Lectures on Naturalness, String Landscape and Multiverse Arthur Hebecker Institute for Theoretical Physics, Heidelberg University, Philosophenweg 19, D-69120 Heidelberg, Germany 24 August, 2020 Abstract The cosmological constant and electroweak hierarchy problem have been a great inspira- tion for research. Nevertheless, the resolution of these two naturalness problems remains mysterious from the perspective of a low-energy effective field theorist. The string theory landscape and a possible string-based multiverse offer partial answers, but they are also controversial for both technical and conceptual reasons. The present lecture notes, suitable for a one-semester course or for self-study, attempt to provide a technical introduction to these subjects. They are aimed at graduate students and researchers with a solid back- ground in quantum field theory and general relativity who would like to understand the string landscape and its relation to hierarchy problems and naturalness at a reasonably technical level. Necessary basics of string theory are introduced as part of the course. This text will also benefit graduate students who are in the process of studying string theory arXiv:2008.10625v3 [hep-th] 27 Jul 2021 at a deeper level. In this case, the present notes may serve as additional reading beyond a formal string theory course. Preface This course intends to give a concise but technical introduction to `Physics Beyond the Standard Model' and early cosmology as seen from the perspective of string theory. Basics of string theory will be taught as part of the course. As a central physics theme, the two hierarchy problems (of the cosmological constant and of the electroweak scale) will be discussed in view of ideas like supersymmetry, string theory landscape, eternal inflation and multiverse. The presentation will include critical points of view and alternative ideas and explanations. Problems with solutions are also provided to facilitate the use of these notes in classroom and for self-study. Basic knowledge of quantum field theory (QFT), general relativity and cosmology will be assumed. Supersymmetry, elements of supergravity and fundamentals of string theory will be taught together with a number of geometrical concepts needed to study string compactifications. However, given the limited scope of a one-semester lecture series, this can clearly not replace a full string theory course or the detailed study of string geometry. The author has taught this course at Heidelberg University with the intention to prepare students who have taken a two-semester QFT and a one-semester relativity course for Master thesis research in string phenomenology. Another goal was to allow students who intend to do research in particle phenomenology, cosmology or formal (mathematical) string theory to develop some basic understanding of the possible relation of string theory to `real world' physics and its most fundamental problems. For students who had the privilege of enjoying a complete graduate-level education (with full lecture courses on strings and on supersymmetry/supergravity) before embarking on research, most of the material in the first part of this course will be familiar. Still, depending on the focus of their string and cosmology courses, they may find useful additional information about landscape, multiverse, eternal inflation and alternative perspectives in the second half of the course. The detailed plan of the lecture notes is as follows: We will start in Sect. 1 with a brief tour of the Standard Model, emphasising the perspective of a low-energy effective field theory, the cou- pling to gravity, and the electroweak hierarchy and cosmological constant problems. Section 2 introduces supersymmetry and supergravity which, however, offer only a partial resolution of the fine-tuning problems of the Standard Model discussed above. It becomes apparent that the highest relevant energy scales, at the cutoff of the effective field theory, have to be involved. This motivates the study of string theory as the best-explored candidate quantum gravity theory in Sects. 3 and 4. We will see that the bosonic string, which suffers from the absence of fermions and from an unstable vacuum, has to be promoted to the superstring. The latter provides all desired ingredients for a consistent theory of gravity and particle physics, albeit in ten spacetime dimen- sions and with far too much supersymmetry. Compactifications to four dimensions are the subject of Sects. 