DOCSLIB.ORG

Some Results in Supergeometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Results in Supergeometry Some results in supergeometry Harmonic maps from super Riemann surfaces and Automorphism supergroups of supermanifolds Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨atzu K¨oln vorgelegt von Dominik Ostermayr aus Dachau 2017 Berichterstatter: PD Dr. Alexander Alldridge Prof. Dr. George Marinescu Prof. Dr. Tilmann Wurzbacher Tag der m¨undlichen Pr¨ufung: 20. Januar 2017 Kurzzusammenfassung Die vorliegende Arbeit besteht aus zwei unabh¨angigenund eigenst¨andigen Teilen. Gegenstand des ersten Teils sind harmonische Abbildungen von super-Riemannschen Fl¨achen nach komplex-projektiven R¨aumenund projektiven R¨aumenbez¨uglich des Super- schiefk¨orpers D: In beiden F¨allenwird die Theorie der Gauß-Transformierten entwickelt und der Begriff der Isotropie studiert, insbesondere mit Hinblick auf den Zusammenhang zu holomorphen Differentialen auf der super-Riemannschen Fl¨ache. Uberdies¨ geben wir eine Definition f¨urharmonische Abbildungen endlichen Typs f¨ureine spezielle Klasse von n n+1 Abbildungen nach CP j und erhalten so eine Klassifikation bestimmter harmonischer super-Tori. Ferner untersuchen wir die Gleichungen, die von den unterliegenden Objek- ten erf¨ulltwerden und geben ein Beispiel eines harmonischen super-Torus in DP 2 dessen unterliegende Abbildung nicht harmonisch ist. Im zweiten Teil studieren wir einen klassischen Satz, der besagt, dass die Gruppe der Automorphismen einer Mannigfaltigkeit, die eine G-Struktur endlichen Typs erhalten, eine Lie-Gruppe bildet, im Kontext von Supermannigfaltigkeiten. Wir verallgemeinern dieses Theorem auf die Kategorie der cs Mannigfaltigkeiten und illustrieren es anhand einiger, sowohl klassische Objekte verallgemeinernder als auch genuin supergeometrischer, Beispiele. Insbesondere ist es n¨otigeine neue Klasse von Supermannigfaltigkeiten einzuf¨uhren - gemischte Supermannigfaltigkeiten. iii Abstract This thesis consists of two independent and self-contained parts. The first part is concerned with harmonic maps form super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field D: In both cases, we develop the theory of Gauß transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a n n+1 definition of finite type harmonic maps for a special class of maps into CP j and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in DP 2 whose underlying map is not harmonic. In the second part, we study a classical theorem stating that the group of automorphisms of a manifold M preserving a G-structure of finite type is a Lie group in the context of supermanifolds. We generalize this statement to the category of cs manifolds and give some examples, some of which being generalizations of classical notions, others being particular to the super case. Notably, we have to introduce a new class of supermanifolds which we call mixed supermanifolds. v Contents Kurzzusammenfassung iii Abstract v Harmonic maps from super Riemann surfaces 1 Automorphism supergroups of supermanifolds 87 Erkl¨arung 125 vii Harmonic maps from super Riemann surfaces Dominik Ostermayr Contents 1 Introduction 3 2 Supergeometry 10 2.1 Recollections on supergeometry . 10 2.1.1 Supermanifolds . 10 2.1.2 Functor of points approach . 10 2.1.3 Tangent bundles . 10 2.1.4 Lie supergroups . 11 2.1.5 The forgetful functor . 11 2.2 The supergroup U(n m)............................. 11 | 2.3 The supergroup Q(n) .............................. 15 n m 2.4 The Grassmannians Grk l(C | ) ........................ 17 | 2.5 The projective spaces DP n ........................... 19 3 Super Riemann surfaces 20 3.1 Basics . 20 3.2 Examples of super Riemann surfaces . 21 3.2.1 The superconformal plane . 21 3.2.2 Split super Riemann surfaces . 22 3.2.3 Genus 0 . 23 3.2.4 Genus 1 . 24 3.3 Points, Divisors, Infinitesimal Neighbourhoods . 26 3.4 Blow ups of JP and parabolic structures . 27 3.4.1 Construction . 28 3.4.2 Lifting the superconformal structure to a parabolic structure . 29 3.5 Holomorphic sections of vector bundles and their regularity . 29 3.6 Connections on super Riemann surfaces . 30 3.6.1 Koszul-Malgrange holomorphic structures . 30 3.6.2 Flat g-valued 1-forms . 32 4 Generalities on harmonic maps 32 5 Harmonic maps into U(n m) 34 | 5.1 Zero curvature representation . 34 5.2 Harmonic maps and framings . 35 5.3 Supersymmetric harmonic maps and elliptic integrable systems . 36 5.3.1 Basics on elliptic integrable systems . 36 5.3.2 The underlying map of a supersymmetric harmonic map in terms of a framing . 37 n m n m 6 Harmonic maps into Grk l(C | ) and the special case CP | 39 | 6.1 The Gauß transform . 