Toric Geometry and String Theory arXiv:hep-th/0609123v1 18 Sep 2006 Vincent Bouchard Magdalen College University of Oxford A thesis submitted for the degree of Doctor of Philosophy July 2005 This thesis is dedicated to my parents. Acknowledgements First and foremost, I would like to thank Philip Candelas for his invaluable help throughout my four years in Oxford. I also owe special thanks to Bogdan Florea, Marcos Mari˜no and Harald Skarke for stimulating and enjoyable collaborations on the research topics discussed in this thesis. Thanks also to Lara Anderson, Xenia de la Ossa, Marco Gualtieri, Wen Jiang, Shabnam Kadir, Lionel Mason, David Skinner, Fonger Ypma and many others for interesting mathematical physics discussions. Many thanks to my parents, who have always supported me in my choices of life and encouraged me to be true and honest with myself. Alexandra, thank you so much for your love and patience; distance relationships are difficult, but you showed me that no matter what the distance is love always exists. Thanks also to my sister Maryse, for your continual support and friendship; and for your invitation to discover Burmese and Malian cultures! I would like to thank all my friends in Oxford, especially James, Stefano, Silje, Lucy, Kezia, Owen, Rahul; you showed me that better worlds exist, and most importantly that we can start creating them ourselves, right here and right now. Thanks to the OSSTW, OSAN, Oxford and UK Indymedia, ZOMBIE, PGA, OCSET and OARC crews, for the inspiration and the fantastic work to make our communities a better place to live. Thanks also to Maarit, Will and Ralph, for the wonderful Sunday nights spent in your good company. Thanks to all my friends from Qu´ebec, particularly David, Sylvie, Hendrick, F´elix, Marc; it is amazing that our friendship is still as intense as ever! Finally, I must not forget to thank all the people I have played music with, for all these special moments where collective creation of music engendered unity through diversity... I must also acknowledge the generous funding of the Rhodes Trust and of the National Science and Engineering Research Council of Canada, without which I could not have pursued my studies in Oxford. Toric Geometry and String Theory Vincent Bouchard Magdalen College University of Oxford A thesis submitted for the degree of Doctor of Philosophy July 2005 In this thesis we probe various interactions between toric geometry and string theory. First, the notion of a top was introduced by Candelas and Font as a useful tool to investigate string dualities. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We classify all tops and give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group. Secondly, we compute all loop closed and open topological string amplitudes on orientifolds of toric Calabi-Yau threefolds, by using geometric transitions involving SO/Sp Chern-Simons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular, we count Klein bottles and projective planes with any number of handles in some Calabi-Yau orientifolds. We determine the BPS structure of the amplitudes, and illustrate our general results in various examples with and without D-branes. We also present an application of our results to the BPS structure of the coloured Kauffman polynomial of knots. Contents 1 Introduction 1 2 Toric Geometry 9 2.1 HomogeneousCoordinates . 10 2.1.1 ToricDivisors ........................... 13 2.2 ToricCalabi-YauThreefolds . 13 2.2.1 DefinitionofaCalabi-YauManifold. 13 2.2.2 Calabi-Yau Manifolds in Toric Geometry . 14 2.3 Toric Diagrams and Symplectic Quotients . .. 17 2.3.1 Toric Manifolds as Symplectic Quotients . 17 2.3.2 T 3 Fibration............................ 18 2.3.3 T 2 × R Fibration ......................... 20 2.4 Examples ................................. 22 2.4.1 ResolvedConifold. .. .. 23 1 2.4.2 Two dP2’s Connected by a CP ................. 26 2.5 HypersurfacesinToricVarieties . .. 27 2.5.1 ReflexivePolytopes . 27 2.5.1.1 LatticeDescription . 28 2.5.1.2 ToricInterpretation . 28 2.5.1.3 Calabi-YauCondition . 29 2.5.1.4 FibrationStructure. 30 2.5.2 Tops................................ 31 2.5.2.1 LatticeDescription . 31 2.5.2.2 ToricInterpretation . 33 3 The Classification of Tops and String Dualities 34 3.1 Toric Properties and Affine Kac-Moody Algebras . .. 34 3.1.1 AffineKac-MoodyAlgebras . 35 i CONTENTS ii 3.1.2 Higher Dimensional Geometries . 38 3.1.3 EllipticCurve........................... 38 3.2 Classification ............................... 39 3.3 GeometricalInterpretation . 44 3.3.1 TwistedAlgebras: Geometry. 44 3.4 StringTheoryInterpretation . 46 3.4.1 Twisted Algebras: String Theory . 47 4 Closed Topological Strings on Orientifolds 49 4.1 A-model Topological Strings on an Orientifold . 