DOCSLIB.ORG

Study on Chohomology of Supermanifolds

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Study on Chohomology of Supermanifolds BRAC University Journal, Vol. III, No. 1, 2006, pp. 59-63 STUDY ON CHOHOMOLOGY OF SUPERMANIFOLDS Khondokar M. Ahmed Department of Mathematics University of Dhaka Dhaka -1000, Bangladesh ABSTRACT In this present paper de Rham cohomology of graded manifolds, cohomology of graded differential forms and cohomology of DeWitt supermanifolds are studied. The structure of a G-supermanifold in general is not acyclic. So, the cohomology is defined via the complex of graded differential forms. This situation is investigated together with the graded Dolbeault cohomology of complex G- supermanifolds. A theorem is established that the structure sheaf of any DeWitt G-supermanifold is acyclic. Key words: Manifolds, chomology, supermanifolds. I. INTRODUCTION d 1 d 0 → Ñ → A → Ω A → 2 d The aim of this paper is to unfold a basic Ω A → L (2.1) cohomology theory for supermanifolds. This is exact. This 'graded Poincare′ Lemma' is most cohomology does not embody only trivial easily proved by working in local coordinates and extensions of results valid for differentiable proceeding on the analogy of the usual Poincare′ manifolds. For instance the natural analogue of the Lemma. By defining the de Rham cohomology of de Rham theorem does not holds in general and • (X,A), denoted by H DR (X , A), as the also in the case of complex super-manifolds there is generally no analogue of the Dolbeault theorem. cohomology of the complex of graded vector As a result the sturecture sheaf of a supermanifolds spaces A • (X ) , from (2.1) and using the ordinary does not need to be cohomology trivial [2]. The de Rham theorem, we obtain the following result fact that the cohomology of the complex of global (cf. [8] Theorem 4.7.). graded differential forms on a G-supermanifold ( M,A) depends on the G-supermanifold structure ( Proposition 2.1. There is a canonical isomorphism M,A) so that super de Rham cohomology is a fine k k DR A)' DR for all invariant of the supermanifold structure [3]. H (X , H (X ) k ≥ 0. k Here H DR (X ) denotes the usual de Rham II. de Rham COHOMOLOGY OF GRADED cohomology of X. MANIFOLDS III. COHOMOLOGY OF GRADED Graded manifolds are not very interesting as far as DIFFERENTIAL FORMS their cohomology is concerned. In the real case, the structure sheaf of a graded manifold (X,A) is fine Let (M,A) be a G-supermanifold. The sheaves k and therefore A and all sheaves Ω A of graded Ω k ⊗ B of smooth B -valued differential differential forms are acyclic. This implies that the Ñ L L • forms on M provide a resolution of the constant cohomology of the complex Ω A (X) coincides sheaf B on M, in the sense that the differential with the de Rham cohomology of X. In the L complex analytic case, a similar argument allows complex of sheaves of graded-commutative BL - one to prove a Dolbeault-type theorem. • 0 ∞ algebras Ω ⊗ Ñ B (with Ω ⊗ Ñ B ´C L ) is a The complex of sheaves A • is exact and moreover, L L it is a resolution of the constant sheaf Ñ on X; i.e., resolution of the constant sheaf BL , i.e. the the sequence of sheaves of Ñ-modules sequence Khondokar M. Ahmed Ak A1 k ∞ ω = dz ∧ ∧ dz ω ∈ Ω H (U) be ∞ d 1 d L A1LAk 0 → B → C L → Ω ⊗ Ñ B → L L an ∞ graded differential k-form on U (k>0); let 2 H Ω ⊗ Ñ BL → L (3.1) us set is exact. The cohomology associated with this ˆ k Ak −1 Kω(z) = (−1) dz ∧L∧ complex via the global section functor Γ(⋅, M ) , t • A1 B k −1 i.e. the cohomology of the complex Ω ⊗ Ñ BL , is dz z t ω (tz)dt. ∫ BA1LAk −1 • 0 denoted by H DR (M , B ) and is called the B - L L k We have an isomorphism [2] Ω G (U ) ' valued de Rham cohomology of M. Since BL is a k ∞ H ; it is therefore possible to introduce a finite dimensional real vector space, the universal Ω Ñ BL coefficient theorem [6] entails the isomorphism homotopy operator • • k k −1 H DR (M , BL ) ' H DR (M ) ⊗ Ñ BL . (3.2) K : Ω G (U ) → Ω G(U), defined by By virtue of the de Rham theorem, equation (3.2) K( a) Kˆ a. can be equivalently written as ω ⊗ = ω ⊗ • • One can indeed verify easily that H DR (M , B ) ' H (M , B ) . (3.3) L L dKλ + Kdλ = λ for any section λ ∈ Gk (U ) , By H • (M ,⋅) we designate interchangeable the so that, if dλ = 0 then λ = d(Kλ) . The case 1ech or sheaf cohomology functor, which coincide since the base space is paracompact. k = 0 has been left out. However, if f ∈ G (U ) , by writing f as f = f ⊗ a with f ∈ In order to gain information not on the topological ∑ i i i or smooth structure of M, but rather on its G- ∞ H (U ) and ai ∈ BL , the condition df = 0 implies supermanifold