DOCSLIB.ORG

Graded Manifolds: Some Issues

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Graded Manifolds: Some Issues Graded Manifolds: Some Issues Jan Vysok´y (joint work with Rudolf Smolka)ˇ Higher Structures and Field Theory ESI Wien, 11 September 2020 Z-graded manifolds: overview Z-graded (or simply graded) manifolds, Z-graded supermanifolds: Suitable mathematical theory of \generalized manifolds", where functions locally depend on Z-graded variables (which do commute accordingly). There are two basic approaches to this problem: 1 Consider (smooth) Z2-supermanifolds with an additional Z-grading. Kontsevich (1997), Severaˇ (2001), Roytenberg, Voronov (2002). 2 Define Z-graded (henceforth just graded) manifolds without an underlying supermanifold. Mehta (2006), Cattaneo & Sch¨atz (2010), Fairon (2017). Optimal scenario Mimic Berezin-Leites approach to supermanifolds (1983) with \purely graded" coordinates of all degrees (not just N-manifolds). Jan Vysok´y Graded Manifolds: Some Issues 1 / 25 Graded algebra: usual approach Definition (graded \object") An object (vector space, ring, algebra) V is graded, if it can be L written as as a direct sum V = Vk of its subspaces Vk ⊆ V . k2Z h Elements of V := [k2ZVk − f0g have defined degree jvj 2 Z and are called homogeneous elements. Algebraic structures (e.g. ring multiplication) satisfy additional grading requirements (e.g. jv · wj = jvj + jwj). Object morphisms ' : V ! W have to preserve the grading, that is j'(v)j = jvj (or shift it by a given number, when required). Example L V = Vk a graded vector space. Symmetric algebra S(V ) is k2Z S(V ) = T (V )=J; J = hfv ⊗ w − (−1)jwjjvjw ⊗ v j v; w 2 V hgi: It is a graded commutative associative algebra. Jan Vysok´y Graded Manifolds: Some Issues 2 / 25 Graded manifolds: usual approach Definition (graded manifold) Let M be a second countable Hausdorff space. We have a structure sheaf of graded commutative associative 1 algebras CM on M. 1 1 M = (M; CM) is a locally ringed space, that is stalks CM;m are local rings for every m 2 M. There is a finite-dimensional graded vector space V , such that 1 V0 = f0g and CM is locally isomorphic to the sheaf n 1 1 U ⊆ R 7! Cn;V (U) := C (U) ⊗R S(V ); Remark (they are just polynomials!) dim(V ) 1 If (ξµ)µ=1 is a homogeneous basis of V , then elements of Cn;V (U) are dim(V ) 1 polynomials in fξµgµ=1 with coefficients in C (U), and jξµjjξν j ξµξν = (−1) ξν ξµ: Jan Vysok´y Graded Manifolds: Some Issues 3 / 25 Issue number one 1 I. Cn;V is not a sheaf. Consider V = V−2 ⊕ V2, where V−2 = V2 = R. Let n = 1 and consider an open subset U ⊆ R in the form U := [m2NUm; Um := (m − 1; m): fUmgm2N is an open cover of U. V has a basis (ξ1; ξ2) with jξ1j = −2 and jξ2j = 2. Define local 1 sections fm 2 C1;V (Um) as m m fm := (ξ1) (ξ2) : 1 1 ffmgm2Z cannot be glued to f 2 C1;V (U), hence C1;V is not a sheaf. The same problem arises even for non-negatively graded V , if we do not restrict the degree of glued local sections. Jan Vysok´y Graded Manifolds: Some Issues 4 / 25 Easy solutions? Issue number two Can we simply relax the gluing axiom to finite open covers? Not a good idea! A lot of constructions require infinite open covers (e.g. those using partition of unity). Can one somehow employ the “sheafification mantra"? However, sheafification does preserve stalks, hence it does not cure the following: 1 II. Stalks of Cn;V are not local rings. 1 For each m 2 M,(Cn;V )m is a local ring, iff its non-invertible elements form an ideal. With n = 1 and V as above, consider sections s± := 1 ± ξ1ξ2 in 1 C1;V (R). Their germs [s±]m are not invertible. 