1 the Superalgebra of the Supersymmetric Quantum Me- Chanics
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Introduction to Supersymmetry(1)
Introduction to Supersymmetry(1) J.N. Tavares Dep. Matem¶aticaPura, Faculdade de Ci^encias,U. Porto, 4000 Porto TQFT Club 1Esta ¶euma vers~aoprovis¶oria,incompleta, para uso exclusivo nas sess~oesde trabalho do TQFT club CONTENTS 1 Contents 1 Supersymmetry in Quantum Mechanics 2 1.1 The Supersymmetric Oscillator . 2 1.2 Witten Index . 4 1.3 A fundamental example: The Laplacian on forms . 7 1.4 Witten's proof of Morse Inequalities . 8 2 Supergeometry and Supersymmetry 13 2.1 Field Theory. A quick review . 13 2.2 SuperEuclidean Space . 17 2.3 Reality Conditions . 18 2.4 Supersmooth functions . 18 2.5 Supermanifolds . 21 2.6 Lie Superalgebras . 21 2.7 Super Lie groups . 26 2.8 Rigid Superspace . 27 2.9 Covariant Derivatives . 30 3 APPENDIX. Cli®ord Algebras and Spin Groups 31 3.1 Cli®ord Algebras . 31 Motivation. Cli®ord maps . 31 Cli®ord Algebras . 33 Involutions in V .................................. 35 Representations . 36 3.2 Pin and Spin groups . 43 3.3 Spin Representations . 47 3.4 U(2), spinors and almost complex structures . 49 3.5 Spinc(4)...................................... 50 Chiral Operator. Self Duality . 51 2 1 Supersymmetry in Quantum Mechanics 1.1 The Supersymmetric Oscillator i As we will see later the \hermitian supercharges" Q®, in the N extended SuperPoincar¶eLie Algebra obey the anticommutation relations: i j m ij fQ®;Q¯g = 2(γ C)®¯± Pm (1.1) m where ®; ¯ are \spinor" indices, i; j 2 f1; ¢ ¢ ¢ ;Ng \internal" indices and (γ C)®¯ a bilinear form in the spinor indices ®; ¯. When specialized to 0-space dimensions ((1+0)-spacetime), then since P0 = H, relations (1.1) take the form (with a little change in notations): fQi;Qjg = 2±ij H (1.2) with N \Hermitian charges" Qi; i = 1; ¢ ¢ ¢ ;N. -
Conformal Killing Superalgebras
CONFORMAL SYMMETRY SUPERALGEBRAS PAUL DE MEDEIROS AND STEFAN HOLLANDS ABSTRACT. We show how the rigid conformal supersymmetries associated with a certain class of pseudo-Riemannian spin manifolds define a Lie superalgebra. The even part of this superalgebra con- tains conformal isometries and constant R-symmetries. The odd part is generated by twistor spinors valued in a particular R-symmetry representation. We prove that any manifold which admits a con- formal symmetry superalgebra of this type must generically have dimension less than seven. Moreover, in dimensions three, four, five and six, we provide the generic data from which the conformal symmetry superalgebra is prescribed. For conformally flat metrics in these dimensions, and compact R-symmetry, we identify each of the associated conformal symmetry superalgebras with one of the conformal su- peralgebras classified by Nahm. We also describe several examples for Lorentzian metrics that are not conformally flat. CONTENTS 1. Introduction 2 2. Preliminaries 3 2.1. Tensors, vectors and conformal Killing vectors 3 2.2. Clifford modules, spinors and twistor spinors 4 3. Conformal structure 6 4. Spinorial bilinear forms 6 4.1. Complex case 8 4.2. Real case 9 5. Conformal symmetry superalgebras 10 5.1. Brackets 10 5.2. Conformal invariance 11 5.3. Jacobi identities 12 5.4. Real forms 12 6. Comparison with Nahm for conformally flat metrics 14 7. Some non-trivial examples in Lorentzian signature 15 arXiv:1302.7269v2 [hep-th] 3 Jun 2013 7.1. Brinkmann waves 16 7.2. Lorentzian Einstein-Sasaki manifolds 17 7.3. Fefferman spaces 17 7.4. Direct products 18 Acknowledgments 19 References 19 Date: 9th October 2018. -
Arxiv:1905.09269V2
Anomaly cancellation in the topological string Kevin Costello∗ Perimeter Institute of Theoretical Physics 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada Si Li† Department of Mathematical Sciences and Yau Mathematical Sciences Center, Jingzhai, Tsinghua University, Beijing 100084, China (Dated: January 7, 2020) We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira- Spencer gravity. We explain a new anomaly cancellation mechanism at all loops in pertur- bation theory for open-closed topological B-model. At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type I version of Kodaira-Spencer theory in complex dimensions 3 and 5. In complex dimension 5, we show that it can only be coupled consistently at the quantum level to holomorphic Chern-Simons theory with gauge group SO(32). This is analogous to the Green-Schwarz mechanism for the physical type I string. This coupled system is conjectured to be a supersymmetric localization of type I string theory. In complex dimension 3, the required gauge group is SO(8). Contents I. Introduction 2 II. Open-closed topological B-model 3 A. The open-string sector 3 arXiv:1905.09269v2 [hep-th] 5 Jan 2020 B. The closed-string sector 6 C. Coupling closed to open strings at leading order 8 D. Kodaira-Spencer gravity in BV formalism 9 E. Green-Schwarz mechanism 13 F. The holomorphic stress-energy tensors 15 ∗Electronic address: [email protected] †Electronic address: [email protected] 2 G. Large N single trace operators 19 H. -
