DOCSLIB.ORG

Orbifolds with ADE-Singularities

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Orbifolds with ADE-Singularities View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Eldorado - Ressourcen aus und für Lehre, Studium und Forschung G2-orbifolds with ADE-singularities HABILITATIONSSCHRIFT Der Fakult¨atf¨urMathematik der Technischen Universit¨atDortmund vorgelegt von Dr. Frank Reidegeld Dortmund, April 2017 Contents 1 Introduction 3 2 The ADE-classification 7 3 K3 surfaces 15 3.1 Introduction to lattices . 15 3.2 Basic facts about K3 surfaces . 17 3.3 Moduli spaces of K3 surfaces . 19 3.4 Singular K3 surfaces . 24 3.5 Non-symplectic automorphisms . 26 4 Introduction to G2-manifolds 39 4.1 Basic facts about G2-manifolds . 39 4.2 Introduction to orbifolds . 44 4.3 Joyce's construction of G2-manifolds . 47 4.4 Twisted connected sums . 50 4.5 G2-orbifolds and physics . 55 5 G2-orbifolds with ADE-singularities 59 5.1 Quotients of the product of a K3 surface and torus . 59 5.2 Twisted connected sums with ADE-singularities . 68 5.3 Torus quotients and their resolutions . 74 1 5.3.1 Basic facts . 74 5.3.2 Known examples . 76 5.3.3 Choice of the group action on T 7 ................... 79 5.3.4 Explicit examples . 85 5.3.5 More complicated examples . 103 6 Conclusion and Outlook 112 2 Chapter 1 Introduction A G2-structure is a 3-form φ on a 7-dimensional manifold M that satisfies certain algebraic conditions. On a manifold with a G2-structure there exists a natural metric that is induced by φ. If φ is parallel with respect to the Levi-Civita connection, the holonomy is a subgroup of the exceptional Lie group G2. If the holonomy acts irreducibly on the tangent space, we call (M; φ) a G2-manifold. The group G2 was one of the last groups on Berger's list of possible holonomy groups, for which it was unknown if metrics with these holonomy groups actually exist. The first local examples of metrics with holonomy G2 have been constructed by Bryant [12], the first complete but non-compact ones by Bryant and Salamon [13] and the first compact examples by Joyce [36]. The idea of Joyce for the construction of G2-manifolds was to 7 divide a torus T that carries a flat G2-structure by a finite group Γ that preserves the 7 G2-structure. The singularities of T =Γ can be resolved and it is possible to show by several steps the existence of a metric with holonomy G2 on the resolved manifold. Another idea for the construction of G2-manifolds has been proposed by Donaldson and was carried out in detail by Kovalev [43]. The starting point of this construction are two asymptotically cylindrical (ACyl) Calabi-Yau manifolds Wi with i = 1; 2 that approach 1 1 1 Di × S × (0; 1) at infinity, where the Di are K3 surfaces. W1 × S and W2 × S can be glued together after cutting off the cylindrical ends. The glueing map interchanges the circle factors of the ends and acts as an isometry with certain properties on the Di. The manifold that we obtain by this method is called a twisted connected sum and carries a metric with holonomy G2. By the twisted connected sum method several authors [17, 43, 44] constructed a large number of G2-manifolds. Most of the known examples of compact G2-manifolds are smooth. In this thesis, we study G2-orbifolds with a certain kind of singularities, namely ADE-singularities. This means that at the singular points the orbifolds are locally diffeomorphic to R3 × C2=∆, where ∆ is a finite subgroup of SU(2), which can be embedded in to G2. The finite subgroups of SU(2) have been classified by Klein [41]. For any finite subgroup ∆ of SU(2) there exists 3 an associated affine Dynkin diagram. This relation is called the McKay correspondence. By deleting one node we obtain a Dynkin diagram of a simple finite-dimensional Lie algebra. The diagrams that can be obtained this way are exactly the A-, D- and E-series from the classification of the simple finite-dimensional Lie algebras, which is the reason why one of the names for singularities of type C2=∆ is ADE-singularities. There are two motivations to study G2-orbifolds with ADE-singularities. One of them is mathematical and the other one is physical. An important object in the theory of G2-manifolds is the moduli space M of parallel G2- structures on a fixed manifold M. Joyce [36] has proven that this moduli space is a smooth manifold of dimension b3(M). Beside this fact very little is known about the geometry of M. It is conjectured that the boundary of M consists of singular G2-manifolds. More explicitly, Karigiannis [39] conjectures that the moduli space of parallel G2-structures with a conical singularity is a boundary component of the moduli space of parallel G2-structures on a desingularized manifold. Halverson and Morrison [32] conjecture that the boundary components of M correspond to G2-manifolds whose singular sets are of codimension 4, 6 and 7. This conjecture implies that for any parallel G2-structure φ on an orbifold with ADE-singularities there exists a one-parameter family of smooth G2-structures that converges in a suitable sense to φ. A proof of the conjecture from [32] and even of the weaker conjecture on the G2-structures converging to an ADE-singularity is probably very hard. For example, a resolution of an ADE-singularity in the above sense is a step in the proposed construction of G2-manifolds in the unpublished paper [40]. Although the conjectures on the moduli space of G2-structures are not yet proven, constructing examples of G2-orbifolds with ADE-singularities is an important step to better understand the shape of M. G2-manifolds do not only play a role in pure mathematics but also in theoretical physics. 1;3 7 7 In M-theory, spacetime is often modeled as R × M , where M is a G2-manifold that is too small to be observed at low energies. If M 7 is smooth, the quantum field theory on the 4-dimensional Minkowski space that we can observe does not fit to our observations. More precisely, we obtain a field theory with gauge group U(1)b2(M) in the low-energy limit, but the weak and the strong interaction are described by Yang-Mills theories with a non-abelian gauge group. Moreover, the existence of chiral fermions cannot be explained by this hypothesis. If M 7 is not smooth but has conical singularities, the existence of chiral fermions can be shown [3, 4, 9]. Moreover, M-theory compactified on a G2-orbifold with ADE-singularities yields, at least if some topological conditions on the singular locus are satisfied, a four-dimensional quantum field theory whose gauge group is determined by the Dynkin diagram that is associated to the singularity by the McKay correspondence. Up to now, there are only few examples of G2-orbifolds with ADE-singularities known. In [3], the author considers a class of non-compact G2-orbifolds that can be obtained by dividing a complete G2-manifold of cohomogeneity one by a subgroup of SU(2). Moreover, Joyce [36] constructs a torus quotient with A1-singularities along 12 three-tori which is the starting point for the construction of a smooth G2-manifold. Further examples of 4 torus quotients with An-singularities can be found in [10]. In this thesis, we construct several examples of compact G2-orbifolds of ADE-singularities. One of our priorities is to construct G2-orbifolds with as many as possible types of ADE-singularities. Our results can be summed up as follows. Let S be a K3 surface with a hyper-K¨ahlermetric and let T 3 be a flat torus. Since the 3 3 holonomy of S × T is Sp(1), which can be embedded into G2, S × T carries a parallel 3 3 G2-structure φ. If Γ is a group that acts freely on S × T and preserves φ,(S × T )=Γ carries a parallel G2-structure, too. If we allow S to have singularities or Γ to act non-freely, we obtain G2-orbifolds with ADE-singularities. We find examples of such quotients that 3 2 have E8-singularities along 2 submanifolds of type T =Z2 and A1-singularities along 3 3 2 submanifolds of type T =Z2. Let G1;:::;Gn be a set of connected Dynkin diagrams that can be obtained by deleting some nodes from the union of 2 diagrams of type E8 and 3 of 3 type A1. There exist further quotients of type (S × T )=Γ whose singular locus consists of 3 2 n submanifolds that are isometric to the flat manifold T =Z2 with singularities of type Gi along them. Another class of examples is constructed with help of twisted connected sums. Kovalev and Lee [44] find many examples of smooth G2-manifolds by applying the twisted connected sum construction to a pair of suitable ACyl Calabi-Yau manifolds W1 and W2. At least one of them is constructed by a procedure that is described in [44] from of a K3 surface S that admits a non-symplectic involution. We modify this construction such that S is a K3 surface with singularities. We obtain a series of G2-orbifolds that have singularities along one or several three-spheres. The number and type of the singularities is described by two copies of an arbitrary subdiagram of E8. Finally, we consider quotients of a torus with a flat G2-structure by a finite group with ADE-singularities.