1 Their sum [s+]m + [s−]m = [2]m is invertible. Whence (C1;V )m is not a local ring. Morphisms of \non-locally" ringed spaces are not so easy to work with. Jan Vysok´y Graded Manifolds: Some Issues 5 / 25 Issue number three III. One cannot calculate pullbacks. Let us again consider the example above. We have coordinates (x; ξ1; ξ2), where jxj = 0, jξ1j = −2 and jξ2j = 2. A (graded) smooth map should be (locally) determined by pullback of coordinate functions. Define e.g. ∗ ∗ ' (x) = x + ξ1ξ2;' (ξ1;2) = ξ1;2: What is a pullback of f = f (x)? The formula (e.g. in supergeometry) would be (for above '): 1 X 1 dmf '∗(f ) = (ξ )m(ξ )m: m! dx m 1 2 m=0 But this is does not converge (in any sense). Jan Vysok´y Graded Manifolds: Some Issues 6 / 25 Almost a solution, issue number four Definition (Graded manifold by Fairon (2017)) 1 Replace Cn;V with a sheaf n ¯1 1 ¯ U ⊆ R 7! Cn;V (U) := C (U) ⊗R S(V ); ¯ Q1 p dim(V ) where S(V ) = p=0 S (V ) (formal power series in variables fξµgµ=1 ). 1 ¯ There is a minor oversight - C (U) ⊗R S(V ) is not a space of formal dim(V ) 1 power series in fξµgµ=1 with coefficients in C (U), it should be 1 ¯1 Y 1 p Cn;V (U) = C (U) ⊗R S (V ): p=0 ¯1 IV: Cn;V (U) is not a graded vector space. It cannot be written as a direct sum of its \elements of degree k" 1 L 1 subspaces, that is C¯ (U) 6= (C¯ (U))k . n;V k2Z n;V Jan Vysok´y Graded Manifolds: Some Issues 7 / 25 Graded algebra revisited There are other inconveniences with the graded algebra: L L For V = Vk and W = Wk , Lin(V ; W ) is not a graded k2Z k2ZQ vector space, instead Lin(V ; W ) = Link (V ; W ). k2Z In particular, V ∗ = Lin(V ; R) is not a graded vector space. Definition (graded \object" again) A graded object V is a sequence V = fVk gk2Z. We write v 2 V , if v 2 Vk for some k 2 Z, and write jvj := k. There is no space for inhomogeneous elements! A graded morphism ' : V ! W is a collection ' = f'k gk2Z where 'k : Vk ! Wk for all k 2 Z. We write '(v) for 'jvj(v). Algebraic structures are introduced naturally. Example (category gVect) V = fVk gk2Z is graded (real) vector space, iff Vk is a vector space for each k 2 Z. ' is graded linear, iff 'k : Vk ! Wk are linear. Jan Vysok´y Graded Manifolds: Some Issues 8 / 25 Graded commutative associative algebras Many constructions are \more natural" from a categorical viewpoint. gVect is a symmetric monoidal category where M (V ⊗R W )k := Vj ⊗R Wk−j ; j2Z where the unit object is R (with the trivial grading), and jvjjwj τVW (v ⊗ w) = (−1) w ⊗ v is the braiding. Lin(V ; W ) = fLink (V ; W )gk2Z where Link (V ; W ) = f' = f'j gj2Z j 'j : Vj ! Wj+k g; is the internal hom in gVect. V ∗ = Lin(V ; R) is a dual object. Definition (category gcAs) Graded (commutative) associative algebras are (commutative) monoids in gVect, that is (A; µ, η) with µ : A ⊗R A ! A and η : R ! A. Jan Vysok´y Graded Manifolds: Some Issues 9 / 25 In plain English, we write v · w := µ(v ⊗ w) and 1A := η(1), finding v · (w · x) = (v · w) · x; v · 1A = 1A · v = v; distributivity: Graded commutativity reads v · w = (−1)jvjjwjw · v. An ideal is a graded abelian subgroup I ⊆ A with µ(A ⊗R I ) ⊆ I . Definition (local graded commutative ring) For any A 2 gcAs, we get a graded ring multiplicationµ ¯ : A ⊗Z A ! A. (A; µ,¯ η) is a graded commutative ring. It is a local graded commutative ring, if it has a unique maximal ideal. Local graded ring morphisms preserve those ideals. This has many equivalent definitions. Let U(A) = fUk (A)gk2Z Uk (A) = fv 2 Ak j (9w 2 A−k )(v · w = w · v = 1A)g: Then A is local, iff A − U(A) is an ideal. In this case J(A) := A − U(A) is the unique maximal ideal, called the Jacobson radical of A. The ring A = ⊕k2ZAk is not necessarily local! Jan Vysok´y Graded Manifolds: Some Issues 10 / 25 Sheaves of graded commutative algebras Let X be a topological space. Op(X ) a category of open subsets. Definition A presheaf of graded commutative algebras on X is a functor F : Op(X )op ! gcAs : For each k 2 Z, we have a presheaf Fk (U) := (F(U))k . Together with natural transformations, we have a category PSh(X ; gcAs). F 2 PSh(X ; gcAs) is a sheaf, iff Fk is a sheaf of vector spaces for every k 2 Z. We have a full subcategory Sh(X ; gcAs). The definition of a sheaf coincides with the \category friendly" definition using products and equalizers. Definition For each x 2 X , a stalk Fx of a presheaf F at x is defined as a filtered colimit of F over the opposite to Opx (X ) = fU 2 Op(X ) j U 3 xg. Jan Vysok´y Graded Manifolds: Some Issues 11 / 25 Graded locally ringed spaces Definition A graded locally ringed space (X ; OX ) is a pair, where X is a topological space. OX 2 Sh(X ; gcAs) is a sheaf of graded commutative algebras. Stalks of OX are local graded rings. A morphism of graded locally ringed spaces (X ; OX ) and (Y ; OY ) is a pair ' = (';'∗), where ' : X ! Y is continuous and 1 '∗ : O ! ' (O ) is a sheaf morphism.
