Contents
1. Introduction
I.
1
Refined BPS State Counting
2. The Refined Topological String
2.1. Twisting superconformal field theories . . . . . . . . .
2.1.1. Topological field theories . . . . . . . . . . . .
2.1.2. N = (2, 2) superconformal field theories . . . .
2.1.3. The chiral ring . . . . . . . . . . . . . . . . . .
2.1.4. Deformations . . . . . . . . . . . . . . . . . . .
2.1.5. The vacuum bundle . . . . . . . . . . . . . . .
2.1.6. The topological twist . . . . . . . . . . . . . . .
2.2. Nonlinear Sigma model realization . . . . . . . . . . .
2.2.1. The special role of Calabi-Yau threefolds . . . .
2.2.2. The moduli space of Calabi-Yau threefolds . .
2.2.3. The A-model . . . . . . . . . . . . . . . . . . .
2.2.4. A-model realization of the vacuum bundle . . .
2.2.5. The B-model . . . . . . . . . . . . . . . . . . .
2.2.6. Hodge filtration and Picard-Fuchs equations . .
2.3. Mirror symmetry and topological string theory . . . .
2.3.1. Topological string theory and mirror symmetry
2.3.2. Coupling the B-model to topological gravity . .
2.3.3. Coupling the A-model to topological gravity . .
2.4. Space-time perspective and refinement . . . . . . . . .
2.4.1. The Gopakumar Vafa invariants . . . . . . . .
2.4.2. The geometrical origin of the spin content . . .
2.4.3. Refinement of the free energy . . . . . . . . . .
2.4.4. Refined stable pair invariants . . . . . . . . . .
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3. Direct Integration of the Refined Holomorphic Anomaly Equations
3.1. Integration of the unrefined holomorphic anomaly equations . . .
3.2. The local and the holomorphic limit . . . . . . . . . . . . . . . .
3.2.1. The local limit . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2. The holomorphic limit . . . . . . . . . . . . . . . . . . . .
3.2.3. Holomorphicity versus modularity . . . . . . . . . . . . .
3.3. Direct integration of the refined topological string . . . . . . . . .
3.3.1. The refined holomorphic anomaly equations . . . . . . . .
3.3.2. The refined free energies at genus one and the propagator
3.3.3. The behavior at the conifold and the gap condition . . . .
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Contents
3.4. Elliptic curve mirrors and closed modular expressions . . . . .
3.4.1. Computing the period and prepotential from the elliptic
3.4.2. Determining the higher genus sector . . . . . . . . . . .
3.4.3. Fixing the holomorphic ambiguity . . . . . . . . . . . .
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4. Geometry of del Pezzo Surfaces and Toric Mirror Symmetry
4.1. The geometry of del Pezzo and half K3 surface . . . . . . . . . . . . .
4.1.1. Algebraic realizations . . . . . . . . . . . . . . . . . . . . . . .
4.2. The Batyrev construction . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. Toric Fano varieties and non-compact Calabi-Yau spaces . . . .
4.2.2. Constructing toric ambient spaces . . . . . . . . . . . . . . . .
4.2.3. Constructing mirror families of Calabi-Yau manifolds . . . . .
4.3. Non-compact mirror symmetry . . . . . . . . . . . . . . . . . . . . . .
4.3.1. Local mirror symmetry as a limit of compact mirror symmetry
4.3.2. Local mirror symmetry without a compact embedding . . . . .
4.4. Constructing the mirror curves of two-dimensional toric Fano varieties
5. Refined BPS Invariants of Toric Calabi-Yau Geometries
5.1. The massless D5 , E6 , E7 and E8 del Pezzo surfaces . . . . . . . .
5.1.1. The E8 del Pezzo surface . . . . . . . . . . . . . . . . . .
5.1.2. The E7 del Pezzo surface . . . . . . . . . . . . . . . . . .
5.1.3. The E6 del Pezzo surface . . . . . . . . . . . . . . . . . .
5.1.4. The D5 del Pezzo surface . . . . . . . . . . . . . . . . . .
5.2. An alternative approach to the massless cases . . . . . . . . . . .
5.3. The toric del Pezzo surfaces . . . . . . . . . . . . . . . . . . . . .
5.3.1. O(−KP2 ) → P2 . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2. O(−KF0 ) → F0 . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3. O(−KB1 ) → B1 . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4. O(−KF2 ) → F2 . . . . . . . . . . . . . . . . . . . . . . . .
5.3.5. O(−KB2 ) → B2 . . . . . . . . . . . . . . . . . . . . . . . .
5.3.6. O(−KB3 ) → B3 . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Almost del Pezzo surfaces . . . . . . . . . . . . . . . . . . . . . .