39 6.2 Isotropic harmonic maps . 41 6.3 Examples for the Gauß transform . 45 n n+1 6.4 2(n + 1)-orthogonal non-isotropic harmonic maps in CP | . 48 6.4.1 The 2(n + 1)-symmetric space P SU(n + 1 n + 1)/P ST . 48 | 6.4.2 Primitive maps . 50 6.4.3 Primitive maps from pseudo-commuting flows . 53 6.4.4 Finite type classification of 2(n+1)-orthogonal non-isotropic harmonic tori . 58 6.4.5 Example: The case n =1 ........................ 63 7 Harmonic maps into DP n 66 7.1 The Gauß transform . 66 7.2 Isotropic harmonic maps . 67 7.3 Examples for the Gauß transform . 70 7.4 Periodic harmonic sequences in DP 2n ...................... 71 7.4.1 The (2n + 1)-symmetric space P SQ(2n + 1)/P SQ(1)2n+1 . 71 7.4.2 Primitive maps . 72 7.4.3 A class of 2n + 1-orthogonal non-isotropic harmonic maps . 74 7.4.4 Example: The case n =1 ........................ 76 8 Outlook 78 2 1 Introduction Harmonic maps and supergeometry The purpose of this study is to prove some foundational results about harmonic maps in supergeometry. More precisely, we study harmonic maps from a super Riemann surface into complex projective spaces and in special cases into general complex Grassmannians. Harmonic maps from Riemann surfaces into various target spaces are by now a classical topic in differential geometry. Such maps occur naturally in surface theory, for instance. The parametrization of a surface in R3 is minimal if and only if it is conformal and harmonic. It has constant mean curvature if and only if its Gauß map is harmonic. We refer to [33] for a treatment of these results. In the context of the anti-self-dual Yang-Mills equation, such maps appear as a symmetry reduction from four to two dimensions [53]. Consequently, a central problem is to develop techniques which allow for a classification and construction of such maps. For a review of this broad subject, we refer the reader to the survey articles [24, 25]. Closer to the specific subject of the present article are [11, 18, 56, 57]. Supergeometry is the extension of ordinary geometry which allows for commuting and anti- commuting coordinate functions. Many notions, constructions, and results from differential geometry carry over to the graded setting directly. In particular, there is a notion of Riemannian supermanifolds. However, the Riemannian structure might be even or odd. Another genuinely supergeometric notion is supersymmetry, the simplest instance of which is the concept of a super Riemann surface. The complex analytic properties have been studied in pioneering works in the 1980s, among others [2, 51], and more recently in [54, 55]. In this setting there exists a natural notion of harmonic maps from super Riemann surfaces to Riemannian supermanifolds [20, 34, 36, 46] which are the central objects of this article. In view of the plethora of results available in the non-graded setup, it is beyond the scope of this thesis to give a comprehensive treatment. Instead, we will concentrate on some selected aspects. Gauß transform, isotropy and harmonic maps of finite type In order to put our results into context, we first give a brief account on the relevant results in the ungraded setting. The energy of a map f : Σ M between a compact Riemann → surface and a Riemannian manifold is defined by E(f) = df (0,1) , df (1,0) , h C|T Σ C|T Σ iC ZΣ where , denotes the complex bilinear extension of the given Riemannian structure to h− −iC TMC. Critical points are called harmonic maps and are characterized in a local complex 3 coordinate z by LC (df )(∂ ) = 0, ∇∂z¯ C z where LC denotes the pullback of the Levi-Civita connection. Due to a result of Koszul ∇ and Malgrange [39], a complex vector bundle with connection (E, ) on a Riemann surface ∇ has a holomorphic structure such that the holomorphic sections are locally characterized by s = 0. ∇∂z¯ Using this result, harmonicity can be stated in a coordinate free manner. The map f is (1,0) harmonic if and only if (df ) (1,0) is a holomorphic section of (T Σ ) TM , where C |T Σ ∗ ⊗ C the second factor is equipped with the Koszul-Malgrange structure. In particular, the differential either vanishes identically or its zeros are isolated. In the case M = CP n, the harmonic map equation is equivalent to LC df (1,0)(∂ ) = 0, (1.1) ∇∂z¯ z (1,0) (0,1) n where dfC = df + df according to the type decomposition on CP . In view of the n (1,0) isomorphism (T CP ) ∼= Hom(γ, γ⊥), where γ is the tautological line bundle, if f is not (1,0) 1+n antiholomorphic, df (∂z) defines a line in C outside a discrete set of points. One n can always extend this to a give a new map f1 : Σ CP , the Gauß transform.