Load more

Recommended publications
  • From Topological Field Theory to Deformation Quantization and Reduction
    Cattaneo, Alberto (2006). From topological field theory to deformation quantization and reduction. In: International Congress of Mathematicians. Vol. III. 339-365. Postprint available at: http://www.zora.uzh.ch University of Zurich Posted at the Zurich Open Repository and Archive, University of Zurich. Zurich Open Repository and Archive http://www.zora.uzh.ch Originally published at: 2006. International Congress of Mathematicians. Vol. III. 339-365. Winterthurerstr. 190 CH-8057 Zurich http://www.zora.uzh.ch Year: 2006 From topological field theory to deformation quantization and reduction Cattaneo, Alberto Cattaneo, Alberto (2006). From topological field theory to deformation quantization and reduction. In: International Congress of Mathematicians. Vol. III. 339-365. Postprint available at: http://www.zora.uzh.ch Posted at the Zurich Open Repository and Archive, University of Zurich. http://www.zora.uzh.ch Originally published at: 2006. International Congress of Mathematicians. Vol. III. 339-365. From topological field theory to deformation quantization and reduction Alberto S. Cattaneo∗ Abstract. This note describes the functional-integral quantization of two-dimensional topolog- ical field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded mani- folds and to the Batalin–Vilkovisky formalism is included. Mathematics Subject Classification (2000). Primary 81T45; Secondary 51P05, 53D55, 58A50, 81T70. Keywords. Topological quantum field theory, BV formalism, graded manifolds, deformation quantization, formality, Poisson reduction, L∞- and A∞-algebras. 1. Introduction: a 2D TFT 0 1.1. The basic setting. Let be a smooth compact 2-manifold. On M1 := ()⊕ 1() one may define the following very simple action functional: S(ξ,η) := η dξ, ξ ∈ 0(), η ∈ 1(), (1.1) which is invariant under the distribution 0 ⊕ dβ, β ∈ 0().
    [Show full text]
  • Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and Their Applications
    ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 415–471 INTRODUCTION TO GRADED GEOMETRY, BATALIN-VILKOVISKY FORMALISM AND THEIR APPLICATIONS Jian Qiu and Maxim Zabzine Abstract. These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV–formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present a proof of the isomorphism between the graph complex and the Chevalley-Eilenberg complex of formal Hamiltonian vectors fields. We briefly discuss how these ideas can be extended to the infinite dimensional setting. These notes should be accessible toboth physicists and mathematicians. Table of Contents 1. Introduction and motivation 416 2. Supergeometry 417 2.1. Idea 418 2.2. Z2-graded linear algebra 418 2.3. Supermanifolds 420 2.4. Integration theory 421 3. Graded geometry 423 3.1. Z-graded linear algebra 423 3.2. Graded manifold 425 4. Odd Fourier transform and BV-formalism 426 4.1. Standard Fourier transform 426 4.2. Odd Fourier transform 427 4.3. Integration theory 430 4.4. Algebraic view on the integration 433 5. Perturbation theory 437 5.1. Integrals in Rn-Gaussian Integrals and Feynman Diagrams 437 2010 Mathematics Subject Classification: primary 58A50; secondary 16E45, 97K30. Key words and phrases: Batalin-Vilkovisky formalism, graded symplectic geometry, graph homology, perturbation theory.
