Structure and Representation Theory of Infinite-Dimensional Lie Algebras
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S-Duality, the 4D Superconformal Index and 2D Topological QFT
Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N S-duality, the 4d Superconformal Index and 2d Topological QFT Leonardo Rastelli with A. Gadde, E. Pomoni, S. Razamat and W. Yan arXiv:0910.2225,arXiv:1003.4244, and in progress Yang Institute for Theoretical Physics, Stony Brook Rutgers, November 9 2010 Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory. Moore-Seiberg groupoid of Σ = (generalized) 4d S-duality • Vast generalization of “ =4 S-duality as modular group of T 2”. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory. -
3D Gauge Theories, Symplectic Duality and Knot Homology I
3d Gauge Theories, Symplectic Duality and Knot Homology I Tudor Dimofte Notes by Qiaochu Yuan December 8, 2014 We want to study some 3d N = 4 supersymmetric QFTs. They won't be topological, but the supersymmetry will still help. We'll look in particular at boundary conditions and compactifications to 2d. Most of this material is relatively old by physics standards (known since the '90s). A motivation for discussing these theories here is that they seem closely related to 1. geometric representation theory, 2. geometric constructions of knot homologies, and 3. symplectic duality. This story gives rise to some interesting mathematical predictions. On Tuesday we'll talk about what symplectic n-ality should mean. On Wednesday we'll give a new construction of projectives in variants of category O. On Thursday we'll talk about category O and knot homologies, and a new approach to both via 2d Landau-Ginzburg models. 1 Geometric representation theory For starters, let G = SL2. The universal enveloping algebra of its Lie algebra is U(sl2) = hE; F; H j [E; F ] = H; [H; E] = 2E; [H; F ] = −2F i: (1) The center is generated by the Casimir operator 1 χ = EF + FE + H2: (2) 2 The full category of U(sl2)-modules is hard to work with. Bernstein, Gelfand, and Gelfand proposed that we work in a nice subcategory O of highest weight representations (on which E acts locally finitely or locally nilpotently, depending on the definition) decomposing as a sum M M = Mµ (3) µ of generalized eigenspaces with respect to the action of the generator of the Cartan L subalgebra H. -
Parabolic Category O for Classical Lie Superalgebras
Parabolic category O for classical Lie superalgebras Volodymyr Mazorchuk Abstract We compare properties of (the parabolic version of) the BGG category O for semi-simple Lie algebras with those for classical (not necessarily simple) Lie superalgebras. 1 Introduction Category O for semi-simple complex finite dimensional Lie algebras, introduced in [BGG], is a central object of study in the modern representation theory (see [Hu]) with many interesting connections to, in particular, combinatorics, alge- braic geometry and topology. This category has a natural counterpart in the super- world and this super version O˜ of category O was intensively studied (mostly for some particular simple classical Lie superalgebra) in the last decade, see e.g. [Br1, Br2, Go2, FM2, CLW, CMW] or the recent books [Mu, CW] for details. The aim of the present paper is to compare some basic but general properties of the category O in the non-super and super cases for a rather general classical super-setup. We mainly restrict to the properties for which the non-super and super cases can be connected using the usual restriction and induction functors and the biadjunction (up to parity change) between these two functors is our main tool. The paper also complements, extends and gives a more detailed exposition for some results which appeared in [AM, Section 7]. The original category O has a natural parabolic version which first appeared in [RC]. We start in Section 2 by setting up an elementary approach (using root system geometry) to the definition of a parabolic category O for classical finite dimensional Lie superalgebras. -
Geometric Representation Theory, Fall 2005
