DOCSLIB.ORG

N = 2 Dualities and 2D TQFT Space of Complex Structures Σ = Parameter Space of the 4D Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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, . ] N Compactify of the 6d (2; 0) AN−1 theory on a 2d surface Σ, with punctures. =)N = 2 superconformal theories in four dimensions. 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 Abhijit Gadde N = 2 Dualities and 2d TQFT 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. 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 Abhijit Gadde N = 2 Dualities and 2d TQFT 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. 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”. Abhijit Gadde N = 2 Dualities and 2d TQFT 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. 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) Abhijit Gadde N = 2 Dualities and 2d TQFT Superconformal Index = twisted partition function on S3 × S1 = Witten index in radial quantization Independent of the gauge theory coupling and invariant under S-duality. Independent of coupling = Independent of complex structure on Σ =) 2d Topological QFT It encodes the protected spectrum of the 4d theory. Useful tool to understand physics. Our aim: To compute the superconformal index of these N = 2 theories (even the strongly coupled ones) by exploiting TQFT structure. Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications In this talk we will discuss the implications of another interesting connection: A protected 4d quantity, the superconformal index, is computed by topological QFT on Σ. Abhijit Gadde N = 2 Dualities and 2d TQFT Our aim: To compute the superconformal index of these N = 2 theories (even the strongly coupled ones) by exploiting TQFT structure. Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications In this talk we will discuss the implications of another interesting connection: A protected 4d quantity, the superconformal index, is computed by topological QFT on Σ. Superconformal Index = twisted partition function on S3 × S1 = Witten index in radial quantization Independent of the gauge theory coupling and invariant under S-duality. Independent of coupling = Independent of complex structure on Σ =) 2d Topological QFT It encodes the protected spectrum of the 4d theory. Useful tool to understand physics. Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications In this talk we will discuss the implications of another interesting connection: A protected 4d quantity, the superconformal index, is computed by topological QFT on Σ. Superconformal Index = twisted partition function on S3 × S1 = Witten index in radial quantization Independent of the gauge theory coupling and invariant under S-duality. Independent of coupling = Independent of complex structure on Σ =) 2d Topological QFT It encodes the protected spectrum of the 4d theory. Useful tool to understand physics. Our aim: To compute the superconformal index of these N = 2 theories (even the strongly coupled ones) by exploiting TQFT structure. Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications Outline 1 Short review of superconformal index 2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials 3 Results and Applications Large N limit Instanton partition function Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications Outline 1 Short review of superconformal index 2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials 3 Results and Applications Large N limit Instanton partition function Abhijit Gadde N = 2 Dualities and 2d TQFT 3 y Consider a 4d SCFT. On S × R (radial quantization), Q = S. ¯ The superconformal algebra implies (taking Q = Q1−_ ) 2 fS; Qg = ∆ − 2 j2 − 2 R + r ≡ H ≥ 0 : where ∆ is the conformal dimension, (j1; j2) the SU(2)1 ⊗ SU(2)2 Lorentz spins, and (R ; r) the quantum numbers under the SU(2)R ⊗ U(1)r R-symmetry. I (p; q ; t ; : : : ) = Tr(−1)F pj1+j2−rq−j1+j2−rtR+r :::: F j1+j2−r −j1+j2−r R+r F1 Fi I (p; q; t; x1; : : : ; xi) = Tr(−1) p