DOCSLIB.ORG

Twisted Loop Groups and Their Affine Flag Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Twisted Loop Groups and Their Affine Flag Varieties TWISTED LOOP GROUPS AND THEIR AFFINE FLAG VARIETIES G. PAPPAS* AND M. RAPOPORT Introduction Loop groups are familiar objects in several branches of mathematics. Let us mention here three variants. The first variant is differential-geometric in nature. One starts with a Lie group G (e.g., a compact Lie group or its complexification). The associated loop group is then the group of (C0-, or C1-, or C∞-) maps of S1 into G, cf. [P-S] and the literature cited there. A twisted version arises from an automorphism α of G. The associated twisted loop group is the group of maps γ : R → G such that γ(θ + 2π) = α(γ(θ)) . The second variant is algebraic and arises in the context of Kac-Moody algebras. Here one constructs an infinite-dimensional algebraic group variety with Lie algebra equal or closely related to a given Kac-Moody algebra. (This statement is an oversimplification and the situation is in fact more complicated: there exist various constructions at a formal, a minimal, and a maximal level which produce infinite-dimensional groups with Lie algebras closely related to the given Kac-Moody Lie algebra, see [Ma2], also [T2], [T3] and the literature cited there). The third variant is algebraic-geometric in nature and is our main concern in this paper. Let us recall the basic definitions in the untwisted case. Let k be a field and let G0 be an algebraic group over Spec (k). We consider the functor LG0 on the category of k-algebras, R 7→ LG0(R) = G0(R((t))). Here R((t)) = R[[t]][1/t] is the ring of Laurent series with coefficients in R. One proves that this functor is representable by an ind-scheme (= inductive limit of k-schemes), called the algebraic loop group associated to G0. At the same time, one considers the associated flag varieties, mostly the affine Grassmannian G and the affine flag variety F, the fpqc-sheaves associated to the functors, R 7→ G0(R((t)))/G0(R[[t]]), resp. R 7→ G0(R((t)))/B(R). Date: August 17, 2006. *Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study). 1 2 G. PAPPAS AND M. RAPOPORT Here B(R) is the inverse image of a Borel subgroup under the reduction map G0(R[[t]]) → G0(R). These sheaves are also representable by ind-schemes. Important results on the structure of algebraic loop groups and their associated flag varieties are due to Beauville, Laszlo, Sorger, Faltings, Beilinson, Drinfeld and Gaitsgory, cf. [B-L], [BLS], [L-S], [Fa1], [B-D], [G]. These results have applications in the theory of vector bundles on algebraic curves [B-L], [L-S], [Fa1], in geometric Langlands theory [B-D], [G], and to local models of Shimura varieties [Go1], [Go2], [P-R]. The aim of the present paper is to develop a similar theory of twisted algebraic loop groups and of their associated flag varieties. As it turns out, this theory gives a geometric interpretation of many aspects of Bruhat-Tits theory in the equal characteristic case. The basic definition is very simple. Let G be a linear algebraic group over K = k((t)) (we always assume that G is connected reductive). Then G defines its associated algebraic loop group LG, which is the ind-scheme representing the functor R 7→ LG(R) = G(R((t))). Note that, since R((t)) is a k((t))-algebra, the right hand side makes sense. When G = G0 ⊗k K, one recovers the definition in the untwisted case. To any facet a in the Bruhat- Tits building of G over K there is associated a unique smooth group scheme with connected fibers Pa over k[[t]] such that Pa(k[[t]]) is equal to the parahoric subgroup of G(K) attached + to a. To Pa corresponds an infinite-dimensional affine group scheme L Pa over Spec (k) with + L Pa(R) = Pa(R[[t]]). + The quotient Fa = LG/L Pa (in the sense of fpqc-sheaves) is representable by an ind- scheme over k which we call the (partial) affine flag variety associated to G and a. In the untwisted case, the affine Grassmannian is associated to the special vertex arising from an isomorphism G ' G0 ⊗k k((t)), and the affine flag variety is associated to an alcove in the Bruhat-Tits building containing this vertex. Note that even in the untwisted case, the definition of Fa is useful, as it re-establishes the natural symmetry attached to the K-group scheme G0 ⊗k K which has no preferred special vertex. Our main results may now be formulated as follows. Let π1(G) denote the algebraic fundamental group of G in the sense of Borovoi. If k is algebraically closed, let I = Gal(K/K) denote the inertia