Dissertations / Theses on the topic 'Supergeometry'

It is shown that H^p-1 (P⁺, 0 (-m-p)) is a Fréchet space, and its dual is H^q-1(P^-, 0 (m-q)), where P⁺ and P^- are the projectivizations of subsets of generalized twistor space (≌ ℂ^p-q) on which the hermitian form (of signature (p,q)) is positive and negative definite respectively, and 0(-m-p) denotes the sheaf of germs of holomorphic functions homogeneous of degree -m-p. It is then proven, for p = 2 and q = 2, that the subspace consisting of all twistor elementary states is dense in H^p-1(P⁺, 0(-m-p)). A supermanifold is a ringed space consisting of an underlying classical manifold and an augmented sheaf of Z₂-graded algebras locally isomorphic to an exterior algebra. The subcategory of the category of ringed spaces generated by such supermanifolds is referred to as the super category. A mathematical framework suitable for describing the generalization of Yang-Mills theory to the super category is given. This includes explicit examples of supercoordinate changes, superline bundles, and superconnections. Within this framework, a definition of the full super Yang-Mills equations is given and the simplest case is studied in detail. A comprehensive account of the generalization of twistor theory to the super category is presented, and it is used in an attempt to formulate a complete description of the super Yang-Mills equations. New concepts are introduced, and several ideas which have previously appeared in the literature at the level of formal calculations are expanded and explained within a consistent framework.

In questa tesi viene studiato l'approccio funtoriale alla supergeometria. In particolare si usano le topologie di Grothendieck per studiare il concetto di rappresentabilità in questo contesto, in analogia a quanto fatto in geometria algebrica classica. Vengono poi introdotti i funtori di Weil-Berezin e lo Schwarz embedding, motivando i legami tra questi concetti e la rappresentabilità nel caso classico.

La présente thèse porte sur le développement d'une théorie d'algèbre linéaire, de géométrie et d'analyse basée sur les algèbres (Z2)n-commutatives, c'est-à-dire des algèbres (Z2)n-graduées associatives unitaires satisfaisant ab = (-1)ba, pour tout couple d'éléments homogènes a, b de degrés deg(a), deg(b) où <.,.> est le produit scalaire usuel). Cette généralisation de la supergéométrie a de nombreuses applications : en mathématiques (l'algèbre de Deligne des superformes différentielles, l'algèbre des quaternions et les algèbres de Clifford en sont des exemples) et même en physique (paraparticules). Dans ce travail, les notions de trace et de (super)déterminant pour des matrices à coefficients dans une algèbre gradué-commutative sont définies et étudiés. Une attention particulière est portée au cas des algèbres de Clifford : ce point de vue gradué fournit une nouvelle approche au problème classique du « bon » déterminant pour des matrices à coefficient non-commutatifs (quaternioniques). En outre, nous entreprenons l'étude de la géométrie différentielle (Z2)n-graduée. Privilégiant l'approche par les espaces annelés, les (Z2)n-supervariétés sont définies en choisissant l'algèbre (Z2)n-commutative des séries formelles en variables graduées comme modèle pour le faisceau de fonctions. Les résultats les plus marquants ainsi obtenus sont : le Berezinien gradué et son interprétation cohomologique (essentielle pour établir une théorie de l'intégration) ; le théorème des morphismes, attestant qu'on peut rétablir un morphisme entre (Z2)n-supervariétés à partir de sa seule expression sur les coordonnées ; le théorème de Batchelor-Gawedzki pour les (Z2)n-supervariétés lisses
The present thesis deals with a development of linear algebra, geometry and analysis based on (Z2)n-superalgebras ; associative unital algebras which are (Z2)n-graded and graded-commutative, i.e. statisfying ab=(-1)ba, for all homogeneous elements a, b of respective degrees deg(a), deg(b) in (Z2)n (<.,.> denoting the usual scalar product). This generalization widens the range of applications of supergeometry to many mathematical structures (quaternions and more generally Clifford algebras, Deligne algebra of superdifferential forms, higher vector bundles) and appears also in physics (for describing paraparticles) proving its worth and relevance. In this dissertation, we first focus on (Z2)n-superalgebra theory ; we define and characterize the notions of trace and (super)determinant of matrices over graded-commutative algebras. Special attention is given to the case of Clifford algebras, where our study gives a new approach to treat the classical problem of finding a “good” determinant for matrices with noncommuting (quaternionic) entries. Further, we undertake the study of (Z2)n-graded differential geometry. Privileging the ringed space approach, we define (smooth) (Z2)n-supermanifolds modeling their algebras of functions on the (Z2)n-commutative algebra of formal power series in graded variables, and develop the theory along the lines of supergeometry. Notable results are : the graded Berezinian and its cohomological interpretation (essential to establish integration theory) ; the theorem of morphism, which states that a morphism of (Z2)n-supermanifolds can be recovered from its coordinate expression ; Batchelor-Gawedzki theorem for (Z2)n-supermanifolds

