DOCSLIB.ORG

Collinear Superspace

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Collinear Superspace Collinear superspace The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Cohen, Timothy; Elor, Gilly and Larkoski, Andrew J. "Collinear superspace." Physical Review D 93, 125013 (June 2016): 1-5 © 2016 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.93.125013 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/110594 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 93, 125013 (2016) Collinear superspace Timothy Cohen,1 Gilly Elor,2 and Andrew J. Larkoski3 1Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA 2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Center for the Fundamental Laws of Nature, Harvard University, Cambridge, Massachusetts 02138, USA (Received 6 May 2016; published 10 June 2016) This paper provides a superfield based approach to constructing a collinear slice of N ¼ 1 superspace. The strategy is analogous to integrating out anticollinear fermionic degrees-of-freedom as was developed in the context of soft-collinear effective theory. The resulting Lagrangian can be understood as an integral over collinear superspace, where half the supercoordinates have been integrated out. The application to N ¼ 1 super Yang-Mills is presented. Collinear superspace provides the foundation for future explorations of supersymmetric soft-collinear effective theory. DOI: 10.1103/PhysRevD.93.125013 Supersymmetry (SUSY) is a powerful framework for SCET. Note that this procedure obscures the underlying exploring the properties of quantum field theory. There Lorentz invariance of the theory, leaving behind the are many examples of extraordinary results derived for constraints known as reparameterization invariance (RPI) SUSY models, for instance the exact NSVZ β-function [1], [35]. Given its spacetime nature, it is unclear that SUSY Seiberg duality [2], Seiberg-Witten theory [3,4], and can be preserved in any meaningful way. Our main result the finiteness of N ¼ 4 SUSY Yang-Mills (SYM) [5]. is to show how collinear superspace packages a SCET Identifying models that manifest SUSY in nontrivial ways Lagrangian in a language that makes the SUSY of the EFT has yielded many fruitful developments, see [6–13] for manifest. recent examples. In this paper, we explore a new class of To derive the collinear limit for a fermion requires N ¼ 1 SUSY effective field theory (EFT) models which integrating out the anticollinear modes, which in practice live on a “collinear slice” of superspace; defining this are half of the full theory fermion helicity degrees of collinear superspace is the subject of this paper. freedom (the momenta of the EFT fields are also con- The connection between collinear superspace and gauge strained). This procedure guides the construction here: the theories becomes apparent in the infrared (IR), where the EFT can be characterized in terms of half the supercharges physics can be largely inferred from the presence of soft for N ¼ 1 SUSY—the other half of the supersymmetries and collinear divergences. There is a rich history associated are nonlinearly realized. We refer to this as “integrating with the IR structure of gauge theories. For example, a out” half of superspace, which leave behind a collinear correspondence between the coefficients of Sudakov logs subsurface of superspace. Our procedure for deriving in Yang-Mills theory and the cusp anomalous dimension of collinear superspace, which should be generally applicable Wilson loops was discovered as early as 1980 [14]. The to a wide class of SUSY EFTs, can be described by the importance of these IR effects helped lead to the discovery following algorithm: of soft-collinear effective theory (SCET) [15–22], which