group. Let π1(G)I be the co-invariants under I. Theorem 0.1. Let k be algebraically closed. The Kottwitz homomorphism ([Ko], see 2.a.2) induces bijections on the set of connected components, π0(LG) = π1(G)I , π0(Fa) = π1(G)I . Note that in the untwisted case G = G0 ⊗k K, the action of I is trivial so that π1(G)I = π1(G) and we recover the result of Beauville, Laszlo and Sorger [BLS], Lemma 1.2., resp. of TWISTED LOOP GROUPS 3 Beilinson and Drinfeld [B-D], Prop. 4.5.4, which is analogous to the corresponding result in topology. Theorem 0.2. Suppose that G is semi-simple and splits over a tamely ramified extension of K and that the order of the fundamental group of the derived group π1(Gder) is prime to the characteristic of k. Then the ind-schemes LG and Fa are reduced. In the case of SLn this is due to Beauville and Laszlo [B-L], and in the case of a general split group and when k has characteristic 0, to Beilinson and Drinfeld [B-D], Thm. 4.5.1. In characteristic p > 0, this is due to Laszlo and Sorger [LS], and to Faltings [Fa1], when G is split and simply connected. We next consider the Schubert varieties contained in a partial affine flag variety Fa. By + definition, these are the reduced closures of orbits of L Pa. They are finite-dimensional projective varieties. Theorem 0.3. Suppose that G splits over a tamely ramified extension of K. Then all Schubert varieties in Fa are normal and have only rational singularities. In positive char- acteristic, they are Frobenius-split. In characteristic zero, normality of Schubert varieties in the context of Kac-Moody partial affine flag varieties is due to Kumar [Ku1], Littelmann [Li] and Mathieu [Ma1]. By an argument of Faltings these coincide with the Schubert varieties in this paper. In positive characteristic the above theorem is due to Faltings [Fa1] in the split simply connected case. (The normality of Schubert varieties in the case of SLn is also proved in [P-R1]). Normality of Schubert varieties in characteristic p has also been shown by Mathieu [Ma1] and Littelmann [Li] (in the context of Kac-Moody theory). Their definition of Schubert varieties is a priori different. They are given using an embedding into the infinite dimensional projective space associated to a highest-weight representation of the Kac-Moody algebra. In contrast, we are essentially defining Schubert varieties as varieties of certain lattices and so [Ma1] and [Li] does not imply our result. Nevertheless, it is a corollary of Theorem 0.3 combined with the results of Mathieu and Littelmann that the two notions of Schubert varieties coincide (see Section 9.h). From this, it also follows a posteriori that (for an absolutely simple, simply connected G that splits over a tame extension) our partial affine flag varieties are isomorphic to the ones defined in the Kac-Moody theory. Let us now comment on our proofs of the above theorems. The proofs of theorems 0.2 and 0.3 are closely intertwined. The proofs go along the lines of the proof of the theorem corresponding to Theorem 0.3 of Faltings [Fa1] (for more details on this proof, cf. [Go2]). The main method is to lift the situation in positive characteristic to characteristic zero and compare the lifted situation with the Kac-Moody setting. It is therefore somewhat reminiscent of the method introduced by Kazhdan to relate the representation theory of a group over a local field of characteristic p to that of a group over a local field of unequal characteristic. This also explains why we have to exclude groups which split over a wild 4 G. PAPPAS AND M. RAPOPORT extension (although we strongly believe that appropriate versions of Theorems 0.2 and 0.3 also hold in this case). We note that in the previous proofs of theorem 0.2 (in [BL] for the case of SLn, in [LS] for a split semi-simple, simply connected group in characteristic 0, in [Fa1, Go2] for a semi-simple split simply connected group in characteristic p) all use the big cell. Our argument, which we learned in Faltings’ course [Fa2] is different. Finally, the proof of Theorem 0.1 generalizes the proof of the corresponding theorem in [B-D], using the construction of the Kottwitz map [Ko]. We would like to stress that interesting problems arise when one tries to generalize to the setting of twisted loop groups classical results in the untwisted case. The theory of perverse sheaves on their partial flag varieties has still to be worked out. Also, we expect a theory of bundles on algebraic curves for corresponding non-constant algebraic groups, of their Picard groups and their theta functions.
