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Graded Geometry, Q-Manifolds, and Microformal Geometry

DOI: 10.1002/prop.201910023

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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

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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. They can be seen as a non-linear analog themselves. They are defined as special type canonical of Lie algebras (in parallel with even and odd Poisson relations or correspondences between the cotangent bun- manifolds), a basis of “non-linear homological algebra”, dles. Canonical relations have long been a standard tool and a powerful tool for describing algebraic and geo- in symplectic or Poisson geometry, perceived as an exten- sion of the notion of a canonical transformation (symplec- metric structures. This language goes together with tomorphism) or a Poisson map. In the context of micro- that of graded manifolds, which are supermanifolds with formal geometry they play a different role as they are used an extra Z-grading in the structure sheaf. “Microformal for replacing ordinary maps of manifolds (the bases of the geometry” is a new notion referring to “thick” or “micro- cotangent bundles). We define pullbacks of functions by formal” morphisms, which generalize ordinary smooth thick morphisms, with the crucial new property of being maps, but whose crucial feature is that the correspond- (in general) non-linear. More precisely, the pullback by a thick morphism Φ: M M is a formal non-linear differ- ing pullbacks of functions are nonlinear. In particular, 1 → 2 ential operator Φ∗ : C∞(M ) C∞(M ), which is a formal “Poisson thick morphisms” of homotopy Poisson super- 2 → 1 perturbation of an ordinary pullback (by some map that manifolds induce L -morphisms of homotopy Poisson ∞ “sits inside” any thick morphism). Such a non-linear trans- brackets. There is a quantum version based on spe- formation has a remarkable feature that its derivative or cial type Fourier integral operators and applicable to variation at every function is the ordinary pullback by an Batalin–Vilkovisky geometry. Though the text is mainly ordinary map (more precisely, a formal perturbation of expository, some results are new or not published previ- an ordinary map) and hence an algebra homomorphism. ously. It remains an open question whether the non-linear pull- backs by thick morphisms can be characterized by this property. The discovery of thick morphisms resulted from our 1 Introduction search of a natural differential-geometric construction that would give non-linear maps of spaces of functions regarded as infinite-dimensional (super)manifolds. This The purpose of this text is to give an overview of graded was necessary for L -morphisms of bracket structures. In- geometry, i.e. the theory of graded manifolds, which are ∞ deed, the most efficient way of describing various bracket a version of supermanifolds (namely, supermanifolds en- structures, particularly homotopy bracket structures, is dowed with an extra grading by integers in the algebra of the language of Q-manifolds, i.e. supermanifolds, possi- functions) that have attracted much attention in recent bly graded manifolds, endowed with an odd vector field years, and an introduction to the new area of microformal geometry, whose main feature is the new notion of “mi- croformal” or “thick” morphisms generalizing ordinary smooth maps. These two topics are related by the type ∗ Corresponding author E-mail: [email protected] of applications, which are structures such as homotopy 1 School of Mathematics, University of Manchester, Manchester, algebras ultimately motivated by physics, in particular by M13 9PL, UK “gauge symmetries” in broad sense. Key for description of 2 Faculty of Physics, Tomsk State University, Tomsk, 634050, homotopy structures is the language of Q-manifolds (see Russia