Load more

Recommended publications
  • Jhep01(2020)007
    Published for SISSA by Springer Received: March 27, 2019 Revised: November 15, 2019 Accepted: December 9, 2019 Published: January 2, 2020 Deformed graded Poisson structures, generalized geometry and supergravity JHEP01(2020)007 Eugenia Boffo and Peter Schupp Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany E-mail: [email protected], [email protected] Abstract: In recent years, a close connection between supergravity, string effective ac- tions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded ge- ometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the 2-graded symplectic manifold T ∗[2]T [1]M that are based on a metric g, a closed Neveu-Schwarz 3-form H (locally expressed in terms of a Kalb-Ramond 2-form B) and a scalar dilaton φ. The derived bracket formalism relates this structure to the generalized differential geometry of a Courant algebroid, which has the appropriate stringy symme- tries, and yields a connection with non-trivial curvature and torsion on the generalized “doubled” tangent bundle E =∼ TM ⊕ T ∗M. Projecting onto TM with the help of a natural non-isotropic splitting of E, we obtain a connection and curvature invariants that reproduce the NS-NS sector of supergravity in 10 dimensions. Further results include a fully generalized Dorfman bracket, a generalized Lie bracket and new formulas for torsion and curvature tensors associated to generalized tangent bundles.
    [Show full text]
  • Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and Their Applications
    ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 415–471 INTRODUCTION TO GRADED GEOMETRY, BATALIN-VILKOVISKY FORMALISM AND THEIR APPLICATIONS Jian Qiu and Maxim Zabzine Abstract. These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV–formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present a proof of the isomorphism between the graph complex and the Chevalley-Eilenberg complex of formal Hamiltonian vectors fields. We briefly discuss how these ideas can be extended to the infinite dimensional setting. These notes should be accessible toboth physicists and mathematicians. Table of Contents 1. Introduction and motivation 416 2. Supergeometry 417 2.1. Idea 418 2.2. Z2-graded linear algebra 418 2.3. Supermanifolds 420 2.4. Integration theory 421 3. Graded geometry 423 3.1. Z-graded linear algebra 423 3.2. Graded manifold 425 4. Odd Fourier transform and BV-formalism 426 4.1. Standard Fourier transform 426 4.2. Odd Fourier transform 427 4.3. Integration theory 430 4.4. Algebraic view on the integration 433 5. Perturbation theory 437 5.1. Integrals in Rn-Gaussian Integrals and Feynman Diagrams 437 2010 Mathematics Subject Classification: primary 58A50; secondary 16E45, 97K30. Key words and phrases: Batalin-Vilkovisky formalism, graded symplectic geometry, graph homology, perturbation theory.
