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TASI Lectures on F-Theory

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TASI Lectures on F-Theory CERN-TH-2018-126 TASI Lectures on F-theory Timo Weigand1,2 1CERN, Theory Division, CH-1211 Geneva 23, Switzerland 2Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at, Philosophenweg 19, 69120, Heidelberg, Germany Abstract F-theory is perhaps the most general currently available approach to study non-perturbative string compactifications in their geometric, large radius regime. It opens up a wide and ever- growing range of applications and connections to string model building, quantum gravity, (non- perturbative) quantum field theories in various dimensions and mathematics. Its computational power derives from the geometrisation of physical reasoning, establishing a deep correspondence between fundamental concepts in gauge theory and beautiful structures of elliptic fibrations. These lecture notes, which are an extended version of my lectures given at TASI 2017, introduce some of the main concepts underlying the recent technical advances in F-theory compactifications and their various applications. The main focus is put on explaining the F-theory dictionary between the local and global data of an elliptic fibration and the physics of 7-branes in Type IIB compactifications to arXiv:1806.01854v1 [hep-th] 5 Jun 2018 various dimensions via duality with M-theory. The geometric concepts underlying this dictionary include the behaviour of elliptic fibrations in codimension one, two, three and four, the Mordell-Weil group of rational sections, and the Deligne cohomology group specifying gauge backgrounds. Contents 1 Introduction 1 2 Setting the stage for F-theory 3 2.1 Type IIB string theory, SL(2, Z) duality and [p,q]7-branes ............... 3 2.2 Elliptic curves, SL(2, Z)bundleandellipticfibrations . 7 2.3 From M-theory to elliptic fibrations . ............ 12 3 F-theory on a smooth elliptic fibration 13 3.1 ThesmoothWeierstrassmodel . ........ 14 3.2 Singular fibers on the smooth Weierstrass model . ............. 17 3.3 Singular fibers due to 7-branes and Type IIB limit . ............. 18 3.4 M-theorypicture(II). .. .. .. .. .. .. .. .. ........ 19 4 Codimension-one singularities and non-abelian gauge algebras 22 4.1 The classification of Kodaira and N´eron . ............ 22 4.2 General structure of codimension-one fibers and relation to group theory . 25 4.3 Non-abelian gauge algebras in M- and F-theory . ............ 29 4.4 TheM-theoryCoulombbranch . ....... 31 4.5 Tatemodelsandresolutions . ......... 33 4.6 Zero-mode counting along the 7-brane . ........... 37 5 Codimension-two singularities and localised charged matter 39 5.1 Codimension-two singularities in the Weierstrass model and Katz-Vafa method . 40 5.2 The relative Mori cone and the weight lattice . ............. 41 5.3 Example:SU(2)Tatemodel. ....... 44 5.4 Countingoflocalisedzeromodes . .......... 46 5.4.1 Localised zero-mode counting for F-theory on R1,5 Yˆ ............. 46 × 3 5.4.2 Localised zero-mode counting for F-theory on R1,3 Yˆ ............. 46 × 4 5.4.3 Localised zero-mode counting for F-theory on R1,1 Yˆ ............. 48 × 5 5.5 Conformal matter in codimension two . .......... 49 5.6 Codimension-two singularities without a crepant resolution ............... 49 6 Higher-codimension singularities 52 6.1 Codimension-three singularities and Yukawa couplings on elliptic 4-folds . 53 6.1.1 Fiber degenerations in codimension three . ........... 53 6.1.2 Yukawa couplings from merging M2-branes . ......... 54 6.1.3 Yukawa couplings in the field theoretic picture . ............ 56 2 6.2 Codimension-three and -four singularities on elliptic 5-folds ............... 57 6.3 A comment on terminal singularities in codimension three................ 58 7 The Mordell-Weil group and abelian gauge symmetries 58 7.1 From rational sections to abelian gauge symmetries via Shioda’smap. 59 7.2 Gauge couplings and the height pairing . ........... 63 1 7.3 Example 1: The Bl P231[6]-fibration (U(1) restricted Tate model) . 65 1 1 7.4 Example 2: The Bl P112[4]-fibration (Morrison-Park model) . 67 7.5 Generalisations and systematics . ............ 72 7.6 Combining abelian and non-abelian gauge algebras . ............... 73 7.7 Torsional sections and the global structure of the gauge group.............. 76 8 Genus-one fibrations and discrete gauge symmetries 78 8.1 Discrete gauge groups and torsional cohomology . .............. 79 8.2 Genus-one fibrations without sections . ............ 81 9 Gauge backgrounds and zero-mode counting 83 9.1 Flux versus Deligne cohomology . ......... 84 9.2 Discrete flux data: Constraints and chirality . ............... 85 9.3 Examplesoffluxes ................................. ..... 87 9.3.1 Horizontalfluxes ............................... 87 9.3.2 Verticalfluxes................................. 