Arxiv:2002.02662V2 [Math.RT]
BLOCKS AND CHARACTERS OF D(2|1; ζ)-MODULES OF NON-INTEGRAL WEIGHTS CHIH-WHI CHEN, SHUN-JEN CHENG, AND LI LUO Abstract. We classify blocks in the BGG category O of modules of non-integral weights for the exceptional Lie superalgebra D(2|1; ζ). We establish various reduction methods, which connect some types of non-integral blocks of D(2|1; ζ) with integral blocks of general linear Lie superalgebras gl(1|1) and gl(2|1). We compute the characters for irreducible D(2|1; ζ)-modules of non-integral weights in O. Contents 1. Introduction 1 2. Preliminaries 3 3. Classification of blocks 10 4. Reduction methods and characters in the generic and 1-integer cases 15 5. Character formulas in 2-integer case 19 6. Primitive spectrum of D(2|1; ζ) 23 References 25 1. Introduction 1.1. Since a Killing-Cartan type classification of finite-dimensional complex simple Lie superalge- bras has been achieved by Kac [Kac77] in 1977 there has been substantial efforts by mathematicians and physicists in the study of their representation theory. The most important class of the simple arXiv:2002.02662v2 [math.RT] 20 Feb 2020 Lie superalgebras is the so-called basic classical, which includes the classical series of types ABCD. Among these basic Lie superalgebras, there are 3 exceptional ones: D(2|1; ζ), G(3) and F (3|1). 1.2. The irreducible character problem is one of the main theme in representation theory of Lie (super)algebras. While complete answers to the irreducible character problem in the BGG cat- egories for basic Lie superalgebras of types ABCD are now known (see [CLW11, CLW15, Ba17, BLW17, BW18]), the BGG category of the exceptional Lie superalgebras were not until recently. -
An Introduction to Supersymmetry
An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems. -
Clifford Algebras, Spinors and Supersymmetry. Francesco Toppan
IV Escola do CBPF – Rio de Janeiro, 15-26 de julho de 2002 Algebraic Structures and the Search for the Theory Of Everything: Clifford algebras, spinors and supersymmetry. Francesco Toppan CCP - CBPF, Rua Dr. Xavier Sigaud 150, cep 22290-180, Rio de Janeiro (RJ), Brazil abstract These lectures notes are intended to cover a small part of the material discussed in the course “Estruturas algebricas na busca da Teoria do Todo”. The Clifford Algebras, necessary to introduce the Dirac’s equation for free spinors in any arbitrary signature space-time, are fully classified and explicitly constructed with the help of simple, but powerful, algorithms which are here presented. The notion of supersymmetry is introduced and discussed in the context of Clifford algebras. 1 Introduction The basic motivations of the course “Estruturas algebricas na busca da Teoria do Todo”consisted in familiarizing graduate students with some of the algebra- ic structures which are currently investigated by theoretical physicists in the attempt of finding a consistent and unified quantum theory of the four known interactions. Both from aesthetic and practical considerations, the classification of mathematical and algebraic structures is a preliminary and necessary require- ment. Indeed, a very ambitious, but conceivable hope for a unified theory, is that no free parameter (or, less ambitiously, just few) has to be fixed, as an external input, due to phenomenological requirement. Rather, all possible pa- rameters should be predicted by the stringent consistency requirements put on such a theory. An example of this can be immediately given. It concerns the dimensionality of the space-time. -