    [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]
  • Geometry of BV Quantization and Mathai-Quillen Formalism
    Alma Mater Studiorum UniversitadiBologna` · Scuola di Scienze Corso di Laurea Magistrale in Fisica Geometry of BV Quantization and Mathai-Quillen Formalism Relatore: Presentata da: Prof. Roberto Zucchini Luigi Tizzano Correlatore: Prof. Maxim Zabzine Sessione II Anno Accademico 2012/2013 Abstract. Il formalismo Mathai-Quillen (MQ) `eun metodo per costruire la classe di Thom di un fibrato vettoriale attraverso una forma differenziale di profilo Gaussiano. Lo scopo di questa tesi ´e quello di formulare una nuova rappresentazione della classe di Thom usando aspetti geometrici della quantizzazione Batalin-Vilkovisky (BV). Nella prima parte del lavoro vengono riassunti i formalismi BV e MQ entrambi nel caso finito dimensionale. Infine sfrutteremo la trasformata di Fourier “odd" considerando la forma MQ come una funzione definita su un opportuno spazio graduato. Contents 1 Introduction 1 2 Super Linear Algebra 4 2.1 Super Vector Spaces 4 2.1.1 Rule of Signs 5 2.2 Superalgebras 6 2.2.1 Supertrace 7 2.2.2 Berezinian 7 2.3 Berezin Integration 10 2.4 Change of Coordinates 10 2.5 Gaussian Integration 13 3 Supermanifolds 17 3.1 Presheaves and Sheaves 17 3.2 Integration Theory 19 4 Graded Geometry 23 4.1 Graded Linear Algebra 23 4.2 Graded Manifold 24 5 Odd Fourier transform and BV-formalism 26 5.1 Odd Fourier Transform 26 5.2 Integration Theory 29 5.3 Algebraic Aspects of Integration 31 5.4 Geometry of BV Quantization 35 5.4.1 The BV Gauge Fixing 37 6 The Mathai-Quillen Formalism 38 6.1 General Remarks on Topological Quantum Field Theories 38 6.2 Euler
    [Show full text]
  • Superspace Conformal Field Theory
    Superspace conformal field theory Thomas Quella1 and Volker Schomerus2 1 Universit¨at zu K¨oln, Institut f¨ur Theoretische Physik, Z¨ulpicher Straße 77, 50937 Cologne, Germany 2 DESY, Theory Group, Notkestrasse 85, Bldg. 2a, 22603 Hamburg, Germany Emails: [email protected], [email protected] Abstract Conformal sigma models and WZW models on coset superspaces provide important exam- ples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We review recent results and developments, including the general construction of WZW models on type I supergroups, the classification of conformal sigma models and their embedding into string theory. Keywords: Review, Logarithmic Conformal Field Theory, Conformal Field Theory, String Theory, Supersymmetry, Lie Superalgebras, Representation Theory, Supergeometry, Harmonic Analysis. Contents 1 Introduction 2 2 Lie superalgebras 4 2.1 Definition and examples of Lie superalgebras . ..... 4 2.2 Classification of Lie superalgebras . .... 6 2.3 Representationtheory .............................. .... 7 2.3.1 Introductionandoverview. 7 2.3.2 Categorytheoreticperspective . ... 11 2.4 Harmonic analysis on supermanifolds . .... 12 2.4.1 Guide to the literature on supergeometry . ... 12 2.4.2 Supergroups.................................... 13 2.4.3 Harmonic analysis on supercosets . .. 16 arXiv:1307.7724v1 [hep-th] 29 Jul 2013 2.5 Cohomologicalreduction.. .. .. .. .. .. .. .. .. .. .. .. .. ... 17 2.5.1 Generalformalism ................................ 17 2.5.2 Application: Function spaces on supercosets . ..... 18 3 Supergroup WZW models 19 3.1 Theactionfunctional................................ ... 19 3.2 Free fermion realization for type I supergroups . ........ 20 3.3 Algebraictreatment ................................. .. 22 3.3.1 Currentalgebra.................................. 22 3.3.2 Free fermion realization . 23 3.3.3 Chiralrepresentationtheory.
    [Show full text]
  • BRST and Superfield Formalism—A Review
    universe Review BRST and Superfield Formalism—A Review Loriano Bonora 1,* and Rudra Prakash Malik 2 1 International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy 2 Center of Advance Studies, Physics Department, Institute of Science, Banaras Hindu University, Varanasi 221 005, India; [email protected] * Correspondence: [email protected] Abstract: This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the N = 1 super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well. Keywords: gauge invariant theories; BRST and anti-BRST symmetry; superfield formalism; diffeo- morphism invariance; consistent gauge anomalies; HS gauge symmetry; N = 1 super-Yang-Mills 1. Introduction Citation: Bonora, L.; Malik, R.P. The discovery of the BRST symmetry in gauge field theories, refs. [1–4], is a funda- BRST and Superfield mental achievement in quantum field theory. This symmetry is not only the building block Formalism—A Review.
