DOCSLIB.ORG

Computational Methods in String Theory and Applications to the Swampland Conjectures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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].
Recommended publications
  • TASI Lectures on String Compactification, Model Building

    TASI Lectures on String Compactification, Model Building

    CERN-PH-TH/2005-205 IFT-UAM/CSIC-05-044 TASI lectures on String Compactification, Model Building, and Fluxes Angel M. Uranga TH Unit, CERN, CH-1211 Geneve 23, Switzerland Instituto de F´ısica Te´orica, C-XVI Universidad Aut´onoma de Madrid Cantoblanco, 28049 Madrid, Spain angel.uranga@cern,ch We review the construction of chiral four-dimensional compactifications of string the- ory with different systems of D-branes, including type IIA intersecting D6-branes and type IIB magnetised D-branes. Such models lead to four-dimensional theories with non-abelian gauge interactions and charged chiral fermions. We discuss the application of these techniques to building of models with spectrum as close as possible to the Stan- dard Model, and review their main phenomenological properties. We finally describe how to implement the tecniques to construct these models in flux compactifications, leading to models with realistic gauge sectors, moduli stabilization and supersymmetry breaking soft terms. Lecture 1. Model building in IIA: Intersecting brane worlds 1 Introduction String theory has the remarkable property that it provides a description of gauge and gravitational interactions in a unified framework consistently at the quantum level. It is this general feature (beyond other beautiful properties of particular string models) that makes this theory interesting as a possible candidate to unify our description of the different particles and interactions in Nature. Now if string theory is indeed realized in Nature, it should be able to lead not just to `gauge interactions' in general, but rather to gauge sectors as rich and intricate as the gauge theory we know as the Standard Model of Particle Physics.
  • Lectures on D-Branes

    Lectures on D-Branes

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server CPHT/CL-615-0698 hep-th/9806199 Lectures on D-branes Constantin P. Bachas1 Centre de Physique Th´eorique, Ecole Polytechnique 91128 Palaiseau, FRANCE [email protected] ABSTRACT This is an introduction to the physics of D-branes. Topics cov- ered include Polchinski’s original calculation, a critical assessment of some duality checks, D-brane scattering, and effective worldvol- ume actions. Based on lectures given in 1997 at the Isaac Newton Institute, Cambridge, at the Trieste Spring School on String The- ory, and at the 31rst International Symposium Ahrenshoop in Buckow. June 1998 1Address after Sept. 1: Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, 24 rue Lhomond, 75231 Paris, FRANCE, email : [email protected] Lectures on D-branes Constantin Bachas 1 Foreword Referring in his ‘Republic’ to stereography – the study of solid forms – Plato was saying : ... for even now, neglected and curtailed as it is, not only by the many but even by professed students, who can suggest no use for it, never- theless in the face of all these obstacles it makes progress on account of its elegance, and it would not be astonishing if it were unravelled. 2 Two and a half millenia later, much of this could have been said for string theory. The subject has progressed over the years by leaps and bounds, despite periods of neglect and (understandable) criticism for lack of direct experimental in- put. To be sure, the construction and key ingredients of the theory – gravity, gauge invariance, chirality – have a firm empirical basis, yet what has often catalyzed progress is the power and elegance of the underlying ideas, which look (at least a posteriori) inevitable.
  • Pitp Lectures

    Pitp Lectures

    MIFPA-10-34 PiTP Lectures Katrin Becker1 Department of Physics, Texas A&M University, College Station, TX 77843, USA [email protected] Contents 1 Introduction 2 2 String duality 3 2.1 T-duality and closed bosonic strings .................... 3 2.2 T-duality and open strings ......................... 4 2.3 Buscher rules ................................ 5 3 Low-energy effective actions 5 3.1 Type II theories ............................... 5 3.1.1 Massless bosons ........................... 6 3.1.2 Charges of D-branes ........................ 7 3.1.3 T-duality for type II theories .................... 7 3.1.4 Low-energy effective actions .................... 8 3.2 M-theory ................................... 8 3.2.1 2-derivative action ......................... 8 3.2.2 8-derivative action ......................... 9 3.3 Type IIB and F-theory ........................... 9 3.4 Type I .................................... 13 3.5 SO(32) heterotic string ........................... 13 4 Compactification and moduli 14 4.1 The torus .................................. 14 4.2 Calabi-Yau 3-folds ............................. 16 5 M-theory compactified on Calabi-Yau 4-folds 17 5.1 The supersymmetric flux background ................... 18 5.2 The warp factor ............................... 18 5.3 SUSY breaking solutions .......................... 19 1 These are two lectures dealing with supersymmetry (SUSY) for branes and strings. These lectures are mainly based on ref. [1] which the reader should consult for original references and additional discussions. 1 Introduction To make contact between superstring theory and the real world we have to understand the vacua of the theory. Of particular interest for vacuum construction are, on the one hand, D-branes. These are hyper-planes on which open strings can end. On the world-volume of coincident D-branes, non-abelian gauge fields can exist.
  • Holography and Compactification

