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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. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
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. -
Electromagnetic Field Theory
Electromagnetic Field Theory BO THIDÉ Υ UPSILON BOOKS ELECTROMAGNETIC FIELD THEORY Electromagnetic Field Theory BO THIDÉ Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden and School of Mathematics and Systems Engineering Växjö University, Sweden Υ UPSILON BOOKS COMMUNA AB UPPSALA SWEDEN · · · Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik Freely downloadable from www.plasma.uu.se/CED This book was typeset in LATEX 2" (based on TEX 3.14159 and Web2C 7.4.2) on an HP Visualize 9000⁄360 workstation running HP-UX 11.11. Copyright c 1997, 1998, 1999, 2000, 2001, 2002, 2003 and 2004 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. Preface The current book is an outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergradu- ate or beginning graduate level, it is hoped that the present book may be useful for research workers -
Lo Algebras for Extended Geometry from Borcherds Superalgebras
Commun. Math. Phys. 369, 721–760 (2019) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-019-03451-2 Mathematical Physics L∞ Algebras for Extended Geometry from Borcherds Superalgebras Martin Cederwall, Jakob Palmkvist Division for Theoretical Physics, Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden. E-mail: [email protected]; [email protected] Received: 24 May 2018 / Accepted: 15 March 2019 Published online: 26 April 2019 – © The Author(s) 2019 Abstract: We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an L∞ algebra. The L∞ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra B(gr+1), which is a double extension of the structure algebra gr . The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra. Contents 1. Introduction ................................. 722 2. The Borcherds Superalgebra ......................... 723 3. Section Constraint and Generalised Lie Derivatives ............. 728 4. Derivatives, Generalised Lie Derivatives and Other Operators ....... 730 4.1 The derivative .............................. 730 4.2 Generalised Lie derivative from “almost derivation” .......... 731 4.3 “Almost covariance” and related operators .............. -
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 -
Perturbative Algebraic Quantum Field Theory
Perturbative algebraic quantum field theory Klaus Fredenhagen Katarzyna Rejzner II Inst. f. Theoretische Physik, Department of Mathematics, Universität Hamburg, University of Rome Tor Vergata Luruper Chaussee 149, Via della Ricerca Scientifica 1, D-22761 Hamburg, Germany I-00133, Rome, Italy [email protected] [email protected] arXiv:1208.1428v2 [math-ph] 27 Feb 2013 2012 These notes are based on the course given by Klaus Fredenhagen at the Les Houches Win- ter School in Mathematical Physics (January 29 - February 3, 2012) and the course QFT for mathematicians given by Katarzyna Rejzner in Hamburg for the Research Training Group 1670 (February 6 -11, 2012). Both courses were meant as an introduction to mod- ern approach to perturbative quantum field theory and are aimed both at mathematicians and physicists. Contents 1 Introduction 3 2 Algebraic quantum mechanics 5 3 Locally covariant field theory 9 4 Classical field theory 14 5 Deformation quantization of free field theories 21 6 Interacting theories and the time ordered product 26 7 Renormalization 26 A Distributions and wavefront sets 35 1 Introduction Quantum field theory (QFT) is at present, by far, the most successful description of fundamental physics. Elementary physics , to a large extent, explained by a specific quantum field theory, the so-called Standard Model. All the essential structures of the standard model are nowadays experimentally verified. Outside of particle physics, quan- tum field theoretical concepts have been successfully applied also to condensed matter physics. In spite of its great achievements, quantum field theory also suffers from several longstanding open problems. The most serious problem is the incorporation of gravity. -
1 the Superalgebra of the Supersymmetric Quantum Me- Chanics
CBPF-NF-019/07 1 Representations of the 1DN-Extended Supersymmetry Algebra∗ Francesco Toppan CBPF, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro (RJ), Brazil. E-mail: [email protected] Abstract I review the present status of the classification of the irreducible representations of the alge- bra of the one-dimensional N− Extended Supersymmetry (the superalgebra of the Supersym- metric Quantum Mechanics) realized by linear derivative operators acting on a finite number of bosonic and fermionic fields. 1 The Superalgebra of the Supersymmetric Quantum Me- chanics The superalgebra of the Supersymmetric Quantum Mechanics (1DN-Extended Supersymme- try Algebra) is given by N odd generators Qi (i =1,...,N) and a single even generator H (the hamiltonian). It is defined by the (anti)-commutation relations {Qi,Qj} =2δijH, [Qi,H]=0. (1) The knowledge of its representation theory is essential for the construction of off-shell invariant actions which can arise as a dimensional reduction of higher dimensional supersymmetric theo- ries and/or can be given by 1D supersymmetric sigma-models associated to some d-dimensional target manifold (see [1] and [2]). Two main classes of (1) representations are considered in the literature: i) the non-linear realizations and ii) the linear representations. Non-linear realizations of (1) are only limited and partially understood (see [3] for recent results and a discussion). Linear representations, on the other hand, have been recently clarified and the program of their classification can be considered largely completed. In this work I will review the main results of the classification of the linear representations and point out which are the open problems. -
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]. -
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]] .