1 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

Q satisfying Q2 0. The superiority of this geometric lan- ogy with a larger class of morphisms. Also, the applica- = guage is proved when morphisms are considered: com- tions require considering graded or super case and this plicated and non-obvious algebraic definitions e.g. for is the most natural framework for us, the construction of L -algebras or Lie algebroids over different bases are de- thick morphisms has nothing particularly super as such ∞ scribed with great simplification and uniformity as noth- and makes perfect sense in an entirely even context. ing but Q-maps of Q-manifolds, i.e. maps of the under- In the super case, there are two parallel versions of lying supermanifolds that intertwine the corresponding thick morphisms, adapted for pullback of even and odd vector fields Q1 and Q2. Non-linearity in such a map is functions respectively (“bosonic” and “fermionic”). In- responsible for “higher homotopies” in the algebraic lan- deed, a non-linear transformation cannot be applied in- guage. discriminately to elements of an algebra satisfying differ- (In physics parlance, a homological vector field Q is ent commutation rules, hence the need to distinguish an infinitesimal “BRST transformation”. In mathematics, between even and odd functions. While the “bosonic” particular instances of homological vector field have been version of thick morphism uses the symplectic geome- known as various differentials, e.g. the de Rham differen- try of cotangent bundles T ∗M, the “fermionic” version tial or Chevalley–Eilenberg differential. The power of the uses the odd symplectic structure on anticotangent bun- Q-manifold language was demonstrated by Kontsevich’s dles ΠT ∗M. Also, the bosonic case can be further lifted formulation and proof of the formality theorem imply- on a quantum level. There are “quantum thick mor- ing the existence of deformation quantization of arbitrary phisms”, which are (up to reversion of arrows) particular Poisson structures, which would be impossible without type Fourier integral operators. The “classical” thick mor- it.) phisms are recovered in the limit 0 (similarly with ħ → Therefore, in the case when a bracket structure is de- Hamilton–Jacobi equation and Schrödinger equation). fined on functions, there is the need for a construction To put the topics of this paper in a broader context, re- naturally leading to non-linear maps between spaces of call that there is a general philosophical principle of a cer- functions. Clearly, ordinary pullbacks cannot serve this tain “duality” between algebraic and geometric languages. purpose as they are algebra homomorphisms and in par- More specifically, there is a duality between commutative ticular linear. We came to the new “non-linear pullbacks” algebras and “spaces” (understood in the broadest sense). of functions and the underlying “thick morphisms” of With every “space” (such as a topological space or a mani- (super)manifolds by solving a very concrete problem con- fold or an algebraic variety) we can associate an algebra, cerning the higher analog of Koszul bracket on differential which is an appropriate algebra of functions, and with a forms (introduced earlier by H. Khudaverdian and the au- map of such spaces we can associate an algebra homo- thor) corresponding to a homotopy Poisson structure. In morphism in the opposite direction, given by the pullback. the classical case of a usual Poisson bracket and the in- Conversely, every commutative algebra can be morally duced by it odd Koszul bracket on forms, the classical fact regarded as an algebra of functions and algebra homo- in Poisson geometry was that raising indices with the help morphisms as morally corresponding to maps of spaces, of the Poisson tensor maps the Koszul bracket on forms with the reversion of arrows. This heuristic principle can to the canonical Schouten bracket (the “antibracket”) on be traced back to the results of Stone and Kolmogorov– multivectors. In the homotopy case, an analog of that Gelfand in 1930s and Gelfand’s duality between compact posed a big problem, since there is only one antibracket Hausdorff spaces and Banach algebras, and is fully real- on multivectors and a whole infinite sequence of “higher ized in Grothendieck’s theory of schemes. Application of Koszul brackets” on forms. Hence only an L -morphism this principle of algebraic-geometric duality to graded al- ∞ linking them would be possible, i.e. a non-linear trans- gebras leads to supergeometry and the theory of graded formation of forms to multivectors. This has been indeed manifolds considered in this paper. Applying it to differ- achieved with the help of thick morphisms and pullbacks ential graded algebras gives Q-manifolds. Therefore the by them. It is absolutely certain now that any theory of ho- theory of the latter is a “non-linear homological algebra”.If motopy brackets on manifolds (super or graded) should one further applies to it the central idea of modern homo- use thick morphisms and will be incomplete otherwise. logical algebra, i.e. that of derived category (considering Although as explained microformal geometry, i.e. the complexes up to quasi-isomorphism), the outcome will theory of (super-, graded) manifolds with thick mor- be “derived geometry” (which we do not consider here; phisms, owes its birth to Poisson and other bracket struc- see e.g. [44]). Now, the emergence of thick morphisms and tures and their homotopy versions, we wish to stress that non-linear pullbacks indicates at a non-linear extension in itself it does not assume on manifolds any additional of algebraic-geometric duality, which needs to be under- structure and as such is an extension of differential topol- stood. Combination of our “microformal geometry” with