    Holography and Compactification

    PUPT-1872 ITFA-99-14 Holography and Compactification Herman Verlinde Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 and Institute for Theoretical Physics, University of Amsterdam Valckenierstraat 65, 1018 XE Amsterdam Abstract arXiv:hep-th/9906182v1 24 Jun 1999 Following a recent suggestion by Randall and Sundrum, we consider string compactification scenarios in which a compact slice of AdS-space arises as a subspace of the compactifica- tion manifold. A specific example is provided by the type II orientifold equivalent to type I theory on (orbifolds of) T 6, upon taking into account the gravitational backreaction of the D3-branes localized inside the T 6. The conformal factor of the four-dimensional metric depends exponentially on one of the compact directions, which, via the holographic corre- spondence, becomes identified with the renormalization group scale in the uncompactified world. This set-up can be viewed as a generalization of the AdS/CFT correspondence to boundary theories that include gravitational dynamics. A striking consequence is that, in this scenario, the fundamental Planck size string and the large N QCD string appear as (two different wavefunctions of) one and the same object. 1 Introduction In string theory, when considered as a framework for unifying gravity and quantum mechan- ics, the fundamental strings are naturally thought of as Planck size objects. At much lower energies, such as the typical weak or strong interaction scales, the strings have lost all their internal structure and behave just as ordinary point-particles. The physics in this regime is therefore accurately described in terms of ordinary local quantum field theory, decoupled from the planckian realm of all string and quantum gravitational physics.
  • Introduction to Complex Analysis Michael Taylor

    Introduction to Complex Analysis Michael Taylor

    Introduction to Complex Analysis Michael Taylor 1 2 Contents Chapter 1. Basic calculus in the complex domain 0. Complex numbers, power series, and exponentials 1. Holomorphic functions, derivatives, and path integrals 2. Holomorphic functions defined by power series 3. Exponential and trigonometric functions: Euler's formula 4. Square roots, logs, and other inverse functions I. π2 is irrational Chapter 2. Going deeper { the Cauchy integral theorem and consequences 5. The Cauchy integral theorem and the Cauchy integral formula 6. The maximum principle, Liouville's theorem, and the fundamental theorem of al- gebra 7. Harmonic functions on planar regions 8. Morera's theorem, the Schwarz reflection principle, and Goursat's theorem 9. Infinite products 10. Uniqueness and analytic continuation 11. Singularities 12. Laurent series C. Green's theorem F. The fundamental theorem of algebra (elementary proof) L. Absolutely convergent series Chapter 3. Fourier analysis and complex function theory 13. Fourier series and the Poisson integral 14. Fourier transforms 15. Laplace transforms and Mellin transforms H. Inner product spaces N. The matrix exponential G. The Weierstrass and Runge approximation theorems Chapter 4. Residue calculus, the argument principle, and two very special functions 16. Residue calculus 17. The argument principle 18. The Gamma function 19. The Riemann zeta function and the prime number theorem J. Euler's constant S. Hadamard's factorization theorem 3 Chapter 5. Conformal maps and geometrical aspects of complex function the- ory 20. Conformal maps 21. Normal families 22. The Riemann sphere (and other Riemann surfaces) 23. The Riemann mapping theorem 24. Boundary behavior of conformal maps 25. Covering maps 26.
  • University of Groningen Seven-Branes and Instantons In

    University of Groningen Seven-Branes and Instantons In

    University of Groningen Seven-branes and instantons in type IIB supergravity Hartong, Jelle IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2008 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Hartong, J. (2008). Seven-branes and instantons in type IIB supergravity. s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 27-09-2021 Introduction The subject of this thesis will be introduced. The studies performed in the coming chapters attempt to improve on the understanding of certain aspects of string theory. Here, I will give a personal point of view on the subject of string theory after which I will briefly sketch the most elementary properties of string theory that allow me to further introduce the subject of this thesis.
  • On Algebraic Functions

