ADE Quivers and Lie Theory Davis Lazowski
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1. Directed Graphs Or Quivers What Is Category Theory? • Graph Theory on Steroids • Comic Book Mathematics • Abstract Nonsense • the Secret Dictionary
1. Directed graphs or quivers What is category theory? • Graph theory on steroids • Comic book mathematics • Abstract nonsense • The secret dictionary Sets and classes: For S = fX j X2 = Xg, have S 2 S , S2 = S Directed graph or quiver: C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) Class C0 of objects, vertices, points, . Class C1 of morphisms, (directed) edges, arrows, . For x; y 2 C0, write C(x; y) := ff 2 C1 j @0f = x; @1f = yg f 2C1 tail, domain / @0f / @1f o head, codomain op Opposite or dual graph of C = (C0;C1;@0;@1) is C = (C0;C1;@1;@0) Graph homomorphism F : D ! C has object part F0 : D0 ! C0 and morphism part F1 : D1 ! C1 with @i ◦ F1(f) = F0 ◦ @i(f) for i = 0; 1. Graph isomorphism has bijective object and morphism parts. Poset (X; ≤): set X with reflexive, antisymmetric, transitive order ≤ Hasse diagram of poset (X; ≤): x ! y if y covers x, i.e., x 6= y and [x; y] = fx; yg, so x ≤ z ≤ y ) z = x or z = y. Hasse diagram of (N; ≤) is 0 / 1 / 2 / 3 / ::: Hasse diagram of (f1; 2; 3; 6g; j ) is 3 / 6 O O 1 / 2 1 2 2. Categories Category: Quiver C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) with: • composition: 8 x; y; z 2 C0 ; C(x; y) × C(y; z) ! C(x; z); (f; g) 7! g ◦ f • satisfying associativity: 8 x; y; z; t 2 C0 ; 8 (f; g; h) 2 C(x; y) × C(y; z) × C(z; t) ; h ◦ (g ◦ f) = (h ◦ g) ◦ f y iS qq <SSSS g qq << SSS f qqq h◦g < SSSS qq << SSS qq g◦f < SSS xqq << SS z Vo VV < x VVVV << VVVV < VVVV << h VVVV < h◦(g◦f)=(h◦g)◦f VVVV < VVV+ t • identities: 8 x; y; z 2 C0 ; 9 1y 2 C(y; y) : 8 f 2 C(x; y) ; 1y ◦ f = f and 8 g 2 C(y; z) ; g ◦ 1y = g f y o x MM MM 1y g MM MMM f MMM M& zo g y Example: N0 = fxg ; N1 = N ; 1x = 0 ; 8 m; n 2 N ; n◦m = m+n ; | one object, lots of arrows [monoid of natural numbers under addition] 4 x / x Equation: 3 + 5 = 4 + 4 Commuting diagram: 3 4 x / x 5 ( 1 if m ≤ n; Example: N1 = N ; 8 m; n 2 N ; jN(m; n)j = 0 otherwise | lots of objects, lots of arrows [poset (N; ≤) as a category] These two examples are small categories: have a set of morphisms. -
An Exercise on Source and Sink Mutations of Acyclic Quivers∗
An exercise on source and sink mutations of acyclic quivers∗ Darij Grinberg November 27, 2018 In this note, we will use the following notations (which come from Lampe’s notes [Lampe, §2.1.1]): • A quiver means a tuple Q = (Q0, Q1, s, t), where Q0 and Q1 are two finite sets and where s and t are two maps from Q1 to Q0. We call the elements of Q0 the vertices of the quiver Q, and we call the elements of Q1 the arrows of the quiver Q. For every e 2 Q1, we call s (e) the starting point of e (and we say that e starts at s (e)), and we call t (e) the terminal point of e (and we say that e ends at t (e)). Furthermore, if e 2 Q1, then we say that e is an arrow from s (e) to t (e). So the notion of a quiver is one of many different versions of the notion of a finite directed graph. (Notice that it is a version which allows multiple arrows, and which distinguishes between them – i.e., the quiver stores not just the information of how many arrows there are from a vertex to another, but it actually has them all as distinguishable objects in Q1. Lampe himself seems to later tacitly switch to a different notion of quivers, where edges from a given to vertex to another are indistinguishable and only exist as a number. This does not matter for the next exercise, which works just as well with either notion of a quiver; but I just wanted to have it mentioned.) • The underlying undirected graph of a quiver Q = (Q0, Q1, s, t) is defined as the undirected multigraph with vertex set Q0 and edge multiset ffs (e) , t (e)g j e 2 Q1gmultiset . -
Higher-Dimensional Instantons on Cones Over Sasaki-Einstein Spaces and Coulomb Branches for 3-Dimensional N = 4 Gauge Theories