2 homological and homotopical ideas seems to us a fruitful After we will have explained the ‘graded’ notions, we will direction of possible future study. be assuming by default that grading can be included in all Note that both graded geometry and microformal ge- our constructions and will make that explicit only when ometry were ultimately motivated by structures coming specifically need it. from physical problems e.g. gauge theory. One of the pur- The author thanks the organizers of the LMS–EPSRC poses of this text is to be read by physicists and I tried to Durham Symposium “Higher Structures in M-Theory” make the paper readable. (Durham, 12–18 August 2018) for the invitation and very The structure of the paper is as follows. We begin from stimulating atmosphere, and in particular for the sugges- standard algebraic preliminaries and then pass to the tion to write this text. definition and constructions of graded manifolds (Sec- tion 2). For those familiar with the subject, we can say that we stress the distinction between grading responsible for 2 Graded geometry signs (Z2-grading or parity) and Z-grading, which we call “weight” (in physics it can be e.g. ghost number). They can be related, but do not have to, as serving different pur- 2.1 Graded notions. Algebraic preliminaries poses. We also distinguish between general Z-graded case and the case of non-negatively graded manifolds. The 2.1.1 Basic definitions latter have a natural structure of a fiber bundle with par- ticular polynomial transformations as the structure group. Recall some general algebraic notions. Let G be an abelian They can be seen as a generalization of vector bundles. group written additively. A “grading” with values in G is (Our general thesis is that a Z-grading is a replacement attaching labels λ G to elements of some object; an el- ∈ of a missing linear structure.) We show that to every such ement to which such a label is attached is called “homo- non-linear fiber bundle there corresponds canonically a geneous”. In such a generality, grading makes sense for graded vector bundle of a larger dimension containing sets. A set S is G-graded if S S (disjoint union). Fix = ∪λ λ all the information about the transition functions of the G and in the future say “graded” for “G-graded” unless original bundle. (We call that “canonical linearization”.) we need to clarify G. Presentation of S in such a form In Section 3, we introduce Q-manifolds and explain how is a “graded structure”. More often the notion of grad- they can be used for describing various structures, in par- ing is applied in the additive situation: to abelian groups, ticular homotopy bracket structures such as L -algebras vector spaces and rings. A vector space or a module M ∞ and L -algebroids. This language is applied in the next over some ring is called graded if it has a form M Mλ, ∞ = ⊕ two sections. In Section 4, we define thick morphisms of where M M are submodules. In the sequel we always λ ⊂ supermanifolds and give their main properties (the most assume that such a presentation is fixed as part of struc- important of which is the formula for the derivative of pull- ture. Notation-wise, the index λ can be written as a lower back). Then we show that a homotopy Poisson thick mor- index or as an upper index depending on convenience and λ phism induces an L -morphism of the corresponding the typical convention is that Mλ M − . Some sources ∞ = homotopy Poisson algebras. On the way we recall S - and promote the idea of defining a graded module just as a ∞ P -structures. We also introduce a “non-linear adjoint” family of modules (Mλ) instead of a direct sum. This is ∞ (an analog of adjoint for a non-linear operator) as a thick more or less the same and amounts to considering only morphism and give an application to L -algebroids. In homogeneous elements instead of sums. The standard ∞ Section 5, we describe “quantum microformal geometry”. notions concerning direct sums, homomorphisms and In particular, we show how quantum thick morphisms tensor products in the graded situation are as follows. The give L -morphisms for “quantum brackets” generated by direct sum of two graded modules (defined as usual) is nat- ∞ a higher-order BV operator. urally graded by (M N) M N .A homomorphism ⊕ λ = λ ⊕ λ There are many other things that we wanted to include of graded modules f : M N of degree µ is a collection → in the paper, but could not because of time and space of homomorphisms f : Mλ Nλ µ for all λ. (Sometimes → + limitations. Also, we do not claim any completeness of the notation f (fλ), where fλ : Mλ Nλ µ, is used.) Hence = → + given bibliography, though we tried to provide accurate there are the family of modules Homµ(M,N) of homomor- historic references. phisms of degree µ and the graded module Hom(M,N) = A note about terminology. We often, but not always, Hom (M,N). We shall often refer to homomorphisms ⊕µ µ drop the prefix ‘super-’ (as well as the adjective ‘graded’) simply as “linear maps”. One may note that homomor- and can speak (for example) about ‘manifolds’ or ‘Lie al- phisms between graded modules can be seen as having gebras’ meaning ‘supermanifolds’ and ‘Lie superalgebras’. naturally a bi-graded structure, i.e. Hom(Mλ,Nν), which