In questa tesi vengono studiati alcuni argomenti in supergeometria algebrica, con particolare attenzione al caso in cui le varietà ridotte delle supervarietà in esame siano spazi proiettivi complessi $\mathbb{P}^n$. Dopo aver introdotto le definizioni di base e alcune nozioni generali della supergeometria, viene studiata in dettaglio la geometria dei superspazi proiettivi $\mathbb{P}^{n|m}$. In questo contesto, vengono dati risultati sulla struttura e la coomologia dei fasci invertibili, sugli automorfismi e le deformazioni infinitesime. Attenzione speciale è riservata al caso della supercurva di Calabi-Yau $\mathbb{P}^{1|2}$. In seguito, vengono studiate le varietà non-projected su $\mathbb{P}^n$ e se ne fornisce una classificazione nel caso la dimensione dispari sia $2$, mostrando che esistono supervarietà non-projected solamente sulla linea proiettiva $\mathbb{P}^1$ e sul piano proiettivo $\mathbb{P}^2$. In particolare, si dimostra che tutte le supervarietà non-projected su $\mathbb{P}^2$ sono Calabi-Yau, cioè hanno fascio Bereziniano banale, ed inoltre sono non proiettive: non possono cioè essere immerse in un superspazio proiettivo $\mathbb{P}^{n|m}$. Si dimostra, invece, che esse possono sempre essere immerse in super Grassmanniane. In questo contesto, alcune immersioni di supervarietà non-projected significative vengono realizzate esplicitamente. Infine, è data una nuova costruzione dei $\Pi$-spazi proiettivi come supervarietà non-projected connesse al fascio cotangente su $\mathbb{P}^n$.
The aim of this thesis is to study some topics in algebraic supergeometry, in particular in the case the supermanifolds have their reduced manifolds given by complex projective spaces $\mathbb{P}^n$. After the main definitions and notions in supergeometry are introduced, the geometry of complex projective superspaces $\mathbb{P}^{n|m}$ is studied in detail. Invertible sheaves and their cohomology, infinitesimal automorphisms and deformations are studied for $\mathbb{P}^{n|m}$. Special attention is paid to the case of the Calabi-Yau supercurve $\mathbb{P}^{1|2}$. The focus is then moved to non-projected supermanifolds over $\mathbb{P}^n$. A complete classification is given in the case the odd dimension is $2$, showing that there exist non-projected supermanifolds only over the projective line $\mathbb{P}^1$ and projective plane $\mathbb{P}^2$. In particular, it is shown that all of the non-projected supermanifolds over $\mathbb{P}^2$ are Calabi-Yau's, i.e.\ they have trivial Berezinian sheaf, and they are all non-projective, i.e.\ they cannot be embedded into any ordinary projective superspace $\mathbb{P}^{n|m}$. Instead, it is shown that there always exist an embedding of these supermanifolds in super Grassmannians, and some meaningful examples are realised explicitly. Finally, a new construction of $\Pi$-projective spaces as non-projected supermanifolds related to the cotangent sheaf over $\mathbb{P}^n $ is given.