Two aspects of gauge theories higher-dimensional instantons on cones over Sasaki-Einstein spaces and Coulomb branches for 3-dimensional N = 4 gauge theories Von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation von Dipl.-Phys. Marcus Sperling, geboren am 11. 10. 1987 in Wismar 2016 Eingereicht am 23.05.2016 Referent: Prof. Olaf Lechtenfeld Korreferent: Prof. Roger Bielawski Korreferent: Prof. Amihay Hanany Tag der Promotion: 19.07.2016 ii Abstract Solitons and instantons are crucial in modern field theory, which includes high energy physics and string theory, but also condensed matter physics and optics. This thesis is concerned with two appearances of solitonic objects: higher-dimensional instantons arising as supersymme- try condition in heterotic (flux-)compactifications, and monopole operators that describe the Coulomb branch of gauge theories in 2+1 dimensions with 8 supercharges. In PartI we analyse the generalised instanton equations on conical extensions of Sasaki- Einstein manifolds. Due to a certain equivariant ansatz, the instanton equations are reduced to a set of coupled, non-linear, ordinary first order differential equations for matrix-valued functions. For the metric Calabi-Yau cone, the instanton equations are the Hermitian Yang-Mills equations and we exploit their geometric structure to gain insights in the structure of the matrix equations. The presented analysis relies strongly on methods used in the context of Nahm equations. For non-Kähler conical extensions, focusing on the string theoretically interesting 6-dimensional case, we first of all construct the relevant SU(3)-structures on the conical extensions and subsequently derive the corresponding matrix equations. -
UTM Naive Lie Theory (John Stillwell) 0387782141.Pdf
Undergraduate Texts in Mathematics Editors S. Axler K.A. Ribet Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Daepp/Gorkin: Reading, Writing, and Proving: Anglin: Mathematics: A Concise History and A Closer Look at Mathematics. Philosophy. Devlin: The Joy of Sets: Fundamentals Readings in Mathematics. of-Contemporary Set Theory. Second edition. Anglin/Lambek: The Heritage of Thales. Dixmier: General Topology. Readings in Mathematics. Driver: Why Math? Apostol: Introduction to Analytic Number Theory. Ebbinghaus/Flum/Thomas: Mathematical Logic. Second edition. Second edition. Armstrong: Basic Topology. Edgar: Measure, Topology, and Fractal Geometry. Armstrong: Groups and Symmetry. Second edition. Axler: Linear Algebra Done Right. Second edition. Elaydi: An Introduction to Difference Equations. Beardon: Limits: A New Approach to Real Third edition. Analysis. Erdos/Sur˜ anyi:´ Topics in the Theory of Numbers. Bak/Newman: Complex Analysis. Second edition. Estep: Practical Analysis on One Variable. Banchoff/Wermer: Linear Algebra Through Exner: An Accompaniment to Higher Mathematics. Geometry. Second edition. Exner: Inside Calculus. Beck/Robins: Computing the Continuous Fine/Rosenberger: The Fundamental Theory Discretely of Algebra. Berberian: A First Course in Real Analysis. Fischer: Intermediate Real Analysis. Bix: Conics and Cubics: A Concrete Introduction to Flanigan/Kazdan: Calculus Two: Linear and Algebraic Curves. Second edition. Nonlinear Functions. Second edition. Bremaud:` An Introduction to Probabilistic Fleming: Functions of Several Variables. Second Modeling. edition. Bressoud: Factorization and Primality Testing. Foulds: Combinatorial Optimization for Bressoud: Second Year Calculus. Undergraduates. Readings in Mathematics. Foulds: Optimization Techniques: An Introduction. Brickman: Mathematical Introduction to Linear Franklin: Methods of Mathematical Programming and Game Theory. Economics. Browder: Mathematical Analysis: An Introduction. Frazier: An Introduction to Wavelets Through Buchmann: Introduction to Cryptography. -
Quiver Indices and Abelianization from Jeffrey-Kirwan Residues Arxiv