3 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

is then converted into a single “total” grading by consider- locally finite-dimensional module as the formal sum Q ing Homµ(M,N): λ ν µ Hom(Mλ,Nν). Likewise, the X = − + = dim(M) m e(λ). tensor product M N appears first as a bi-graded object, λ = λ G ⊗ ∈ Mλ Nν, and then the total degree is defined as the sum ⊗ It is an element of the formal group ring Z[[G]] and the of the two degrees, so that (M N)µ : λ ν µMλ Nν. In ⊗ = ⊕ + = ⊗ the same way grading is defined for tensor products with function dimg (M) is the “Fourier transform” of dim(M). any finite number of factors. We have consequently If there is a bilinear multiplication of any kind, it is dim(M N) dim(M) dim(N) naturally translated as a homomorphism M N L (of ⊕ = + ⊗ → some degree). Such are, in particular, multiplications in and graded associative algebras and a left module structure dim(M N) dim(M)dim(N), over such an algebra. By default such product structures ⊗ = are assumed of degree zero, i.e. Aλ Aµ Aλ µ and AλMµ where the product is in the formal group ring. ⊂ + ⊂ Mλ µ. (Dimension of a graded module is very close to the + The tensor algebra of a graded module is naturally bi- notion of a formal character used in representation theory, n n graded by (Z,G): T (M) n+∞0T (M), where T (M) where they consider formal exponents, see e.g. Kac [19].) n n = ⊕ = = M ⊗ λ(M ⊗ )λ λ1 ... λn λMλ1 ... Mλn . Defini- = ⊕ = ⊕ + + = ⊗ ⊗ tions of the symmetric and exterior algebras depend on an extra piece of structure and will be discussed below. 2.1.3 Grading and commutativity

Grading in mathematics plays two different roles. One is 2.1.2 Dimension a counting device, e.g. for “vanishing by dimensional con- siderations” type arguments and arguments by induction. Suppose a graded module M is free, i.e. there is a basis Another is about the form of “commutation rules”. This consisting of free homogeneous generators. More pre- concerns linearity if the ground ring has a non-trivial grad- cisely, a basis is E E , where E M , so that the ing, relation between left and right modules, and identi- = ∪λ λ λ ⊂ λ basis elements can be written as e E where i I fication of M N with N M. In all these cases there are λiλ ∈ λ λ ∈ λ ⊗ ⊗ for some chosen set of indices I , I E . A free mod- elements moving past each other and one needs rules λ | λ| = | λ| ule is locally finite-dimensional if E for all λ G as to what happens, referred to as a “commutativity con- | λ| < ∞ ∈ and finite-dimensional if it is locally finite-dimensional straint”. (Abstract algebraic theory is “braidings in tensor and bounded (i.e. M 0 only for a finite number of λ). categories”,but we do not have to go that far.) As it was dis- λ 6= (The forgetful functor maps free and finite-dimensional covered experimentally in topology, differential geometry modules to the same type ungraded objects.) For a lo- and algebra, in the graded situation there arises a non- cally finite-dimensional module M, there are numbers trivial “commutativity constraint” expressed by the fol- 0 m dimM ZÊ , so there is a function G Z, λ m , lowing “sign rule”. Fix a group homomorphism ε: G Z2, λ = λ ∈ → 7→ λ → which we shall denote dimg (M) for the reasons which will called parity. For homogeneous elements x of degree λ be clear soon, dim(M)(λ) m . Then we write ε(x) ε(λ). Then the sign rule says that if in the g λ = = ungraded setting in a formula there is a swap of adjacent dim(g M N) dim(g M) dim(g N) elements x and y, then in the graded case it should be ⊕ = + modified by inserting the sign ( 1)ε(x)ε(y) (and any change − and of order is reduced to swapping of neighbors). The obser- vation is that if this rule is applied in the definitions, it will dim(g M N) dim(g M) dim(g N), ⊗ = ∗ then appear in the theorems. As Manin rightly notes, this cannot be made a meta-theorem because it fails for the the convolution of functions, i.e. graded analog of determinant; but it works in many cases. X We quote some examples, which should be undoubtedly dim(g M N)(λ) dim(g M)(µ)dim(g N)(λ µ). ⊗ = µ − familiar, but we need them for reference. Note that there is no change in the notions of a left or (The formula for the tensor product makes sense if one right module structures for the graded case (i.e. a(bx) = of the modules is finite-dimensional.) It is convenient to (ab)x or (xa)b x(ab) is required) since there is no = introduce formal exponentials e(λ) as symbols satisfying change in associativity laws. If there are left module struc- e(λ µ) e(λ)e(µ) and define the (formal) dimension of a tures over a graded algebra A, a homomorphism of graded + =