In questa tesi vengono presentati i piu recenti risultati relativi all'estensione della teoria dei campi localmente covariante a geometrie che permettano di descrivere teorie di campo supersimmetriche. In particolare, si mostra come la definizione assiomatica possa essere generalizzata, mettendo in evidenza le problematiche rilevanti e le tecniche utilizzate in letteratura per giungere ad una loro risoluzione. Dopo un'introduzione alle strutture matematiche di base, varieta Lorentziane e operatori Green-iperbolici, viene definita l'algebra delle osservabili per la teoria quantistica del campo scalare. Quindi, costruendo un funtore dalla categoria degli spazio-tempo globalmente iperbolici alla categoria delle *-algebre, lo stesso schema viene proposto per le teorie di campo bosoniche, purche definite da un operatore Green-iperbolico su uno spazio-tempo globalmente iperbolico. Si procede con lo studio delle supervarieta e alla definizione delle geometrie di background per le super teorie di campo: le strutture di super-Cartan. Associando canonicamente ad ognuna di esse uno spazio-tempo ridotto, si introduce la categoria delle strutture di super-Cartan (ghsCart) il cui spazio-tempo ridotto e globalmente iperbolico. Quindi, si mostra, in breve, come e possibile costruire un funtore da una sottocategoria di ghsCart alla categoria delle super *-algebre e si conclude presentando l'applicazione dei risultati esposti al caso delle strutture di super-Cartan in dimensione 2|2.

The following thesis will be concerned with various aspects of supergravity theories in a superspace setting, focusing mainly on maximal and half-maximal theories in three dimensions and maximal theories in ten-dimensions. For the three-dimensional theories it is convenient to start from an off-shell superconformal geometry valid for any number of supersymmetries. We first apply this formalism to show that it is consistent to couple ABJM and BLG theory to conformal supergravity, in doing so we find that N = 8 superconformal matter can also be charged under the gauge group SO(N). By imposing further constraints on the off-shell superconformal geometry, we obtain half-maximal and maximal Poincare supergravity. We solve for the geometry at dimension one in the half-maximal case with sigma models of the form (SO(8) x SO(n))\SO(8, n), and for the complete geometry in the maximal theory, where the scalar fields live in the coset SO(16)\E8. Using the Ricci identity, we also derive the equations of motion for the scalar and fermion fields in the latter theory. Using supersymmetry and duality we derive the form spectrum of the above Poincare supergravity theories and of type IIA and IIB supergravity in ten dimensions. Particular we show that the consistent Bianchi identities, which are not guaranteed to be satisfied from cohomology, determine a Lie super co-algebra. We derive the Cartan matrices of the dual algebras which are Borcherds algebras. The Cartan matrices can be used to generate the entire form field spectrum. We study gaugings of half-maximal and maximal Poincare supergravity in three dimensions by introducing a non-abelian gauged subgroup of the duality group and making use of the gauged Maurer-Cartan form. The differential forms can also be studied in the gauged theory by deforming the Bianchi identities. The closure of the full system of forms requires the presence of D + 2-form field strengths in the supergravity limit. In superspace, the Borcherds algebras predict an infinite number of form fields of degree larger than that of space-time. Indeed all those of degree larger than D + 2 are zero in supergravity, although this might change in string theory. We provide some evidence that a six-form, in half-maximal supergravity in three dimensions can become non-zero in the presence of α'-corrections.

