenius manifold      III.4.10.4
Darboux — Egoroff’s equations      1.3.4
Degeneration Axiom for GW-invariants      III.5.2 VI.
Deligne — Mumford (DM) stacks      V.5.5
Differential Gerstenhaber — Batalin — Vilkovyski (dGBV) algebra      III.9.5
Dilaton equation      V.5.3
Dimension axiom for GW-invariants      III.5.3
Direct sum diagram (of Saito’s frameworks)      III.8.5
Direct sum of singularities      III.8.6
Divisor axiom for GW-invariants      III.5.3
Dual modular graph of prestable curve      III.2.5
Dualizing sheaf      V.l.l
Edge (of graph)      III.2.1
Effectivity axiom for GW-invariants      III.5.2 VI.1.2
Equivalence groupoid      V.5.5.2
Euler field
Euler field in quantum cohomology      1.4.4 III.5.3.4
Evaluation morphism      V.4.2.2
Excess intersection formula      V.8.8
Extended structure connection
F-algebra      1.5.5
F-manifold      1.5.1
Flag (of graph)      III.2.1
Flat families      V.2.1 V.2.2
Flat functions and forms
Flat pullback      V.6.1.5 V.7.1.5
Formal Frobenius manifold      0.4.1 III.
Formal Frobenius manifold of qc-type      IH.5.4.1
Formal Laplace transform      II.1.3
Frobenius manifold      0.4.1 1.1.3
Generalized correlators      VI.6.1
Gepner’s Frobenius manifolds, III.8.4.1      
Gerstenhaber — Batalin — Vilkovyski (GBV) algebra      III.9.4
Gluing along pairs of sections      V.2.3
Good monomials      III.3.5.1
Graph      III.2.1
Gravitational descendants      VI.2.1
Gravity algebra      III.1.9
Gromov — Witten (GW) correspondences      III.5 VI.1.3
Gromov — Witten (GW) invariants      III.5
Groupoid      V.3.2
Groupoid of prestable curves      V.3.2.1
Groupoid of prestable maps      V.3.2.2
Groupoid of universal curves      V.3.2.3
Gysin maps      V.6.2 V.7.2
Gysin pushforward for operational Chow groups      V.8.9.2
Hamiltonian structure of Schlesinger’s equations      II.2.4 Homological Chow groups of schemes
Homological Chow groups of DM-stacks, V.7.1.3 Homological Chow groups of Artin stacks, V.7.3      
Identity Axiom for GW-invariants, III.5.3 Identity on pre-FVobenius manifold, 1.2.1 Induced Frobenius structure, 1.1.7      
Landin transform, II.5.5.6      
Large phase space, VI.2.4      
Legendre-type transformation,      
Local regular imbedding (l.r.i.) of stacks, V.7.2      
Logarithmic CohFT of rank one, III.6.1.4      
Mapping to a Point Axiom for GW-invariants, III.5.3, VI.1.2      
Markl’s operad, IV.2.5      
Maurer-Cartan equations, III.9.1      
Metric potential, 1.3.3      
Modified dualizing sheaf, V.1.2      
Modular graph, III.2.4      
Moduli space Mo,n, 0.3      
Morphism of groupoids, V.4.2, V.4.3      
Morphism of operads, IV. 1.4      
Motivic correlators, VI.2.2      
Mumford classes, III.6.2      
Normal bundle, V.6.2.1      
Normal cone, V.6.2.1      
Normalized CohFT of rank one, III.6.1.5      
Novikov ring, III.5.2.1      
Operational Chow groups, V.8.1 Orientation classes, V.8.1.1, V.8.1.2      
Painleve VI equation, II.5.4      
Partition function, IV.3.2.2      
Perturbation series, V.3      
Potential of qc-type, III.5.4.1      
Potential pre-FVobenius manifold, 1.1.3      
Potential, 0.2      
Pre-FVobenius manifold, 1.1.3      
Prestable curve, III.2.1      
Prestable map, V.l.3.1      
Primary fields, VI.2.1      
Projection formulas, V.8.5, V.8.9.4      
Proper pushforward for operational Chow groups, V.8.9.1      
Proper pushforward, V.6.1.4, V.7.1.4, V.8.3      
Pullback for operational Chow groups, V.8.4      
Puncture equation, VI.5.1      
Pushforward of operational Chow groups, V.8.9      
Quantum cohomology      0.1 III.5
Quantum cohomology of projective spaces      II.4
Rank one CohFT      III.6.1
Rational equivalence of cycles on DM-stacks      V.7.1.2
Rational equivalence of cycles on schemes      V.6.1.2
Representable morphism of groupoids      V.5.3
Rotation coefficients      1.3.4
Saito’s framework      III.8.2.1
Schlesinger’s equations      III.2.3.1
Second structure connection      II.1.1
Semisimple (pre-)Frobenius manifold      1.3.1
Semisimple Euler field      1.2.4
Singularities of meromorphic connections      II.2.1
Small quantum multiplication      III.5.4.2
Special coordinates of tame semisimple germ      III.7.1.1
Special initial conditions      II.3.4
Special solutions to Schlesinger’s equations      II.3.1.1
Species of algebras      IV.6.1.4
Spectral cover of Frobenius manifold      III.8.1
Spectrum of Frobenius manifold      1.2.4
Split identity III.7.5.8
Split semisimple (pre-) Frobenius manifold      1.3.1
Stabilization morphism of moduli stacks      V.4.4
Stabilization of prestable curve      V.1.6
Stabilization of prestable map      V.1.7
Stable curve      III.2.5
Stable map      V. 1.3.2
Stable modular graph      III.2.4
STACK      V.5.1
Standard weight      IV.3.2.1
Strictly special solutions to Schlesinger’s equations      II.3.1.3
String equation      VI.5.2
Structure connection of pre-Frobenius manifold      1.1.4
Supermanifold      I.1.1.1
Tame semisimple germ of Frobenius manifold      III.7.1
Tame singularity of a connection      II.2.1
Tau-function of solution      II.2.3.2
Tensor product diagram      III.7.5.9
Tensor product of analytic Frobenius manifolds      III.7
Tensor product of formal Frobenius manifolds      III.4.4 III.4.10 III.6.6
Theta-divisor of Schlesinger’s equations      II.2.3
Twisted Frobenius manifold
Unfolding singularities      III.8.4
Versal deformation of meromorphic connection      II.2.2
Vertex (of graph)      III.2.1
Virasoro constraints      Vl.2.5 VI.5
Virtual dimension      V.1.9
Virtual fundamental class      VI.1.1
Virtual Poincare polynomial      IV.4.1
WDW-equations (= associativity equations) 1.1.9
Weak Euler field
Weak Frobenius manifold      1.5.3
Weight of Euler field      III.4.10
Weight of identity      III.4.10
Weight of weak Euler field
Weil — Petersson volumes      III.6.4
Wick’s lemma      IV.3.4.2
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