GEOMETRIC REPRESENTATION THEORY, FALL 2005 Some references 1) J. -P. Serre, Complex semi-simple Lie algebras. 2) T. Springer, Linear algebraic groups. 3) A course on D-modules by J. Bernstein, availiable at www.math.uchicago.edu/∼arinkin/langlands. 4) J. Dixmier, Enveloping algebras. 1. Basics of the category O 1.1. Refresher on semi-simple Lie algebras. In this course we will work with an alge- braically closed ground field of characteristic 0, which may as well be assumed equal to C Let g be a semi-simple Lie algebra. We will fix a Borel subalgebra b ⊂ g (also sometimes denoted b+) and an opposite Borel subalgebra b−. The intersection b+ ∩ b− is a Cartan subalgebra, denoted h. We will denote by n, and n− the unipotent radicals of b and b−, respectively. We have n = [b, b], and h ' b/n. (I.e., h is better to think of as a quotient of b, rather than a subalgebra.) The eigenvalues of h acting on n are by definition the positive roots of g; this set will be denoted by ∆+. We will denote by Q+ the sub-semigroup of h∗ equal to the positive span of ∆+ (i.e., Q+ is the set of eigenvalues of h under the adjoint action on U(n)). For λ, µ ∈ h∗ we shall say that λ ≥ µ if λ − µ ∈ Q+. We denote by P + the sub-semigroup of dominant integral weights, i.e., those λ that satisfy hλ, αˇi ∈ Z+ for all α ∈ ∆+. + For α ∈ ∆ , we will denote by nα the corresponding eigen-space. -
Arxiv:1907.06685V1 [Math.RT]
CATEGORY O FOR TAKIFF sl2 VOLODYMYR MAZORCHUK AND CHRISTOFFER SODERBERG¨ Abstract. We investigate various ways to define an analogue of BGG category O for the non-semi-simple Takiff extension of the Lie algebra sl2. We describe Gabriel quivers for blocks of these analogues of category O and prove extension fullness of one of them in the category of all modules. 1. Introduction and description of the results The celebrated BGG category O, introduced in [BGG], is originally associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra. The definition of O is naturally generalized to all Lie algebras admitting some analogue of a triangular decomposition, see [MP]. These include, in particular, Kac-Moody algebras and Virasoro algebra. Category O has a number of spectacular properties and applications to various areas of mathematics, see for example [Hu] and references therein. The paper [DLMZ] took some first steps in trying to understand structure and prop- erties of an analogue of category O in the case of a non-reductive finite dimensional Lie algebra. The investigation in [DLMZ] focuses on category O for the so-called Schr¨odinger algebra, which is a central extension of the semi-direct product of sl2 with its natural 2-dimensional module. It turned out that, for the Schr¨odinger algebra, the behavior of blocks of category O with non-zero central charge is exactly the same as the behavior of blocks of category O for the algebra sl2. At the same, block with zero central charge turned out to be significantly more difficult. -
N = 2 Dualities and 2D TQFT Space of Complex Structures Σ = Parameter Space of the 4D Theory
Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications = 2 Dualities and 2d TQFT N Abhijit Gadde with L. Rastelli, S. Razamat and W. Yan arXiv:0910.2225 arXiv:1003.4244 arXiv:1104.3850 arXiv:1110:3740 ... California Institute of Technology Indian Israeli String Meeting 2012 Abhijit Gadde N = 2 Dualities and 2d TQFT Space of complex structures Σ = parameter space of the 4d theory. Mapping class group of Σ = (generalized) 4d S-duality Punctures: SU(2) ! SU(N) = Flavor symmetry: Commutant We will focus on "maximal" puncture: with SU(N) flavor symmetry Sphere with 3 punctures = Theories without parameters Free hypermultiplets Strongly coupled fixed points Vast generalization of “N = 4 S-duality as modular group of T 2”. 6=4+2: beautiful and unexpected 4d/2d connections. For ex., Correlators of Liouville/Toda on Σ compute the 4d partition functions (on S4) Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications A new paradigm for 4d = 2 SCFTs [Gaiotto, . ] N Compactify of the 6d (2; 0) AN−1 theory on a 2d surface Σ, with punctures. =)N = 2 superconformal theories in four dimensions. Abhijit Gadde N = 2 Dualities and 2d TQFT We will focus on "maximal" puncture: with SU(N) flavor symmetry Sphere with 3 punctures = Theories without parameters Free hypermultiplets Strongly coupled fixed points Vast generalization of “N = 4 S-duality as modular group of T 2”. 6=4+2: beautiful and unexpected 4d/2d connections. For ex., Correlators of Liouville/Toda on Σ compute the 4d partition functions (on S4) Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications A new paradigm for 4d = 2 SCFTs [Gaiotto, . -