    [Show full text]
  • [Math.AG] 2 Apr 2018 Supergeometry in Mathematics and Physics
    Supergeometry in mathematics and physics Mikhail Kapranov April 3, 2018 Contents 0 Introduction 2 1 Supergeometry as understood by mathematicians 3 1.1 Commutativesuperalgebras . ... 3 1.2 The symmetric monoidal category of super-vector spaces ........... 4 1.3 Superschemes and supermanifolds . ..... 6 1.4 Liesupergroupsandsuperalgebras . ..... 7 2 Supergeometry as understood by physicists 10 2.1 The idea of non-observable square roots . ....... 10 2.2 A square root of d dt andtheta-functions . 12 2.3 Squarerootsofspacetimetranslations{ . ....... 14 2.4 Quadratic spaces and intersections of quadrics . .......... 15 2.5 Supersymmetry, superspace and constraints . ........ 17 2.6 Constraints and complete intersection slices . .......... 19 3 Homotopy-theoretic underpinnings of supergeometry 21 3.1 The skeleton of the Koszul sign rule . ..... 21 3.2 (Higher)Picardgroupoidsandspectra . ...... 23 3.3 The sphere spectrum and the free Picard n-groupoid ............. 24 arXiv:1512.07042v2 [math.AG] 2 Apr 2018 3.4 Towardshighersupergeometry. .... 26 1 0 Introduction In die Traum- und Zaubersphäre Sind wir, scheint es, eingegangen. Goethe, Faust I Supergeometry is a geometric tool which is supposed to describe supersymmetry (“sym- metry between bosons and fermions”) in physics. In the impressive arsenal of physico- geometric tools, it occupies a special, somewhat mysterious place. To an outside observer, it projects a false sense of accessibility, presenting itself as a simple modification of the familiar formalism. However, this is only an appearance. Like a bewitched place, it is protected not by barriers but by something less tangible and therefore much more powerful. It is not even so clear where this place is localized. A naive attempt to “forge ahead” may encounter little resistance but may also end up missing the real point.
    [Show full text]
  • Superstring Field Theory, Superforms and Supergeometry
    DISIT-2018 ARC-18-05 SUPERSTRING FIELD THEORY, SUPERFORMS AND SUPERGEOMETRY ROBERTO CATENACCI, PIETRO ANTONIO GRASSI, AND SIMONE NOJA Abstract. Inspired by superstring field theory, we study differential, integral, and inverse forms and their mutual relations on a supermanifold from a sheaf-theoretical point of view. In particular, the formal distributional properties of integral forms are recovered in this scenario in a geometrical way. Further, we show how inverse forms \extend" the ordinary de Rham complex on a supermanifold, thus providing a mathematical foundation of the Large Hilbert Space used in superstrings. Last, we briefly discuss how the Hodge diamond of a supermanifold looks like, and we explicitly compute it for super Riemann surfaces. Contents 1. Introduction 1 2. The Large Hilbert Space, PCO's and New Superforms 3 3. Elements of Supermanifolds 8 4. Locally-Free Sheaves on Supermanifolds: Tangent, Cotangent and Berezinian Sheaves 9 5. Superforms and Integral Forms Complex on a Supermanifold 16 6. Negative Degree Superforms and their Complex 21 6.1. Large Hilbert Space and Cechˇ Cohomology 25 6.2. Large Hilbert Space and Calabi-Yau Supermanifolds 27 7. Superforms and Pseudo-forms for Higher Odd Dimensions 29 8. Serre Duality and Hodge Diamond of a Supermanifold 33 9. Conclusions and Outlooks 38 Acknowledgements 39 References 39 1. Introduction Supergeometry is a fascinating branch of mathematics that prompted from the physical motivation of describing fermionic degrees of freedom. As is well known since the first years of quantum mechanics, identical particles can appear in two types: bosons and fermions. They have different properties, but essentially their wave functions, describing the states of those particles, have to be either fully symmetrized under the exchange of two identical particles in the case of bosons, or fully anti-symmetrized in the case of fermions.