Load more

Recommended publications
  • Affine Springer Fibers and Affine Deligne-Lusztig Varieties
    Affine Springer Fibers and Affine Deligne-Lusztig Varieties Ulrich G¨ortz Abstract. We give a survey on the notion of affine Grassmannian, on affine Springer fibers and the purity conjecture of Goresky, Kottwitz, and MacPher- son, and on affine Deligne-Lusztig varieties and results about their dimensions in the hyperspecial and Iwahori cases. Mathematics Subject Classification (2000). 22E67; 20G25; 14G35. Keywords. Affine Grassmannian; affine Springer fibers; affine Deligne-Lusztig varieties. 1. Introduction These notes are based on the lectures I gave at the Workshop on Affine Flag Man- ifolds and Principal Bundles which took place in Berlin in September 2008. There are three chapters, corresponding to the main topics of the course. The first one is the construction of the affine Grassmannian and the affine flag variety, which are the ambient spaces of the varieties considered afterwards. In the following chapter we look at affine Springer fibers. They were first investigated in 1988 by Kazhdan and Lusztig [41], and played a prominent role in the recent work about the “fun- damental lemma”, culminating in the proof of the latter by Ngˆo. See Section 3.8. Finally, we study affine Deligne-Lusztig varieties, a “σ-linear variant” of affine Springer fibers over fields of positive characteristic, σ denoting the Frobenius au- tomorphism. The term “affine Deligne-Lusztig variety” was coined by Rapoport who first considered the variety structure on these sets. The sets themselves appear implicitly already much earlier in the study of twisted orbital integrals. We remark that the term “affine” in both cases is not related to the varieties in question being affine, but rather refers to the fact that these are notions defined in the context of an affine root system.
    [Show full text]
  • Dynamics for Discrete Subgroups of Sl 2(C)
    DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]).
    [Show full text]
  • ADELIC VERSION of MARGULIS ARITHMETICITY THEOREM Hee Oh 1. Introduction Let R Denote the Set of All Prime Numbers Including
    ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Hee Oh Abstract. In this paper, we generalize Margulis’s S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one ∞ and set Q∞ = R. Let S be any subset of R. For each p ∈ S, let Gp be a connected semisimple adjoint Qp-group without any Qp-anisotropic factors and Dp ⊂ Gp(Qp) be a compact open subgroup for almost all finite prime p ∈ S. Let (GS , Dp) denote the restricted topological product of Gp(Qp)’s, p ∈ S with respect to Dp’s. Note that if S is finite, (GS , Dp) = Qp∈S Gp(Qp). We show that if Pp∈S rank Qp (Gp) ≥ 2, any irreducible lattice in (GS , Dp) is a rational lattice. We also present a criterion on the collections Gp and Dp for (GS , Dp) to admit an irreducible lattice. In addition, we describe discrete subgroups of (GA, Dp) generated by lattices in a pair of opposite horospherical subgroups. 1. Introduction Let R denote the set of all prime numbers including the infinite prime ∞ and Rf the set of finite prime numbers, i.e., Rf = R−{∞}. We set Q∞ = R. For each p ∈ R, let Gp be a non-trivial connected semisimple algebraic Qp-group and for each p ∈ Rf , let Dp be a compact open subgroup of Gp(Qp). The adele group of Gp, p ∈ R with respect to Dp, p ∈ Rf is defined to be the restricted topological product of the groups Gp(Qp) with respect to the distinguished subgroups Dp.
