Computational Methods in String Theory and Applications to the Swampland Conjectures Lorenz Schlechter M¨unchen2021 Computational Methods in String Theory and Applications to the Swampland Conjectures Lorenz Schlechter Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Lorenz Schlechter aus Ravensburg M¨unchen, den 09.04.2021 Erstgutachter: Priv.-Doz. Dr. Ralph Blumenhagen Zweitgutachter: Prof. Dr. Dieter L¨ust Tag der m¨undlichen Pr¨ufung:28.05.2021 Zusammenfassung Das Ziel des Swampland Programms ist die Klassifizierung effektiver, zu Quantengravita- tionstheorien vervollst¨andigbarerTheorien. Aufgrund der enormen Anzahl an m¨oglichen Stringvacua, zusammengefasst in der sogenannten Stringlandschaft, sind die meisten der bisherigen Resultate des Programms Vermutungen. Jedoch existiert ein best¨andigwach- sendes dichtes Netz aus Abh¨angigkeiten zwischen diesen Vermutungen. Ein besseres Verst¨andnisoder ein Beweis dieser Vermutungen w¨urdedie erlaubten Niederenergietheorien einschr¨anken. Das Ziel dieser Arbeit ist deshalb die Entwicklung mathematischer Methoden, die explizite Tests der Swampland Vermutungen in stringthe- oretischen Modellen erm¨oglichen. Insbesondere werden Perioden von Calabi-Yau Mannig- faltigkeiten auf numerischem und analytischem Weg berechnet. Dar¨uber hinaus werden Methoden zur Berechnung von Calabi-Yau Metriken, Linienb¨undelkohomologien und Stre- beldifferentialen behandelt. Diese werden zur Uberpr¨ufung¨ zweier Vermutungen eingesetzt, zum Test der Swamp- land Distanzvermutung sowie zum Test der dS Vermutung. Erstere besagt, dass eine effek- tive Theorie nur bis zu bestimmten Feldwerten g¨ultig sein kann. Werden diese ¨uberschritten werden unendlich viele nicht ber¨ucksichtigte Zust¨andeexponentiell leicht und die verwen- dete effektive Beschreibung bricht zusammen. Diese Vermutung wird durch eine explizite Berechnung von Distanzen zwischen effektiven Theorien in Calabi-Yau Modulir¨aumen getestet. Die dS Vermutung verbietet hingegen stabile Vacua mit positiver kosmologischer Konstante. Um diese Vermutung zu ¨uberpr¨ufen,wird ein Teil der KKLT-Konstruktion explizit durchgef¨uhrt. Dar¨uber hinaus wird die Validit¨atder zugrundeliegenden effek- tiven Theorie in einem warped throat\ analysiert. Neben diesen traditionellen Herange- " hensweisen werden exotischere Ans¨atzef¨urdie Konstruktion von dS R¨aumenuntersucht. Dies umfasst Tachyonenkondensation sowie andere Raumzeitsignaturen. Abstract The goal of the swampland program is the classification of low energy effective theories which can be consistently coupled to quantum gravity. Due to the vastness of the string landscape most results of the swampland program are still conjectures, yet the web of conjectures is ever growing and many interdependencies between different conjectures are known. A better understanding or even proof of these conjectures would result in restrictions on the allowed effective theories. The aim of this thesis is to develop the necessary math- ematical tools to explicitly test the conjectures in a string theory setup. To this end the periods of Calabi-Yau manifolds are computed numerically as well as analytically. Further- more, tools applicable to general string phenomenological models are discussed, including the computation of target space Calabi-Yau metrics, line bundle cohomologies and Strebel differentials. These periods are used to test two conjectures, the refined swampland distance conjec- ture as well as the dS conjecture. The first states that an effective field theory is only valid up to a certain value of field excursions. If larger field values are included, the effective description breaks down due to an infinite tower of states becoming exponentially light. The conjecture is tested explicitly by computing the distances in the moduli space of CY manifolds. Challenging this conjecture requires the computation of the periods of different Calabi-Yau spaces. The dS conjecture on the other hand forbids vacua with positive cos- mological constant. To test this conjecture, the KKLT construction is examined in detail and some steps of the construction are carried out explicitly. Moreover, the validity of the assumed effective theory in a warped throat is investigated. Besides these traditional approaches more exotic ones are followed, including the construction of dS theories using tachyons as well as modifying the signature of space time. Contents Zusammenfassung iv 1 Introduction 1 2 Basics of String Theory 7 2.1 Conceptual Ideas . .7 2.2 Bosonic String Theory . 10 2.3 Superstring Theory . 17 2.4 Compactifications . 30 2.4.1 Geometric Preliminaries . 30 2.4.2 Kaluza Klein Compactifications . 32 2.4.3 Preliminaries: Calabi-Yau Geometry . 33 2.4.4 Calabi-Yau Compactifications . 