49 4.1.1 Type IIA Superstrings and Topological Strings on an Orientifold 49 4.1.2 Structure of the Topological String Amplitudes . ... 51 4.2 GeometricTransitions . .. .. 53 4.2.1 Orientifold of the Resolved Conifold and its Geometric Transition 53 4.2.2 OurMainExample ........................ 55 4.3 Closed String Amplitudes from Chern-Simons Theory . ..... 57 4.3.1 Results from Chern-Simons Theory with Classical Gauge Groups 57 4.3.2 Computation of Open String Amplitudes . 63 4.3.3 Duality Map and Closed String Amplitudes . 66 4.3.4 TheOrientedContribution. 68 4.4 UnorientedLocalization . .. .. 75 4.4.1 Unoriented Localization at 2 Crosscaps and Degree 2 RP2 .. 76 4.4.2 Unoriented Localization at 2 Crosscaps and Degree 4 RP2 .. 78 4.5 TopologicalVertexComputation. 82 4.5.1 GeneralPrescription . 82 4.5.2 Examples ............................. 83 4.5.2.1 Orientifold of the Resolved Conifold . 83 4.5.2.2 Local CP2 Attached to RP2 .............. 84 4.5.3 A Theorem Relating the Topological Vertex and SO/Sp Chern- SimonsInvariants......................... 85 5 Open Topological Strings on Orientifolds 88 5.1 Open Topological String Amplitudes in Orientifolds . ....... 88 5.1.1 BPS Structure of Topological String Amplitudes . ... 88 5.1.2 BPS Structure of Topological Strings on Orientifolds ..... 90 5.2 Examples of Open String Amplitudes . 95 5.2.1 The topologicalvertex onorientifolds . .. 95 CONTENTS iii 5.2.2 The SO/Sp framedunknot ................... 96 5.2.3 CP2 Attached to RP2 ....................... 99 5.2.4 SO/Sp HopfLinkInvariant . 101 5.2.5 LocalizationComputations. 103 5.2.5.1 The SO/Sp FramedUnknot. 105 5.2.5.2 CP2 Attached to RP2 .................. 107 5.2.5.3 SO/Sp HopfLinkInvariant . 109 5.3 Application: the BPS Structure of the Coloured Kauffman Polynomial 110 5.3.1 Chern-Simons Invariants and Knot Polynomials . 110 5.3.2 BPS Structure: Statement and Examples . 111 6 Conclusions and Future Directions 117 A Useful Formulae 120 A.1 SubsetsofYoungTableaux. 120 A.2 The Topological Vertex, Chern-Simons Invariants and Schur Functions 121 B Various Results 124 B.1 FullClassificationofTops . 124 B.2 Full Results for the Closed Topological String Amplitudes ...... 131 B.3 BPSInvariantsfortheTrefoilKnot . 132 Bibliography 134 List of Figures 2.1 The fan Σof CP2.............................. 12 2.2 The Γ and Γ˜ graphs for O(−3) → CP2. The toric diagram Γ is the normal diagram drawn in thick lines. The points (νi, 1) give the fan Σ, where the νi are the vertices of Γ˜ and are shown in the figure. 17 2.3 Toric diagram Γ of O(−3) → CP2 visualized as a three dimensional graph. It encodes the degeneration loci of the T 3 fiber. ........ 20 3 2.4 Trivalent vertex associated to C , drawn in the rα-rβ plan. The vectors represent the generating cycles over the lines. ... 22 2.5 The Γ and Γ˜ graphs for O(−1) ⊕ O(−1) → CP1. The toric diagram Γ is the normal diagram drawn in thick lines. The points (νi, 1) give the fan Σ, where the νi are the vertices of Γ˜ and are shown in the figure. 23 2.6 Toric diagram Γ of O(−1) ⊕ O(−1) → CP1 visualized as a three di- mensional graph. It encodes the degeneration loci of the T 3 fiber. 24 1 2.7 Toric diagram of O(−1) ⊕ O(−1) → CP , drawn in the rα-rβ plan. The vectors represent the generating cycles over the lines. The origin of the second patch U2 is shifted to (−t, −t)............... 25 2.8 The Γ and Γ˜ graphs for the Calabi-Yau threefold X whose compact 1 locus consists of two dP2’s connected by a CP . The toric diagram Γ is the normal diagram drawn in thick lines. 26 2.9 A top, its dual and the minimal point notation. ... 32 3.1 Dynkin diagrams of the self-dual untwisted ADE Kac-Moody algebras. 36 3.2 An edge of ♦∗ and the intersection pattern to which it corresponds. 36 iv LIST OF FIGURES v 3.3 Dynkin diagrams of the duals of untwisted non-simply laced Kac- Moodyalgebras............................... 37 3.4 Dynkin diagrams of the duals of twisted Kac-Moody algebras that can bereadofffromtops............................ 38 3.5 The 16 two-dimensional reflexive polygons. The polygons 1, 2,..., 6 are respectively dual to the polygons 16, 15,..., 11, and the polygons 7,..., 10areself-dual. .......................... 40 3.6 Possible halves of reflexive polygons. ... 42 3.7 Three maximal tops (the significance of • vs. ◦ will be explained in section3.3). ................................ 43 3.8 Afamilyoftopsoversquares. 45 3.9 Two families of tops over the dual pair (4, 13).............. 45 4.1 Geometric transition for the orientifold of the conifold. The cross in the figure to the left represents an RP2 obtained by quotienting a CP1 by the involution I, and the dashed line in the figure on the right represents an S3 with SO/Sp gaugegroup...............