88 2,2 9.3.3 Fluxes in Hrem(Yˆ4).................................. 90 9.4 Chowgroupsandgaugebackgrounds . ......... 91 9.5 Cohomology formulae counting zero-modes . ............ 92 9.5.1 Localised charged matter . ...... 93 9.5.2 Chargedbulkmatter ............................. 95 10 Applications 96 10.1 F-theory model building and landscape reasoning . ................ 97 10.2 Non-perturbative Quantum Field Theories from F-theory ................ 99 10.3 From physics back to mathematics . .......... 101 A Divisors, cycles, and equivalence relations 102 B Notation and Conventions 103 1 Introduction F-theory [1–3] occupies a remarkable sweetspot in the landscape of geometric, large radius compac- tifications of string theory: It is general enough to incorporate the regime non-perturbative in the string coupling gs while at the same time it is sufficiently well-controlled for reliable computations to be performed. As a result, F-theory offers a number of fascinating connections between physics in various dimensions and mathematics, most notably algebraic (and even arithmetic) geometry. Recent years have seen significant progress in our understanding of this correspondence between geometry and physics, along with numerous applications ranging from particle physics model building to quantum gravity to non-perturbative quantum field theory and back to mathematics. It is the purpose of these lectures to give a pedagogical introduction to some of the techniques to describe compactifications of F-theory and its manifold applications. From its outset, F-theory can be understood as a supersymmetric compactification of Type IIB string theory with 7-branes. The backreaction of the 7-branes generates a holomorphically varying profile of the axio-dilaton τ = C0 + i/gs on the compactification space. Since the imaginary part of τ is the inverse string coupling, the resulting compactifications inevitably include regions that are inherently strongly coupled. The variation of the string coupling can be made sense of thanks to the non-perturbative SL(2, Z) invariance of Type IIB string theory [4]. This formulation naturally leads to the structure of an elliptic fibration over the compactification space and hence makes contact with most beautiful concepts in algebraic geometry. Duality with M-theory [1,5] is a crucial tool to analyze the resulting compactification at a quantitative level much in the spirit of the geometric engineering programme of Type II string theory. The dual M-theory probes an a priori singular geometry, which is identified with the same elliptic fibration over the physical compactification space of Type IIB string theory. Wrapped M2-branes along vanishing cycles in the geometry engineer massless matter states which give rise to non-abelian degrees of freedom. These complete the spectrum obtained from the supergravity modes in the long wavelength limit. In special cases, the F-theory compactification enjoys yet another duality with the heterotic string [1–3]. Indeed, it is often quoted that F-theory combines two attractive properties of both Type II compactifications with branes and compactifications of the heterotic string: From the first the hierarchy of localisation of gauge degrees of freedom along the branes compared to gravitational physics in the bulk of spacetime, and from the second the appearance of exceptional gauge symmetry. This has been exploited heavily amongst other things in the context of string model building [6–9], and was one of the motivations for the revived interest in F-theory in the past decade which has lead to significant, ongoing activities with numerous far-reaching insights. It is probably fair to say that its - in many ways ideal - location at the intersection of various dual M-theory corners makes F-theory the most generic currently controllable framework for studying string vacua in their geometric regime. At the same time, it must be kept in mind that F-theory is by definition most powerful in the long wavelength limit; here truly stringy effects, which appear at higher order in an expansion of ℓs/R (with R a typical radius and ℓs the string length), are suppressed and other techniques are required to analyze this parameter regime. In this sense, F-theory addresses string vacua in their supergravity limit, and furthermore in their geometric regime. Interestingly enough, though, such compactifications can encapsulate properties of non-geometric string vacua by duality with the heterotic string [10–13]. Among the many fascinating aspects of F-theory is the emergence of a clear dictionary, summarized in Table 1.1, between fundamental concepts in theoretical physics and beautiful structures in algebraic geometry. Further developing this dictionary has been a source of continuous inspiration both for phys- ics and mathematics: Many challenging considerations about the
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