In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields.
In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannigfaltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermionische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassmannalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von Räumen durch Funktoren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologischen Räume, Mannigfaltigkeiten, ...). Nach einer Einführung in das allgemeine Konzept dieses Zugangs verwenden wir es um eine Beschreibung der resultierenden Supermannigfaltigkeit der Felder bzw. Abbildungen anzugeben. Wir zeigen, dass jede Abbildung eindeutig durch eine Familie von Differentialoperatoren geeigneter Ordnung charakterisiert wird. Darüber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. von ungeraden Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschränkung umgeht: Durch das Vergrößern der betreffenden Supermannigfaltigkeit ist es immer möglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super-Version der Geodäte und eine Verallgemeinerung von harmonischen Abbildungen auf Supermannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialfällen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gewöhnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die Lösungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzt.

Le but de ce travail est d'etendre a la situation supergeometrique la formule de localisation de berline-vergne. Tout d'abord on a etendu la theorie de la cohomologie equivariante a la situation supergeometrique. En particulier on a obtenu un equivalent supergeometrique de la classe de thom et de la classe d'euler pour la cohomologie equivariante a support compact. Pour cela on est amene a considerer non pas la cohomologie equivariante traditionnelle a coefficients polynomiaux mais plutot a coefficients dans les fonctions generalisees. Une fois ce travail effectue, on a trouve une generalisation supergeometrique de la formule de localisation de berline-vergne pour calculer des integrales de formes equivariantes fermees a support compact. Enfin, on a montre a travers divers exemples que cette theorie est bien adaptee au calcul de la transformee de fourier d'une orbite coadjointe d'un supergroupe de lie et que l'on retrouve ainsi certaines formule de caracteres a la kirillov.

Le but initial de cette thèse était d'étudier les espaces de superlacets, version géométrique des espaces de supercordes en Physique. Le point de départ était alors d'étendre les résultats de classifications de l'article de Oleg Mokhov : Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems au cadre de la supergéométrie. Dans cet article l'auteur établit une classification des formes symplectiques locales homogènes d'ordre 0, 1 et 2 sur l'espace des lacets LM = C1(S1;M) à partir d'objets géométriques sur la variété différentiable M. Dans cette thèse, on remplace la variété M par une supervariété Mpjq et le cercle S1 par un supercercle S1jn et l'on étudie l'espace des morphismes de supervariétésMor(S1jn;Mpjq). Dans les deux premières parties, l'on définit les structures géométriques classiques et super des espaces de superlacets. Pour ce faire, l'on se restreint aux deux supercercles S1j1 et en s'inspirant des travaux sur LM, l'on détermine une structure de variété de Fréchet des espaces de superlacets SLM = Mor(S1j1;M). Puis l'on introduit la structure super qui nous a semblé la plus naturelle sur SLM en terme de faisceaux. Afin de pouvoir travailler en coordonnées, l'on introduit la structure super par un autre point de vue en considérant l'espace de superlacets SLM comme le foncteur de points SLM. De plus, en interprétant les calculs de Mokhov en terme de jets, ceci nous permet d'une part d'apporter une justification rigoureuse aux-dits calculs et d'autre part, d'obtenir une généralisation directe des méthodes de calculs en coordonnées ("à la physicienne"). Le troisième chapitre expose les résultats de classification obtenus. Comme dans le cas classique, on obtient un théorème de dépendance limitée de l'ordre des jets qui interviennent dans les formes d'ordre 0 et 1. Puis, on obtient une classification des formes d'ordre 0 au moyen de formes différentielles sur la supervariété Mpjq. Une classification des formes homogènes d'ordre 1 et 2 au moyen de métriques Riemaniennes et de connexions sur Mpjq. Enfin le quatrième chapitre est consacré à la généralisation des résultats d'un autre article de O. Mokhov : Complex homogeneous forms on loop spaces of smooth manifolds and their cohomology groups. De par la présence de la variable impaire, on précise tout d'abord la définition des formes homogènes locales sur SLM, puis on démontre que muni de la différentielle extérieure, l'espace des formes homogènes sur SLM d'ordre m 2 N donné définit un complexe. On calcule alors complètement les espaces de cohomologie pour les ordres m = 0 et 1, partiellement pour les ordres 2 et 3 et on explicite ainsi les formes symplectiques exactes obtenues au troisième chapitre.