    [Show full text]
  • 1. Introduction There Is a Deep Connection Between Algebras and the Categories of Their Modules
    Pr´e-Publica¸c~oesdo Departamento de Matem´atica Universidade de Coimbra Preprint Number 09{15 SEMI-PERFECT CATEGORY-GRADED ALGEBRAS IVAN YUDIN Abstract: We introduce the notion of algebras graded over a small category and give a criterion for such algebras to be semi-perfect. AMS Subject Classification (2000): 18E15,16W50. 1. Introduction There is a deep connection between algebras and the categories of their modules. The interplay between their properties contributes to both the structure theory of algebras and the theory of abelian categories. It is often the case that the study of the category of modules over a par- ticular algebra can lead to the use of other abelian categories, which are non-equivalent to module categories over any algebra. One of such exam- ples is the category of modules over a C-graded algebra, where C is a small category. Such categories appear in the joint work of the author and A. P. Santana [18] on homological properties of Schur algebras. Algebras graded over a small category generalise the widely known group graded algebras (see [16, 12, 8, 15, 14]), the recently introduced groupoid graded algebras (see [10, 11, 9]), and Z-algebras, used in the theory of operads (see [17]). One of the important properties of some abelian categories, that consider- ably simplifies the study of their homological properties, is the existence of projective covers for finitely generated objects. Such categories were called semi-perfect in [7]. We give in this article the characterisation of C-graded algebras whose categories of modules are semi-perfect.
    [Show full text]
  • Z3-Graded Geometric Algebra 1 Introduction
    Adv. Studies Theor. Phys., Vol. 4, 2010, no. 8, 383 - 392 Z3-Graded Geometric Algebra Farid Makhsoos Department of Mathematics Faculty of Sciences Islamic Azad University of Varamin(Pishva) Varamin-Tehran-Iran Majid Bashour Department of Mathematics Faculty of Sciences Islamic Azad University of Varamin(Pishva) Varamin-Tehran-Iran majidbashour@yahoo.com Abstract By considering the Z2 gradation structure, the aim of this paper is constructing Z3 gradation structure of multivectors in geometric alge- bra. Mathematics Subject Classification: 16E45, 15A66 Keywords: Graded Algebra, Geometric Algebra 1 Introduction The foundations of geometric algebra, or what today is more commonly known as Clifford algebra, were put forward already in 1844 by Grassmann. He introduced vectors, scalar products and extensive quantities such as exterior products. His ideas were far ahead of his time and formulated in an abstract and rather philosophical form which was hard to follow for contemporary math- ematicians. Because of this, his work was largely ignored until around 1876, when Clifford took up Grassmann’s ideas and formulated a natural algebra on vectors with combined interior and exterior products. He referred to this as an application of Grassmann’s geometric algebra. Due to unfortunate historic events, such as Clifford’s early death in 1879, his ideas did not reach the wider part of the mathematics community. Hamil- ton had independently invented the quaternion algebra which was a special 384 F. Makhsoos and M. Bashour case of Grassmann’s constructions, a fact Hamilton quickly realized himself. Gibbs reformulated, largely due to a misinterpretation, the quaternion alge- bra to a system for calculating with vectors in three dimensions with scalar and cross products.
    [Show full text]
  • Dimension Formula for Graded Lie Algebras and Its Applications
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 11, Pages 4281–4336 S 0002-9947(99)02239-4 Article electronically published on June 29, 1999 DIMENSION FORMULA FOR GRADED LIE ALGEBRAS AND ITS APPLICATIONS SEOK-JIN KANG AND MYUNG-HWAN KIM Abstract. In this paper, we investigate the structure of infinite dimensional Lie algebras L = α Γ Lα graded by a countable abelian semigroup Γ sat- isfying a certain finiteness∈ condition. The Euler-Poincar´e principle yields the denominator identitiesL for the Γ-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces Lα (α Γ). Our dimen- sion formula enables us to study the structure of the Γ-graded2 Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also dis- cuss the relation of graded Lie algebras and the product identities for formal power series. Introduction In [K1] and [Mo1], Kac and Moody independently introduced a new class of Lie algebras, called Kac-Moody algebras, as a generalization of finite dimensional simple Lie algebras over C. These algebras are ususally infinite dimensional, and are defined by the generators and relations associated with generalized Cartan matrices, which is similar to Serre’s presentation of complex semisimple Lie algebras. In [M], by generalizing Weyl’s denominator identity to the affine root system, Macdonald obtained a new family of combinatorial identities, including Jacobi’s triple product identity as the simplest case.