Quantizations of Regular Functions on Nilpotent Orbits 3
QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS IVAN LOSEV Abstract. We study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa. 1. Introduction 1.1. Nilpotent orbits and their quantizations. Let G be a connected semisimple algebraic group over C and let g be its Lie algebra. Pick a nilpotent orbit O ⊂ g. This orbit is a symplectic algebraic variety with respect to the Kirillov-Kostant form. So the algebra C[O] of regular functions on O acquires a Poisson bracket. This algebra is also naturally graded and the Poisson bracket has degree −1. So one can ask about quantizations of O, i.e., filtered algebras A equipped with an isomorphism gr A −→∼ C[O] of graded Poisson algebras. We are actually interested in quantizations that have some additional structures mir- roring those of O. Namely, the group G acts on O and the inclusion O ֒→ g is a moment map for this action. We want the G-action on C[O] to lift to a filtration preserving action on A. -
Classification of Holomorphic Framed Vertex Operator Algebras of Central Charge 24
Classification of holomorphic framed vertex operator algebras of central charge 24 Ching Hung Lam, Hiroki Shimakura American Journal of Mathematics, Volume 137, Number 1, February 2015, pp. 111-137 (Article) Published by Johns Hopkins University Press DOI: https://doi.org/10.1353/ajm.2015.0001 For additional information about this article https://muse.jhu.edu/article/570114/summary Access provided by Tsinghua University Library (22 Nov 2018 10:39 GMT) CLASSIFICATION OF HOLOMORPHIC FRAMED VERTEX OPERATOR ALGEBRAS OF CENTRAL CHARGE 24 By CHING HUNG LAM and HIROKI SHIMAKURA Abstract. This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism. 1. Introduction. The classification of holomorphic vertex operator alge- bras (VOAs) of central charge 24 is one of the fundamental problems in vertex operator algebras and mathematical physics. In 1993 Schellekens [Sc93] obtained a partial classification by determining possible Lie algebra structures for the weight one subspaces of holomorphic VOAs of central charge 24. There are 71 cases in his list but only 39 of the 71 cases were known explicitly at that time. It is also an open question if the Lie algebra structure of the weight one subspace will determine the VOA structure uniquely when the central charge is 24. -
Supersymmetric Nonlinear Sigma Models
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server OU-HET 348 TIT/HET-448 hep-th/0006025 June 2000 Supersymmetric Nonlinear Sigma Models a b Kiyoshi Higashijima ∗ and Muneto Nitta † aDepartment of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan bDepartment of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan Abstract Supersymmetric nonlinear sigma models are formulated as gauge theo- ries. Auxiliary chiral superfields are introduced to impose supersymmetric constraints of F-type. Target manifolds defined by F-type constraints are al- ways non-compact. In order to obtain nonlinear sigma models on compact manifolds, we have to introduce gauge symmetry to eliminate the degrees of freedom in non-compact directions. All supersymmetric nonlinear sigma models defined on the hermitian symmetric spaces are successfully formulated as gauge theories. 1Talk given at the International Symposium on \Quantum Chromodynamics and Color Con- finement" (Confinement 2000) , 7-10 March 2000, RCNP, Osaka, Japan. ∗e-mail: [email protected]. †e-mail: [email protected] 1 Introduction Two dimensional (2D) nonlinear sigma models and four dimensional non-abelian gauge theories have several similarities. Both of them enjoy the property of the asymptotic freedom. They are both massless in the perturbation theory, whereas they acquire the mass gap or the string tension in the non-perturbative treatment. Although it is difficult to solve QCD in analytical way, 2D nonlinear sigma models can be solved by the large N expansion and helps us to understand various non- perturbative phenomena in four dimensional gauge theories. -
Scientific Report for the Year 2000