is General Algorithm a powerful formalism developed for resumming the IR (i) Find projection operators that separate the divergences occurring for processes that are dominated by superfield into collinear/anti-collinear superfields soft (low momentum) and collinear degrees of freedom; see [e.g., Eq. (10)]. [23,24] for reviews. There exists an ever growing literature (ii) Starting with the superspace action for the full exploring practical applications of SCET to heavy meson theory, integrate out the entire anticollinear decays [21,22,25,26], LHC collisions [27–31], and even superfield. This will yield a constraint equation WIMP dark matter systems [32–34]. Our purpose here is to [e.g., Eq. (14)]. lay the groundwork for supersymmetrizing SCET, in hopes (iii) Based on the constraint equation, guess an ansatz of further illuminating nontrivial aspects of field theory. for the equation of motion for the anticollinear SCET can be understood in terms of a mode expansion, superfield in terms of collinear degrees-of-freedom where a power-counting parameter λ is used to separate [e.g., Eq. (16)]. degrees-of-freedom that are “near” a lightlike direction, (iv) Plug the ansatz into the full theory action to yield the thereby capturing the IR dynamics as an expansion in λ, superspace action of the effective theory [e.g., (19)]. from the “far” modes. Integrating out these “anticollinear” In what follows, we will apply this procedure to the explicit degrees-of-freedom yields the effective Lagrangian of case of N ¼ 1 SYM. 2470-0010=2016=93(12)=125013(5) 125013-1 © 2016 American Physical Society COHEN, ELOR, and LARKOSKI PHYSICAL REVIEW D 93, 125013 (2016) To begin, we will provide some conventions. The contractions the gauge boson Lagrangian density does scale SUSY EFT is defined in Minkowski space with signature homogeneously: F2 ∼ λ4. In what follows, we focus on the gμν ¼ diagðþ1; −1; −1; −1Þ. The collinear direction is collinear modes, as the soft modes can be decoupled at taken along the zˆ light-cone direction: nμ ¼ð1; 0; 0; 1Þ. leading power by a field redefinition [23,24]. The anticollinear direction is defined by n2 ¼ 0 ¼ n¯ 2 and Collinear superspace is on-shell, i.e., only physical n · n¯ ¼ 2. It is usually convenient to make the explicit degrees-of-freedom will be present in the Lagrangian. To choice n¯ μ ¼ð1; 0; 0; −1Þ. Four vectors are expanded as this end, it is convenient to work in light cone gauge (LCG) μ p ¼ðn · p; n¯ · p; p~ ⊥Þ, where “⊥” refers to the two direc- which corresponds to the non-(space-time)-covariant gauge tions perpendicular to both n and n¯. A state is collinear to choice n¯ · A ¼ 0, see e.g., [37] for a review. Additionally, the light-cone when it lives within a momentum shell which the mode n · A is nonpropagating in this gauge (with μ 2 scales as pn ∼ ðλ ; 1; λÞ, where λ ≪ 1 is the SCET power respect to light-cone time)—it can be integrated out by counting parameter. The virtuality for the collinear modes solving the classical equation of motion. The two remain- in the effective theory p2 ∼ λ2 can be interpreted as ing bosonic physical degrees of freedom, the transverse closeness to the light cone. Similarly, an anticollinear components of the gauge field, can be recast as a complex μ ∼ 1 λ2 λ scalar A, defined by momenta scales as pn¯ ð ; ; Þ. Fields also scale as powers of λ; the power counting rules can be inferred from ∂ ≡ −∂ÃA − ∂Aà the appropriate kinetic terms, and must be necessarily ⊥ · An⊥ ; ð4Þ tracked when determining the order of a given operator. where ∂ and ∂à are also implicitly defined by this As discussed previously, studying the collinear fermion equation [37]. Then L ¼ − 1 F2 → AÃ□A þ L . EFT will provide insight for the derivation of collinear 4 n int