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Table of contents
1 Introduction
1.1 String theory as a candidate for a “theory of everything”
1.2 Topological string theory as a bridge between mathematics and physics
2 Topological Field Theory
2.1 Two dimensional N = (2, 2) supersymmetry
2.2 Topological field theories of Witten type
2.3 Topological twisting
2.4 Interlude: moduli space of complex structures
3 Topological String Theory: Coupling to Gravity
3.1 Topological string theory
3.2 Mirror symmetry
4 Resurgence and Quantum Mirror Curve: A Case Study
4.1 The Harper-Hofstadter problem
4.2 Trans-series expansion and one-instanton sector
4.3 Two-instanton sector
4.4 Instanton fluctuation from topological string
4.5 Interlude: holomorphic anomaly at work
5 Elliptic Genera and Topological Strings: Overview
5.1 Elliptic fibrations and four-cycles
5.2 The base degree k partition function as Jacobi form
6 Geometries without codimension-one singular fibers: Reconstruction
6.1 The structure of the topological string free energy
6.2 Zk of negative index
6.3 Partition function from genus zero GW Invariants
7 Geometries with codimension-one singular fibers: Higgsing Trees
7.1 Higgsing trees
7.2 a2 model
7.3 g2 model
8 Conclusion and Outlook
A Modular Forms
A.1 Elliptic modular forms
A.2 Jacobi modular forms
A.3 Weyl-invariant Jacobi forms
B Toric Geometries
B.1 Fans
B.2 Polytopes
B.3 Blow-ups
B.4 Toric Diagrams
C Computations in One-Instanton Sector
C.1 Instanton solution
C.2 The moduli-space metric
C.3 The one-instanton determinant
D Principal Parts of Z3 and Z4 for the Massless E-string
D.1 Base degree 3
D.2 Base degree 4
E Some Enumerative Invariants
E.1 Genus zero GW invariants for massless E strings
E.2 GV invariants for a2 and g2 models
Bibliography