Duality and Integrability in Topological String Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by SISSA Digital Library International School of Advanced Studies Mathematical Physics Sector Academic Year 2008–09 PhD Thesis in Mathematical Physics Duality and integrability in topological string theory Candidate Andrea Brini Supervisors External referees Prof. Boris Dubrovin Prof. Marcos Marino˜ Dr. Alessandro Tanzini Dr. Tom Coates 2 Acknowledgements Over the past few years I have benefited from the interaction with many people. I would like to thank first of all my supervisors: Alessandro Tanzini, for introducing me to the subject of this thesis, for his support, and for the fun we had during countless discussions, and Boris Dubrovin, from whose insightful explanations I have learned a lot. It is moreover a privilege to have Marcos Mari˜no and Tom Coates as referees of this thesis, and I would like to thank them for their words and ac- tive support in various occasions. Special thanks are also due to Sara Pasquetti, for her help, suggestions and inexhaustible stress-energy transfer, to Giulio Bonelli for many stimulating and inspiring discussions, and to my collaborators Renzo Cava- lieri, Luca Griguolo and Domenico Seminara. I would finally like to acknowledge the many people I had discussions/correspondence with: Murad Alim, Francesco Benini, Marco Bertola, Vincent Bouchard ;-), Ugo Bruzzo, Mattia Cafasso, Hua-Lian Chang, Michele Cirafici, Tom Claeys, Andrea Collinucci, Emanuel Diaconescu, Bertrand Ey- nard, Jarah Evslin, Fabio Ferrari Ruffino, Barbara -
SUPERSYMMETRY 1. Introduction the Purpose
SUPERSYMMETRY JOSH KANTOR 1. Introduction The purpose of these notes is to give a short and (overly?)simple description of supersymmetry for Mathematicians. Our description is far from complete and should be thought of as a first pass at the ideas that arise from supersymmetry. Fundamental to supersymmetry is the mathematics of Clifford algebras and spin groups. We will describe the mathematical results we are using but we refer the reader to the references for proofs. In particular [4], [1], and [5] all cover spinors nicely. 2. Spin and Clifford Algebras We will first review the definition of spin, spinors, and Clifford algebras. Let V be a vector space over R or C with some nondegenerate quadratic form. The clifford algebra of V , l(V ), is the algebra generated by V and 1, subject to the relations v v = v, vC 1, or equivalently v w + w v = 2 v, w . Note that elements of· l(V ) !can"b·e written as polynomials· in V · and this! giv"es a splitting l(V ) = l(VC )0 l(V )1. Here l(V )0 is the set of elements of l(V ) which can bCe writtenC as a linear⊕ C combinationC of products of even numbers ofCvectors from V , and l(V )1 is the set of elements which can be written as a linear combination of productsC of odd numbers of vectors from V . Note that more succinctly l(V ) is just the quotient of the tensor algebra of V by the ideal generated by v vC v, v 1. -
Infinitely Generated Clifford Algebras and Wedge Representations of Gl∞|∞ Bradford J
University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations May 2015 Infinitely Generated Clifford Algebras and Wedge Representations of gl∞|∞ Bradford J. Schleben University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Mathematics Commons Recommended Citation Schleben, Bradford J., "Infinitely Generated Clifford Algebras and Wedge Representations of gl∞|∞" (2015). Theses and Dissertations. 919. https://dc.uwm.edu/etd/919 This Dissertation is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. Infinitely Generated Clifford Algebras and Wedge Representations of gl1j1 by Bradford J. Schleben A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics at The University of Wisconsin{Milwaukee May 2015 ABSTRACT Infinitely Generated Clifford Algebras and Wedge Representations of gl1j1 by Bradford J. Schleben The University of Wisconsin-Milwaukee, 2015 Under the Supervision of Dr. Yi Ming Zou The goal of this dissertation is to explore representations of gl1j1 and associ- ated Clifford superalgebras. The machinery utilized is motivated by developing an alternate superalgebra analogue to the Lie algebra theory developed by Kac [17]. Kac and van de Leur first developed a super analogue, but it had various departures from a natural extension of their Lie algebra approach, made most certainly for the physics consequences. In an effort to establish a natural mathematical analogue, we construct a theory distinct from the super analogue developed by Kac and van de Leur [16]. -
SUSY and XD Notes