    [Show full text]
  • A Quasideterminantal Approach to Quantized Flag Varieties
    A QUASIDETERMINANTAL APPROACH TO QUANTIZED FLAG VARIETIES BY AARON LAUVE A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Vladimir Retakh & Robert L. Wilson and approved by New Brunswick, New Jersey May, 2005 ABSTRACT OF THE DISSERTATION A Quasideterminantal Approach to Quantized Flag Varieties by Aaron Lauve Dissertation Director: Vladimir Retakh & Robert L. Wilson We provide an efficient, uniform means to attach flag varieties, and coordinate rings of flag varieties, to numerous noncommutative settings. Our approach is to use the quasideterminant to define a generic noncommutative flag, then specialize this flag to any specific noncommutative setting wherein an amenable determinant exists. ii Acknowledgements For finding interesting problems and worrying about my future, I extend a warm thank you to my advisor, Vladimir Retakh. For a willingness to work through even the most boring of details if it would make me feel better, I extend a warm thank you to my advisor, Robert L. Wilson. For helpful mathematical discussions during my time at Rutgers, I would like to acknowledge Earl Taft, Jacob Towber, Kia Dalili, Sasa Radomirovic, Michael Richter, and the David Nacin Memorial Lecture Series—Nacin, Weingart, Schutzer. A most heartfelt thank you is extended to 326 Wayne ST, Maria, Kia, Saˇsa,Laura, and Ray. Without your steadying influence and constant comraderie, my time at Rut- gers may have been shorter, but certainly would have been darker. Thank you. Before there was Maria and 326 Wayne ST, there were others who supported me.
    [Show full text]
  • On Some Recent Developments in the Theory of Buildings
    On some recent developments in the theory of buildings Bertrand REMY∗ Abstract. Buildings are cell complexes with so remarkable symmetry properties that many groups from important families act on them. We present some examples of results in Lie theory and geometric group theory obtained thanks to these highly transitive actions. The chosen examples are related to classical and less classical (often non-linear) group-theoretic situations. Mathematics Subject Classification (2010). 51E24, 20E42, 20E32, 20F65, 22E65, 14G22, 20F20. Keywords. Algebraic, discrete, profinite group, rigidity, linearity, simplicity, building, Bruhat-Tits' theory, Kac-Moody theory. Introduction Buildings are cell complexes with distinguished subcomplexes, called apartments, requested to satisfy strong incidence properties. The notion was invented by J. Tits about 50 years ago and quickly became useful in many group-theoretic situations [75]. By their very definition, buildings are expected to have many symmetries, and this is indeed the case quite often. Buildings are relevant to Lie theory since the geometry of apartments is described by means of Coxeter groups: apartments are so to speak generalized tilings, where a usual (spherical, Euclidean or hyper- bolic) reflection group may be replaced by a more general Coxeter group. One consequence of the existence of sufficiently large automorphism groups is the fact that many buildings admit group actions with very strong transitivity properties, leading to a better understanding of the groups under consideration. The beginning of the development of the theory is closely related to the theory of algebraic groups, more precisely to Borel-Tits' theory of isotropic reductive groups over arbitrary fields and to Bruhat-Tits' theory of reductive groups over non-archimedean valued fields.