For concreteness, our focus here is on-shell N ¼ 1 superspace. A two-component left-handed Weyl spinor can SYM. The fermionic degree of freedom u (the single be decomposed into collinear and anticollinear momentum n remaining spin state in the EFT after the SCET projection) modes using projection operators; u ¼ðP þ P¯ Þu ¼ n n is the collinear gaugino whose superpartner is the bosonic u þ u¯ , where n n light cone scalar A.In[36], we will provide a detailed n · σ n¯ · σ¯ n¯ · σ n · σ¯ derivation of the corresponding collinear SCET Lagrangian Pn ¼ 2 2 Pn¯ ¼ 2 2 : ð1Þ along with a demonstration that it passes checks necessary for EFT consistency, e.g., RPI. These also correspond to chiral projection operators that The N ¼ 1 supercharges are defined by the graded distinguish the fermion’s spin states in the collinear limit † μ algebra Qα;Q 2σ Pμ, where the spinor and anti- (a detailed discussion of two-component collinear fermions f α_ g¼ αα_ α α_ 1 will be given in a forthcoming paper [36]). The anticol- spinor indices run over ; ¼ , 2. Power counting the 2 generator of translations Pμ ¼ i∂μ as appropriate for collin- linear modes u¯ , which scale as Oðλ Þ, are power sup- n ear momenta, yields the scaling of the algebra in the EFT: pressed relative to the collinear ones un ∼ OðλÞ. Therefore, pffiffiffi u¯ should be integrated out using the classical equation of à 2 n † n · ∂ 2∂ Oðλ Þ OðλÞ motion: fQα;Qα_ g¼2i pffiffiffi ∼ ; 2∂ n¯ · ∂ αα_ OðλÞ Oð1Þ n¯ · σ 1 u¯ ¼ − ðσ¯ · D⊥Þu ; ð2Þ ð5Þ n 2 n¯ · D n yielding the following Lagrangian for a charged collinear from which we can infer fermion Q2 ∼ Oð1Þ;Q1 ∼ OðλÞ; 1 ¯ σ¯ † † † n · Q_ ∼ Oð1Þ;Q_ ∼ OðλÞ: ð6Þ L ¼ iu n · D þ σ¯ · D⊥ σ · D⊥ u ; ð3Þ 2 1 u n n¯ · D 2 n To leading power, only one supercharge (Q2) is present in the where D is the covariant derivative appropriately power EFT.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Interpreting Supersymmetry
    Interpreting Supersymmetry David John Baker Department of Philosophy, University of Michigan [email protected] October 7, 2018 Abstract Supersymmetry in quantum physics is a mathematically simple phenomenon that raises deep foundational questions. To motivate these questions, I present a toy model, the supersymmetric harmonic oscillator, and its superspace representation, which adds extra anticommuting dimensions to spacetime. I then explain and comment on three foundational questions about this superspace formalism: whether superspace is a sub- stance, whether it should count as spatiotemporal, and whether it is a necessary pos- tulate if one wants to use the theory to unify bosons and fermions. 1 Introduction Supersymmetry{the hypothesis that the laws of physics exhibit a symmetry that transforms bosons into fermions and vice versa{is a long-standing staple of many popular (but uncon- firmed) theories in particle physics. This includes several attempts to extend the standard model as well as many research programs in quantum gravity, such as the failed supergravity program and the still-ascendant string theory program. Its popularity aside, supersymmetry (SUSY for short) is also a foundationally interesting hypothesis on face. The fundamental equivalence it posits between bosons and fermions is prima facie puzzling, given the very different physical behavior of these two types of particle. And supersymmetry is most naturally represented in a formalism (called superspace) that modifies ordinary spacetime by adding Grassmann-valued anticommuting coordinates. It 1 isn't obvious how literally we should interpret these extra \spatial" dimensions.1 So super- symmetry presents us with at least two highly novel interpretive puzzles. Only two philosophers of science have taken up these questions thus far.