    On Algebraic Functions

    On Algebraic Functions N.D. Bagis Stenimahou 5 Edessa Pella 58200, Greece [email protected] Abstract In this article we consider functions with Moebius-periodic rational coefficients. These functions under some conditions take algebraic values and can be recovered by theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers- Ramanujan continued fraction, Eisenstein series and functions asso- ciated with Jacobi symbol coefficients. Keywords: Theta functions; Algebraic functions; Special functions; Peri- odicity; 1 Known Results on Algebraic Functions The elliptic singular moduli kr is the solution x of the equation 1 1 2 2F1 , ;1;1 x 2 2 − = √r (1) 1 1 2 2F1 2 , 2 ; 1; x where 1 2 π/2 ∞ 1 1 2 2 n 2n 2 2 dφ 2F1 , ; 1; x = 2 x = K(x)= (2) 2 2 (n!) π π 2 n=0 Z0 1 x2 sin (φ) X − q The 5th degree modular equation which connects k25r and kr is (see [13]): 5/3 1/3 krk25r + kr′ k25′ r +2 (krk25rkr′ k25′ r) = 1 (3) arXiv:1305.1591v3 [math.GM] 27 Mar 2014 The problem of solving (3) and find k25r reduces to that solving the depressed equation after named by Hermite (see [3]): u6 v6 +5u2v2(u2 v2)+4uv(1 u4v4) = 0 (4) − − − 1/4 1/4 where u = kr and v = k25r . The function kr is also connected to theta functions from the relations 2 θ (q) ∞ 2 ∞ 2 k = 2 , where θ (q)= θ = q(n+1/2) and θ (q)= θ = qn r θ2(q) 2 2 3 3 3 n= n= X−∞ X−∞ (5) 1 π√r q = e− .
  • Symmetries and Polynomials

    Symmetries and Polynomials

    Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we’ll learn how to solve a cubic. We’ll also sketch how to solve a quartic. We’ll explore the connections between these solutions and group theory. Towards the end, we’ll hint towards the deep connection between polynomials and groups via Galois Theory. This connection allows one to convert the classical question of whether a general quintic can be solved by radicals to a group theory problem, which can be relatively easily answered. The 5 day outline is as follows: 1. The first two days are dedicated to studying polynomials. On the first day, we’ll study a simple invariant associated to every polynomial, the discriminant. 2. On the second day, we’ll solve the cubic equation by a method motivated by Galois theory. 3. Days 3 and 4 are a “practical” introduction to group theory. On day 3 we’ll learn about groups as symmetries of geometric objects. 4. On day 4, we’ll learn about the commutator subgroup of a group. 5. Finally, on the last day we’ll connect the two theories and explain the Galois corre- spondence for the cubic and the quartic. There are plenty of optional problems along the way for those who wish to explore more. Tricky optional problems are marked with ∗, and especially tricky optional problems are marked with ∗∗. 1 1 The Discriminant Today we’ll introduce the discriminant of a polynomial. The discriminant of a polynomial P is another polynomial Q which tells you whether P has any repeated roots over C.
  • Non-Supersymmetric Ads and the Swampland

    Non-Supersymmetric Ads and the Swampland

    CALT-TH 2016-027, IPMU16-0139 Non-supersymmetric AdS and the Swampland Hirosi Ooguria,b and Cumrun Vafac a Walter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena, CA 91125, USA b Kavli Institute for the Physics and Mathematics of the Universe University of Tokyo, Kashiwa, 277-8583, Japan c Center for the Fundamental Laws of Nature Harvard University, Cambridge, MA 02138, USA Abstract We propose to sharpen the weak gravity conjecture by the statement that, except for BPS states in a supersymmetric theory, the gravitational force is strictly weaker than any electric force and provide a number of evidences for this statement. Our conjecture implies that any non-supersymmetric anti-de Sitter vacuum supported by fluxes must be unstable, as is the case for all known attempts at such holographic constructions. arXiv:1610.01533v3 [hep-th] 25 Aug 2017 1 Introduction The weak gravity conjecture (WGC) [1] is an example of how seemingly consistent low energy gravitational theories can fail to have ultra-violet completion and instead belong to the swampland [2]; see [3] for more swampland constraints. Recently, there have been a number of papers checking and extending the WGC and applying it to cosmology [4–7,9–12]. The conjecture has also been related to the holographic principle in [13]. In this brief note, we propose sharpening of the WGC and point out that it implies that non-supersymmetric anti-de Sitter (AdS) vacua supported by fluxes are unstable. The WGC states that the mass of a particle carrying a basic unit of charge or its small multiple is less than or equal to that predicted if the particle were an extremal black hole.
  • Notes for a Talk on Resolvent Degree