structure, we therefore need to define a new cohomology, obtained via a resolution directly that f is a constant in BL . of BL , different from the differential complex k Definition 3.2. Given a G-supermanifold (M,A), (3.1). We consider the sheaves Ω A of graded the cohomology of the complex differential forms. The following result is a d 1 d A(M) → Ω A generalization of the usual Poincare′ lemma (cf. (M ) → 2 [5]). Ω A (M ) → L (3.5) • Proposition 3.1. Given a G supermanifold (M,A), denoted by H SDR (M , A), is called the super de the differential complex of sheaves of graded B - Rham cohomology of (M,A). L The operation of taking the SDR cohomology of a algebras on M G-supermanifold is functorial. Indeed, given a G- d 1 d 0 → BL → A → Ω A → morphism ( f ,φ) : (M , A) → (N, B), it is easily 2 • Ω A → L (3.4) proved that the morphism Ω B (N) → is a resolution of B . • L f∗Ω A(M) induced byφ commutes with the Proof. Since the claim to be proved is a local exterior differential and therefore yields a m,n ¤ matter, we may assume that (M,A) = (BL ,G); morphism of graded BL -modules φ : moreover, it is enough to • • H SDR (N,B ) → H SDR (M ,A ) . It should be show that, if U is an open ball around the origin in • m,n noticed that the functor H SDR (⋅) does not fulfill BL then any closed graded differential k-form k the Eilenberg-Steenrod [10] axiomatics for λ ∈ Ω G(U) is exact; i.e., there exists a graded cohomology (if it did, it would coincide with the k−1 differential (k -1)-form η ∈ Ω G (U) such that BL -valued de Rham cohomology functor) since it λ = dη . Given coordinates (z1 , , z m+n ) in U, does not satisfy the excision axiom. Moreover, the L • let functor H SDR (⋅) does not give rise to 60 Study on Chohomology of Supermanifolds topological invariants. On the other hand, it is Proposition 3.3. provides a usefull tool for investigating the cohomological properties of the easily varified that the graded BL -modules k structure sheaf of a G- supermanifold. For instance, H SDR (M , A)are invariants associated with the it suffices to exhibit a G-supermanifold (M,A) such G-supermanifold structure of M. Indeed, if 1 1 that H SDR (M , A) ≠ H DR (M , BL ) to deduce ( f ,φ) : (M , A) → (N, B) is a G-isomorphism, it that, in general, the sheaf A can not be expected to ¤ • is easily proved that φ : H SDR (N, B) be acyclic. • → H SDR (M , A). is an isomorphism. IV. COHOMOLOGY OF DEWITT SUPERMANIFOLDS The most natural thing to do to gain insight into the geometric significance of the groups Considering in M the fine topology we study the k cohomology of a De Witt supermanifold (M,A); H SDR (M , A) - which, as a matter of fact, are this is advantageous because in this way M is para- graded BL -modules - is to compare them with the compact. Thus we continue to confuse the sheaf cohomology groups H k (M , B ) , which have a and 1ech cohomologies with coefficients in L sheaves on M. natural structure of graded BL -modules as well. k k We need the following Lemma, which is obtained The morphisms Ω A (M ) → Ω from a result given in [19] by strengthening certain induced by the morphism (M ) ⊗ Ñ BL hypotheses. ∞ δ : A → C L give rise to a morphism of Lemma 4.1. Let X and Y be topological spaces, differential complexes, which induces in with Y locally euclidean and F a sheaf of abelian cohomology a morphism of graded BL -modules groups on X; let we assume that all groups k k k · : H SDR (M , A) → H (X , F) are finitely generated. Then for all k n ≥ 0 there is an exact sequence of abelian H DR (M , BL ) ∀k ≥ 0. (3.6) groups In degree zero, · 0 is an isomorphism, in that one has manifestly j k 0 C 0 0 → ⊕ H (X , F) ⊗ Ù H (Y, Ù) → H SDR (M , A)' (BL ) ' H DR (M , BL ) , j+k=n where C is the number of connected components of H n (X ×Y, π −1 F) → M, which we assume to be finite. In degree higher → ⊕ Tor[H j (X , F), H k (X , Ù)] → 0 than zero, we have, as a straightforward application j+k =n+1 of the abstract de Rham theorem, the following result.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [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]
  • 2. the Concept of a Supermanifold
