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E-grāmata: Classical Mirror Symmetry

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Classical Mirror SymmetryPalielināt
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This book furnishes a brief introduction to classical mirror symmetry, a term that denotes the process of computing Gromov–Witten invariants of a Calabi–Yau threefold by using the Picard–Fuchs differential equation of period integrals of its mirror Calabi–Yau threefold. The book concentrates on the best-known example, the quintic hypersurface in 4-dimensional projective space, and its mirror manifold.
First, there is a brief review of the process of discovery of mirror symmetry and the striking result proposed in the celebrated paper by Candelas and his collaborators. Next, some elementary results of complex manifolds and Chern classes needed for study of mirror symmetry are explained. Then the topological sigma models, the A-model and the B-model, are introduced. The classical mirror symmetry hypothesis is explained as the equivalence between the correlation function of the A-model of a quintic hyper-surface and that of the B-model of its mirror manifold.
On the B-model side, the process of construction of a pair of mirror Calabi–Yau threefold using toric geometry is briefly explained. Also given are detailed explanations of the derivation of the Picard–Fuchs differential equation of the period integrals and on the process of deriving the instanton expansion of the A-model Yukawa coupling based on the mirror symmetry hypothesis.
On the A-model side, the moduli space of degree d quasimaps from CP^1 with two marked points to CP^4 is introduced, with reconstruction of the period integrals used in the B-model side as generating functions of the intersection numbers of the moduli space. Lastly, a mathematical justification for the process of the B-model computation from the point of view of the geometry of the moduli space of quasimaps is given.
The style of description is between that of mathematics and physics, with the assumption that readers have standard graduate student backgrounds in both disciplines.

Recenzijas

Graduate students or other researchers in theoretical physics with an interest in the mathematical aspects of mirror symmetry form a natural audience for this text. (Thomas Prince, zbMATH 1431.14034, 2020)

1 Brief History of Classical Mirror Symmetry
1(26)
1.1 Grand Unified Theory and Superstring Theory
1(4)
1.2 Compactification of Heterotic String Theory
5(7)
1.3 Discovery of Mirror Symmetry of N = 2 Superconformal Field Theory
12(8)
1.4 First Striking Prediction of Mirror Symmetry
20(7)
References
26(1)
2 Basics of Geometry of Complex Manifolds
27(28)
2.1 Complex Manifold
27(3)
2.2 Vector Bundles of Complex Manifold
30(6)
2.2.1 Definition of Holomorphic Vector Bundles, Covariant Derivatives, Connections and Curvatures
30(6)
2.3 Chern Classes
36(8)
2.3.1 Calculus of Holomorphic Vector Bundles and Chern Classes
40(4)
2.4 Kahler Manifolds and Projective Spaces
44(5)
2.4.1 Definition of Kahler Manifolds
44(3)
2.4.2 Dolbeault Cohomology of Kahler Manifolds
47(2)
2.5 Complex Projective Space as an Example of Compact Kahler Manifold
49(6)
Reference
53(2)
3 Topological Sigma Models
55(28)
3.1 N = 2 Supersymmetric Sigma Model
55(2)
3.2 Topological Sigma Model (A-Model)
57(18)
3.2.1 Lagrangian and Saddle Point Approximation
57(6)
3.2.2 Observable Satisfying {Q, O} = 0 and Topological Selection Rule
63(2)
3.2.3 Geometrical Interpretation of Degree d Correlation Function
65(2)
3.2.4 Degree d Correlation Function of Degree 5 Hypersurface in CP4
67(8)
3.3 Topological Sigma Model (B-Model)
75(8)
3.3.1 Lagrangian
75(3)
3.3.2 Observable that Satisfies {Q, O} =0 and Topological Selection Rule
78(1)
3.3.3 Correlation Function
79(2)
References
81(2)
4 Details of B-Model Computation
83(26)
4.1 Toric Geometry
83(7)
4.1.1 Outline of Toric Geometry
83(1)
4.1.2 An Example: Complex Projective Space
84(6)
4.2 Formulation of Mirror Symmetry by Toric Geometry
90(9)
4.2.1 An Example: Quintic Hypersurface in CP4
90(4)
4.2.2 Blow-Up (Resolution of Singularity) and Hodge Number
94(5)
4.3 Details of B-Model Computation
99(10)
4.3.1 Derivation of Differential Equations Satisfied by Period Integrals
99(4)
4.3.2 Derivation of B-Model Yukawa Coupling
103(2)
4.3.3 Instanton Expansion of A-Model Yukawa Couplings
105(3)
References
108(1)
5 Reconstruction of Mirror Symmetry Hypothesis from a Geometrical Point of View
109
5.1 Simple Compactification of Holomorphic Maps from CP1 to CP4
109(7)
5.1.1 A-Model Correlation Functions as Intersection Numbers
109(3)
5.1.2 Evaluation of Yukawa Coupling of Quintic Hypersurface in CP4 by Using Simple Compactification of the Moduli Space
112(4)
5.2 Toric Compactification of the Moduli Space of Degree d Quasi Maps with Two Marked Points
116(4)
5.3 Construction of Two Point Intersection Numbers on Mp0,2(CP4, d)
120(2)
5.4 Fixed Point Theorem and Computation of w (Oha Ohb)0,d
122(8)
5.4.1 Fixed Point Theorem
122(3)
5.4.2 Computation of w(Oha Ohb)0,d
125(5)
5.5 Reconstruction of Mirror Symmetry Computation
130
References
140
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