Representations of the Lie Superalgebra Osp(1L2n)
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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. -
Social Choice Theory Christian List
1 Social Choice Theory Christian List Social choice theory is the study of collective decision procedures. It is not a single theory, but a cluster of models and results concerning the aggregation of individual inputs (e.g., votes, preferences, judgments, welfare) into collective outputs (e.g., collective decisions, preferences, judgments, welfare). Central questions are: How can a group of individuals choose a winning outcome (e.g., policy, electoral candidate) from a given set of options? What are the properties of different voting systems? When is a voting system democratic? How can a collective (e.g., electorate, legislature, collegial court, expert panel, or committee) arrive at coherent collective preferences or judgments on some issues, on the basis of its members’ individual preferences or judgments? How can we rank different social alternatives in an order of social welfare? Social choice theorists study these questions not just by looking at examples, but by developing general models and proving theorems. Pioneered in the 18th century by Nicolas de Condorcet and Jean-Charles de Borda and in the 19th century by Charles Dodgson (also known as Lewis Carroll), social choice theory took off in the 20th century with the works of Kenneth Arrow, Amartya Sen, and Duncan Black. Its influence extends across economics, political science, philosophy, mathematics, and recently computer science and biology. Apart from contributing to our understanding of collective decision procedures, social choice theory has applications in the areas of institutional design, welfare economics, and social epistemology. 1. History of social choice theory 1.1 Condorcet The two scholars most often associated with the development of social choice theory are the Frenchman Nicolas de Condorcet (1743-1794) and the American Kenneth Arrow (born 1921). -
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. -
Mathematics and Materials
IAS/PARK CITY MATHEMATICS SERIES Volume 23 Mathematics and Materials Mark J. Bowick David Kinderlehrer Govind Menon Charles Radin Editors American Mathematical Society Institute for Advanced Study Society for Industrial and Applied Mathematics 10.1090/pcms/023 Mathematics and Materials IAS/PARK CITY MATHEMATICS SERIES Volume 23 Mathematics and Materials Mark J. Bowick David Kinderlehrer Govind Menon Charles Radin Editors American Mathematical Society Institute for Advanced Study Society for Industrial and Applied Mathematics Rafe Mazzeo, Series Editor Mark J. Bowick, David Kinderlehrer, Govind Menon, and Charles Radin, Volume Editors. IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program 2010 Mathematics Subject Classification. Primary 82B05, 35Q70, 82B26, 74N05, 51P05, 52C17, 52C23. Library of Congress Cataloging-in-Publication Data Names: Bowick, Mark J., editor. | Kinderlehrer, David, editor. | Menon, Govind, 1973– editor. | Radin, Charles, 1945– editor. | Institute for Advanced Study (Princeton, N.J.) | Society for Industrial and Applied Mathematics. Title: Mathematics and materials / Mark J. Bowick, David Kinderlehrer, Govind Menon, Charles Radin, editors. Description: [Providence] : American Mathematical Society, [2017] | Series: IAS/Park City math- ematics series ; volume 23 | “Institute for Advanced Study.” | “Society for Industrial and Applied Mathematics.” | “This volume contains lectures presented at the Park City summer school on Mathematics and Materials in July 2014.” – Introduction. | Includes bibliographical references. Identifiers: LCCN 2016030010 | ISBN 9781470429195 (alk. paper) Subjects: LCSH: Statistical mechanics–Congresses. -
Msc in Applied Mathematics
MSc in Applied Mathematics Title: Past, Present and Challenges in Yang-Mills Theory Author: José Luis Tejedor Advisor: Sebastià Xambó Descamps Department: Departament de Matemàtica Aplicada II Academic year: 2011-2012 PAST, PRESENT AND CHALLENGES IN YANG-MILLS THEORY Jos´eLuis Tejedor June 11, 2012 Abstract In electrodynamics the potentials are not uniquely defined. The electromagnetic field is un- changed by gauge transformations of the potential. The electromagnestism is a gauge theory with U(1) as gauge group. C. N. Yang and R. Mills extended in 1954 the concept of gauge theory for abelian groups to non-abelian groups to provide an explanation for strong interactions. The idea of Yang-Mills was