    2. THE CONCEPT OF A SUPERMANIFOLD 2.1. Geometry of physical space. 2.2. The mathematical evolution of the concept of space as a geometrical ob- ject. 2.3. Geometry and algebra. 2.4. Supermanifolds and their supersymmetries. 2.1. Geometry of physical space. Someone who is already familiar with the theory of differentiable manifolds or algebraic varieties can be very quickly intro- duced to the notion of a supermanifold and the concept of supersymmetry. Just as the manifolds and varieties are defined by first starting with local pieces on which the coordinate functions are defined, and then gluing these local pieces together, a supermanifold may be defined as a space on which locally one has coordinates x1; : : : ; xn; θ1; : : : θr where the xi are the usual commuting coordinates and the θj, the anticommuting (fermionic) coordinates, with the various sets of local chats be- ing related by transformations of the appropriate smoothness type. Everything is then done exactly as in the classical theory. Supersymmetries are diffeomorphisms of such spaces and these form super Lie groups. One can construct a theory of differentiation and integration on such spaces and write down equations of motions of particles and fields starting from suitable Lagrangians. If one starts with a super- symmetric Lagrangian then one obtains an action of the supersymmetric group on the solutions of the field equations thus defined. The stage on which supersymmetic quantum field theory lives is then a super spacetime, either flat or curved. How- ever, such a treatment, in spite of being very practical and having the advantage of getting into the heart of matters very quickly, does not do full justice either to the understanding of the concepts at a deeper level or to comprehending the boldness of the generalization of conventional geometry that is involved here.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • (Work in Progress On) Quantum Field Theory on Curved Supermanifolds
    Introduction Supermanifolds DiffGeo on supermanifolds Wess-Zumino Model (Work in progress on) Quantum Field Theory on Curved Supermanifolds Paul-Thomas Hack Universit¨atHamburg LQP 31, Leipzig, November 24, 2012 in collaboration with Andreas Schenkel Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiffGeo on supermanifolds Wess-Zumino Model Motivation supergravity theories can be described as field theories on supermanifolds = \curved superspace", invariant under \superdiffeomorphisms" want to obtain a systematic treatment of supergravity theories as \locally superinvariant" algebraic QFT on curved supermanifolds (CSM) long term goal: understand SUSY non-renormalisation theorems in the algebraic language and extend them to curved spacetimes first: understand Wess-Zumino model as QFT on CSM covariant under \superdiffeomorphisms" ! locally supercovariant QFT Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiffGeo on supermanifolds Wess-Zumino Model Supersymmetry and superspace for pedestrians a simple SUSY model (Wess-Zumino model) in d = 2 + 1 Minkowski 3 spacetime R : φ, ; F invariant under : φ 7! φ+" 7! +6@φε+F " F 7! F +"6@ on shell : φ = 0 6@ = 0 F = 0 idea: consider Φ = (φ, ; F ) as a single \superfield” on \superspace" 3j2 α R 3 (x ; θa) 1 Φ(x; θ) = φ(x) + (x)θ + F (x)θ2 (θ2 = θθ) 2 Φ0(x; θ) = Φ(x 0; θ0) α 0α α α 0 \special supertranslations" x 7! x = x +εγ θ θa 7! θa = θa+"a Paul-Thomas Hack (Work in progress
    [Show full text]
  • Integration and Bordism on Supermanifolds
    NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 125-140 INTEGRATION AND BORDISM ON SUPERMANIFOLDS C. V ic t o r ia (Received July 1996) Abstract. An integration scheme for Z2-forms on Berezin-Leites-Kostant su­ permanifolds together with a bordism theory is introduced. Its relationship with the berezinian integration is stated. Stokes formula is proved for this integration scheme, thus supplying a pairing between the cohomology of Z2- forms and bordism cycles. It is proved that the bordism groups of a given supermanifold depend only on its split (Batchelor) structure. Introduction The de Rham complex of forms on manifolds was generalized in [28] to complexes of Z2-forms on real or complex differentiable Berezin-Leites-Kostant supermani­ folds, and a cohomology ring for them was obtained. In this paper we define an integration scheme for these Z2-forms (Section 3) and we prove that it verifies Stokes theorem (Theorem 4.2). Thus, the integration of cohomology classes of Z2-forms over bordism cycles of Berezin-Leites-Kostant supermanifolds - a no­ tion introduced in Section 5 - is well defined (Proposition 5.1). Whence, a pairing between cohomology and bordism is established. We also prove (Theorem 5.2) that the bordism groups, 0 (V)(X), of a supermanifold X are isomorphic to the bordism groups fi(.,.) (Gr(X)), where Gr is the functor which associates with any supermanifold its underlying linear supermanifold (also called its split (Batchelor) supermanifold). In order to better locate this paper in the existing literature, a few words should be said about the types of supermanifolds and the various relationships among them.
    [Show full text]