Prepared for submission to JHEP arXiv:1907.01354v2 Quiver indices and Abelianization from Jeffrey-Kirwan residues Guillaume Beaujard,1 Swapnamay Mondal,2 Boris Pioline1 1 Laboratoire de Physique Th´eoriqueet Hautes Energies (LPTHE), UMR 7589 CNRS-Sorbonne Universit´e,Campus Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris, France 2 International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Shivakote, Hesaraghatta, Bangalore 560089, India e-mail: fpioline,[email protected], [email protected] Abstract: In quiver quantum mechanics with 4 supercharges, supersymmetric ground states are known to be in one-to-one correspondence with Dolbeault cohomology classes on the moduli space of stable quiver representations. Using supersymmetric localization, the refined Witten index can be expressed as a residue integral with a specific contour prescription, originally due to Jeffrey and Kirwan, depending on the stability parameters. On the other hand, the physical picture of quiver quantum mechanics describing interactions of BPS black holes predicts that the refined Witten index of a non-Abelian quiver can be expressed as a sum of indices for Abelian quivers, weighted by `single-centered invariants'. In the case of quivers without oriented loops, we show that this decomposition naturally arises from the residue formula, as a consequence of applying the Cauchy-Bose identity to the vector multiplet contributions. For quivers with loops, the same procedure produces a natural decomposition of the single-centered invariants, which remains to be elucidated. In the process, we clarify some under-appreciated aspects of the localization formula. Part of the results reported herein have been obtained by implementing the Jeffrey-Kirwan residue formula in a public arXiv:1907.01354v2 [hep-th] 15 Oct 2019 Mathematica code. -
132 Lie's Theory of Continuous Groups Plays Such an Important Part in The
132 SHORTER NOTICES. [Dec, Lie's theory of continuous groups plays such an important part in the elementary problems of the theory of differential equations, and has proved to be such a powerful weapon in the hands of competent mathematicians, that a work on differential equations, of even the most modest scope, appears decidedly incomplete without some account of it. It is to be regretted that Mr. Forsyth has not introduced this theory into his treatise. The introduction of a brief account of Runge's method for numerical integration is a very valuable addition to the third edition. The treatment of the Riccati equation has been modi fied. The theorem that the cross ratio of any four solutions is constant is demonstrated but not explicitly enunciated, which is much to be regretted. From the point of view of the geo metric applications, this is the most important property of the equations of the Riccati type. The theory of total differential equations has been discussed more fully than before, and the treatment on the whole is lucid. The same may be said of the modified treatment adopted by the author for the linear partial differential equations of the first order. A very valuable feature of the book is the list of examples. E. J. WlLCZYNSKI. Introduction to Projective Geometry and Its Applications. By ARNOLD EMOH. New York, John Wiley & Sons, 1905. vii + 267 pp. To some persons the term projective geometry has come to stand only for that pure science of non-metric relations in space which was founded by von Staudt. -
Arxiv:2012.08669V1 [Math.CT] 15 Dec 2020 2 Preface
Sheaf Theory Through Examples (Abridged Version) Daniel Rosiak December 12, 2020 arXiv:2012.08669v1 [math.CT] 15 Dec 2020 2 Preface After circulating an earlier version of this work among colleagues back in 2018, with the initial aim of providing a gentle and example-heavy introduction to sheaves aimed at a less specialized audience than is typical, I was encouraged by the feedback of readers, many of whom found the manuscript (or portions thereof) helpful; this encouragement led me to continue to make various additions and modifications over the years. The project is now under contract with the MIT Press, which would publish it as an open access book in 2021 or early 2022. In the meantime, a number of readers have encouraged me to make available at least a portion of the book through arXiv. The present version represents a little more than two-thirds of what the professionally edited and published book would contain: the fifth chapter and a concluding chapter are missing from this version. The fifth chapter is dedicated to toposes, a number of more involved applications of sheaves (including to the \n- queens problem" in chess, Schreier graphs for self-similar groups, cellular automata, and more), and discussion of constructions and examples from cohesive toposes. Feedback or comments on the present work can be directed to the author's personal email, and would of course be appreciated. 