5.4.1. A mass deformation of the local E8 del Pezzo surface . . .
5.5. Solving the topological string on C3 /Z5 . . . . . . . . . . . . . .
5.5.1. Genus two curves and Igusa invariants . . . . . . . . . . .
5.5.2. The geometry and its mirror . . . . . . . . . . . . . . . .
5.5.3. Extracting the complex structure moduli from the mirror
5.5.4. Periods and free energies at genus zero and one . . . . . .
5.5.5. The propagator . . . . . . . . . . . . . . . . . . . . . . . .
6. The
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6.4.
iv
Refined BPS Invariants for the Half K3
The refined G¨
ottsche formula and the unrefined
Refined BPS invariants for the massless half K3
The massive half K3 . . . . . . . . . . . . . . .
Flow to the del Pezzo models . . . . . . . . . .
HST
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Contents
II. Corrections and Nonperturbative Phenomena in F-theory
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7. Basic Concepts of F-Theory
7.1. F-theory as the strong coupling regime of Type IIB .
7.2. Embedding F-theory into the web of string dualities
7.2.1. M/F-Theory duality . . . . . . . . . . . . . .
7.2.2. Heterotic/F-theory duality . . . . . . . . . .
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8. Counting Non-Perturbative States in F-Theory
8.1. Counting refined BPS states of the E-string . . . . . .
8.1.1. Zero-sized Heterotic instantons . . . . . . . . .
8.1.2. The E-String in six and five dimensions . . . .
8.1.3. The Green-Schwarz string . . . . . . . . . . . .
8.1.4. The F-theory perspective . . . . . . . . . . . .
8.1.5. Refined stable pair invariants solve the problem
8.2. Counting [p, q]-strings using refined invariants . . . . .
8.2.1. The Seiberg-Witten description of the Sen limit
8.2.2. Generalizations . . . . . . . . . . . . . . . . . .
8.2.3. Counting [p, q]-strings in the half K3 . . . . . .
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9. The Sevenbrane Gauge Coupling Function in F-Theory
9.1. Motivation: D7-brane gauge coupling function . . . . . . . . . .
9.1.1. D7-brane gauge couplings in 4d: Calabi-Yau orientifolds
9.1.2. Dimensional reduction to three dimensions . . . . . . .
9.2. M-theory compactifications and Taub-NUT geometries . . . . .
9.2.1. Kaluza-Klein-monopoles: TNk -spaces in M-theory . . .
9.2.2. S 1 -compactification of TNk : TN∞
k -space in M-theory . .
9.3. 7-brane gauge coupling functions in warped F-theory . . . . . .
9.3.1. The effective action of F-theory . . . . . . . . . . . . . .
9.3.2. Leading 7-brane gauge coupling functions . . . . . . . .
9.3.3. On dimensional reduction with fluxes and warp factor .
9.3.4. Calculation of corrections to the gauge kinetic function
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10.Conclusion and Outlook
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A. Appendix
A.1. The general Weierstrass forms for the cubic, the quartic and the bi-quadratic
A.1.1. The Weierstrass normal form of cubic curves . . . . . . . . . . . . .
A.1.2. The Weierstrass normal form of quartic curves . . . . . . . . . . . .
A.1.3. The Weierstrass normal form for a bi-quadratic curve . . . . . . . .
A.2. Some more details on del Pezzo surfaces . . . . . . . . . . . . . . . . . . . .
A.2.1. En -curves as Cubic curves . . . . . . . . . . . . . . . . . . . . . . . .
A.2.2. The third order differential operator for B2 . . . . . . . . . . . . . .
A.3. Jacobi and Siegel modular forms . . . . . . . . . . . . . . . . . . . . . . . .
A.3.1. Weyl invariant Jacobi modular forms for E8 lattice . . . . . . . . . .
A.3.2. Siegel modular forms of genus two . . . . . . . . . . . . . . . . . . .
A.4. The BPS invariants for the half K3 and the diagonal classes of P × P1 . . .
A.4.1. The diagonal P × P1 model . . . . . . . . . . . . . . . . . . . . . . .
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Contents
A.4.2. The massless half K3 . . . . . . . . . . .
A.4.3. The massive half K3 . . . . . . . . . . .
A.5. Data for C3 /Z5 . . . . . . . . . . . . . . . . . .
A.5.1. The Gopakumar Vafa invariants . . . .
A.6. Conventions of N = 1 actions and dimensionful
A.7. Linear multiplets and gauge couplings . . . . .
A.8. Details of T Nk . . . . . . . . . . . . . . . . . .
A.9. Details of T Nk∞ . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
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