q t x1 : : : xi ≡ I (p; q; t; x): Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005] The SC index is the Witten index I = T r(−1)F e−βH+M Here M is a generic combination of charges (weighted by chemical potentials) which commutes with S and Q. States with H > 0 come in pairs, boson + fermion, and cancel out,so I is β-independent. The SC Index counts (with signs) the (semi)short multiplets, up to equivalence relations that sets to zero ⊕i Shorti =Long. Abhijit Gadde N = 2 Dualities and 2d TQFT F j1+j2−r −j1+j2−r R+r F1 Fi I (p; q; t; x1; : : : ; xi) = Tr(−1) p q t x1 : : : xi ≡ I (p; q; t; x): Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005] The SC index is the Witten index I = T r(−1)F e−βH+M Here M is a generic combination of charges (weighted by chemical potentials) which commutes with S and Q. States with H > 0 come in pairs, boson + fermion, and cancel out,so I is β-independent. The SC Index counts (with signs) the (semi)short multiplets, up to equivalence relations that sets to zero ⊕i Shorti =Long. 3 y Consider a 4d SCFT. On S × R (radial quantization), Q = S. ¯ The superconformal algebra implies (taking Q = Q1−_ ) 2 fS; Qg = ∆ − 2 j2 − 2 R + r ≡ H ≥ 0 : where ∆ is the conformal dimension, (j1; j2) the SU(2)1 ⊗ SU(2)2 Lorentz spins, and (R ; r) the quantum numbers under the SU(2)R ⊗ U(1)r R-symmetry.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • S-Duality-And-M5-Mit
    N . More on = 2 S-dualities and M5-branes .. Yuji Tachikawa based on works in collaboration with L. F. Alday, B. Wecht, F. Benini, S. Benvenuti, D. Gaiotto November 2009 Yuji Tachikawa (IAS) November 2009 1 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 2 / 47 Montonen-Olive duality • N = 4 SU(N) SYM at coupling τ = θ=(2π) + (4πi)=g2 equivalent to the same theory coupling τ 0 = −1/τ • One way to ‘understand’ it: start from 6d N = (2; 0) theory, i.e. the theory on N M5-branes, put on a torus −1/τ τ 0 1 0 1 • Low energy physics depends only on the complex structure S-duality! Yuji Tachikawa (IAS) November 2009 3 / 47 S-dualities in N = 2 theories • You can wrap N M5-branes on a more general Riemann surface, possibly with punctures, to get N = 2 superconformal field theories • Different limits of the shape of the Riemann surface gives different weakly-coupled descriptions, giving S-dualities among them • Anticipated by [Witten,9703166], but not well-appreciated until [Gaiotto,0904.2715] Yuji Tachikawa (IAS) November 2009 4 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 5 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 6 / 47 S-duality in N = 2 . SU(2) with Nf = 4 .
    [Show full text]
  • Exceptional Field Theory and Supergravity Arnaud Baguet
    Exceptional Field Theory and Supergravity Arnaud Baguet To cite this version: Arnaud Baguet. Exceptional Field Theory and Supergravity. High Energy Physics - Experiment [hep-ex]. Université de Lyon, 2017. English. NNT : 2017LYSEN022. tel-01563063 HAL Id: tel-01563063 https://tel.archives-ouvertes.fr/tel-01563063 Submitted on 17 Jul 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Numéro National de Thèse : 2017LYSEN022 Thèse de Doctorat de l’Université de Lyon opérée par l’École Normale Supérieure de Lyon École Doctorale 52 École Doctorale de Physique et d’Astrophysique de Lyon Soutenue publiquement le 30/06/2017, par : Arnaud Baguet Exceptional Field Theory and Supergravity - Théorie des Champs Exceptionnels et Supergravité Devant le jury composé de : Martin Cederwall Chalmers University of Technology Rapporteur François Delduc École Normale Supérieure de Lyon Examinateur Axel Kleinschmidt Max Planck Institute Postdam Rapporteur Michela Petrini Université Pierre et Marie Curie Examinatrice Henning Samtleben École Normale Supérieure de Lyon Directeur Contents 1 Introduction 9 1.1 String theory, dualities and supergravity . 11 1.2 The bosonic string . 13 1.3 Compactification and T-duality . 18 1.4 Double Field Theory . 21 1.5 Exceptional Field Theory .