    Notes for a Talk on Resolvent Degree

    Roots of topology Tobias Shin Abstract These are notes for a talk in the Graduate Student Seminar at Stony Brook. We discuss polynomials, covering spaces, and Galois theory, and how they all relate through the unifying concept of \resolvent degree", following Farb and Wolfson[1]. We will also see how this concept relates Hilbert's 13th problem (among others) to classical enumerative problems in algebraic geometry, such as 27 lines on a smooth cubic, 28 bitangents on a planar quartic, etc. Algebraic functions and roots of topology First, a historial note: the motivation for the concept of the fundamental group comes from studying differential equations on the complex plane (i.e., the theory of Riemann surfaces); namely, in the study of the monodromy of multi-valued complex functions. Consider for example, the path integral R 1=z around the punctured plane. This is nonzero and in fact, its value changes by multiples of 2πi based on how many times you wind around the puncture. This behavior arises since the path integral of this particular function is multivalued; it is only well defined as a function up to branch p cuts. In a similar spirit, we can consider the function z which, as we go around the punctured plane once, sends a specified value to its negative. More specifically, suppose we have a branched covering space π : Y ! X, i.e., a map and a pair of spaces such that away from a nowhere dense subset of X, called the branched locus of X, we have that π is a covering map.
  • Lecture 1. Model Building in IIA: Intersecting Brane Worlds

    Lecture 1. Model Building in IIA: Intersecting Brane Worlds

    Compactification, Model Building, and Fluxes Angel M. Uranga IFT, Madrid, and CERN, Geneva Lecture 1. Model building in IIA: Intersecting brane worlds We review the construction of chiral four-dimensional compactifications of type IIA string theory with intersecting D6-branes. Such models lead to four-dimensional the- ories with non-abelian gauge interactions and charged chiral fermions. We discuss the application of these techniques to building of models with spectrum as close as possible to the Standard Model, and review their main phenomenological properties. 1Introduction String theory has the remarkable property that it provides a description of gauge and gravitational interactions in a unified framework consistently at the quantum level. It is this general feature (beyond other beautiful properties of particular string models) that makes this theory interesting as a possible candidate to unify our description of the different particles and interactions in Nature. Now if string theory is indeed realized in Nature, it should be able to lead not just to ‘gauge interactions’ in general, but rather to gauge sectors as rich and intricate as the gauge theory we know as the Standard Model of Particle Physics. In these lecture we describe compactifications of string theory where sets of D-branes lead to gauge sectors close to the Standard Model. We furthermore discuss the interplay of such D-brane systems with flux compactifications, recently introduced to address the issues of moduli stabilization and supersymmetry breaking. Before starting, it is important to emphasize that there are other constructions in string theory which are candidates to reproduce the physics of the Standard Model at low energies, which do not involve D-branes.
  • Recent Developments in Physics Far Beyond the Standard Model

    Recent Developments in Physics Far Beyond the Standard Model

    FTUAM-99/02; IFT-UAM/CSIC-99-02 Recent Developments in Physics Far Beyond the Standard Model 1 L.E. Ib´a˜nez Departamento de F´ısica Te´orica C-XI and Instituto de F´ısica Te´orica C-XVI, Universidad Aut´onoma de Madrid Cantoblanco, 28049 Madrid, Spain Abstract String theory is the leading candidate for a unified theory of the standard model and gravity. In the last few years theorists have realized that there is a unique structure underlying string theory. In this unification a prominent role is played by the duality symmetries which relate different theories. I present a brief overview of these developements and discuss their possible impact in low-energy physics. One of the lessons learned is that the string scale, usually asumed to be of order of the Planck scale, could be arbitrarily low, even close to accelerator energies. arXiv:hep-ph/9901292v1 13 Jan 1999 1Contribution to the proceedings of the XXVI International Meeting on Fundamental Physics, La Toja, Spain (June 1998). 1 1 Far Beyond the Standard Model (FBSM) You have already heard many times that the Standard Model (SM) is fine and describes everything (except, maybe, neutrino masses) very well. And you have also heard almost the same number of times that in spite of that there are plenty of aspects of the SM that we do not yet understand. I am not going to repeat the well known list of questions without answers in the SM, it has not changed very much in the last decade. Instead of that I am going to talk today about one particular problem: the problem of combining the SM interactions with gravity.