    [Show full text]
  • Arxiv:Math/9810056V1 [Math.GN] 9 Oct 1998 Itrauiest Fwligo,Otbr1998
    October 23, 2018 WHAT IS SUPERTOPOLOGY? ‡ Ugo Bruzzo §,‡ and Vladimir Pestov ¶ § Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Beirut 2-4, 34014 Trieste, Italy ‡ Dipartimento di Matematica, Universit`adegli Studi di Genova, Via Dodecaneso 35, 16146 Genova, Italy ¶ School of Mathematical and Computing Sciences, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand E-mail: [email protected], [email protected] Home (V.P.): http://www.vuw.ac.nz/∼ vova Abstract. We discuss the problem of finding an analogue of the concept of a topo- logical space in supergeometry, motivated by a search for a procedure to compactify a supermanifold along odd coordinates. In particular, we examine the topologies naturally arising on the sets of points of locally ringed superspaces, and show that in the presence of a nontrivial odd sector such topologies are never compact. The main outcome of our discussion is that not only the usual framework of supergeometry (the theory of locally ringed spaces), but the more general approach of the functor of points, need to be further enlarged. arXiv:math/9810056v1 [math.GN] 9 Oct 1998 1. Introduction Geometries with anticommuting variables (supergeometries) have been introduced in connection with several issues in theoretical physics, notably to study supersym- metric field theories; physical motivations to introduce such geometries are briefly discussed in the next section. Supergeometries have been quite intensively studied in the 70s and 80s. (It is impossible to provide here any exhaustive bibliography; we only cite [5] as a general reference, and the basic works [11, 12, 39, 43, 26, 56, 37, 58, 7].
    [Show full text]
  • 2012 Lectures on Twisted K-Theory and Orientifolds
    LECTURES ON TWISTED K-THEORY AND ORIENTIFOLDS DANIEL S. FREED This is an improved version of informal supplementary notes for lectures delivered in June, 2012 at the Erwin Schr¨odinger International Institute for Mathematical Physics as part of a graduate workshop in the program K-Theory and Quantum Fields. This is not a record of the actual lectures, but rather is in large part supplementary reading and exercises. These notes will also be the basis of a formal paper on some of this material, but that will necessarily contain less exposition, so we are making these informal notes available. We do not make any attempt to assemble even a representative list of references in these lecture notes; we do so in the published papers. There were four lectures, but these notes are divided into three lectures as is more natural. The goal of the lectures is to give an overview of some aspects of ongoing joint work with Jacques Distler and Greg Moore about topological aspects of superstring theory. The inclusion of “orientifolds” has especially subtle topological features, and it is these that we focus on. There are two papers1,2 about this work so far and several more to follow. These theories are known in the physics literature as “Type II”; the “Type I” theories are a special case of the orientifold construction, so are included here. The “heterotic” string is not part of this discussion, nor is “M-theory”, “F-theory”, or other variants of string theory with supersymmetry. Lecture 1 is a purely mathematical discussion of twistings of K-theory, with a little about twisted K-theory.
    [Show full text]
  • The Geometry of Supergravity Torsion Constraints
    THE GEOMETRY OF SUPERGRAVITY TORSION CONSTRAINTS JOHN LOTT 1. Introduction This paper, prepared for the 2001 Park City Research Program in Supergeometry, is an exposition of [4]. References to relevant earlier papers can be found in the bibliography of [4]. As explained in Jim Gates' lectures, an essential ingredient of the superspace formulation of supergravity is a nonzero torsion tensor. Furthermore, there are torsion constraints in the sense that only certain components of the torsion tensor are allowed to be nonzero. These torsion constraints must be stringent enough to give physically relevant solutions, but flexible enough to allow for nonflat solutions. For example, for N = 1 supergravity on c c γ a Lorentzian 3-manifold, Jim Gates motivated the choice that Tαβ = i(γ )αβ, Tαb can be γ nonzero, Tab can be nonzero and the other torsion components must be zero. The existence of a nonzero torsion tensor in supergravity theories, and the constraints on the torsion tensor, are perhaps surprising from the viewpoint of standard differential geometry. In particular, their geometric meaning is not immediately clear. To give a somewhat analogous situation from conventional geometry, suppose that M 2n is an almost complex manifold with a Hermitian metric. Complexifying TM and using notation that will be explained later, suppose that we are told that a desirable set of torsion conditions is Tijk = 0;T ¯ = T¯ T¯ : (1.1) ijk kij − kji The geometric meaning of (1.1) may also not be immediately clear. If fact, (1.1) holds if and only if M is K¨ahler. The first equation in (1.1) is the integrability condition for the almost complex structure and the second equation expresses the K¨ahlercondition, in terms of a unitary basis.
    [Show full text]