4 modules f : M N of degree λ is (graded) A-linear or T (M) by the ideals generated by x y ( 1)ε(x)ε(y) y x → ⊗ + − ⊗ A-linear from the left or is an A-homomorphism (a homo- and x y ( 1)ε(x)ε(y) y x respectively (where x, y M). ⊗ − − ⊗ ∈ morphism over A) if Recall that T (M) is bi-graded by tensor degree and de- gree induced from M. Since these ideals are homoge- f (ax) ( 1)ε(a)ε(λ)a f (x) = − neous in the “bi-” sense, both Λ(M) and S(M) inherit bi-grading: Λ(M) P Λn(M) P Λn(M) and S(M) for a homogeneous a A. (In the sequel we shall always = n = n,λ λ = ∈ P Sn(M) P Sn(M) . The symmetric algebra S(M) is write formulas for homogeneous objects.) If there is a left n = n,λ λ commutative with respect to the induced grading, module structure over A, then the multiplication from the right is defined by a b ( 1)ε(a)ε(b) b a , · = − · xa : ( 1)ε(x)ε(a)ax . = − (tensor degree playing no role), while the exterior algebra One can immediately see that this formula gives a right Λ(M) is not commutative. The multiplication in Λ(M) module structure if A is considered with the new product (called the exterior product) satisfies

ε(a)ε(b) pq a b : ( 1)ε(a)ε(b)ba . a b ( 1) + b a , • = − ∧ = − ∧ This is called the opposite algebra Aop . Further, if a ho- where a Λp (M) and b Λq (M). Such a property is called ∈ ∈ momorphism f is A-linear for left A-modules as defined skew-commutativity. above, this converts into f (xa) f (x)a 2.1.4 Choice of grading group: G Z Z = = × 2 (no sign!), which should be taken as the definition of So far we have worked with a general group G endowed (graded) linearity from the right. The commutator of two with a parity ε. Favorite choices are: G Z with ε(n) n elements in an algebra is defined by = = mod 2 and G Z where ε is the identity. The former was = 2 [a,b]: ab ( 1)ε(a)ε(b)ba . a classical choice in algebraic topology. The latter is the = − − choice in superalgebra and supergeometry. We will use If if vanishes, the elements commute. If all elements in an the following combination: G Z Z and ε: Z Z Z = × 2 × 2 → 2 associate algebra commute, it is called commutative. (We is the projection on the second factor. (This includes other suppress any adjectives and do not say “graded commuta- mentioned choices as special cases.) For the first factor tive”.) If an algebra is commutative, it coincides with the Z we use the term weight. Notation: w(x) and ε(x) for the opposite algebra defined as above and hence left modules weight and parity of a homogeneous object x. We shall over a commutative algebra are also right modules, and also use the tilde notation for parity, x˜ : ε(x). Objects of = conversely. Properties of the commutator in an associa- parity 0 are referred to as even and of parity 1, odd. We tive algebra turned into axioms define what is traditionally stress that Z and Z2 gradings are in general independent; called a “graded Lie algebra”. The name may raise some this does not exclude particular cases where parity of a objections and we shall come back to that later (when given object coincides with its weight modulo 2. It may we also elaborate the definition). Finally, a derivation of also happen that weights take values in Z+ only (i.e., there a graded algebra A is defined as a homomorphism (lin- are no objects with negative weights). We have chosen ear map) D : A A of degree λ (called the degree of a the term ‘weight’ as generic; in particular situations the → derivation) satisfying Z-grading may be called ‘degree’, ‘ghost number’, etc. Specifying the notion of graded dimension from 2.1.2 ε(λ)ε(a) D(ab) D(a)b ( 1) aD(b). for the group G Z Z , we arrive at graded dimensions = + − = × 2 of the form Similarly are defined derivations for other possible set- tings (e.g. over an algebra homomorphism A B). The X w ε 1 → dimM nw,εq Π Z[q,q− ](Π) guiding principle in all cases is that derivations are per- = w Z,ε 0,1 ∈ turbations of maps respecting multiplication (e.g algebra ∈ = homomorphisms) and the rest follows from the formula where Π2 1. Recall that dimensions of Z -graded ob- = 2 for linearity. jects take values in Z[Π]/(Π2 1) Z ZΠ. Denote for − = + The exterior algebra Λ(M) and the symmetric algebra convenience the latter ring by Zˆ . Its elements are written S(M) are defined as the quotients of the tensor algebra as p qΠ or p q, where Π 0 1 and 1 1 0. So we can + | = | = |