    [Show full text]
  • Lectures on Super Analysis
    Lectures on Super Analysis —– Why necessary and What’s that? Towards a new approach to a system of PDEs arXiv:1504.03049v4 [math-ph] 15 Dec 2015 e-Version1.5 December 16, 2015 By Atsushi INOUE NOTICE: COMMENCEMENT OF A CLASS i Notice: Commencement of a class Syllabus Analysis on superspace —– a construction of non-commutative analysis 3 October 2008 – 30 January 2009, 10.40-12.10, H114B at TITECH, Tokyo, A. Inoue Roughly speaking, RA(=real analysis) means to study properties of (smooth) functions defined on real space, and CA(=complex analysis) stands for studying properties of (holomorphic) functions defined on spaces with complex structure. On the other hand, we may extend the differentiable calculus to functions having definition domain in Banach space, for example, S. Lang “Differentiable Manifolds” or J.A. Dieudonn´e“Trea- tise on Analysis”. But it is impossible in general to extend differentiable calculus to those defined on infinite dimensional Fr´echet space, because the implicit function theorem doesn’t hold on such generally given Fr´echet space. Then, if the ground ring (like R or C) is replaced by non-commutative one, what type of analysis we may develop under the condition that newly developed analysis should be applied to systems of PDE or RMT(=Random Matrix Theory). In this lectures, we prepare as a “ground ring”, Fr´echet-Grassmann algebra having count- ably many Grassmann generators and we define so-called superspace over such algebra. On such superspace, we take a space of super-smooth functions as the main objects to study. This procedure is necessary not only to associate a Hamilton flow for a given 2d 2d system × of PDE which supports to resolve Feynman’s murmur, but also to make rigorous Efetov’s result in RMT.
    [Show full text]
  • Torsion Constraints in Supergeometry
    Commun. Math. Phys. 133, 563-615(1990) Communications ΪΠ Mathematical Physics ©Springer-Verlagl990 Torsion Constraints in Supergeometry John Lott* I.H.E.S., F-91440 Bures-sur-Yvette, France Received November 9, 1989; in revised form April 2, 1990 Abstract. We derive the torsion constraints for superspace versions of supergravity theories by means of the theory of G-stmctures. We also discuss superconformal geometry and superKahler geometry. I. Introduction Supersymmetry is a now well established topic in quantum field theory [WB, GGRS]. The basic idea is that one can construct actions in ordinary spacetime which involve both even commuting fields and odd anticommuting fields, with a symmetry which mixes the two types of fields. These actions can then be interpreted as arising from actions in a superspace with both even and odd coordinates, upon doing a partial integration over the odd coordinates. A mathematical framework to handle the differential topology of supermanifolds, manifolds with even and odd coordinates, was developed by Berezin, Kostant and others. A very readable account of this theory is given in the book of Manin [Ma]. The right notion of differential geometry for supermanifolds is less clear. Such a geometry is necessary in order to write supergravity theories in superspace. One could construct a supergeometry by ΊL2 grading what one usually does in (pseudo) Riemannian geometry, to have supermetrics, super Levi-Civita connections, etc. The local frame group which would take the place of the orthogonal group in standard geometry would be the orthosymplectic group. However, it turns out that this would be physically undesirable. Such a program would give more fields than one needs for a minimal supergravity theory, i.e.
    [Show full text]
  • Analysis on Superspace: an Overview
    BULL. AUSTRAL. MATH. SOC. 58C50, 15A75, 17B70, 46S20, 81Q60 VOL. 50 (1994) [135-165] ANALYSIS ON SUPERSPACE: AN OVERVIEW VLADIMIR G. PESTOV The concept of superspace is fundamental for some recent physical theories, notably supersymmetry, and a mathematical feedback for it is provided by superanalysis and supergeometry. We survey the state of affairs in superanalysis, shifting our attention from supermanifold theory to "plain" superspaces. The two principal existing approaches to superspaces are sketched and links between them discussed. We examine a problem by Manin of representing even geometry (analysis) as a collective effect in infinite-dimensional purely odd geometry (analysis), by applying the technique of nonstandard (infinitesimal) analysis. FOREWORD "Superspace is the greatest invention since the wheel" (see [59]). I don't know how many of us would agree with this radical statement, and it was only intended to be a jocular epigraph, of course. However, the concept of superspace is thriving indeed in theoretical physics of our days. Under superspace physicists understand our space-time labeled not only with usual coordinates (x11), but also with anticommuting coordinates (0a), and endowed with a representation of a certain group or, better still, supergroup, whatever it means. The anticommuting coordinates represent certain internal degrees of freedom of the physical system. This concept was actually known in physics since the early 60s or late 50s [138, 138, 91, 28], but what made it so important was the discovery of supersymmetry [144, 131, 145], which is an essentially new form of symmetry unifying fermions (particles of matter) with bosons (particles carrying interaction between bosons).
