DOCSLIB.ORG

TASI Lectures on F-Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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
Load more

Recommended publications
  • Arxiv:0907.0298V3 [Math.AG] 9 Jul 2010 E Oefrmn Rtmtcadgoerccnieain.Wieth While Considerations
    ELLIPTIC SURFACES MATTHIAS SCHUTT¨ AND TETSUJI SHIODA Abstract. This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of Mordell-Weil lattices and address arithmetic questions. Contents 1. Introduction 1 2. Elliptic curves 3 3. Elliptic surfaces 7 4. Singular fibres 10 5. Base change and quadratic twisting 17 6. Mordell-Weil group and N´eron-Severi lattice 22 7. Torsion sections 32 8. Rational elliptic surfaces 38 9. Extremal rational elliptic surfaces 42 10. Semi-stable elliptic surfaces 45 11. Mordell-Weil lattice 48 12. Elliptic K3 surfaces 58 13. Arithmetic of elliptic K3 surfaces 72 14. Ranks of elliptic curves 84 References 88 arXiv:0907.0298v3 [math.AG] 9 Jul 2010 1. Introduction Elliptic surfaces are ubiquitous in the theory of algebraic surfaces. They play a key role for many arithmetic and geometric considerations. While this feature Date: February 22, 2010. 2000 Mathematics Subject Classification. Primary 14J27; Secondary 06B05, 11G05, 11G07, 11G50, 14J20, 14J26, 14J28. Key words and phrases. Elliptic surface, elliptic curve, N´eron-Severi group, singular fibre, Mordell-Weil group, Mordell-Weil lattice, Tate algorithm, K3 surface. Partial funding from DFG under grant Schu 2266/2-2 and JSPS under Grant-in-Aid for Scientific Research (C) No. 20540051 is gratefully acknowledged. 1 2 MATTHIAS SCHUTT¨ AND TETSUJI SHIODA has become ever more clear during the last two decades, extensive survey papers and monographs seem to date back exclusively to the 80’s and early 90’s.
    [Show full text]
  • Multicanonical Systems of Elliptic Surfaces in Small Characteristics Compositio Mathematica, Tome 97, No 1-2 (1995), P
    COMPOSITIO MATHEMATICA TOSHIYUKI KATSURA Multicanonical systems of elliptic surfaces in small characteristics Compositio Mathematica, tome 97, no 1-2 (1995), p. 119-134 <http://www.numdam.org/item?id=CM_1995__97_1-2_119_0> © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 97: 119-134, 1995. 119 © 1995 Kluwer Academic Publishers. Printed in the Netherlands. Multicanonical systems of elliptic surfaces in small characteristics* Dedicated to Frans Oort on the occasion of his 60th birthday TOSHIYUKI KATSURA Department of Mathematical Sciences, University of Tokyo, Tokyo, Japan Received 4 January 1995; accepted in final form 12 April 1995 Abstract. We determine the smallest integer M such that the multicanonical system |mKS| gives the structure of elliptic surface for any elliptic surface S’ with Kodaira dimension 03BA(S) = 1 and for any integer m M. We also investigate the structure of elliptic surfaces of type (p, p,..., p). 0. Introduction Let k be an algebraically closed field of characteristic p, and f : S ~ C an elliptic surface over with a nonsingular complete curve C. As is well known, for an elliptic surface S with Kodaira dimension 03BA(S) = 1, there exists a positive integer m such that the multicanonical system |mKS| gives a unique structure of elliptic surface.
    [Show full text]
  • Density of Rational Points on Elliptic K3 Surfaces
    Density of rational points on elliptic K3 surfaces F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: [email protected] and Yu. Tschinkel Dept. of Mathematics, U.I.C. 851 South Morgan str. Chicago, (IL) 60607-7045, U.S.A. e-mail: [email protected] 1 Introduction Let X be a smooth projective algebraic variety defined over a number field K. We will say that rational points on X are potentially dense if there exists a finite extension K0=K such that the set X(K0) of K0-rational points is Zariski dense. What are possible strategies to propagate rational points on an algebraic variety? We thought of two: using the group of automorphisms Aut(X) and using additional geometric structures - like elliptic fibrations. The class of K3 surfaces is an ideal test case for both methods. One of our main results is: Theorem 1.1 Let X be a K3 surface defined over a number field K. As- sume that X has a structure of an elliptic fibration or an infinite group of automorphisms. Then rational points on X are potentially dense. Here is a more detailed list of what we learned: We don't know if ratio- nal points are potentially dense on a general K3 surface with Picard group Pic(XC) = Z. In particular, we don't know if rational points are dense on a double cover of P2 ramified in a general curve of degree 6. However, we can prove potential density for a divisor in the space of all such K3 surfaces, cor- responding to the case when, for example, the ramification curve is singular (cf.