5 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

re-write plex or mixed complex-real cases.) The following classes a X w 1 of functions of variables x are natural to consider: dimM nˆw q Zˆ [q,q− ]. = w Z ∈ – polynomial in all variables; ∈ – smooth (C ∞) in the variables of weight 0 and polyno- mial in the variables of weights 0; 6= 2.1.5 Weight as generalization of linear structure – smooth (C ∞) in the variables of weight 0 and formal power series in the variables of weights 0. 6= If we start from a vector space V and consider an algebra (Since the expansion in a finite number of odd variables generated by V such as T (V ), S(V ) or Λ(V ), they all have always terminates due to their nilpotence, and polynomi- a natural Z-grading by powers of V : n-fold products have als and smooth functions in odd variables are the same, degree n, etc. Linear maps of vector spaces induce alge- the difference arises only for even variables.) bra homomorphisms preserving degrees. Conversely, any Why one may need formal power expansions, is clear algebra homomorphism of one of these algebras preserv- from the following example. ing degrees is naturally induced by a linear map of vector Example 2.1. Let w(x) 0, w(y) 1 and w(z) 1. The spaces. The key moment here is that all free generators = = − = + element x yz is of weight 0 and the substitution of it into of an algebra have the same grading, hence they cannot + any smooth function, e.g. sinx, must be legitimate: “mix” under grading-preserving homomorphisms. At the same time, allowing algebra homomorphisms of T (V ), 1 sin(x yz) sinx (yz)cosx (yz)2 sinx ... S(V ) or Λ(V ) without preservation of natural grading ef- + = + − 2 + fectively destroys a memory of the original vector space V . Suppose however that free generators of one these al- It transforms a smooth function of a variable of weight gebras as assigned weights not necessarily equal to each 0 into a power series with respect to the variables y,z. other. Then a homomorphism preserving such a weight is (This should be contrasted with the case of odd variables ξ and η, e.g. sin(x ξη) sinx (ξη)cosx, where the Taylor in general non-linear and a geometric object associated + = + with such an algebra is no longer a vector space, but some expansion always terminates.) “graded space”. Therefore, we are forced to consider formal power se- Such a situation materializes, for example, for multiple ries in the variables of non-zero weights if we wish to use vector bundles, say, double vector bundles such as TE or arbitrary smooth functions, not only polynomials, in the T ∗E for a given vector bundle E M. They are not vector variables of weight zero, and if general transformations → bundles over the original base M (no linear structure) and of variables preserving weights and parities are allowed. considering ‘total weight’ for them leads to coordinates Hence the two exceptions. having weights 0,1, and 2. Example 2.2 (Restriction of admissible transformations). If we agree to restrict admissible transformations of vari- ables so that: (1) the variables of weight 0 are allowed 2.2 Graded manifolds. Definition and constructions to transform only between themselves (no admixture of variables of weights 0) and (2) within the variables of 2.2.1 Local models 6= non-zero weights any homogeneous polynomial transfor- mations are allowed, then the class of functions of xa that In the sequel, ‘polynomials’ will refer to a free commuta- are arbitrary smooth in xa of wa 0 and polynomial in xa tive algebra, which is an ordinary polynomial algebra if all = of wa 0 will be stable under the corresponding substi- generators are even, a Grassmann algebra if all generators 6= tutions. (Geometrically this corresponds to considering are odd, and the tensor product of an ordinary polynomial fiber bundles where the variables of weight zero are coor- algebra and a Grassmann algebra in general. dinates on the base and the variables of non-zero weights To describe a local model of a graded manifold, con- are coordinates in the fibers.) sider a finite number of variables xa (practical needs my require considering the infinite-dimensional case as well, Example 2.3 (Non-negative weights only). Suppose for but here we confine ourselves to finite dimensions) to all variables xa, w(xa) 0. Then there is only a finite num- Ê which are assigned both parities and weights. Notation: ber of homogeneous monomials in xa, wa 0, of any 6= wa w(xa), a˜ ε(xa). The variables xa are assumed to given total weight n N and there are no monomials of = = ˜ ∈ be commuting: xa xb ( 1)a˜b xb xa. We also assume that weight 0. Hence all homogeneous formal power series in = − they are real, since we are going to define real graded man- the variables of non-zero weights are polynomials, and ifolds. (Necessary modifications can be made for the com- there can be no transformations admixing variables of