    [Show full text]
  • FINITE GROUP ACTIONS on REDUCTIVE GROUPS and BUILDINGS and TAMELY-RAMIFIED DESCENT in BRUHAT-TITS THEORY by Gopal Prasad Dedicat
    FINITE GROUP ACTIONS ON REDUCTIVE GROUPS AND BUILDINGS AND TAMELY-RAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to Guy Rousseau Abstract. Let K be a discretely valued field with Henselian valuation ring and separably closed (but not necessarily perfect) residue field of characteristic p, H a connected reductive K-group, and Θ a finite group of automorphisms of H. We assume that p does not divide the order of Θ and Bruhat-Tits theory is available for H over K with B(H=K) the Bruhat-Tits building of H(K). We will show that then Bruhat-Tits theory is also available for G := (HΘ)◦ and B(H=K)Θ is the Bruhat-Tits building of G(K). (In case the residue field of K is perfect, this result was proved in [PY1] by a different method.) As a consequence of this result, we obtain that if Bruhat-Tits theory is available for a connected reductive K-group G over a finite tamely-ramified extension L of K, then it is also available for G over K and B(G=K) = B(G=L)Gal(L=K). Using this, we prove that if G is quasi-split over L, then it is already quasi-split over K. Introduction. This paper is a sequel to our recent paper [P2]. We will assume fa- miliarity with that paper; we will freely use results, notions and notations introduced in it. Let O be a discretely valued Henselian local ring with valuation !. Let m be the maximal ideal of O and K the field of fractions of O.
    [Show full text]
  • Matrices Lie: an Introduction to Matrix Lie Groups and Matrix Lie Algebras
    Matrices Lie: An introduction to matrix Lie groups and matrix Lie algebras By Max Lloyd A Journal submitted in partial fulfillment of the requirements for graduation in Mathematics. Abstract: This paper is an introduction to Lie theory and matrix Lie groups. In working with familiar transformations on real, complex and quaternion vector spaces this paper will define many well studied matrix Lie groups and their associated Lie algebras. In doing so it will introduce the types of vectors being transformed, types of transformations, what groups of these transformations look like, tangent spaces of specific groups and the structure of their Lie algebras. Whitman College 2015 1 Contents 1 Acknowledgments 3 2 Introduction 3 3 Types of Numbers and Their Representations 3 3.1 Real (R)................................4 3.2 Complex (C).............................4 3.3 Quaternion (H)............................5 4 Transformations and General Geometric Groups 8 4.1 Linear Transformations . .8 4.2 Geometric Matrix Groups . .9 4.3 Defining SO(2)............................9 5 Conditions for Matrix Elements of General Geometric Groups 11 5.1 SO(n) and O(n)........................... 11 5.2 U(n) and SU(n)........................... 14 5.3 Sp(n)................................. 16 6 Tangent Spaces and Lie Algebras 18 6.1 Introductions . 18 6.1.1 Tangent Space of SO(2) . 18 6.1.2 Formal Definition of the Tangent Space . 18 6.1.3 Tangent space of Sp(1) and introduction to Lie Algebras . 19 6.2 Tangent Vectors of O(n), U(n) and Sp(n)............. 21 6.3 Tangent Space and Lie algebra of SO(n).............. 22 6.4 Tangent Space and Lie algebras of U(n), SU(n) and Sp(n)..
    [Show full text]
  • Tessellations: Lessons for Every Age
    TESSELLATIONS: LESSONS FOR EVERY AGE A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Kathryn L. Cerrone August, 2006 TESSELLATIONS: LESSONS FOR EVERY AGE Kathryn L. Cerrone Thesis Approved: Accepted: Advisor Dean of the College Dr. Linda Saliga Dr. Ronald F. Levant Faculty Reader Dean of the Graduate School Dr. Antonio Quesada Dr. George R. Newkome Department Chair Date Dr. Kevin Kreider ii ABSTRACT Tessellations are a mathematical concept which many elementary teachers use for interdisciplinary lessons between math and art. Since the tilings are used by many artists and masons many of the lessons in circulation tend to focus primarily on the artistic part, while overlooking some of the deeper mathematical concepts such as symmetry and spatial sense. The inquiry-based lessons included in this paper utilize the subject of tessellations to lead students in developing a relationship between geometry, spatial sense, symmetry, and abstract algebra for older students. Lesson topics include fundamental principles of tessellations using regular polygons as well as those that can be made from irregular shapes, symmetry of polygons and tessellations, angle measurements of polygons, polyhedra, three-dimensional tessellations, and the wallpaper symmetry groups to which the regular tessellations belong. Background information is given prior to the lessons, so that teachers have adequate resources for teaching the concepts. The concluding chapter details results of testing at various age levels. iii ACKNOWLEDGEMENTS First and foremost, I would like to thank my family for their support and encourage- ment. I would especially like thank Chris for his patience and understanding.