3 4 Contents Introduction 7 0.1 An Invitation . .7 0.2 A First Pass at the Idea of a Sheaf . 11 0.3 Outline of Contents . 20 1 Categorical Fundamentals for Sheaves 23 1.1 Categorical Preliminaries . -
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. -
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 -
Lectures on Representations of Quivers by William Crawley-Boevey
Lectures on Representations of Quivers by William Crawley-Boevey Contents §1. Path algebras. 3 §2. Bricks . 9 §3. The variety of representations . .11 §4. Dynkin and Euclidean diagrams. .15 §5. Finite representation type . .19 §6. More homological algebra . .21 §7. Euclidean case. Preprojectives and preinjectives . .25 §8. Euclidean case. Regular modules. .28 §9. Euclidean case. Regular simples and roots. .32 §10. Further topics . .36 1 A 'quiver' is a directed graph, and a representation is defined by a vector space for each vertex and a linear map for each arrow. The theory of representations of quivers touches linear algebra, invariant theory, finite dimensional algebras, free ideal rings, Kac-Moody Lie algebras, and many other fields. These are the notes for a course of eight lectures given in Oxford in spring 1992. My aim was the classification of the representations for the ~ ~ ~ ~ ~ Euclidean diagrams A , D , E , E , E . It seemed ambitious for eight n n 6 7 8 lectures, but turned out to be easier than I expected. The Dynkin case is analysed using an argument of J.Tits, P.Gabriel and C.M.Ringel, which involves actions of algebraic groups, a study of root systems, and some clever homological algebra. The Euclidean case is treated using the same tools, and in addition the Auslander-Reiten translations - , , and the notion of a 'regular uniserial module'. I have avoided the use of reflection functors, Auslander-Reiten sequences, and case-by-case analyses. The prerequisites for this course are quite modest, consisting of the basic 1 notions about rings and modules; a little homological algebra, up to Ext ¡ n and long exact sequences; the Zariski topology on ; and maybe some ideas from category theory. -
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................ -
Introduction to Representation Theory by Pavel Etingof, Oleg Golberg
Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina with historical interludes by Slava Gerovitch Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Contents Chapter 1. Introduction 1 Chapter 2. Basic notions of representation theory 5 x2.1. What is representation theory? 5 x2.2. Algebras 8 x2.3. Representations 9 x2.4. Ideals 15 x2.5. Quotients 15 x2.6. Algebras defined by generators and relations 16 x2.7. Examples of algebras 17 x2.8. Quivers 19 x2.9. Lie algebras 22 x2.10. Historical interlude: Sophus Lie's trials and transformations 26 x2.11. Tensor products 30 x2.12. The tensor algebra 35 x2.13. Hilbert's third problem 36 x2.14. Tensor products and duals of representations of Lie algebras 36 x2.15. Representations of sl(2) 37 iii Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms iv Contents x2.16. Problems on Lie algebras 39 Chapter 3. General results of representation theory 41 x3.1. Subrepresentations in semisimple representations 41 x3.2. The density theorem 43 x3.3. Representations of direct sums of matrix algebras 44 x3.4. Filtrations 45 x3.5. Finite dimensional algebras 46 x3.6. Characters of representations 48 x3.7. The Jordan-H¨oldertheorem 50 x3.8. The Krull-Schmidt theorem 51 x3.9.