2. the Concept of a Supermanifold
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Cohomological Approach to the Graded Berezinian
J. Noncommut. Geom. 9 (2015), 543–565 Journal of Noncommutative Geometry DOI 10.4171/JNCG/200 © European Mathematical Society Cohomological approach to the graded Berezinian Tiffany Covolo n Abstract. We develop the theory of linear algebra over a .Z2/ -commutative algebra (n N), which includes the well-known super linear algebra as a special case (n 1). Examples of2 such graded-commutative algebras are the Clifford algebras, in particularD the quaternion algebra H. Following a cohomological approach, we introduce analogues of the notions of trace and determinant. Our construction reduces in the classical commutative case to the coordinate-free description of the determinant by means of the action of invertible matrices on the top exterior power, and in the supercommutative case it coincides with the well-known cohomological interpretation of the Berezinian. Mathematics Subject Classification (2010). 16W50, 17A70, 11R52, 15A15, 15A66, 16E40. Keywords. Graded linear algebra, graded trace and Berezinian, quaternions, Clifford algebra. 1. Introduction Remarkable series of algebras, such as the algebra of quaternions and, more generally, Clifford algebras turn out to be graded-commutative. Originated in [1] and [2], this idea was developed in [10] and [11]. The grading group in this case is n 1 .Z2/ C , where n is the number of generators, and the graded-commutativity reads as a;b ab . 1/hQ Qiba (1.1) D n 1 where a; b .Z2/ C denote the degrees of the respective homogeneous elements Q Q 2 n 1 n 1 a and b, and ; .Z2/ C .Z2/ C Z2 is the standard scalar product of binary .n 1/h-vectorsi W (see Section 2). -
SUPERSYMMETRY 1. Introduction the Purpose
SUPERSYMMETRY JOSH KANTOR 1. Introduction The purpose of these notes is to give a short and (overly?)simple description of supersymmetry for Mathematicians. Our description is far from complete and should be thought of as a first pass at the ideas that arise from supersymmetry. Fundamental to supersymmetry is the mathematics of Clifford algebras and spin groups. We will describe the mathematical results we are using but we refer the reader to the references for proofs. In particular [4], [1], and [5] all cover spinors nicely. 2. Spin and Clifford Algebras We will first review the definition of spin, spinors, and Clifford algebras. Let V be a vector space over R or C with some nondegenerate quadratic form. The clifford algebra of V , l(V ), is the algebra generated by V and 1, subject to the relations v v = v, vC 1, or equivalently v w + w v = 2 v, w . Note that elements of· l(V ) !can"b·e written as polynomials· in V · and this! giv"es a splitting l(V ) = l(VC )0 l(V )1. Here l(V )0 is the set of elements of l(V ) which can bCe writtenC as a linear⊕ C combinationC of products of even numbers ofCvectors from V , and l(V )1 is the set of elements which can be written as a linear combination of productsC of odd numbers of vectors from V . Note that more succinctly l(V ) is just the quotient of the tensor algebra of V by the ideal generated by v vC v, v 1. -
Equivariant De Rham Cohomology and Gauged Field Theories
Equivariant de Rham cohomology and gauged field theories August 15, 2013 Contents 1 Introduction 2 1.1 Differential forms and de Rham cohomology . .2 1.2 A commercial for supermanifolds . .3 1.3 Topological field theories . .5 2 Super vector spaces, super algebras and super Lie algebras 11 2.1 Super algebras . 12 2.2 Super Lie algebra . 14 3 A categorical Digression 15 3.1 Monoidal categories . 15 3.2 Symmetric monoidal categories . 17 4 Supermanifolds 19 4.1 Definition and examples of supermanifolds . 19 4.2 Super Lie groups and their Lie algebras . 23 4.3 Determining maps from a supermanifold to Rpjq ................ 26 5 Mapping supermanifolds and the functor of points formalism 31 5.1 Internal hom objects . 32 5.2 The functor of points formalism . 34 5.3 The supermanifold of endomorphisms of R0j1 .................. 36 5.4 Vector bundles on supermanifolds . 37 5.5 The supermanifold of maps R0j1 ! X ...................... 41 5.6 The algebra of functions on a generalized supermanifold . 43 1 1 INTRODUCTION 2 6 Gauged field theories and equivariant de Rham cohomology 47 6.1 Equivariant cohomology . 47 6.2 Differential forms on G-manifolds. 49 6.3 The Weil algebra . 53 6.4 A geometric interpretation of the Weil algebra . 57 1 Introduction 1.1 Differential forms and de Rham cohomology Let X be a smooth manifold of dimension n. We recall that Ωk(X) := C1(X; ΛkT ∗X) is the vector space of differential k-forms on X. What is the available structure on differential forms? 1. There is the wedge product Ωk(M) ⊗ Ω`(M) −!^ Ωk+`(X) induced by a vector bundle homomorphism ΛkT ∗X ⊗ Λ`T ∗X −! Λk+`T ∗X ∗ Ln k This gives Ω (X) := k=0 Ω (X) the structure of a Z-graded algebra. -