L’obiettivo di questa tesi è quello di definire e analizzare le varie tipologie di supervarietà su R e su C, dando spazio alla discussione approfondita di un esempio molto importante, la supervarietà grassmanniana Gr^ch. Nel capitolo 1 proporremo un ripasso delle nozioni di algebra, geometria e topologia. Per poter arrivare successivamente a parlare di supervarietà, introdurremo nel capitolo 2 i concetti di base di superalgebra lineare, quali le definizioni di superspazio vettoriale, superalgebra, supermodulo e matrice a valori in un superspazio, e le loro proprietà fondamentali. Dedicheremo poi il capitolo 3 all’analisi della varietà grassmanniana ordinaria G(2,4) dei sottospazi 2-dimensionali di C^4, mostrando come questa assuma la struttura di varietà algebrica analitica, e anche di varietà proiettiva: identificheremo infatti G(2,4) con una sottovarietà algebrica dello spazio proiettivo P^5(C), detta quadrica di Klein, tramite la cosiddetta immersione di Plücker. Nel capitolo 4 tratteremo la teoria delle supervarietà. Parleremo di fasci di algebre e superalgebre, strumenti molto utili per trattare concettualmente le varietà e le supervarietà, azioni di supergruppi e superspazi omogenei. Utilizzeremo anche il linguaggio del funtore dei punti, per analizzare gli oggetti del nostro studio dal punto di vista della teoria delle categorie. Svilupperemo poi dettagliatamente il caso della supervarietà grassmanniana Gr^ch, l’estensione supergeometrica della varietà G(2; 4). Vedremo Gr^ch come supervarietà analitica e come superspazio omogeneo, studieremo il suo funtore dei punti e mostreremo come, attraverso la super immersione di Plücker, questa sia isomorfa ad una supervarietà proiettiva dentro al superspazio P^6|4, detta super quadrica di Klein. Infine nel capitolo 5 vedremo come lo spaziotempo di Minkowski, oggetto molto importante nella teoria fisica della relatività, possa essere identificato con la grande cella U_12, un particolare aperto di G(2,4).

The interplay between geometry and supergeometry, from an algebraic point of view, sets the theme guiding the considerations in this thesis. In the smooth setting there is a sense in which these geometries can be identified and, in the complex (i.e., holomorphic) setting, such an identification no longer holds. As such, for at least this reason, complex algebraic supergeometry can find interest in its own right. It is the subject of this thesis and we study it here under two broad headings: obstruction theory and deformation theory. Under the umbrella of obstruction theory, we focus largely on foundational aspects of supermanifolds and their description by means of supersymmetric thickenings. We start from the general principle that: any supermanifold will define a supersymmetric thickening but not necessarily conversely. One of the key objectives in this part of the thesis is in precisely formulating and proving this principle by elementary methods. We complement the proof given with examples of obstructed thickenings on the complex projective plane. To illustrate obstruction theory more generally for complex supermanifolds, we include and comment on a collection of examples from the literature, in addition to providing some new examples. Moreover, we will also consider the splitting problem for complex supermanifolds. Upon obtaining a characterisation of the obstruction classes to splitting via the grading vector field, we present a new proof of the Koszul splitting theorem for supermanifolds. Regarding deformation theory, we concern ourselves with the construction of (odd) infinitesimal deformations of superconformal structures. These are structures on supermanifolds and, in the one dimensional case, arise under the guise of super Riemann surfaces. Explicit and elementary constructions of (odd) deformations are given for N = 1 and N = 2 super Riemann surfaces. One of the key objectives in this part of the thesis is on establishing precise relations between (1) the deformation theory of these super Riemann surfaces and (2) their obstruction theory as supermanifolds. We conclude this thesis with a brief sketch on the state of supermoduli spaces, in both the N = 1 and N = 2 setting, as it presently stands in the literature. These discussions lead naturally toward directions for future research and paint a grander scheme in which this thesis sits.