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Relevant bibliographies by topics / Supergeometry

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Academic literature on the topic 'Supergeometry'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 March 2023

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Journal articles on the topic "Supergeometry"

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Goertsches, O. "Riemannian supergeometry." Mathematische Zeitschrift 260, no. 3 (December 9, 2007): 557–93. http://dx.doi.org/10.1007/s00209-007-0288-z.

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Tsimpis, Dimitrios. "Curved 11D Supergeometry." Journal of High Energy Physics 2004, no. 11 (December 2, 2004): 087. http://dx.doi.org/10.1088/1126-6708/2004/11/087.

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Fioresi, R., and F. Zanchetta. "Representability in supergeometry." Expositiones Mathematicae 35, no. 3 (September 2017): 315–25. http://dx.doi.org/10.1016/j.exmath.2016.10.001.

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CATTANEO, ALBERTO S., and FLORIAN SCHÄTZ. "INTRODUCTION TO SUPERGEOMETRY." Reviews in Mathematical Physics 23, no. 06 (July 2011): 669–90. http://dx.doi.org/10.1142/s0129055x11004400.

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These notes are based on a series of lectures given by the first author at the school of "Poisson 2010", held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super- and graded manifolds, cohomological vector fields, graded symplectic structures, reduction and the AKSZ-formalism.

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Voronov, A. A., Yu I. Manin, and I. B. Penkov. "Elements of supergeometry." Journal of Soviet Mathematics 51, no. 1 (August 1990): 2069–83. http://dx.doi.org/10.1007/bf01098184.

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Schwarz, A., and I. Shapiro. "Supergeometry and arithmetic geometry." Nuclear Physics B 756, no. 3 (November 2006): 207–18. http://dx.doi.org/10.1016/j.nuclphysb.2006.08.024.

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Hoker, E. D., and D. H. Phong. "Complex geometry and supergeometry." Current Developments in Mathematics 2005, no. 1 (2005): 1–40. http://dx.doi.org/10.4310/cdm.2005.v2005.n1.a1.

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Lott, John. "Torsion constraints in supergeometry." Communications in Mathematical Physics 133, no. 3 (November 1990): 563–615. http://dx.doi.org/10.1007/bf02097010.

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Schmitt, Thomas. "Supergeometry and hermitian conjugation." Journal of Geometry and Physics 7, no. 2 (January 1990): 141–69. http://dx.doi.org/10.1016/0393-0440(90)90009-r.

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Schwarz, Albert. "Noncommutative supergeometry, duality and deformations." Nuclear Physics B 650, no. 3 (February 2003): 475–96. http://dx.doi.org/10.1016/s0550-3213(02)01088-x.

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Dissertations / Theses on the topic "Supergeometry"

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Kleppe, Anne Friederike. "Supersymmetry, spinors and supergeometry." Thesis, University of Cambridge, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.613938.

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Pilato, Alejandro Miguel. "Elementary states, supergeometry and twistor theory." Thesis, University of Oxford, 1986. http://ora.ox.ac.uk/objects/uuid:d86c78d7-2e6e-4a5c-a37a-81d8dbf3ccd8.

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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.

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Zanchetta, Ferdinando. "Supergeometry: a categorical point of view." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9027/.

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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.

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Covolo, Tiffany. "(Z2)n-Superalgebra and (Z2)n-Supergeometry." Thesis, Lyon 1, 2014. http://www.theses.fr/2014LYO10203.

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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

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NOJA, SIMONE. "TOPICS IN ALGEBRAIC SUPERGEOMETRY OVER PROJECTIVE SPACES." Doctoral thesis, Università degli Studi di Milano, 2018. http://hdl.handle.net/2434/554352.

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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.

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Cuzzola, Angelo. "Aspects of supergeometry in locally covariant quantum field theory." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10391/.

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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.

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Greitz, Jesper. "Supergravity in superspace : supergeometry, differential forms and algebraic structure." Thesis, King's College London (University of London), 2012. https://kclpure.kcl.ac.uk/portal/en/theses/supergravity-in-superspace(4ef77d1c-bdc8-4d1c-aa99-254929a3c14b).html.

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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.

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Hanisch, Florian. "Variational problems on supermanifolds." Phd thesis, Universität Potsdam, 2011. http://opus.kobv.de/ubp/volltexte/2012/5975/.

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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.

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Papantonis, Theocharis [Verfasser]. "Z-graded supergeometry: Differential graded modules, higher algebroid representations, and linear structures / Theocharis Papantonis." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2021. http://d-nb.info/1236401727/34.

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Ostermayr, Dominik [Verfasser], Alexander [Gutachter] Alldridge, George [Gutachter] Marinescu, and Tilmann [Gutachter] Wurzbacher. "Some results in supergeometry: Harmonic maps from super Riemann surfaces and Automorphism supergroups of supermanifolds / Dominik Ostermayr ; Gutachter: Alexander Alldridge, George Marinescu, Tilmann Wurzbacher." Köln : Universitäts- und Stadtbibliothek Köln, 2017. http://d-nb.info/1129872475/34.

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Books on the topic "Supergeometry"

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Keßler, Enno. Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8.

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Keßler, Enno. Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional. Springer, 2019.

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Book chapters on the topic "Supergeometry"

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Manin, Yuri Ivanovich. "Introduction to Supergeometry." In Grundlehren der mathematischen Wissenschaften, 181–232. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-662-07386-5_5.

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Julia, B. "Supergeometry and Kac-Moody Algebras." In Mathematical Sciences Research Institute Publications, 393–409. New York, NY: Springer US, 1985. http://dx.doi.org/10.1007/978-1-4613-9550-8_19.

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Grosse, H., and G. Reiter. "Noncommutative Supergeometry of Graded Matrix Algebras." In Geometry and Quantum Physics, 386. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/3-540-46552-9_11.

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Keßler, Enno. "Introduction." In Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, 1–9. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8_1.

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Keßler, Enno. "Connections on Super Riemann Surfaces." In Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, 169–83. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8_10.

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Keßler, Enno. "Metrics and Gravitinos." In Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, 185–213. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8_11.

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Keßler, Enno. "The Superconformal Action Functional." In Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, 215–34. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8_12.

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Keßler, Enno. "Computations in Wess–Zumino Gauge." In Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, 235–78. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8_13.

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Keßler, Enno. "Linear Superalgebra." In Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, 13–40. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8_2.

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Keßler, Enno. "Supermanifolds." In Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, 41–66. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13758-8_3.

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Conference papers on the topic "Supergeometry"

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Heller, Marc Andre, Noriaki Ikeda, and Satoshi Watamura. "Courant algebroids from double field theory in supergeometry." In Workshop on Strings, Membranes and Topological Field Theory. WORLD SCIENTIFIC, 2017. http://dx.doi.org/10.1142/9789813144613_0008.

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