A Proof of the Multiplicative Property of the Berezinian ∗
Morfismos, Vol. 12, No. 1, 2008, pp. 45–61 A proof of the multiplicative property of the Berezinian ∗ Manuel Vladimir Vega Blanco Abstract The Berezinian is the analogue of the determinant in super-linear algebra. In this paper we give an elementary and explicative proof that the Berezinian is a multiplicative homomorphism. This paper is self-conatined, so we begin with a short introduction to super- algebra, where we study the category of supermodules. We write the matrix representation of super-linear transformations. Then we can define a concept analogous to the determinant, this is the superdeterminant or Berezinian. We finish the paper proving the multiplicative property. 2000 Mathematics Subject Classification: 81R99. Keywords and phrases: Berezinian, superalgebra, supergeometry, super- calculus. 1 Introduction Linear superalgebra arises in the study of elementary particles and its fields, specially with the introduction of super-symmetry. In nature we can find two classes of elementary particles, in accordance with the statistics of Einstein-Bose and the statistics of Fermi, the bosons and the fermions, respectively. The fields that represent it have a parity, there are bosonic fields (even) and fermionic fields (odd): the former commute with themselves and with the fermionic fields, while the latter anti-commute with themselves. This fields form an algebra, and with the above non-commutativity property they form an algebraic structure ∗Research supported by a CONACyT scholarship. This paper is part of the autor’s M. Sc. thesis presented at the Department of Mathematics of CINVESTAV-IPN. 45 46 Manuel Vladimir Vega Blanco called a which we call a superalgebra. -
Topics in Algebraic Supergeometry Over Projective Spaces
Dipartimento di Matematica \Federigo Enriques" Dottorato di Ricerca in Matematica XXX ciclo Topics in Algebraic Supergeometry over Projective Spaces MAT/03 Candidato: Simone Noja Relatore: Prof. Bert van Geemen Correlatore: Prof. Sergio Luigi Cacciatori Coordinatore di Dottorato: Prof. Vieri Mastropietro A.A. 2016/2017 Contents Introduction 4 1 Algebraic Supergeometry 8 1.1 Main Definitions and Fundamental Constructions . .8 1.2 Locally-Free Sheaves on a Supermanifold and Even Picard Group . 14 1.3 Tangent and Cotangent Sheaf of a Supermanifold . 16 1.4 Berezinian Sheaf, First Chern Class and Calabi-Yau Condition . 21 2 Supergeometry of Projective Superspaces 24 2.1 Cohomology of O n|m ` ................................. 24 P p q n|m 2.2 Invertible Sheaves and Even Picard Group Pic0pP q ................ 26 2.3 Maps and Embeddings into Projective Superspaces . 31 2.4 Infinitesimal Automorphisms and First Order Deformations . 32 2.4.1 Supercurves over P1 and the Calabi-Yau Supermanifold P1|2 ......... 35 2.5 P1|2 as N “ 2 Super Riemann Surface . 37 2.6 Aganagic-Vafa's Mirror Supermanifold for P1|2 .................... 41 3 N “ 2 Non-Projected Supermanifolds over Pn 45 3.1 Obstruction to the Splitting of a N “ 2 Supermanifold . 45 3.2 N “ 2 Non-Projected Supermanifolds over Projective Spaces . 49 3.3 Non-Projected Supermanifolds over P1 ......................... 50 1 3.3.1 Even Picard Group of P!pm; nq ......................... 52 1 3.3.2 Embedding of P!p2; 2q: an Example by Witten . 54 3.4 Non-Projected Supermanifolds over P2 ......................... 56 2 3.4.1 P!pFM q is a Calabi-Yau Supermanifold . -
Superfield Equations in the Berezin-Kostant-Leites Category