    [Show full text]
  • Buildings, Group Homology and Lattices
    Mathematik Buildings, Group Homology and Lattices Jan Essert Inaugural-Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften im Fachbereich Mathematik und Informatik der Westfälischen Wilhelms-Universität Münster vorgelegt von Jan Essert aus Seeheim-Jugenheim 2010 arXiv:1008.4908v1 [math.GR] 29 Aug 2010 Dekan Prof. Dr. Dr. h.c. Joachim Cuntz Erster Gutachter Prof. Dr. Linus Kramer Zweiter Gutachter Prof. Dr. Benson Farb Tag der mündlichen Prüfung 18.05.2010 Tag der Promotion 18.05.2010 Abstract This thesis discusses questions concerning the homology of groups related to buildings. We also give a new construction of lattices in such groups and investigate their group homology. Specifically, we construct a simplicial complex called the Wagoner complex associated to any group of Kac-Moody type. For a 2-spherical group G of Kac-Moody type, we show that the fundamental group of the Wagoner complex is almost always isomorphic to the Schur multiplier of the little projective group of G. Furthermore, we present a general method to prove homological stability for groups with weak spherical Tits systems, that is, groups acting strongly transitively on weak spherical buildings. We use this method to prove strong homological stability results for special linear groups over infinite fields and for unitary groups over division rings, improving the previously best known results in many cases. ˜ ˜ Finally, we give a new construction method for buildings of types A2 and C2 with cocompact lattices in their full automorphism groups. Almost all of these buildings are ˜ ˜ exotic in the case C2. In the A2-case, it is not known whether these buildings are classical.
    [Show full text]
  • Arxiv:1711.08410V2
    LATTICE ENVELOPES URI BADER, ALEX FURMAN, AND ROMAN SAUER Abstract. We introduce a class of countable groups by some abstract group- theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing ℓ2- Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Γ in this class we determine the general structure of the possible lattice embeddings of Γ, i.e. of all compactly generated, locally compact groups that contain Γ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possi- ble lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature. 1. Introduction 1.1. Motivation and background. Let G be a locally compact second countable group, hereafter denoted lcsc1. Such a group carries a left-invariant Radon measure unique up to scalar multiple, known as the Haar measure. A subgroup Γ < G is a lattice if it is discrete, and G/Γ carries a finite G-invariant measure; equivalently, if the Γ-action on G admits a Borel fundamental domain of finite Haar measure. If G/Γ is compact, one says that Γ is a uniform lattice, otherwise Γ is a non- uniform lattice. The inclusion Γ ֒→ G is called a lattice embedding. We shall also say that G is a lattice envelope of Γ. The classical examples of lattices come from geometry and arithmetic. Starting from a locally symmetric manifold M of finite volume, we obtain a lattice embedding π1(M) ֒→ Isom(M˜ ) of its fundamental group into the isometry group of its universal covering via the action by deck transformations.