Load more

Recommended publications
  • Resolution of ADE Singularities
    Resolution of Kleinian Singularities James J. Green April 25, 2014 abstract Firstly, the classification of finite subgroups of SL(2; C), a result of Felix Klein in 1884, is presented. The polynomial invariant subrings of these groups are then found. The generators of these subrings 3 satisfy a polynomial relation in three variables, which can be realised as a hypersurface in C . Each of these surfaces have a singularity at the origin; these are the Kleinian singularities. These singularities are blown-up, and their resolution graphs are shown to be precisely the Coxeter- Dynkin diagrams ADE. The target readership of this project is intended to be undergraduates with a foundational knowledge of group theory, topology and algebraic geometry. 1 1 Classifying the Finite Subgroups of SL(2; C) 1.1 Important Subgroups of the General Linear Group Recall: The general linear group of a vector space V over a field F is given by GL(V ) = ff : V −! V j f is linear and invertibleg: n In particular, we denote GL(F ) by GL(n; F). Since we can view linear maps as matrices, GL(n; F) can also be viewed as the set of invertible n × n matrices with entries in F. The next few definitions include important subgroups of GL(n; F). Definition 1.1. The special linear group over F is given by SL(n; F) = fA 2 GL(n; F) j det A = 1g: Definition 1.2. The orthogonal group over F is given by T O(n; F) = fA 2 GL(n; F) j AA = Ig where AT denotes the transpose of A, and I denotes the n × n identity matrix.
    [Show full text]
  • Swampland Constraints on 5D N = 1 Supergravity Arxiv:2004.14401V2
    Prepared for submission to JHEP Swampland Constraints on 5d N = 1 Supergravity Sheldon Katz, a Hee-Cheol Kim, b;c Houri-Christina Tarazi, d Cumrun Vafa d aDepartment of Mathematics, MC-382, University of Illinois at Urbana-Champaign, Ur- bana, IL 61801, USA bDepartment of Physics, POSTECH, Pohang 790-784, Korea cAsia Pacific Center for Theoretical Physics, Postech, Pohang 37673, Korea dDepartment of Physics, Harvard University, Cambridge, MA 02138, USA Abstract: We propose Swampland constraints on consistent 5-dimensional N = 1 supergravity theories. We focus on a special class of BPS magnetic monopole strings which arise in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the low-energy particle spectrum and the effective action of the 5d supergravity based on unitarity of the worldsheet CFT. In M-theory, where these theories are realized by compactification on Calabi-Yau 3-folds, the spe- cial monopole strings arise from wrapped M5-branes on special (\semi-ample") 4- cycles in the threefold. We identify various necessary geometric conditions for such cycles to lead to requisite BPS strings and translate these into constraints on the low-energy theories of gravity. These and other geometric conditions, some of which can be related to unitarity constraints on the monopole worldsheet, are additional arXiv:2004.14401v2 [hep-th] 13 Jul 2020 candidates for Swampland constraints on 5-dimensional N = 1 supergravity
    [Show full text]
  • Little String Defects and Nilpotent Orbits by Christian Schmid A
    Little String Defects and Nilpotent Orbits by Christian Schmid A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Mina Aganagic, Chair Professor Ori Ganor Professor Nicolai Reshetikhin Spring 2020 Little String Defects and Nilpotent Orbits Copyright 2020 by Christian Schmid 1 Abstract Little String Defects and Nilpotent Orbits by Christian Schmid Doctor of Philosophy in Physics University of California, Berkeley Professor Mina Aganagic, Chair In this thesis, we first derive and analyze the Gukov{Witten surface defects of four- dimensional N = 4 Super Yang-Mills (SYM) theory from little string theory. The little string theory arises from type IIB string theory compactified on an ADE singularity. Defects are introduced as D-branes wrapping the 2-cycles of the singularity. In a suitable limit, these become defects of the six-dimensional superconformal N = (2; 0) field theory, which reduces to SYM after further compactification. We then use this geometric setting to connect to the complete nilpotent orbit classification of codimension-two defects, and find relations to ADE-type Toda CFT. We highlight the differences between the defect classification in the little string theory and its (2; 0) CFT limit, and find physical insights into nilpotent orbits and their classification by Bala{Carter labels and weighted Dynkin diagrams. i Contents Contents i 1 Introduction1 1.1 Little string theory................................ 2 1.2 Six-Dimensional (2; 0) Superconformal Field Theory ............. 3 1.3 Gukov{Witten Surface Defects of N = 4 SYM................