    [Show full text]
  • Introduction to Supersymmetry PHYS-F-417
    Introduction to Supersymmetry PHYS-F-417 Riccardo Argurio Universit´eLibre de Bruxelles December 18, 2020 2 Contents 1 Introduction 7 1.1 Motivating supersymmetry . 7 1.2 Plan of the lecture notes . 14 1.3 Further reading . 14 2 The superalgebra 17 2.1 A graded extension of the Poincar´ealgebra . 17 2.2 Representations of the Lorentz group . 18 2.3 A first look at the superalgebra . 25 2.3.1 Basic consequences of the superalgebra . 27 2.4 Representations of the superalgebra . 29 2.4.1 Massless supermultiplets . 29 2.4.2 Supermultiplets of extended supersymmetry . 31 2.4.3 Massive supermultiplets . 33 3 Supersymmetric field theories 37 3.1 The theory for the scalar multiplet . 37 3.1.1 Supersymmetry variations on scalar and fermionic fields 38 3.1.2 Introducing auxiliary fields . 43 3.1.3 Supersymmetric mass terms . 45 3.1.4 Supersymmetric interactions . 47 3.2 The theory for the vector multiplet . 48 4 Superspace and superfields 53 4.1 Introducing superspace and superfields . 53 4.1.1 Supersymmetry transformations as translations in su- perspace . 56 4.1.2 Manifestly supersymmetry invariant action . 59 3 4 CONTENTS 4.2 Chiral superfields . 61 4.2.1 Action for a chiral superfield . 64 4.2.2 Superpotential . 67 4.2.3 Equations of motion for chiral superfields . 69 4.3 Real superfields . 71 4.3.1 Gauge symmetry and the vector multiplet . 72 4.3.2 The action and the gaugino chiral superfield . 78 4.3.3 Equations of motion for real superfields . 83 4.3.4 Fayet-Iliopoulos term .
    [Show full text]
  • On Supersymmetric Quantum Mechanics Maurice Kibler, Mohammed Daoud
    On supersymmetric quantum mechanics Maurice Kibler, Mohammed Daoud To cite this version: Maurice Kibler, Mohammed Daoud. On supersymmetric quantum mechanics. E. Brandas and E.S. Kryachko. Fundamental World of Quantum Chemistry, A Tribute to the Memory of Per-Olov Lowdin, Volume 3, Springer-Verlag, pp.67-96, 2004. hal-00002942 HAL Id: hal-00002942 https://hal.archives-ouvertes.fr/hal-00002942 Submitted on 25 Sep 2004 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. ON SUPERSYMMETRIC QUANTUM MECHANICS M.R. KIBLER Institut de Physique Nucl´eaire de Lyon IN2P3-CNRS et Universit´eClaude Bernard F-69622 Villeurbanne Cedex, France and M. DAOUD D´epartement de Physique Facult´edes Sciences Universit´eIbnou Zohr B.P. 28/S, Agadir, Morocco Abstract. This paper constitutes a review on = 2 fractional supersymmetric Quantum Mechanics of order k. The presentationN is based on the introduction of a generalized Weyl-Heisenberg algebra Wk. It is shown how a general Hamil- tonian can be associated with the algebra Wk. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, P¨oschl- Teller system, fractional supersymmetric oscillator of order k, etc.).