    [Show full text]
  • A Supersymmetry Primer
    hep-ph/9709356 version 7, January 2016 A Supersymmetry Primer Stephen P. Martin Department of Physics, Northern Illinois University, DeKalb IL 60115 I provide a pedagogical introduction to supersymmetry. The level of discussion is aimed at readers who are familiar with the Standard Model and quantum field theory, but who have had little or no prior exposure to supersymmetry. Topics covered include: motiva- tions for supersymmetry, the construction of supersymmetric Lagrangians, superspace and superfields, soft supersymmetry-breaking interactions, the Minimal Supersymmetric Standard Model (MSSM), R-parity and its consequences, the origins of supersymmetry breaking, the mass spectrum of the MSSM, decays of supersymmetric particles, experi- mental signals for supersymmetry, and some extensions of the minimal framework. Contents 1 Introduction 3 2 Interlude: Notations and Conventions 13 3 Supersymmetric Lagrangians 17 3.1 The simplest supersymmetric model: a free chiral supermultiplet ............. 18 3.2 Interactions of chiral supermultiplets . ................ 22 3.3 Lagrangians for gauge supermultiplets . .............. 25 3.4 Supersymmetric gauge interactions . ............. 26 3.5 Summary: How to build a supersymmetricmodel . ............ 28 4 Superspace and superfields 30 4.1 Supercoordinates, general superfields, and superspace differentiation and integration . 31 4.2 Supersymmetry transformations the superspace way . ................ 33 4.3 Chiral covariant derivatives . ............ 35 4.4 Chiralsuperfields................................ ........ 37 arXiv:hep-ph/9709356v7 27 Jan 2016 4.5 Vectorsuperfields................................ ........ 38 4.6 How to make a Lagrangian in superspace . .......... 40 4.7 Superspace Lagrangians for chiral supermultiplets . ................... 41 4.8 Superspace Lagrangians for Abelian gauge theory . ................ 43 4.9 Superspace Lagrangians for general gauge theories . ................. 46 4.10 Non-renormalizable supersymmetric Lagrangians . .................. 49 4.11 R symmetries........................................
    [Show full text]
  • Construction and Classification of Holomorphic Vertex Operator Algebras
    Construction and classification of holomorphic vertex operator algebras Jethro van Ekeren1, Sven Möller2, Nils R. Scheithauer2 1Universidade Federal Fluminense, Niterói, RJ, Brazil 2Technische Universität Darmstadt, Darmstadt, Germany We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens’ classification of V1-structures of meromorphic confor- mal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds. 1 Introduction Vertex algebras give a mathematically rigorous description of 2-dimensional quantum field the- ories. They were introduced into mathematics by R. E. Borcherds [B1]. The most prominent example is the moonshine module V \ constructed by Frenkel, Lepowsky and Meurman [FLM]. This vertex algebra is Z-graded and carries an action of the largest sporadic simple group, the monster. Borcherds showed that the corresponding twisted traces are hauptmoduls for genus 0 groups [B2]. The moonshine module was the first example of an orbifold in the theory of vertex algebras. The (−1)-involution of the Leech lattice Λ lifts to an involution g of the associated lattice vertex g algebra VΛ. The fixed points VΛ of VΛ under g form a simple subalgebra of VΛ. Let VΛ(g) be the unique irreducible g-twisted module of VΛ and VΛ(g)Z the subspace of vectors with integral g g L0-weight. Then the vertex algebra structure on VΛ can be extended to VΛ ⊕ VΛ(g)Z. This sum \ g is the moonshine module V .
    [Show full text]
  • Fusion Rules of Virasoro Vertex Operator Algebras
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 9, September 2015, Pages 3765–3776 http://dx.doi.org/10.1090/proc/12552 Article electronically published on May 1, 2015 FUSION RULES OF VIRASORO VERTEX OPERATOR ALGEBRAS XIANZU LIN (Communicated by Kailash C. Misra) Abstract. In this paper we prove the fusion rules of Virasoro vertex operator algebras L(c1,q, 0), for q ≥ 1. Roughly speaking, we consider L(c1,q, 0) as the limit of L(cn,nq−1, 0), for n →∞, and the fusion rules of L(c1,q, 0) follow as the limits of the fusion rules of L(cn,nq−1, 0). 1. Introduction In classical representation theories (of compact groups or semi-simple Lie al- gebras), we can define the tensor product for two modules. The famous Clebsch- Gordan problem is: given irreducible representations U and V , describe the decom- position, with multiplicities, of the representation U ⊗ V . For the representation theory of vertex operator algebras, the difference is that we cannot define the tensor product of two modules over a vertex operator algebra A. Nevertheless, we still have the analogue of the Clebsch-Gordan problem via the notion of intertwining operator. In particular, for three irreducible modules U, V and W over a vertex N W operator algebra A, we can define the fusion rule U,V , the analogue of the Clebsch- Gordan coefficient. For the representation theory of vertex operator algebras, it is N W also very important to determine the fusion rules U,V . The Virasoro vertex operator algebras constitute one of the most important classes of vertex operator algebras.