    [Show full text]
  • Representations of Quivers & Gabriel's Theorem
    Level 5M Project: Representations of Quivers & Gabriel's Theorem Emma Cummin (0606574) March 24, 2011 Abstract Gabriel's theorem, first proved by Peter Gabriel in 1972 [1], comes in two parts. Part (i) states that \a quiver Q is of finite orbit type if and only if each component of its underlying undirected graph Q^ is a simply{laced Dynkin diagram" and part (ii) states \let Q be a quiver such that Q^ is a simply{laced Dynkin diagram; then n is the dimension of a (unique) indecomposable representation of Q if and only if n 2 Φ+" [2]. This surprising theorem gives a deceptively simply result which links isomorphism classes of representations of a given quiver with an assigned dimension vector, with the root system of the geometric object underlying such a quiver. This project introduces quivers and representation theory to the reader, with the intention of heading towards proving Gabriel's theorem. This is the main result in the project, and so a proof of both parts is provided. On the way, the reader is introduced to quivers, the category of quiver representations and its equivalence with the category of KQ{modules, Dynkin diagrams, root systems and the Weyl group and the Coxeter functors. 1 Contents 1 Introduction 3 2 Quiver Representations and Categories 4 2.1 Quivers and Representations . .4 2.1.1 Introduction to Quivers . .4 2.1.2 The Kroenecker Quiver . .5 2.2 Path Algebras . .7 2.3 Introduction to Category Theory . .8 2.4 Equivalence of Categories . 11 3 Geometric Interpretation of Isomorphism Classes of Quiver Representa- tions 15 3.1 The Representation Space .
    [Show full text]
  • Heterotic Little String Theories and Holography
    Received: November 10, 1999, Accepted: November 12, 1999 HYPER VERSION Heterotic little string theories and holography Martin Gremm∗ JHEP11(1999)018 Joseph Henry Laboratories, Princeton University Princeton, NJ 08544 E-mail: [email protected] Anton Kapustin School of Natural Sciences, Institute for Advanced Study Olden Lane, Princeton, NJ 08540 E-mail: [email protected] Abstract: It has been conjectured that little string theories in six dimensions are holographic to critical string theory in a linear dilaton background. We test this con- jecture for theories arising on the worldvolume of heterotic fivebranes. We compute the spectrum of chiral primaries in these theories and compare with results following from type-I-heterotic duality and the AdS/CFT correspondence. We also construct holographic duals for heterotic fivebranes near orbifold singularities. Finally we find several new little string theories which have Spin(32)/Z2 or E8 E8 global symmetry × but do not have a simple interpretation either in heterotic or M-theory. Keywords: Superstrings and Heterotic Strings, p-branes, Conformal Field Models in String Theory. ∗On leave of absence from MIT, Cambridge, MA 02139 Contents 1. Introduction 1 2. Holographic description of heterotic fivebranes in flat space 4 2.1 Supergravity solutions 4 2.2 Worldsheet partition function 6 2.3 Symmetries of the worldsheet CFT 7 2.4 The spectrum of space-time chiral primaries 8 JHEP11(1999)018 3. More general heterotic LSTs with SU(2)L SU(2)R G symmetry 11 × × 3.1 Worldsheet partition functions 11 3.2 Examples of chiral primaries 13 4. Fivebranes near orbifold singularities 14 5.
    [Show full text]
  • Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier
    Preprint typeset in JHEP style. - HYPER VERSION MIT-CTP-3026 hep-th/0010023 Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier Bo Feng, Amihay Hanany, Yang-Hui He and Nikolaos Prezas ∗ Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. fengb, hanany, yhe, [email protected] Abstract: Armed with the explicit computation of Schur Multipliers, we offer a classifica- tion of SU(n) orbifolds for n =2, 3, 4 which permit the turning on of discrete torsion. This is in response to the host of activity lately in vogue on the application of discrete torsion to D-brane orbifold theories. As a by-product, we find a hitherto unknown class of N = 1 orbifolds with non-cyclic discrete torsion group. Furthermore, we supplement the status quo ante by investigating a first example of a non-Abelian orbifold admitting discrete torsion, arXiv:hep-th/0010023v3 8 Dec 2000 namely the ordinary dihedral group as a subgroup of SU(3). A comparison of the quiver theory thereof with that of its covering group, the binary dihedral group, without discrete torsion, is also performed. Keywords: Discrete Torsion, non-Abelian Orbifolds, Schur Multiplier, Discrete Subgroups of SU(2), SU(3) and SU(4). ∗Research supported in part by the Reed Fund Award, the CTP and the LNS of MIT and the U.S. De- partment of Energy under cooperative research agreement # DE-FC02-94ER40818. A. H. is also supported by an A. P. Sloan Foundation Fellowship, and a DOE OJI award. Contents 1. Introduction 2 2. Some Mathematical Preliminaries 4 2.1 Projective Representations of Groups 4 2.2 Group Cohomology and the Schur Multiplier 5 2.3 The Covering Group 6 3.