    [Show full text]
  • Antum Black Holes
    antum Black Holes Some static and dynamical aspects Navaneeth Krishna Gaddam Reading committee Prof. dr. G. ’t Hooft, Universiteit Utrecht Prof. dr. B. Q. P. J. de Wit, Universiteit Utrecht Prof. dr. H. T. C. Stoof, Universiteit Utrecht dr. T. W. Grimm, Universiteit Utrecht Prof. dr. K. E. Schalm, Universiteit Leiden Cover design: Vishnu Pasupathy (www.vishnupasupathy.com) isbn 978-90-393-6737-7 Copyright © 2016 by Navaneeth Krishna Gaddam Printed by Ipskamp Printing, Enschede. Quantum Black Holes Some static and dynamical aspects Kwantum Zwarte Gaten Sommige statische en dynamische aspecten (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnicus, prof. dr. G. J. van der Zwaan, ingevolge het besluit van het college van promoties in het openbaar te verdedigen op woensdag 15 maart 2017 des middags te 12.45 uur door Navaneeth Krishna Gaddam geboren op 21 augustus 1988 te Hyderabad, India Promotor: Prof. dr. S. J. G. Vandoren This thesis was nancially supported by the Netherlands Organisation for Scientic Re- search (nwo) under the vici grant 680-47-603, and the Delta-Institute for Theoretical Physics (d-itp) that is funded by the Dutch Ministry of Education, Culture and Science (ocw). To people of all genders, the genderless and queer, to those who can but mostly to those who cannot, to those who believe and those who do not, to the downtrodden and the hopeless, to the powerless and the oppressed, to the mentally ill, to the layman. But most of all, to you ..
    [Show full text]
  • Lectures on Geometry of Supersymmetry
    LECTURES ON GEOMETRY OF SUPERSYMMETRY SI LI ABSTRACT. To be polished. This is note for my course on supersymmetry in the fall 2017 at Tsinghua university. We discuss supersymmetric gauge theories in various dimensions, their geometric structures and dualities. CONTENTS 1. Clifford algebra 2 2. Spin group 5 3. Spinor 8 3.1. Real spin representation 8 3.2. Complex spin representation 10 3.3. Spinors in physics 11 3.4. Unitarity 12 3.5. Charge conjugation 14 4. Poincare´ group 17 4.1. Poincare´ algebra 18 4.2. Unitary representation 18 4.3. Coleman-Mandula Theorem 19 5. SUSY algebra 19 5.1. Super Lie algebra 19 5.2. Super Poincare´ algebra 20 5.3. Physics degree of freedom 21 5.4. d=2 and R× 22 5.5. d=3 and SL(2, R) 22 5.6. d=4 and SL(2, C) 23 5.7. d=6 and SL(2, H) 25 5.8. d=10 27 6. N=1 Super Yang-Mills 28 6.1. Normed division algebra 28 6.2. N=1 Super Yang-Mills 30 1 2 SI LI 6.3. Berkovits construction 31 7. Supersymmetry in D=4 32 7.1. SUSY representations 32 7.2. Superspace 35 7.3. N=1 chiral multiplet 36 7.4. N=1 vector multiplet 39 7.5. N=1 gauge theory with matter 41 7.6. N=2 vector multiplet 41 7.7. N=2 hyper multiplet 42 7.8. N=2 gauge theory with matter 42 8. Seiberg-Witten theory 42 8.1. Electro-Magnetic duality 42 8.2.
    [Show full text]
  • SUSY Localization and Ads Black Holes
    Imperial College London Department of Physics SUSY Localization and AdS Black Holes Author: Supervisor: Tymothy Mangan Prof. Daniel Waldram A thesis submitted for the degree of MSc Quantum Fields and Fundamental Forces October 23, 2020 Abstract Equivariant localization has a natural extension to supersymmetric field theories, which directly ex- ploits the particular nature of supermanifolds to reduce dimensionality of an otherwise intractable path integral to a 1-loop exact form. This paper walks through pedagogically and historically inter- esting examples where localization naturally appears, performing calculations such as the twisted Dirac index. Subsequently after the general ideas have been revealed naturally using physics tech- niques in field theories, the specific description of the mathematics of equivariant cohomology, the cartan model, localization, and the Atiyah-Bott-Berline-Vergne formula are discussed. With the relevant structure of localization presented a direct analogy and extension to supersymmet- ric localization is presented. Concluding is a digression on AdS black holes with the intent of establishing readily apparent application of supersymmetric localization to make exact quantum entropy calculations or perform analysis of holography. Bekenstein-Hawking radiation, Wald en- tropy, AdS/CFT, and a quantum entropy formula for supersymmetric extremal black holes are reviewed. Contents 1 Localization in 0 and 1 dimensions, a heuristic introduction 1 1.1 Localization in a simple 0-d Supersymmetric QFT . .3 1.1.1 Deforming the path integral . .5 1.1.2 The 0-d analog of the Witten/Supersymmetric Index . .6 1.1.3 Landau-Ginzburg and the localization principle . .8 1.2 Brief overview of SUSY QM, the Witten Index, and our first Index theorem .
