The Quantum Mechanics of Supersymmetry
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On Supergravity Domain Walls Derivable from Fake and True Superpotentials
On supergravity domain walls derivable from fake and true superpotentials Rommel Guerrero •, R. Omar Rodriguez •• Unidad de lnvestigaci6n en Ciencias Matemtiticas, Universidad Centroccidental Lisandro Alvarado, 400 Barquisimeto, Venezuela ABSTRACT: We show the constraints that must satisfy the N = 1 V = 4 SUGRA and the Einstein-scalar field system in order to obtain a correspondence between the equations of motion of both theories. As a consequence, we present two asymmetric BPS domain walls in SUGRA theory a.ssociated to holomorphic fake superpotentials with features that have not been reported. PACS numbers: 04.20.-q, 11.27.+d, 04.50.+h RESUMEN: Mostramos las restricciones que debe satisfacer SUGRA N = 1 V = 4 y el sistema acoplado Einstein-campo esca.lar para obtener una correspondencia entre las ecuaciones de movimiento de ambas teorias. Como consecuencia, presentamos dos paredes dominic BPS asimetricas asociadas a falsos superpotenciales holomorfos con intersantes cualidades. I. INTRODUCTION It is well known in the brane world [1) context that the domain wa.lls play an important role bece.UIIe they allow the confinement of the zero mode of the spectra of gravitons and other matter fields (2], which is phenomenologically very attractive. The domain walls are solutions to the coupled Einstein-BCa.lar field equations (3-7), where the scalar field smoothly interpolatesbetween the minima of the potential with spontaneously broken of discrete symmetry. Recently, it has been reported several domain wall solutions, employing a first order formulation of the equations of motion of coupled Einstein-scalar field system in terms of an auxiliaryfunction or fake superpotential (8-10], which resembles to the true superpotentials that appear in the supersymmetry global theories (SUSY). -
Operator Theory in the C -Algebra Framework
Operator theory in the C∗-algebra framework S.L.Woronowicz and K.Napi´orkowski Department of Mathematical Methods in Physics Faculty of Physics Warsaw University Ho˙za74, 00-682 Warszawa Abstract Properties of operators affiliated with a C∗-algebra are studied. A functional calculus of normal elements is constructed. Representations of locally compact groups in a C∗-algebra are considered. Generaliza- tions of Stone and Nelson theorems are investigated. It is shown that the tensor product of affiliated elements is affiliated with the tensor product of corresponding algebras. 1 0 Introduction Let H be a Hilbert space and CB(H) be the algebra of all compact operators acting on H. It was pointed out in [17] that the classical theory of unbounded closed operators acting in H [8, 9, 3] is in a sense related to CB(H). It seems to be interesting to replace in this context CB(H) by any non-unital C∗- algebra. A step in this direction is done in the present paper. We shall deal with the following topics: the functional calculus of normal elements (Section 1), the representation theory of Lie groups including the Stone theorem (Sections 2,3 and 4) and the extensions of symmetric elements (Section 5). Section 6 contains elementary results related to tensor products. The perturbation theory (in the spirit of T.Kato) is not covered in this pa- per. The elementary results in this direction are contained the first author’s previous paper (cf. [17, Examples 1, 2 and 3 pp. 412–413]). To fix the notation we remind the basic definitions and results [17]. -
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. -
One-Loop and D-Instanton Corrections to the Effective Action of Open String
One-loop and D-instanton corrections to the effective action of open string models Maximilian Schmidt-Sommerfeld M¨unchen 2009 One-loop and D-instanton corrections to the effective action of open string models Maximilian Schmidt-Sommerfeld Dissertation der Fakult¨at f¨ur Physik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Maximilian Schmidt-Sommerfeld aus M¨unchen M¨unchen, den 18. M¨arz 2009 Erstgutachter: Priv.-Doz. Dr. Ralph Blumenhagen Zweitgutachter: Prof. Dr. Dieter L¨ust M¨undliche Pr¨ufung: 2. Juli 2009 Zusammenfassung Methoden zur Berechnung bestimmter Korrekturen zu effektiven Wirkungen, die die Niederenergie-Physik von Stringkompaktifizierungen mit offenen Strings er- fassen, werden erklaert. Zunaechst wird die Form solcher Wirkungen beschrieben und es werden einige Beispiele fuer Kompaktifizierungen vorgestellt, insbeson- dere ein Typ I-Stringmodell zu dem ein duales Modell auf Basis des heterotischen Strings bekannt ist. Dann werden Korrekturen zur Eichkopplungskonstante und zur eichkinetischen Funktion diskutiert. Allgemeingueltige Verfahren zu ihrer Berechnung werden skizziert und auf einige Modelle angewandt. Die explizit bestimmten Korrek- turen haengen nicht-holomorph von den Moduli der Kompaktifizierungsmannig- faltigkeit ab. Es wird erklaert, dass dies nicht im Widerspruch zur Holomorphie der eichkinetischen Funktion steht und wie man letztere aus den errechneten Re- sultaten extrahieren kann. Als naechstes werden D-Instantonen und ihr Einfluss auf die Niederenergie- Wirkung detailliert analysiert, wobei die Nullmoden der Instantonen und globale abelsche Symmetrien eine zentrale Rolle spielen. Eine Formel zur Berechnung von Streumatrixelementen in Instanton-Sektoren wird angegeben. Es ist zu erwarten, dass die betrachteten Instantonen zum Superpotential der Niederenergie-Wirkung beitragen. Jedoch wird aus der Formel nicht sofort klar, inwiefern dies moeglich ist. -