    [Show full text]
  • Full Text (PDF Format)
    © 2000 International Press Adv. Theor. Math. Phys. 4 (2000) 747-789 Of McKay Correspondence, Non-linear Sigma-mo del and Conformal Field Theory Yang-Hui He and Jun S. Song Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, MA 02139 [email protected] [email protected] Abstract The ubiquitous A-D-E classification has induced many pro- posals of often mysterious correspondences both in mathemat- ics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group rep- resentation theory to Lie algebras as well as crepant resolutions e-print archive: http://xxx.lanl.gov/hep-th/9903056 Research supported in part by the NSF and the U.S. Department of Energy under cooperative research agreement #DE-FC02-94ER40818. 748 Y.-H. HE AND J.S. SONG of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlin- ear cr-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these cor- respondences of the A-D-E type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possi- ble guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions. 1 Introduction WZW Non-Linear Sigma Modular Invariants Model (B=0) (Fusion Rules) IV String Theory Quiver On Orbifolds and Tensor (B*0) Categories Graph Theory Finite Group Representation Theory Figure 1: The Myriad of Correspondences: it is the purpose of this pa- per to elucidate these inter-relations in 2-dimensions, so as to motivate a similar coherent picture in higher dimensions.
    [Show full text]
  • Fixed Points and D-Branes
    PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 169–180 DOI: 10.2298/PIM1308169B FIXED POINTS AND D-BRANES Elaine Beltaos Abstract. The affine Kac–Moody algebras give rise to rational conformal field theories (RCFTs) called the Wess–Zumino–Witten (WZW) models. An important component of an RCFT is its fusion ring, whose structure con- stants are given by the associated S-matrix. We apply a fixed point property possessed by the WZW models (“fixed point factorization") to calculate non- negative integer matrix representations of the fusion ring, allowing for the calculation of D-brane charges in string theory. 1. Introduction To each quantum field theory is associated an S-matrix. This matrix expresses amplitudes and thus is a fundamental component of the theory. In string theory, a particle is a finite curve of length approximately 10−33cm. Modern string theories contain both open (e.g., photon) and closed (e.g. graviton) strings. Topologically, a closed string is a circle S1 and an open string is the interval [0, 1]. As a string evolves through time, it traces out a surface called a worldsheet. For example, an incoming closed string travelling from t = −∞ to t = 0 traces out a semi-infinite cylinder. An important discovery, in 1989, was of higher dimensional objects (‘membranes’) called Dirichlet-branes, or D-branes for short (co-discoverers were Polchinski, Dai, Leigh, and Hořava). Physically, D-branes are the membranes where the endpoints of open strings reside. In 1995, Polchinski proved that a consistent theory requires branes. D-branes are physical entities having tension and charge.
    [Show full text]
  • Lattice Polytopes and Orbifolds
    Lattice Polytopes and Orbifolds Matthew DeCross October 4, 2015 1 Introduction Recent work [1{4] in the context of quiver gauge theories has focused on enumeration of Abelian orbifolds n n−1 3 of C whose toric diagrams are lattice simplices in R . For example, Abelian orbifolds of C have toric diagrams that are lattice triangles with corresponding symmetry group S3. In this work, we consider a larger set of orbifolds with geometrically interesting toric diagrams. First, we will examine a member of a class of toric Calabi-Yau varieties which are surfaces of positive curvature, the del Pezzo surfaces. Specifically, we enumerate the orbifolds of the affine cone over the del 2 Pezzo 3 surface (dP3), which is described by the projective two-space P blown up at 3 generic points [5,6]. Such orbifolds have hexagonal toric diagrams with corresponding symmetry group D6 [7]. We will show that regardless of the larger order of the symmetry group, the number of orbifolds of the dP3 toric diagram 3 up to GL(2; Z) equivalence is the same as the number of inequivalent Abelian orbifolds of C . Secondly, we will enumerate orbifolds whose toric diagrams correspond to the Platonic solids. In the case 4 of the tetrahedron, we are enumerating Abelian orbifolds of C , with symmetry group corresponding to S4. More exotic orbifolds may have toric diagrams given by the other Platonic solids: the cube, octahedron, dodecahedron, and icosahedron. Since the cube and octahedron diagrams are dual to each other, as are the dodecahedron and icosahedron, we need only additionally consider the cube and dodecahedron to enumerate orbifolds of all independent diagrams (the tetrahedron is self-dual).