    [Show full text]
  • Graded-And-Microformal-Prop
    The University of Manchester Research Graded Geometry, Q-Manifolds, and Microformal Geometry DOI: 10.1002/prop.201910023 Document Version Accepted author manuscript Link to publication record in Manchester Research Explorer Citation for published version (APA): Voronov, T. (2019). Graded Geometry, Q-Manifolds, and Microformal Geometry: LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. Fortschritte der Physik. https://doi.org/10.1002/prop.201910023 Published in: Fortschritte der Physik Citing this paper Please note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscript or Proof version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version. General rights Copyright and moral rights for the publications made accessible in the Research Explorer are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Takedown policy If you believe that this document breaches copyright please refer to the University of Manchester’s Takedown Procedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providing relevant details, so we can investigate your claim. Download date:06. Oct. 2021 Graded geometry, Q-manifolds, and microformal geometry 1,2 Theodore Th. Voronov ¤ below), to which we give a brief introduction as well. We give an exposition of graded and microformal ge- “Thick morphisms” (defined for ordinary manifolds, ometry, and the language of Q-manifolds. Q-manifolds supermanifolds or graded manifolds) generalize ordinary are supermanifolds endowed with an odd vector field of maps or supermanifold morphisms, but are not maps square zero.
    [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]
  • Determinants of Complexes
    APPENDIX A Determinants of Complexes Complexes Let k be a field. A complex of k-vector spaces is a graded vector space W· = EBieZ Wi together with a family of linear operators di+l Odi = o. The spaces Wi are called the terms of W·. Throughout this Appendix we shall assume that all but finitely many tenns of a complex are zero. Cohomology spaces of a complex w· are defined as Hi (W·) = Ker(di)/Im(di_l). A complex w· is called exact if all Hi (W·) are equal to zero. Example 1. Any linear operator between two vector spaces, say, w- 1 and Wo, can be regarded as a complex with only two non-zero tenns: Such a complex is exact if and only if d_1 is invertible. For any graded vector space w· and any m E Z we shall denote by W·[m] the same vector space but with the grading shifted by m: The same notation will be used for complexes. Let v· and w· be two complexes of vector spaces. A morphism of complexes J : V· -+ w· is a collection of linear operators /; : Vi -+ Wi which commute with the differentials in v· and w·. The cone of a morphism J is a new complex Cone(f) with tenns Cone(fi = Wi 6) Vi+l and the differential given by d(w, v) = (dw(w) + (-li+1J(v), dv(v»), WE Wi, V E Vi+l. (I) Any morphism of complexes J : V· -+ w· induces a morphism of cohomology spaces Hi (f) : Hi(V.) -+ Hi(W·).
    [Show full text]
  • Introduction to Graded Geometry
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector European Journal of Mathematics DOI 10.1007/s40879-017-0138-4 RESEARCH/REVIEW ARTICLE Introduction to graded geometry Maxime Fairon1 Received: 7 November 2016 / Revised: 1 March 2017 / Accepted: 8 March 2017 © The Author(s) 2017. This article is an open access publication Abstract This paper aims at setting out the basics of Z-graded manifolds theory. We introduce Z-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures. Keywords Supergeometry · Graded manifold · Differential graded manifold · Q-manifold Mathematics Subject Classification 58A50 · 51-02 1 Introduction In the 70s, supermanifolds were introduced and studied to provide a geometric back- ground to the developing theory of supersymmetry. In the Berezin–Leites [2] and Kostant [11] approach, they are defined as Z2-graded locally ringed spaces. More pre- cisely, this means that a supermanifold is a pair (|N|, ON ) such that |N| is a topological space and ON is a sheaf of Z2-graded algebras which satisfies, for all sufficiently small open subsets U, ∞ m ON (U) CRn (U)⊗ R , B Maxime Fairon mmmfai@leeds.ac.uk 1 School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom 123 M. Fairon m where R is endowed with its canonical Z2-grading.