Dr. Joseph Conlon Supersymmetry: Example
DR. JOSEPH CONLON Hilary Term 2013 SUPERSYMMETRY: EXAMPLE SHEET 3 1. Consider the Wess-Zumino model consisting of one chiral superfield Φ with standard K¨ahler potential K = Φ†Φ and tree-level superpotential. 1 1 W = mΦ2 + gΦ3. (1) tree 2 3 Consider the couplings g and m as spurion fields, and define two U(1) symmetries of the tree-level superpotential, where both U(1) factors act on Φ, m and g, and one of them also acts on θ (making it an R-symmetry). Using these symmetries, infer the most general form that any loop corrected su- perpotential can take in terms of an arbitrary function F (Φ, m, g). Consider the weak coupling limit g → 0 and m → 0 to fix the form of this function and thereby show that the superpotential is not renormalised. 2. Consider a renormalisable N = 1 susy Lagrangian for chiral superfields with F-term supersymmetry breaking. By analysing the scalar and fermion mass matrix, show that 2 X 2j+1 2 STr(M ) ≡ (−1) (2j + 1)mj = 0, (2) j where j is particle spin. Verfiy that this relation holds for the O’Raifertaigh model, and explain why this forbids single-sector susy breaking models where the MSSM superfields are the dominant source of susy breaking. 3. This problem studies D-term susy breaking. Consider a chiral superfield Φ of charge q coupled to an Abelian vector superfield V . Show that a nonvanishing vev for D, the auxiliary field of V , cam break supersymmetry, and determine the goldstino in this case. -
TASI 2008 Lectures: Introduction to Supersymmetry And
TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg -
Introduction to Supersymmetry
Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × . -
Computing the Singular Value Decomposition to High Relative Accuracy
Computing the Singular Value Decomposition to High Relative Accuracy James Demmel Department of Mathematics Department of Electrical Engineering and Computer Science University of California - Berkeley [email protected] Plamen Koev Department of Mathematics University of California - Berkeley [email protected] Structured Matrices In Operator Theory, Numerical Analysis, Control, Signal and Image Processing Boulder, Colorado June 27-July 1, 1999 Supported by NSF and DOE INTRODUCTION • High Relative Accuracy means computing the correct SIGN and LEADING DIGITS • Singular Value Decomposition (SVD): A = UΣV T where U, V are orthogonal, σ1 σ2 Σ = and σ1 ≥ σ2 ≥ . σn ≥ 0 .. . σn • GOAL: Compute all σi with high relative accuracy, even when σi σ1 • It all comes down to being able to compute determi- nants to high relative accuracy. Example: 100 by 100 Hilbert Matrix H(i, j) = 1/(i + j − 1) • Singular values range from 1 down to 10−150 • Old algorithm, New Algorithm, both in 16 digits Singular values of Hilb(100), blue = accurate, red = usual 0 10 −20 10 −40 10 −60 10 −80 10 −100 10 −120 10 −140 10 0 10 20 30 40 50 60 70 80 90 100 • D = log(cond(A)) = log(σ1/σn) (here D = 150) • Cost of Old algorithm = O(n3D2) • Cost of New algorithm = O(n3), independent of D – Run in double, not bignums as in Mathematica – New hundreds of times faster than Old • When does it work? Not for all matrices ... • Why bother? Why do we want tiny singular values accurately? 1. When they are determined accurately by the data • Hilbert: H(i, j) = 1/(i + j − 1) • Cauchy: C(i, j) = 1/(xi + yj) 2. -
Variational Calculus of Supervariables and Related Algebraic Structures1
Variational Calculus of Supervariables and Related Algebraic Structures1 Xiaoping Xu Department of Mathematics, The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong2 Abstract We establish a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gel’fand-Dikii and the fermionic theory in our earlier work. Certain interesting new algebraic structures are found in connection with Hamiltonian superoperators in terms of our theory. In particular, we find connections between Hamiltonian superoperators and Novikov- Poisson algebras that we introduced in our earlier work in order to establish a tensor theory of Novikov algebras. Furthermore, we prove that an odd linear Hamiltonian superoperator in our variational calculus induces a Lie superalgebra, which is a natural generalization of the Super-Virasoro algebra under certain conditions. 1 Introduction Formal variational calculus was introduced by Gel’fand and Dikii [GDi1-2] in studying Hamiltonian systems related to certain nonlinear partial differential equation, such as the arXiv:math/9911191v1 [math.QA] 24 Nov 1999 KdV equations. Invoking the variational derivatives, they found certain interesting Pois- son structures. Moreover, Gel’fand and Dorfman [GDo] found more connections between Hamiltonian operators and algebraic structures. Balinskii and Novikov [BN] studied sim- ilar Poisson structures from another point of view. The nature of Gel’fand and Dikii’s formal variational calculus is bosonic. In [X3], we presented a general frame of Hamiltonian superoperators and a purely fermionic formal variational calculus. Our work [X3] was based on pure algebraic analogy. In this paper, we shall present a formal variational calculus of supervariables, which is a combination 11991 Mathematical Subject Classification. -