Superfield equations in the Berezin-Kostant-Leites category Michel Egeileh∗ and Daniel Bennequin† ∗Conservatoire national des arts et m´etiers (ISSAE Cnam Liban) P.O. Box 113 6175 Hamra, 1103 2100 Beirut, Lebanon E-mail: [email protected] †Institut de Math´ematiques de Jussieu-Paris Rive Gauche, Bˆatiment Sophie Germain, 8 place Aur´elie Nemours, 75013 Paris, France E-mail: [email protected] Abstract Using the functor of points, we prove that the Wess-Zumino equations for massive chiral superfields in dimension 4|4 can be represented by supersymmetric equations in terms of superfunctions in the Berezin-Kostant-Leites sense (involving ordinary fields, with real and complex valued components). Then, after introducing an appropriate supersymmetric extension of the Fourier transform, we prove explicitly that these su- persymmetric equations provide a realization of the irreducible unitary representations with positive mass and zero superspin of the super Poincar´egroup in dimension 4|4. 1 Introduction From the point of view of quantum field theory, 1-particle states of a free elementary par- ticle constitute a Hilbert space which, by the requirement of relativistic invariance, must carry an irreducible unitary representation of the Poincar´egroup V ⋊ Spin(V ), where V is a Lorentzian vector space. In signature (1, 3), it is well-known since [Wig] that the irreducible representations of the Poincar´egroup that are of physical interest are classified 1 3 by a nonnegative real number m (the mass), and a half-integer s ∈ {0, 2 , 1, 2 , ...} (the spin)1. It is also known, cf. [BK], that all the irreducible unitary representations of the Poincar´e group of positive (resp. -
Quotient Supermanifolds
BULL. AUSTRAL. MATH. SOC. 58A50 VOL. 58 (1998) [107-120] QUOTIENT SUPERMANIFOLDS CLAUDIO BARTOCCI, UGO BRUZZO, DANIEL HERNANDEZ RUIPEREZ AND VLADIMIR PESTOV A necessary and sufficient condition for the existence of a supermanifold structure on a quotient defined by an equivalence relation is established. Furthermore, we show that an equivalence relation it on a Berezin-Leites-Kostant supermanifold X determines a quotient supermanifold X/R if and only if the restriction Ro of R to the underlying smooth manifold Xo of X determines a quotient smooth manifold XQ/RQ- 1. INTRODUCTION The necessity of taking quotients of supermanifolds arises in a great variety of cases; just to mention a few examples, we recall the notion of supergrassmannian, the definition of the Teichmiiller space of super Riemann surfaces, or the procedure of super Poisson reduction. These constructions play a crucial role in superstring theory as well as in supersymmetric field theories. The first aim of this paper is to prove a necessary and sufficient condition ensuring that an equivalence relation in the category of supermanifolds gives rise to a quotient supermanifold; analogous results, in the setting of Berezin-Leites-Kostant (BLK) super- manifolds, were already stated in [6]. Secondly and rather surprisingly at that, it turns out that for Berezin-Leites-Kostant supermanifolds any obstacles to the existence of a quotient supermanifold can exist only in the even sector and are therefore purely topological. We demonstrate that an equivalence relation Ron a. BLK supermanifold X determines a quotient BLK supermanifold X/R if and only if the restriction of R to the underlying manifold Xo of X determines a quotient smooth manifold. -
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. -
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. -
Super Quantum Mechanics in the Integral Form Formalism 3
Super Quantum Mechanics in the Integral Form Formalism L. Castellani, R. Catenacci and P.A. Grassi Abstract. We reformulate Super Quantum Mechanics in the context of integral forms. This framework allows to interpolate between different actions for the same theory, connected by different choices of Picture Changing Operators (PCO). In this way we retrieve component and superspace actions, and prove their equivalence. The PCO are closed integral forms, and can be interpreted as super Poincar´eduals of bosonic submanifolds embedded into a supermanifold.. We use them to construct Lagrangians that are top integral forms, and therefore can be integrated on the whole supermanifold. The D = 1, N = 1 and the D = 1, N = 2 cases are studied, in a flat and in a curved supermanifold. In this formalism we also consider coupling with gauge fields, Hilbert