5 and 6: First, we consider pure Calabi-Yau geometries, leading to highly supersymmet- ric and unrealistic 4d Minkowski-space models. Then, the inclusion of non-perturbative objects and fluxes leads to the proper string landscape with supersymmetric and non-supersymmetric models with non-zero cosmological constants of either sign. The key insight is the enormous number of such solutions (a recent analysis arriving at ∼ 10272;000), each corresponding to a dif- 2 ferent 4d effective theory. We now see that, assuming the non-trivial constructions with broken supersymmetry and positive cosmological constant stand up to further scrutiny, the ‘fine-tuned’ or `unnatural' parameters of the Standard Model may indeed be accommodated by the string landscape. Section 7 deals with the important but complicated and speculative question of how the landscape gets populated during eternal inflation and whether statistical predictions for fu- ture observations can be derived. A number of alternative perspectives on the hierarchy problems and on quantum gravity are discussed in Sect. 8 before, in Sect. 9, we end by summarising the overall picture and the challenges that should have crystallised during the study of this course. While useful references for background material and deeper exploration will be provided as we go along, it may not hurt to give some essential literature right away: Good sources for the background knowledge in QFT and relativity are [1] and [2] respectively. For more de- tails on Standard Model and particle-physics related topics, Refs. [3,4] represent useful sources. For supersymmetry and supergravity see [5, 6]. Two of the most complete modern string the- ory textbooks are [7, 8]. Concerning string phenomenology, [9] represents a very comprehensive monograph which, in particular, covers the important subjects of how specific gauge and matter sectors are realised in string compactifications - a topic which we treat very superficially in this course. A very useful set of notes emphasising the geometric side of how the landscape arises from string theory is [10]. For a detailed review of string landscape physics see [11]. Contents 1 The Standard Model and its Hierarchy Problem(s) 7 1.1 Standard Model - the basic structure . .7 1.2 Standard Model - parameter count . 10 1.3 Effective field theories - cutoff perspective . 13 1.4 Effective field theories - QFTUV vs. QFTIR .................... 14 1.5 The Standard Model as an effective field theory . 17 1.6 The electroweak hierarchy problem . 20 1.7 Fine tuning . 22 1.8 Gravity and the cosmological constant problem . 25 1.9 Problems . 29 1.9.1 Electroweak symmetry breaking . 29 1.9.2 The Standard Model is anomaly free . 30 1.9.3 The Standard Model and SU(5) . 33 1.9.4 Weyl spinors . 36 1.9.5 Covariant expression for the 1-loop vacuum energy . 38 3 2 Supersymmetry and Supergravity 40 2.1 SUSY algebra and superspace . 40 2.2 Superfields . 43 2.3 Chiral superfields . 44 2.4 SUSY-invariant lagrangians . 45 2.5 Wess-Zumino-type models . 46 2.6 Real Superfields . 47 2.7 SUSY breaking . 49 2.8 Supersymmetrizing the Standard Model . 50 2.9 Supersymmetric and SUSY breaking masses and non-renormalisation . 52 2.10 The Minimal Supersymmetric Standard Model (MSSM) . 54 2.11 Supergravity - superspace approach . 57 2.12 Supergravity - component approach . 60 2.13 Problems . 62 2.13.1 Simple manipulations within the superspace approach . 62 2.13.2 Deriving component actions . 64 2.13.3 Fierz identities for Weyl spinors . 67 2.13.4 SUSY in components . 68 2.13.5 Gauge coupling unification . 70 2.13.6 Graviton spin (helicity) . 74 3 String Theory: Bosonic String 76 3.1 Strings { basic ideas . 76 3.2 Symmetries, equations of motion, gauge choice . 79 3.3 Open string . 82 3.4 Quantisation . 84 3.5 Explicit construction of physical states { open string . 89 3.6 Explicit construction of physical states { closed string . 91 3.7 The 26d action . 93 3.8 Problems . 96 3.8.1 Point particle action . 96 3.8.2 Commutation relations of oscillator modes . 97 4 3.8.3 Trace of the energy-momentum tensor . 98 3.8.4 Virasoro algebra . 98 3.8.5 Normal ordering constant as Casimir energy . 100 3.8.6 Kalb-Ramond field from the worldsheet perspective . 101 4 String Theory: Interactions and Superstring 101 4.1 State-operator correspondence . 102 4.2 Scattering amplitudes . 103 4.3 Worldsheet supersymmetry . 105 4.4 Worldsheet supergravity . 106 4.5 Quantisation of the superstring . 108 4.6 GSO or Gliozzi-Scherk-Olive projection . 112 4.7 Consistent type II superstring theories . 113 4.8 Other 10d theories . 115 4.9 Problems . 117 4.9.1 Explicit state-operator mapping in the free case . 117 4.9.2 Euler number and genus of Riemann surfaces . 119 4.9.3 Dilaton vs. String Coupling . 120 4.9.4 Elementary exercises with 2d spinors . 121 4.9.5 SUSY algebra in 2d . 123 5 10d actions and compactification 124 5.1 10d supergravities and Type IIB as an example . 124 5.2 Kaluza-Klein compactification . 126 5.3 Towards Calabi-Yau manifolds . 129 5.4 Homology and cohomology . 134 5.5 Calabi-Yau moduli spaces . 139 5.6 Explicit parameterization of Calabi-Yau moduli spaces . 143 5.7 An aside on string model building: From heterotic compactifications to orientifold models with branes and F-theory . 147 5.8 Problems . 152 5.8.1 Dimensional reduction .