    [Show full text]
  • Structure and Representation Theory of Infinite-Dimensional Lie Algebras
    Structure and Representation Theory of Infinite-dimensional Lie Algebras David Mehrle Abstract Kac-Moody algebras are a generalization of the finite-dimensional semisimple Lie algebras that have many characteristics similar to the finite-dimensional ones. These possibly infinite-dimensional Lie algebras have found applica- tions everywhere from modular forms to conformal field theory in physics. In this thesis we give two main results of the theory of Kac-Moody algebras. First, we present the classification of affine Kac-Moody algebras by Dynkin di- agrams, which extends the Cartan-Killing classification of finite-dimensional semisimple Lie algebras. Second, we prove the Kac Character formula, which a generalization of the Weyl character formula for Kac-Moody algebras. In the course of presenting these results, we develop some theory as well. Contents 1 Structure 1 1.1 First example: affine sl2(C) .....................2 1.2 Building Lie algebras from complex matrices . .4 1.3 Three types of generalized Cartan matrices . .9 1.4 Symmetrizable GCMs . 14 1.5 Dynkin diagrams . 19 1.6 Classification of affine Lie algebras . 20 1.7 Odds and ends: the root system, the standard invariant form and Weyl group . 29 2 Representation Theory 32 2.1 The Universal Enveloping Algebra . 32 2.2 The Category O ............................ 35 2.3 Characters of g-modules . 39 2.4 The Generalized Casimir Operator . 45 2.5 Kac Character Formula . 47 1 Acknowledgements First, I’d like to thank Bogdan Ion for being such a great advisor. I’m glad that you agreed to supervise this project, and I’m grateful for your patience and many helpful suggestions.
    [Show full text]
  • Polyvector Deformations in Eleven-Dimensional Supergravity
    PHYSICAL REVIEW D 103, 066021 (2021) Polyvector deformations in eleven-dimensional supergravity † Kirill Gubarev * and Edvard T. Musaev Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny 141700, Russia (Received 8 December 2020; accepted 3 March 2021; published 26 March 2021) We consider 3- and 6-vector deformations of 11-dimensional supergravity backgrounds of the form M5 × M6 admitting at least three Killing vectors. Using flux formulation of the E6ð6Þ exceptional field theory, we derive (sufficient) conditions for the deformations to generate a solution. In the group manifold case, these generalizations of the classical Yang-Baxter equation for the case of r-matrices with three and six indices are shown to reproduce those obtained from exceptional Drinfeld algebra for E6ð6Þ. In general, we see an additional constraint, which might be related to higher exceptional Drinfeld algebras. DOI: 10.1103/PhysRevD.103.066021 I. INTRODUCTION transformations that keep the string sigma-model in a consistent vacuum, however changing it in a controllable Vacua of string theory understood as a perturbative way. Particularly interesting examples are based on mani- formulation of the nonlinear two-dimensional sigma-model folds with an AdS factor known to be holographically dual to are known to be represented by a vast landscape of 10-dimensional manifolds equipped by various gauge superconformal field theories. While in general, CFTs are an fields: Kalb-Ramond 2-form and Ramond-Ramond p-form isolated point in the space of couplings corresponding to a gauge fields. In the full nonperturbative formulation, one fixed point of renormalization group flow, SCFTs belong to finds that the space of string vacua is mostly populated a family of theories connected by varying couplings.