Supersymmetric Dualities and Quantum Entanglement in Conformal Field Theory
Supersymmetric Dualities and Quantum Entanglement in Conformal Field Theory Jeongseog Lee A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Igor R. Klebanov September 2016 c Copyright by Jeongseog Lee, 2016. All rights reserved. Abstract Conformal field theory (CFT) has been under intensive study for many years. The scale invariance, which arises at the fixed points of renormalization group in rela- tivistic quantum field theory (QFT), is believed to be enhanced to the full conformal group. In this dissertation we use two tools to shed light on the behavior of certain conformal field theories, the supersymmetric localization and the quantum entangle- ment Renyi entropies. The first half of the dissertation surveys the infrared (IR) structure of the = 2 N supersymmetric quantum chromodynamics (SQCD) in three dimensions. The re- cently developed F -maximization principle shows that there are richer structures along conformal fixed points compared to those in = 1 SQCD in four dimensions. N We refer to one of the new phenomena as a \crack in the conformal window". Using the known IR dualities, we investigate it for all types of simple gauge groups. We see different decoupling behaviors of operators depending on the gauge group. We also observe that gauging some flavor symmetries modifies the behavior of the theory in the IR. The second half of the dissertation uses the R`enyi entropy to understand the CFT from another angle. At the conformal fixed points of even dimensional QFT, the entanglement entropy is known to have the log-divergent universal term depending on the geometric invariants on the entangling surface. -
An Introduction to Supersymmetric Quantum Mechanics
MIT Physics 8.05 Nov. 12, 2007 SUPPLEMENTARY NOTES ON SOLVING THE RADIAL WAVE EQUATION USING OPERATOR METHODS Alternative title: An Introduction to Supersymmetric Quantum Mechanics 1 Introduction In lecture this week we reduced the problem of finding the energy eigenstates and eigenvalues of hydrogenic atoms to finding the eigenstates and eigenvalues of the Coulomb Hamiltonians d2 HCoul = + V (x) (1) ℓ −dx2 ℓ where ℓ(ℓ + 1) 1 V (x)= . (2) ℓ x2 − x (You will review this procedure on a Problem Set.) That is, we are looking for the eigenstates u (x) and eigenvalues κ2 satisfying νℓ − νℓ HCoul u (x)= κ2 u (x) . (3) ℓ νℓ − νℓ νℓ Here the dimensionless variable x and dimensionless eigenvalue constants κνℓ are related to r and the energy by a 2µe4Z2 r = x , E = κ2 , (4) 2Z νℓ − h¯2 νℓ 2 2 −1 where a =¯h /(me ) is the Bohr radius, the nuclear charge is Ze, and µ = (1/m +1/mN ) is the reduced mass in terms of the electron mass m and nuclear mass mN . Even though Coul the Hamiltonians Hℓ secretly all come from a single three-dimensional problem, for our present purposes it is best to think of them as an infinite number of different one-dimensional Hamiltonians, one for each value of ℓ, each with different potentials Vℓ(x). The Hilbert space for these one-dimensional Hamiltonians consists of complex functions of x 0 that vanish ≥ for x 0. The label ν distinguishes the different energy eigenstates and eigenvalues for a →Coul given Hℓ . -
Notes on N = 1 QCD with Baryon Superpotential
Notes on N = 1 QCD3 with baryon superpotential Vladimir Bashmakov1, Hrachya Khachatryan2 1 Dipartimento di Fisica, Universit`adi Milano-Bicocca and INFN, Sezione di Milano-Bicocca, I-20126 Milano, Italy 2 International School of Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy [email protected], [email protected] Abstract We study the infrared phases of N = 1 QCD3 with gauge group SU(Nc) and with the superpotential containing the mass term and the baryon opera- tor. We restrict ourselves to the cases, when the number of flavors is equal to the number of colours, such that baryon operator is neutral under the global SU(NF ). We also focus on the cases with two, three and four colours, such that the theory is perturbatively renormalizable. On the SU(2) phase diagram we find two distinct CFTs with N = 2 supersymmetry, already known from the phase diagram of SU(2) theory with one flavor. As for the SU(3) phase diagram, consistency arguments lead us to conjecture that also there super- symmetry enhancement to N = 2 must occur. Finally, similar arguments lead us to conclude that also for the SU(4) case we should observe supersymmetry arXiv:1911.10034v1 [hep-th] 22 Nov 2019 enhancement at the fixed point. Yet, we find that this conclusion would be in contradiction with the renormalization group flow analysis. We relate this inconsistency between the two kinds of arguments to the fact that the theory is not UV complete. 1 Introduction One of the cornerstones in understanding of any quantum field theory (QFT) is the question about its infrared (IR) phase.