    [Show full text]
  • Heterotic Orbifolds in Blowup
    Heterotic Orbifolds in Blowup Stefan Groot Nibbelink ITP, Heidelberg University Stefan Groot Nibbelink (Heidelberg) Heterotic Orbifolds in Blowup Warsaw,19Jun2009 1/33 Based on a collection of works: JHEP 03 (2007) 035 [hep-th/0701227], Phys. Lett. B652 (2007) 124 [hep-th/0703211], Phys. Rev. D77 (2008) 026002 [0707.1597], JHEP (2008) 060 [0802.2809], JHEP03(2009)005 [arXiv:0901.3059 [hep-th]] and work in progress in collaboration with Michael Blaszczyk, Tae-Won Ha,Johannes Held, Dennis Klever, Hans-Peter Nilles, Filipe Paccetti, Felix Ploger,¨ Michael Ratz, Fabian R¨uhle, Michele Trapletti, Patrick Vaudrevange, Martin Walter Stefan Groot Nibbelink (Heidelberg) Heterotic Orbifolds in Blowup Warsaw,19Jun2009 2/33 1 Introduction and motivation Calabi–Yau model building Orbifold model building 2 Blowups: Connecting orbifolds with smooth CYs 3 Constructing gauge backgrounds on blowups 6 4 Blowing up T /Z6–II MSSM orbifolds ′ 5 6 Z Z Blowing up T / 2 × 2 GUT Orbifolds 6 Conclusions Stefan Groot Nibbelink (Heidelberg) Heterotic Orbifolds in Blowup Warsaw,19Jun2009 3/33 Introduction and motivation Introduction and motivation One of the aims of String Phenomenology is to find the Standard Model of Particle Physics from String constructions : The E8×E8 Heterotic Strings naturally incorporates properties of GUT theories , and could lead to the Supersymmetric Standard Model (MSSM) . Two approaches are often considered to achieve this goal: smooth Calabi–Yau compactifications with gauge bundles Candelas,Horowitz,Strominger,Witten’85 singular Orbifold constructions Dixon,Harvey,Vafa,Witten’85 Stefan Groot Nibbelink (Heidelberg) Heterotic Orbifolds in Blowup Warsaw,19Jun2009 4/33 Introduction and motivation Calabi–Yau model building Calabi–Yau model building Calabi–Yau manifolds can be constructed as: a bundle M→ B with an elliptically fibered torus , complete Intersections CY : hypersurfaces in projective spaces.
    [Show full text]
  • Textures, Model Building, and Orbifold Gauge Anomalies: Research in Three Topics in Physics Beyond the Standard Model
    TEXTURES, MODEL BUILDING, AND ORBIFOLD GAUGE ANOMALIES: RESEARCH IN THREE TOPICS IN PHYSICS BEYOND THE STANDARD MODEL DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Leslie J. Schradin III, B.S., M.S. ***** The Ohio State University 2006 Dissertation Committee: Approved by Professor Stuart Raby, Adviser Professor Richard Kass Adviser Professor Junko Shigemitsu Graduate Program in Professor Terrence Walker Physics ABSTRACT We introduce the Standard Model, list a large sector of the low energy data, and present extensions to the Standard Model including grand unification, supersymme- try, and orbifold extra dimensions. These foundations underly the research presented in this dissertation, which is from three separate projects. Texture models are Ans¨atze for the undiagonalized Yukawa matrices in which some of the matrix elements have been chosen to vanish. Recent precise measurements of sin 2β from the B-factories (BABAR and BELLE) and a better known strange quark mass from lattice QCD make precision tests of predictive texture models possible. We show that in a set of these models, their maximal sin 2β values rule them out at the 3σ level. While at present sin 2β and V /V are equally good for testing N-zero | ub cb| texture models, in the near future the former will surpass the latter in constraining power. We construct a supersymmetric SO(10) D grand unified model with an orbifold × 3 1 extra dimension S /(Z Z′ ). The model uses 11 parameters to fit the 13 independent 2 × 2 low energy observables of the charged fermion Yukawa matrices and predicts the val- ues of two quark mass combinations, mu/mc and mdmsmb, to each be approximately 1σ above their experimental values.
    [Show full text]