The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Scientific Report for the Year 2000 Vienna, ESI-Report 2000 March 1, 2001 Supported by Federal Ministry of Education, Science, and Culture, Austria ESI–Report 2000 ERWIN SCHRODINGER¨ INTERNATIONAL INSTITUTE OF MATHEMATICAL PHYSICS, SCIENTIFIC REPORT FOR THE YEAR 2000 ESI, Boltzmanngasse 9, A-1090 Wien, Austria March 1, 2001 Honorary President: Walter Thirring, Tel. +43-1-4277-51516. President: Jakob Yngvason: +43-1-4277-51506. [email protected] Director: Peter W. Michor: +43-1-3172047-16. [email protected] Director: Klaus Schmidt: +43-1-3172047-14. [email protected] Administration: Ulrike Fischer, Eva Kissler, Ursula Sagmeister: +43-1-3172047-12, [email protected] Computer group: Andreas Cap, Gerald Teschl, Hermann Schichl. International Scientific Advisory board: Jean-Pierre Bourguignon (IHES), Giovanni Gallavotti (Roma), Krzysztof Gawedzki (IHES), Vaughan F.R. Jones (Berkeley), Viktor Kac (MIT), Elliott Lieb (Princeton), Harald Grosse (Vienna), Harald Niederreiter (Vienna), ESI preprints are available via ‘anonymous ftp’ or ‘gopher’: FTP.ESI.AC.AT and via the URL: http://www.esi.ac.at. Table of contents General remarks . 2 Winter School in Geometry and Physics . 2 Wolfgang Pauli und die Physik des 20. Jahrhunderts . 3 Summer Session Seminar Sophus Lie . 3 PROGRAMS IN 2000 . 4 Duality, String Theory, and M-theory . 4 Confinement . 5 Representation theory . 7 Algebraic Groups, Invariant Theory, and Applications . 7 Quantum Measurement and Information . 9 CONTINUATION OF PROGRAMS FROM 1999 and earlier . 10 List of Preprints in 2000 . 13 List of seminars and colloquia outside of conferences . -
Geometries, the Principle of Duality, and Algebraic Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Expo. Math. 24 (2006) 195–234 www.elsevier.de/exmath Geometries, the principle of duality, and algebraic groups Skip Garibaldi∗, Michael Carr Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322, USA Received 3 June 2005; received in revised form 8 September 2005 Abstract J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits’s geometry and works through several examples. We also give a criterion for recognizing the auto- morphism of the geometry induced by an automorphism of the group. The E6 geometry is studied in depth. ᭧ 2005 Elsevier GmbH. All rights reserved. MSC 2000: Primary 22E47; secondary 20E42, 20G15 Contents 1. Tits’s geometry P ........................................................................197 2. A concrete geometry V , part I..............................................................198 3. A concrete geometry V , part II .............................................................199 4. Example: type A (projective geometry) .......................................................203 5. Strategy .................................................................................203 6. Example: type D (orthogonal geometry) ......................................................205 7. Example: type E6 .........................................................................208 -
On Dynkin Diagrams, Cartan Matrices
Physics 220, Lecture 16 ? Reference: Georgi chapters 8-9, a bit of 20. • Continue with Dynkin diagrams and the Cartan matrix, αi · αj Aji ≡ 2 2 : αi The j-th row give the qi − pi = −pi values of the simple root αi's SU(2)i generators acting on the root αj. Again, we always have αi · µ 2 2 = qi − pi; (1) αi where pi and qi are the number of times that the weight µ can be raised by Eαi , or lowered by E−αi , respectively, before getting zero. Applied to µ = αj, we know that qi = 0, since E−αj jαii = 0, since αi − αj is not a root for i 6= j. 0 2 Again, we then have AjiAij = pp = 4 cos θij, which must equal 0,1,2, or 3; these correspond to θij = π=2, 2π=3, 3π=4, and 5π=6, respectively. The Dynkin diagram has a node for each simple root (so the number of nodes is 2 2 r =rank(G)), and nodes i and j are connected by AjiAij lines. When αi 6= αj , sometimes it's useful to darken the node for the smaller root. 2 2 • Another example: constructing the roots for C3, starting from α1 = α2 = 1, and 2 α3 = 2, i.e. the Cartan matrix 0 2 −1 0 1 @ −1 2 −1 A : 0 −2 2 Find 9 positive roots. • Classify all simple, compact Lie algebras from their Aji. Require 3 properties: (1) det A 6= 0 (since the simple roots are linearly independent); (2) Aji < 0 for i 6= j; (3) AijAji = 0,1, 2, 3.