    [Show full text]
  • A Modular Compactification of the Space of Elliptic K3 Surfaces By
    A Modular Compactification of the Space of Elliptic K3 Surfaces by Adrian Brunyate (Under the Direction of Valery Alexeev) Abstract We describe work towards a compact moduli space of elliptic K3 surfaces with marked divisor given by a small multiple of the sum of rational curves in an ample class of sufficiently large degree. Index words: K3 Moduli, Elliptic surfaces. A Modular Compactification of the Space of Elliptic K3 Surfaces by Adrian Brunyate BS, Towson University, 2006 A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2015 c 2015 Adrian Brunyate All Rights Reserved A Modular Compactification of the Space of Elliptic K3 Surfaces by Adrian Brunyate Approved: Major Professor: Valery Alexeev Pete L. Clark Mitchell Rothstein Committee: Robert Varley Electronic Version Approved: Suzanne Barbour Dean of the Graduate School The University of Georgia August 2015 Acknowledgments I would like to thank my advisor, Valery Alexeev, for introducing me to this material and without whom this work would have been impossible. I would like to thank my family, friends and colleagues for their support, even when I didn't deserve it. iv Contents Acknowledgements iv 1 Introduction 1 I Background 4 2 K3 Surfaces 5 2.1 Introduction and Examples . 5 2.2 Basic Properties . 6 3 Toric Geometry 9 3.1 Affine Toric Varieties . 10 3.2 Non-Affine and Projective Toric Varieties. 11 4 The Minimal Model Program and Moduli of Stable Pairs 16 4.1 Prehistory: Moduli of Pointed Curves .
    [Show full text]
  • Elliptic Surfaces
    ELLIPTIC SURFACES MATTHIAS SCHUTT¨ AND TETSUJI SHIODA Abstract. This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of Mordell-Weil lattices and address arithmetic questions. Contents 1. Introduction 1 2. Elliptic curves 3 3. Elliptic surfaces 7 4. Singular fibres 10 5. Base change and quadratic twisting 17 6. Mordell-Weil group and N´eron-Severi lattice 21 7. Torsion sections 31 8. Rational elliptic surfaces 37 9. Extremal rational elliptic surfaces 41 10. Semi-stable elliptic surfaces 44 11. Mordell-Weil lattice 47 12. Elliptic K3 surfaces 56 13. Arithmetic of elliptic K3 surfaces 69 14. Ranks of elliptic curves 80 References 84 1. Introduction Elliptic surfaces are ubiquitous in the theory of algebraic surfaces. They play a key role for many arithmetic and geometric considerations. While this feature Date: July 21, 2009. 2000 Mathematics Subject Classification. Primary 14J27; Secondary 06B05, 11G05, 11G07, 11G50, 14J20, 14J26, 14J28. Key words and phrases. Elliptic surface, elliptic curve, N´eron-Severi group, singular fibre, Mordell-Weil lattice, Tate algorithm, K3 surface. Partial funding from DFG under grant Schu 2266/2-2 and JSPS under Grant-in-Aid for Scientific Research (C) No. 20540051 is gratefully acknowledged. 1 2 MATTHIAS SCHUTT¨ AND TETSUJI SHIODA has become ever more clear during the last two decades, extensive survey papers and monographs seem to date back exclusively to the 80’s and early 90’s. Hence we found it a good time to comprise a detailed overview of the theory of elliptic surfaces that includes the recent developments.