space of quantum states and observables. Keywords. Supersymmetry, Supermanifold, Super Quantum Mechanics. arXiv:1706.04704v2 [hep-th] 6 Nov 2017 1. Introduction Since the invention of supersymmetry, several authors provided useful math- ematical tools for its geometrical formulation, based essentially on the in- terpretation of supersymmetry as a coordinate transformation in fermionic directions, described by Grassmann coordinates θ. Still, there remained sev- eral problems, mostly related to integration theory on supermanifolds. The first formulations of supersymmetric models were given in terms of a component action, containing bosonic and fermionic fields, and invariant under supersymmetry transformations mixing bosons and fermions. The same dynamics can be derived in a more efficient way from an action which is manifestly invariant under supersymmetry. This framework is known as superspace approach and the various fields of the spectrum are contained in some superfields (or superforms). -
Supersymmetric Field-Theoretic Models on a Supermanifold
Supersymmetric Field-Theoretic Models on a Supermanifold Daniel H.T. Franco∗†(a) and Caio M.M. Polito(b) (a) Centro de Estudos de F´ısica Te´orica Rua Rio Grande do Norte 1053/302 - Funcion´arios CEP:30130-131 - Belo Horizonte - MG - Brasil. (b) Centro Brasileiro de Pesquisas F´ısicas – (CBPF) Coordena¸c˜ao de Campos e Part´ıculas – (CCP) Rua Dr. Xavier Sigaud 150 - Urca CEP:22290-180 - Rio de Janeiro - RJ - Brasil. Abstract We propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general spacetime manifold so as to include superfield models on a su- permanifold. We only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure. Our considerations are based on the Catenacci-Reina-Teofillatto-Bryant approach to supermani- arXiv:hep-th/0212038v3 18 Jan 2004 folds. In particular, we show that the class of supermanifolds constructed by Bonora-Pasti-Tonin satisfies the criterions which guarantee that a superma- nifold admits a Hausdorff body manifold. This construction is the closest to the physicist’s intuitive view of superspace as a manifold with some anticom- muting coordinates, where the odd sector is topologically trivial. The paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects. Moreover, a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistribu- tions, which is equivalent to the requirement that all of the component fields satisfy, on the body manifold, a microlocal spectral condition proposed by Brunetti-Fredenhagen-K¨ohler. -
Some Things Physicists Do
SOME THINGS PHYSICISTS DO MANU CHAO Contents 1. Some things physicists do 2 1.0. A bit of QFT terminology 2 1.1. Quantum Field Theories in Lagrangian formalism 2 1.2. Perturbative (asymptotic) expansions of Feynman integrals 7 1.3. Dualities 7 1.4. Quantum Field Theories in Hamiltonian formalism 7 1.5. Supersymmetry 9 1.6. Simplifications: topological twists 10 1.7. Example: sigma models 10 1.8. Branes 11 1.9. Operators 13 1.10. Dimensional reduction via invariance and compactifications 15 1.11. Example: gauge theory and S-duality 16 1.12. Renormalization 19 Appendix A. Electro-magnetic Duality and S-duality 19 A.1. Classical theory of electromagnetism: fields and Maxwell’s equations 19 A.2. Quantum mechanical setting for electromagnetism 20 A.3. Quantum Electro-Magnetic Duality 21 A.4. S-duality 22 1 2 MANU CHAO 1. Some things physicists do To a QFT physicists associate a category of branes Br( ) and they study Br( ) by as- T T T sociating to each brane Q Br( )a boundary QFT Q. In simplest cases category Br( ) is something that mathematicians∈ T have already considered,T such as coherent sheavesT on complex manifold or the Fukaya category on a symplectic manifold. In such cases we are interested in results and predictions of physicists resulting from their method of study of a mathematical theory Br( ). Moreover, the theories that are mathematically viable are usually simplifications (“Ttopological twists”) of physicalT theories which are at the mo- ment beyond our grasp and therefore all the more interesting. For instance, the physical theory of 2-dimensional sigma models has two such topological twists called A-model and B-model and these produce the Fukaya category and the category coherent sheaves.