criticized by Pauli, as the quanta of the Yang-Mills field must be massless in order to maintain gauge invariance. The idea was set aside until 1960, when the concept of particles acquiring mass through symmetry breaking in massless theories was put forward. This prompted a significant restart of Yang-Mills theory studies that proved succesful in the formula- tion of both electroweak unification and quantum electrodynamics (QCD). The Standard Model combines the strong interaction with the unified electroweak interaction through the symmetry group SU(3) ⊗ SU(2) ⊗ U(1). Modern gauge theory includes lattice gauge theory, supersymme- try, magnetic monopoles, supergravity, instantons, etc. Concerning the mathematics, the field of Yang-Mills theories was included in the Clay Mathematics Institute's list of \Millenium Prize Problems". This prize-problem focuses, especially, on a proof of the conjecture that the lowest excitations of a pure four-dimensional Yang-Mills theory (i.e. -
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 -
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]. -
The Extended Relativity Theory in Clifford Spaces
THE EXTENDED RELATIVITY THEORY IN CLIFFORD SPACES C. Castroa and M. Pav·si·cb May 21, 2004 aCenter for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta bJo·zef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia; Email: [email protected] Abstract A brief review of some of the most important features of the Extended Rela- tivity theory in Cli®ord-spaces (C-spaces) is presented whose " point" coordinates are noncommuting Cli®ord-valued quantities and which incorporate the lines, ar- eas, volumes,.... degrees of freedom associated with the collective particle, string, membrane,... dynamics of p-loops (closed p-branes) living in target D-dimensional spacetime backgrounds. C-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, noncommuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable di- mensions/signatures that allows to study the dynamics of all (closed) p-branes, for all values of p, on a uni¯ed footing. It resolves the ordering ambiguities in QFT and the problem of time in Cosmology. A discussion of the maximal-acceleration Rela- tivity principle in phase-spaces follows along with the study of the invariance group of symmetry transformations in phase-space that allows to show why Planck areas are invariant under acceleration-boosts transformations and which seems to suggest that a maximal-string tension principle may be operating in Nature. We continue by pointing out how the relativity of signatures of the underlying n-dimensional spacetime results from taking di®erent n-dimensional slices through C-space. The conformal group emerges as a natural subgroup of the Cli®ord group and Relativity in C-spaces involves natural scale changes in the sizes of physical objects without the introduction of forces nor Weyl's gauge ¯eld of dilations. -
Applied Analysis & Scientific Computing Discrete Mathematics
CENTER FOR NONLINEAR ANALYSIS The CNA provides an environment to enhance and coordinate research and training in applied analysis, including partial differential equations, calculus of Applied Analysis & variations, numerical analysis and scientific computation. It advances research and educational opportunities at the broad interface between mathematics and Scientific Computing physical sciences and engineering. The CNA fosters networks and collaborations within CMU and with US and international institutions. Discrete Mathematics & Operations Research RANKINGS DOCTOR OF PHILOSOPHY IN ALGORITHMS, COMBINATORICS, U.S. News & World Report AND OPTIMIZATION #16 | Applied Mathematics Carnegie Mellon University offers an interdisciplinary Ph.D program in Algorithms, Combinatorics, and #7 | Discrete Mathematics and Combinatorics Optimization. This program is the first of its kind in the United States. It is administered jointly #6 | Best Graduate Schools for Logic by the Tepper School of Business (Operations Research group), the Computer Science Department (Algorithms and Complexity group), and the Quantnet Department of Mathematical Sciences (Discrete Mathematics group). #4 | Best Financial Engineering Programs Carnegie Mellon University does not CONTACT discriminate in admission, employment, or Logic administration of its programs or activities on Department of Mathematical Sciences the basis of race, color, national origin, sex, handicap or disability, age, sexual orientation, 5000 Forbes Avenue gender identity, religion, creed, ancestry,