    [Show full text]
  • Three Dimensional Conformal Sigma Models
    Lectures on exact renormalization group-2 Etsuko Itou (YITP, Japan) Progress of Theoretical Physics 109 (2003) 751 Progress of Theoretical Physics 110 (2003) 563 Progress of Theoretical Physics 117 (2007)1139 Collaborated with K.Higashijima and T.Higashi (Osaka U.) 2008/2/20, NCTS Plan to talk First lecture: Wilsonian renormalization group Second lecture: Scalar field theory (sigma model) Third lecture: Gauge theory There are some Wilsonian renormalization group equations. Wegner-Houghton equation (sharp cutoff) K-I. Aoki, H. Terao, K.Higashijima… local potential, Nambu-Jona-Lasinio, NLσM Polchinski equation (smooth cutoff) Lecture-3 T.Morris, K. Itoh, Y. Igarashi, H. Sonoda, M. Bonini,… YM theory, QED, SUSY… Exact evolution equation ( for 1PI effective action) C. Wetterich, M. Reuter, N. Tetradis, J. Pawlowski,… quantum gravity, Yang-Mills theory, higher-dimensional gauge theory… The WRG equation (Wegner-Houghton equation) describes the variation of effective action when energy scale Λ is changed to Λ(δt)=Λ exp[−δt] . Sigma model approximation We expand the most generic action as In today talk, we expand the action up to second order in derivative and constraint it =2 supersymmetry. N-components Lagrangian : the metric of target spaces :functional of field variables and coupling constants The point of view of non-linear sigma model: Two-dimensional case In perturbative analysis, the 1-loop beta- function for 2-dimensional non-linear sigma model proportional to Ricci tensor of target spaces. Alvarez-Gaume, Freedman and Mukhi Ann. of Phys. 134 (1982) 392 The perturbative results Ricci flat (Calabi-Yau) Non-perturbative? D=2 (3) N =2 supersymmetric non linear sigma model i=1～N：N is the dimensions of target spaces Where K is Kaehler potential and Φ is chiral superfield.
    [Show full text]
  • Instanton Operators and the Higgs Branch at Infinite Coupling Arxiv
    IMPERIAL-TP-15-AH-03 CERN-PH-TH-2015-115 Instanton Operators and the Higgs Branch at Infinite Coupling Stefano Cremonesi,a Giulia Ferlito,b Amihay Hanany,b Noppadol Mekareeyac aDepartment of Mathematics, King's College London, The Strand, London WC2R 2LS, United Kingdom bTheoretical Physics Group, Imperial College London, Prince Consort Road, London, SW7 2AZ, UK cTheory Division, Physics Department, CERN, CH-1211, Geneva 23, Switzerland E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: The richness of 5d N = 1 theories with a UV fixed point at infinite cou- pling is due to the existence of local disorder operators known as instanton operators. By considering the Higgs branch of SU(2) gauge theories with Nf ≤ 7 flavours at finite and infinite coupling, we write down the explicit chiral ring relations between instanton operators, the glueball superfield and mesons. Exciting phenomena appear at infinite coupling: the glueball superfield is no longer nilpotent and the classical chiral ring relations are quantum corrected by instanton operators bilinears. We also find expressions for the dressing of instanton operators of arbitrary charge. The same arXiv:1505.06302v2 [hep-th] 16 Jun 2017 analysis is performed for USp(2k) with an antisymmetric hypermultiplet and pure SU(N) gauge theories. Contents 1 Introduction1 2 2 SU(2) with Nf flavours: one ENf +1 instanton on C 3 3 USp(4) with one antisymmetric hypermultiplet 31 4 USp(2k) with one antisymmetric hypermultiplet 34 5 Pure super Yang-Mills theories 36 6 Discussion 37 A Hilbert series of chiral rings with gaugino superfields 38 B Nf = 6 in representations of SO(12) × SU(2) 41 1 Introduction The dynamics of five-dimensional supersymmetric gauge theories has many interest- ing features.
    [Show full text]