Supersymmetry and Extra Dimensions Flip Tanedo LEPP, Cornell University Ithaca, New York A pedagogical set of notes based on lectures by Fernando Quevedo, Adrian Signer, and Csaba Cs´aki,as well as various books and reviews. Last updated: January 9, 2009 ii iii Abstract This is a set of combined lecture notes on supersymmetry and extra dimensions based on various lectures, textbooks, and review articles. The core of these notes come from Professor Fernando Quevedo's 2006- 2007 Lent Part III lecture course of the same name [1]. iv v Acknowledgements Inspiration to write up these notes in LATEX came from Steffen Gielen's excel- lent notes from the Part III Advanced Quantum Field Theory course and the 2008 ICTP Introductory School on the Gauge Gravity Correspondence. Notes from Profes- sor Quevedo's 2005-2006 Part III Supersymmetry and Extra Dimensions course exist in TEXform due to Oliver Schlotterer. The present set of notes were written up indepen- dently, but simliarities are unavoidable. It is my hope that these notes will provide a broader pedagogical introduction supersymmetry and extra dimensions. vi vii Preface These are lecture notes. Version 1 of these notes are based on Fernando Quevedo's lecture notes and structure. I've also incorporated some relevant topics from my research that I think are important to round-out the course. Version 2 of these notes will also incorporate Csaba Cs´aki'sAdvanced Particle Physics notes. Framed text. Throughout these notes framed text will include parenthetical dis- cussions that may be omitted on a first reading. They are meant to provide a broader picture or highlight particular applications that are not central to the main purpose of the chapter. -
Lectures on Supersymmetry and Superstrings 1 General Superalgebra
Lectures on supersymmetry and superstrings Thomas Mohaupt Abstract: These are informal notes on some mathematical aspects of su- persymmetry, based on two ‘Mathematics and Theoretical Physics Meetings’ in the Autumn Term 2009.1 First super vector spaces, Lie superalgebras and su- percommutative associative superalgebras are briefly introduced together with superfunctions and superspaces. After a review of the Poincar´eLie algebra we discuss Poincar´eLie superalgebras and introduce Minkowski superspaces. As a minimal example of a supersymmetric model we discuss the free scalar superfield in two-dimensional N = (1, 1) supersymmetry (this is more or less copied from [1]). We briefly discuss the concept of chiral supersymmetry in two dimensions. None of the material is original. We give references for further reading. Some ‘bonus material’ on (super-)conformal field theories, (super-)string theories and symmetric spaces is included. These are topics which will play a role in future meetings, or which came up briefly during discussions. 1 General superalgebra Definition: A super vector space V is a vector space with a decomposition V = V V 0 ⊕ 1 and a map (parity map) ˜ : (V V ) 0 Z · 0 ∪ 1 −{ } → 2 which assigns even or odd parity to every homogeneous element. Notation: a V a˜ =0 , ‘even’ , b V ˜b = 1 ‘odd’ . ∈ 0 → ∈ 1 → Definition: A Lie superalgebra g (also called super Lie algebra) is a super vector space g = g g 0 ⊕ 1 together with a bracket respecting the grading: [gi , gj] gi j ⊂ + (addition of index mod 2), which is required to be Z2 graded anti-symmetric: ˜ [a,b]= ( 1)a˜b[b,a] − − (multiplication of parity mod 2), and to satisfy a Z2 graded Jacobi identity: ˜ [a, [b,c]] = [[a,b],c] + ( 1)a˜b[b, [a,c]] . -
Adinkras for Mathematicians
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 6, June 2014, Pages 3325–3355 S 0002-9947(2014)06031-5 Article electronically published on February 6, 2014 ADINKRAS FOR MATHEMATICIANS YAN X. ZHANG Abstract. Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown non-trivial connections with coding the- ory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computa- tional nature. We survey these topics for a mathematical audience, make new connections to other areas (homological algebra and poset theory), and solve some of these said problems, including the enumeration of all hyper- cube adinkras up through dimension 5 and the enumeration of odd dashings of adinkras for any dimension. 1. Introduction In a series of papers starting with [11], different subsets of the “DFGHILM collaboration” (Doran, Faux, Gates, H¨ubsch, Iga, Landweber, Miller) have built and extended the machinery of adinkras. Following the ubiquitous spirit of visual diagrams in physics, adinkras are combinatorial objects that encode information about the representation theory of supersymmetry algebras. Adinkras have many intricate links with other fields such as graph theory, Clifford theory, and coding theory. Each of these connections provide many problems that can be compactly communicated to a (non-specialist) mathematician. This paper is a humble attempt to bridge the language gap and generate communication. In short, adinkras are chromotopologies (a class of edge-colored bipartite graphs) with two additional structures, a dashing condition on the edges and a ranking condition on the vertices.