    [Show full text]
  • Strings, Conformal Fields, and M-Theory Second Edition
    Michio Kaku Strings, Conformal Fields, and M-Theory Second Edition With 48 Illustrations Springer Contents Preface vii Acknowledgments ix I Conformal Field Theory and Perturbation Theory 1 1 Introduction to Superstrings 3 1.1 Quantizing the Relativistic String 9 1.2 Scattering Amplitudes 17 1.3 Supersymmetry . 22 1.4 2D SUSY Versus 10D SUSY 25 1.5 Types of Strings 30 1.6 Summary 32 2 BPZ Bootstrap and Minimal Models 38 2.1 Conformal Symmetry in D Dimensions 38 2.2 Conformal Group in Two Dimensions 41 2.3 Representations of the Conformal Group 45 2.4 Fusion Rules and Correlations Function 47 2.5 Minimal Models 51 2.6 Fusion Rules for Minimal Models 58 2.7 Superconformal Minimal Series 60 2.8 Summary 64 3 WZW Model, Cosets, and Rational Conformal Field Theory 69 3.1 Compactification and the WZW Model 69 3.2 Frenkel-Kac Construction 75 xii -Contents 3.3 GKO Coset Construction 79 3.4 Conformal and Current Blocks 81 3.5 Racah Coefficients for Rational Conformal Field Theory 84 3.6 Summary 91 4 Modular Invariance and the A-D-E Classification 96 4.1 Dehn Twists 96 4.2 Free Fermion and Boson Characters 99 4.3 GSO and Supersymmetry 105 4.4 Minimal Model Characters 106 4.5 Affine Characters 108 4.6 A-D-E Classification 113 4.7 Higher Invariants and Simple Currents 116 4.8 Diagonalizing the Fusion Rules 119 4.9 RCFT: Finite Number of Primary Fields 122 4.10 Summary 125 5 N = 2 SUSY and Parafermions 130 5.1 Calabi-Yau Manifolds 130 5.2 N = 2 Superconformal Symmetry 137 5.3 N - 2 Minimal Series 141 5.4 N = 2 Minimal Models and Calabi-Yau Manifolds ...
    [Show full text]
  • ADE Quivers and Lie Theory Davis Lazowski
    ADE quivers and Lie theory An honours thesis presented by Davis Lazowski Presented to the Department of Mathematics, in partial fulfillment of the requirements for the degree of Bachelor of Arts with joint concentration in Mathematics and Physics Advised by Professor Arthur Jaffe Department of Mathematics Harvard University Cambridge, MA March 2019 3 Contents 1 Introduction5 1.1 Motivation...................................5 1.1.1 ADE Classification..........................5 1.1.2 Inverting the ADE classification for Lie algebras..........7 1.2 Overview....................................7 1.2.1 Assumed mathematics........................8 1.2.2 Acknowledgments...........................8 2 Background on quiver representations and higher categories9 2.1 Definition of quiver representations......................9 2.2 The Auslander-Reiten quiver......................... 10 2.3 Dynkin quivers and an example....................... 11 2.4 Quiver representations are modules over the path algebra......... 12 2.5 Higher categories, cofibres and fibres.................... 14 3 The root system via the category of quiver representations 17 3.1 Classical BGP reflection functors and Gabriel’s theorem.......... 17 3.2 Moving to the derived category: homological prerequisites........ 20 3.3 Derived BGP reflection functors....................... 22 3.4 The Categorical Root System Inner Product................ 24 3.5 The Auslander-Reiten quiver and combinatorial Lie theory........ 25 3.6 Auslander-Reiten translation......................... 26 3.7 Proof of the combinatorial description of the Auslander-Reiten quiver, and the theory of height functions........................ 29 3.8 Combinatorial theory of the Auslander-Reiten quiver........... 31 3.8.1 Root system in terms of the Auslander-Reiten quiver....... 32 3.8.2 A quick introduction to height functions............... 33 3.9 A purely formal theory of BGP functors.................
    [Show full text]
  • Durham Research Online
    Durham Research Online Deposited in DRO: 21 February 2017 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Beem, Christopher and Lemos, Madalena and Rastelli, Leonardo and van Rees, Balt C. (2016) 'The (2, 0) superconformal bootstrap.', Physical review D., 93 (2). 025016. Further information on publisher's website: https://doi.org/10.1103/PhysRevD.93.025016 Publisher's copyright statement: This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk PHYSICAL REVIEW D 93, 025016 (2016) The (2, 0) superconformal bootstrap Christopher Beem,1 Madalena Lemos,2 Leonardo Rastelli,2 and Balt C. van Rees3 1Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA 2C.
    [Show full text]