    [Show full text]
  • Notes for 483-3: Kodaira Dimension of Algebraic Varieties
    NOTES FOR 483-3: KODAIRA DIMENSION OF ALGEBRAIC VARIETIES Contents 1. Plurigenera2 2. Kodaira dimension3 3. Projective bundles6 4. Intersection numbers and Riemann-Roch-type theorems8 5. Nef and big line bundles 16 6. Birational classification of surfaces 23 7. Iitaka's conjecture 30 8. Vanishing theorems 33 9. Castelnuovo-Mumford regularity 40 10. Log-resolutions, birational transformations, Kawamata-Viehweg 42 11. Vanishing for direct images of pluricanonical bundles 47 12. Positivity for vector bundles and torsion-free sheaves 50 13. Multiplication maps 60 14. Iitaka's conjecture for a base of general type 61 15. Variation of families of varieties 63 16. Bigness of the determinant implies Viehweg's conjecture 65 17. Vector bundle constructions, variation and positivity 68 18. Positivity for families of varieties of general type 72 References 74 1 2 Mihnea Popa 1. Plurigenera Let X be a smooth projective variety over an algebraically closed field k. The crucial invariant of X we will repeatedly refer to is its canonical bundle dim X 1 !X := ^ ΩX : Definition 1.1. The plurigenera of X are the non-negative integers 0 ⊗m 0 ⊗m Pm(X) = h (X; !X ) := dimk H (X; !X ); 8 m ≥ 0: n Example 1.2 (Projective space). If X = P , then !X = OPn (−n−1), and so Pm(X) = 0 for all m ≥ 0. Example 1.3 (Curves). If X = C, a smooth projective curve if genus g, then by definition P1(X) = g. Moreover: 1 • If C = P , i.e. g = 0, then !C = OP1 (−2), and so Pm(C) = 0 for all m ≥ 0.
    [Show full text]
  • K3 Surfaces and Elliptic Fibrations in Number Theory
    K3 surfaces and elliptic fibrations in number theory Banff Workshop 18w5190: Geometry and Physics of F-theory 23 January 2018 Noam D. Elkies, Harvard University 1. Examples of K3's and ell. fibrations in number theory 2. Shioda's \excellent families" of rational elliptic surfaces 3. An \excellent family" of K3 elliptic surfaces 4. A family of elliptically fibered CY3's of rank 10 1 1. Examples of K3's and elliptic fibrations in number theory Overview: Like elliptic curves (and more familiar building blocks such as rational functions, calculus, and Fourier analysis), K3 surfaces and elliptic fibrations are a central enough mathe- matical structure that \of course" they figure prominently also in some parts of number theory. Number theorists need to know about varieties such as K3's, and invariants such as N´eron-Severigroups, not just over C but also over small fields such as Q and even over finite fields such as Z=pZ. WARNING: CONTAINS EXPLICIT MATERIAL 2 1. Examples of K3's and elliptic fibrations in number theory Overview: Like elliptic curves (and more familiar building blocks such as rational functions, calculus, and Fourier analysis), K3 surfaces and elliptic fibrations are a central enough mathe- matical structure that \of course" they figure prominently also in some parts of number theory. Number theorists need to know about varieties such as K3's, and invariants such as N´eron-Severigroups, not just over C but also over small fields such as Q and even over finite fields such as Z=pZ. WARNING: CONTAINS EXPLICIT MATERIAL 2 i) Diophantine equations In general: rational solution of Diophantine equation () rational point on alg.
    [Show full text]
  • The Liftability of Elliptic Surfaces
    The Liftability of Elliptic Surfaces Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨at D¨usseldorf vorgelegt von Holger Partsch aus Augsburg D¨usseldorf, Dezember 2010 Aus dem Mathematischen Institut der Heinrich-Heine-Universit¨at D¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨at D¨usseldorf Referent: Prof. Dr. Stefan Schr¨oer Koreferent: Dr. habil. Cristiana Bertolin Tag der m¨undlichen Pr¨ufung: 21.01.2011 Erstellungsdatum der elektronischen Version: 25.01.2011 Contents Acknowledgments v Zusammenfassung vii Summary ix Introduction xi Background and motivation xiv Notations and conventions xvi Chapter 1. Smooth elliptic fibrations and elliptic fibre bundles 1 1. Preliminaries: Deformation theory and thickened schemes 1 2. Elliptic fibre bundles: Definition and examples 3 3. Elliptic curves and their moduli 4 4. Deformations of Jacobian elliptic fibre bundles 5 5. Non-liftable elliptic fibre bundles 8 6. Deformations of elliptic torsors 9 7. Elliptic fibre bundles of Kodaira dimension one 12 8. Bielliptic surfaces 15 Chapter 2. Liftings of semistable elliptic fibrations 27 1. Semistable genus-1 curves and generalized elliptic curves 27 2. Modular invariants and isomorphism types over complete bases 29 3. Moduli of generalized elliptic curves with level structures 31 4. Level structures and liftability 32 5. Liftings of generalized elliptic curves with separable modular invariants 34 6. The constructions 38 7. Liftings of principal homegenous spaces 44 Chapter 3. Liftability under tameness assumptions 49 1. Tame coverings 49 2. Tame Jacobian fibrations 50 3. Non-Jacobian fibrations of period prime to p 56 4.