37 3 Mathematical Tools 39 3.1 Construction of Calabi-Yau manifolds . 39 3.1.1 An Example: The (Mirror) Quintic . 45 3.1.2 Constructing General CICYs . 46 3.2 Periods of CY Manifolds . 47 3.2.1 Local Solutions . 49 3.2.2 Computation of the Transition Matrix . 50 4 3.2.3 Example: P112812[24] . 58 3.3 Gauged Linear Sigma Models . 61 3.3.1 The Gauged Linear Sigma Model . 62 3.3.2 The Partition Function . 63 3.4 Numerical CY Metrics . 73 3.5 Line Bundle Cohomologies of Toric Varieties . 77 3.6 Strebel Differentials . 83 4 The Swampland Conjectures 88 4.1 The Swampland Program . 88 4.2 The Refined Swampland Distance Conjecture . 91 Contents vii 5 KKLT and the dS Conjecture 103 5.1 The KKLT Construction . 103 5.2 Small Superpotentials in the Quintic . 105 5.3 Small Superpotentials Close to a Conifold . 107 5.3.1 jW0j 1 in the Coni-LCS Regime . 113 5.3.2 Example: P1;1;2;8;12[24] . 115 5.4 Warped Throats . 119 6 Other Ways to dS 128 6.1 dS Spaces from Tachyons . 128 6.1.1 Tachyon Condensation . 130 6.1.2 Heterotic String Tachyon Condensation . 131 6.1.3 Closed Bosonic String Tachyon . 134 6.2 Exotic String Theories . 135 6.2.1 Fluxed AdS×dS Solutions . 136 6.2.2 Exotic Superstring Theories . 138 6.2.3 Ghosts in Exotic String Theories . 141 6.2.4 CFT Description of Euclidean Exotic Strings . 148 6.2.5 The Influence of Space-time Signature . 158 6.2.6 Exotic Brane Phenomenology . 162 6.2.7 Conclusions . 164 7 Outlook 165 A Appendix 168 A.1 (A)dS Spaces of Signature (p; q)........................ 168 A.2 First Order Systems . 169 A.2.1 Fermionic Systems . 169 A.2.2 Bosonic Systems . 170 A.3 Number Theory . 171 A.3.1 The ζ-function . 171 A.3.2 Colored Multiple Zeta Values . 172 A.4 Modularity and L-functions . 172 A.4.1 Examples of Modular Functions . 173 A.5 Hypergeometric Functions . 175 A.6 Line Bundle Cohomologies . 176 4 A.7 Periods of P112812[24] . 178 A.7.1 Local Periods at the LCS . 178 A.7.2 Local Periods at the Conifold . 180 List of Figures 2.1 Comparison of a QFT amplitude on the left and a closed string amplitude on the right. There is no pointlike interaction in the string case. .8 2.2 The web of string theories and their interconnections. Gauge groups rep- resent heterotic strings, T- and S- denote the respective dualities and t denotes tachyon condensations. Geometric notations like S1 or T 2 denote compactifications on the geometry. 28 2.3 A plot of the Teichm¨ullerspace. It extends infinitely in the imaginary di- rection. 36 3.1 The relations between fans and cones. 40 3.2 The fan for P2.................................. 42 3.3 The fan for O(−3)................................ 43 3.4 The different bases involved in the computation and the relations between them. 52 3.5 The divisor structure for P11222[8] at two different scales. On the left hand side the cones are clearly visible. On the right hand side the region around the origin is shown, making the polyhedron around the origin visible. 68 3.6 The four GLSM phases and the contributing pairs in the partition function for P11222[8]. I corresponds to the geometric phase while III is the LG phase. 69 1 3.7 h (O(m; n)) of dP1................................ 80 1 3.8 Classification result for h (O(m; n)) of dP1.................. 80 0 3 3.9 h (O(m; n)) of P1112[5]. 81 0 3 3.10 h (O(m; n)) of P1112[5] separated into phases. 82 0 4 3.11 h (O(m; n)) of P11222[8]. 82 0 4 3.12 Classification result for h (O(m; n)) of P11222[8]. 83 4.1 The metric on the K¨ahlermoduli space of the quintic. 92 4.2 Sketch of the K¨ahlermoduli space of the quintic. 93 4.3 Geodesics for the initial data (r; r;_ θ; θ_) = (0; 1; i · π=50; 0), for i = 1;:::; 10. The orange geodesics are the Z2 images. 96 4.4 The moduli space of P11222[8]. Figure taken from [1]. .............. 98 4.5 Definitions of curves in the moduli space of P11222. The dotted blue lines represent a sketch of the phase boundaries. Adapted from [2]. ............... 99 List of Figures ix 5.1 The effective scalar potential for the real part of S shows the existence of a Minkowski minimum. 117 5.2 Full scalar potential around the minimum. 118 5.3 A sketch of a Calabi-Yau with a KS-throat. At the tip of the throat the S2 3 shrinks to zero size while the S remains finite. yUV marks the cutoff where the throat meets the bulk. 122 5.4 Eigenfunction of the numerical solution (blue) and analytical approximation (red) of the first (left) and second (right) radial mode. 125 5.5 Left: The first eigenvalue of the numerical solution for different yUV . Right: Eigenvalues of the numerical solution (blue) and analytical approximation (red) of the first five radial modes. The numerical evaluation of the spherical Bessel equation is shown in black. 125 6.1 Cobordism groups appearing in string theory. Table taken from the ap- pendix of [3].