    [Show full text]
  • Factorization Algebras in Quantum Field Theory Volume 2 (28 April 2016)
    Factorization algebras in quantum field theory Volume 2 (28 April 2016) Kevin Costello and Owen Gwilliam Contents Chapter 1. Overview1 1.1. Classical field theory and factorization algebras1 1.2. Quantum field theory and factorization algebras2 1.3. The quantization theorem3 1.4. The rigid quantization conjecture4 Chapter 2. Structured factorization algebras and quantization7 2.1. Structured factorization algebras7 2.2. Commutative factorization algebras9 2.3. The P0 operad9 2.4. The Beilinson-Drinfeld operad 13 Part 1. Classical field theory 17 Chapter 3. Introduction to classical field theory 19 3.1. The Euler-Lagrange equations 19 3.2. Observables 20 3.3. The symplectic structure 20 3.4. The P0 structure 21 Chapter 4. Elliptic moduli problems 23 4.1. Formal moduli problems and Lie algebras 24 4.2. Examples of elliptic moduli problems related to scalar field theories 28 4.3. Examples of elliptic moduli problems related to gauge theories 30 4.4. Cochains of a local L¥ algebra 34 4.5. D-modules and local L¥ algebras 36 Chapter 5. The classical Batalin-Vilkovisky formalism 45 5.1. The classical BV formalism in finite dimensions 45 5.2. The classical BV formalism in infinite dimensions 47 5.3. The derived critical locus of an action functional 50 5.4. A succinct definition of a classical field theory 55 5.5. Examples of field theories from action functionals 57 5.6. Cotangent field theories 58 Chapter 6. The observables of a classical field theory 63 iii iv CONTENTS 6.1. The factorization algebra of classical observables 63 6.2.
    [Show full text]
  • 1604.00711.Pdf
    LIE ALGEBROIDS AS L∞ SPACES RYAN GRADY AND OWEN GWILLIAM Abstract. In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid—and the natural generalization to dg Lie algebroids— provides an (essentially unique) L∞ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of L∞ spaces. Then we show that for each Lie algebroid L, there is a fully faithful functor from the category of representations up to homotopy of L to the category of vector bundles over the associated L∞ space. Indeed, this functor sends the adjoint complex of L to the tangent bundle of the L∞ space. Finally, we show that a shifted-symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated L∞ space. Contents 1. Introduction 1 2. Recollections on L∞ spaces 3 3. Recollections on Lie algebroids 11 4. Lie algebroids as L∞ spaces 13 5. Representations up to homotopy and vector bundles on L∞ spaces 20 6. Symplectic structures 32 Appendix A. Jets and connections 37 References 43 1. Introduction Lie algebroids appear throughout geometry and physics, and they provide a fertile transfer of ideas and intuition between geometry and Lie theory. (The literature is vast. See, for instance, arXiv:1604.00711v2 [math.DG] 18 Jan 2018 [Mac05, CdSW99, ELW99, Fer02, CRvdB10, CF03], among many other papers.) Recently the language of derived geometry has provided another perspective on the relationship between Lie theory and geometry, emphasizing the idea that a deformation problem is describable by a dg Lie algebra (e.g., [Lur, Hin01]).
    [Show full text]
  • Graded Polynomial Identities of Matrices Yuri Bahturin A,B,∗,1, Vesselin Drensky C Adepartment of Mathematics and Statistics, Memorial University of Newfoundland, St
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 357 (2002) 15–34 www.elsevier.com/locate/laa Graded polynomial identities of matrices Yuri Bahturin a,b,∗,1, Vesselin Drensky c aDepartment of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Canada NF A1A 5K9 bDepartment of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow 119899, Russia cInstitute of Mathematics and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria Received 17 October 2001; accepted 9 March 2002 Submitted by R. Guralnick Abstract We consider G-graded polynomial identities of the p × p matrix algebra Mp(K) over a field K of characteristic 0 graded by an arbitrary group G. We find relations between the G- graded identities of the G-graded algebra Mp(K) and the (G × H)-graded identities of the tensor product of Mp(K) and the H-graded algebra Mq (K) with a fine H-grading. We also find a basis of the G-graded identities of Mp(K) with an elementary grading such that the identity component coincides with the diagonal of Mp(K). © 2002 Elsevier Science Inc. All rights reserved. AMS classification: 16R20; 16W50 Keywords: Full matrix algebras; Graded algebras; Identical relations 1. Introduction In this paper we study graded polynomial identities of the p × p matrix algebra Mp(K) over a field K of characteristic 0. Concerning the ordinary polynomial iden- tities, the picture is completely clear only for 2 × 2 matrices.
    [Show full text]