    [Show full text]
  • Path Integrals for Fermions, Susy Quantum Mechanics, Etc.. (Appunti Per Il Corso Di Fisica Teorica 2 – 2012/13) Fiorenzo Bastianelli
    Path integrals for fermions, susy quantum mechanics, etc.. (Appunti per il corso di Fisica Teorica 2 { 2012/13) Fiorenzo Bastianelli Fermions at the classical level can be described by Grassmann variables, also known as anticommuting or fermionic variables. These variables allow to introduce \classical" models that upon quantization produce the degrees of freedom associated to spin. Actually, rather than \classical" models one should call them \pseudoclassical", since the description of spin at the classical level is a formal construction (spin vanish for ~ ! 0). However, quantization of Grassmann variables gives rise to spin degrees of freedom, and for simplicity we keep using the term classical for such models. In worldline approaches to quantum field theories, one often describes relativistic point particles with spin by using worldline coordinates for the position of the particle in space-time and Grassmann variables to account for the associated spin degrees of freedom. From the worldline point of view these Grassmann variables have equations of motion that are first order in time, and give rise to a Dirac equation in one dimension (the worldline can be considered as a 0+1 dimensional space-time, with the one dimension corresponding to the time direction). They are fermions on the worldline and obey the Pauli exclusion principle, even though there is no real spin in a zero dimensional space. In the following we study the path integral quantization of models with Grassmann variables, and refer to them as path integrals for fermions, or fermionic path integrals. In a hypercondensed notation the resulting formulae describe the quantization of a Dirac field in higher dimensions as well.
    [Show full text]
  • Supersymmetric Localization in Two Dimensions
    SISSA 27/2016/FISI Imperial/TP/2016/FB/01 Supersymmetric localization in two dimensions Francesco Benini1,2 and Bruno Le Floch3 1International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy 2Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom 3Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA [email protected], [email protected] Abstract This is an introductory review to localization techniques in supersymmetric two-dimensional gauge theories. In particular we describe how to construct Lagrangians of =(2, 2) theories on curved spaces, and how to compute their partition functions and certainN correlators on the sphere, the hemisphere and other curved backgrounds. We also describe how to evaluate the partition function of =(0, 2) theories on the torus, known as the elliptic genus. Finally we summarize some of theN applications, in particular to probe mirror symmetry and other non-perturbative dualities. This is a contribution to the review volume “Localization techniques in quantum field theories” (eds. V. Pestun and M. Zabzine) which contains 17 Chapters available at [1] Contents 1 Introduction 2 2 N =(2, 2) gauge theories on spheres 5 2.1 Multiplets, Lagrangians and supersymmetry . ........ 6 2.2 BPSequations................................... 8 arXiv:1608.02955v4 [hep-th] 15 Oct 2016 2.3 Coulombbranchlocalization . .... 10 2.4 TheA-system................................... 13 2.5 Higgsbranchlocalization. .... 15 2.6 One-loopdeterminants . .. 18 3 Other N =(2, 2) curved-space results 21 3.1 Localoperatorinsertions . .... 21 3.2 Twisted multiplets on the sphere . .... 23 3.3 Localizationonthehemisphere . .... 26 3.4 Ω-deformedA-twistonthesphere . .... 29 3.5 General supersymmetric backgrounds .
    [Show full text]