    [Show full text]
  • View PDF Datastream
    A tale of two moduli spaces: Hilbert schemes of singular curves and moduli of elliptic surfaces by Dori Bejleri B. Sc., California Institute of Technology; Pasadena, CA 2013 M. Sc., Brown University; Providence, RI, 2016 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University Providence, Rhode Island May 2018 © Copyright 2018 by Dori Bejleri This dissertation by Dori Bejleri is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Dan Abramovich, Ph. D., Advisor Recommended to the Graduate Council Date Brendan Hassett, Ph. D., Reader Date Melody Chan, Ph. D., Reader Approved by the Graduate Council Date Andrew G. Campbell Dean of the Graduate School iii Curriculum Vitae Dori Bejleri grew up in Connecticut and graduated salutatorian from Wolcott High School in 2009. He earned a Bachelor of Science degree in mathematics from the California Institute of Technology in 2013 and a Master of Science degree in mathematics from Brown University in 2016. He completed his PhD in algebraic geometry at Brown University in May 2018 under the supervision of Dan Abramovich. Publication list (1) K. Ascher and D. Bejleri, Stable pair compactifications of the moduli space of degree one del pezzo surfaces via elliptic fibrations, I. arXiv:1802.00805. (2) D. Bejleri, D. Ranganathan and R. Vakil, Motivic Hilbert zeta functions of curves are rational. arXiv:1710.04198. In revision at the Journal of the Institute of Mathematics of Jussieu.
    [Show full text]
  • Global Moduli for Polarized Elliptic Surfaces Compositio Mathematica, Tome 62, No 2 (1987), P
    COMPOSITIO MATHEMATICA WOLFGANG K. SEILER Global moduli for polarized elliptic surfaces Compositio Mathematica, tome 62, no 2 (1987), p. 187-213 <http://www.numdam.org/item?id=CM_1987__62_2_187_0> © Foundation Compositio Mathematica, 1987, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 62: 187-213 (1987) © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands 187 Global moduli for polarized elliptic surfaces WOLFGANG K. SEILER Universitât Mannheim, Fakultâtfür Mathematik und Informatik, Seminargebâude A5, 6800 Mannheim 1, FRG Received 30 May 1986; accepted 7 October 1986 Table of contents Introduction This paper is a continuation of [23], where 1 showed the existence of moduli schemes for elliptic surfaces with a section; now this result is applied to show the existence of moduli schemes for polarized elliptic surfaces not necessarily admitting a section. The idea is to show that the moduli functor to be represented is finite over a coarsely representable moduli functor, which is an extension of the moduli functor for elliptic surfaces with a section, and to show - using ideas of Seshadri - that one has coarse representability in such a situation. Since 1 need finiteness in the algebrogeometric sense (including properness), it turns out that even if one is only interested in nonsingular surfaces, one still has to consider surfaces with rational double points, too, and in some constructions even worse singularities have to be 188 admitted.
    [Show full text]
  • Flopping and Slicing: SO(4) and Spin(4)-Models
    i \2-Esole" | 2020/1/7 | 1:44 | page 1003 | #1 i i i ADV. THEOR. MATH. PHYS. Volume 23, Number 4, 1003{1066, 2019 Flopping and slicing: SO(4) and Spin(4)-models Mboyo Esole and Monica Jinwoo Kang We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graphs Ae 1. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell{Weil group of the elliptic fibration is required to be trivial for Spin(4) and Z=2Z for SO(4). Each of these models has two possible crepant resolutions con- nected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of ar- bitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of Calabi-Yau threefolds, we also compute their Hodge numbers and check the cancellations of anomalies in a six-dimensional su- pergravity theory. 1 Introduction 1004 2 Summary of results 1011 3 Preliminaries 1027 4 SO(4)-model 1043 5 Spin(4)-model 1049 References 1061 1003 i i i i i \2-Esole" | 2020/1/7 | 1:44 | page 1004 | #2 i i i 1004 M. Esole and M. J. Kang 1. Introduction Crepant resolutions of Weierstrass models and their network of flops is a topic of interest for both mathematicians and physicists [4, 53, 57, 64].
    [Show full text]