6 non-zero weights to the variables of weight zero. (Geomet- graded coordinate domains ϕ: U W as a morphism of → rically this corresponds to a fiber bundle structure that local ringed spaces over R, comes about automatically.) ϕ (ϕ00,ϕ∗):(U00,EU ) (W00,EW ), Every algebra homomorphism from functions of xa = → to real numbers sends the variables that are of non-zero with the algebra homomorphisms weight or odd, to zero. (They cannot ‘take values’ apart 1 ϕ∗ : EW (V00) EU (ϕ− (V00)) from zero.) On the other hand, all even xa such that → a w(x ) 0 can be considered as coordinates in the ordi- for all open subsets V00 W00 being of described type. = n00 ⊂ nary sense on some open domain (open set) U00 R Such morphisms are in a one-one correspondence with a ⊂ a that we can choose (n00 is the number of x with w(x ) homomorphisms of algebras of “global” functions a = ε(x ) 0). We interpret U00 taken together with all our = ϕ∗ : E (W ) E (U) variables xa, of all weights and parities, as a graded coor- → dinate domain, and use the same letter U but without the (The role of locality in the definition of a morphism is subscript to denote this new object. The (graded) dimen- to ensure that the map ϕ00 of the underlying topological sion of a graded coordinate domain U is spaces is exactly the one obtained from the homomor-

X w 1 phism of algebras with the help of the augmentation.) dimU nˆw q Zˆ [q,q− ], = w ∈ where nˆ n n and n (resp., n ) is the number w = w,0| w,1 w,0 w,1 2.2.2 Definition of a graded manifold of even (resp., odd) coordinates xa of weight w Z. For- ∈ mally, a graded coordinate domain U can be seen as a pair The definition of graded manifold is now completely consisting of an open domain U00 and the algebra E (U), straightforward; it mimics definitions of ordinary smooth

a a manifolds and supermanifolds. Recall that for a ringed E (U): C∞(U00)[[x w 0 or a˜ 1]] = | 6= = space X (X00,AX ), an open subset U X is the ringed = ⊂ space U (U ,A U ), for an open U X . In the (formal power series in variables of non-zero weights). By = 00 X | 00 00 ⊂ 00 same sense we understand preimages and intersections a slight modification, we can replace a single algebra E (U) by a sheaf of algebras on U (by taking (V ): of open subsets. An open cover consists of open subsets EU 00 EU 00 S = (Uα) such that Uα00 X00. Fix a collection of numbers E (V ) for all open subsets V00 U00) and define, finally, = ⊂ of variables with prescribed parities and weights, i.e., a

U : (U00,EU ). graded dimension. Consider graded coordinate domains = of this graded dimension. Suppose X is a local ringed The sheaf EU is a sheaf of Z Z2-graded commutative space over R, so in particular is a sheaf of R-algebras. × AX algebras over R with a unit and there is a natural augmen- A local chart for X is an isomorphism ϕ: V U of local → tation ringed spaces over R, where U X is n open subset and V ⊂ is a graded coordinate domain. An atlas for X is a collec- ε: EU C ∞ U00 tion of charts ϕ : V U such that (U ) make an open → α α → α α cover. We require that the resulting transformations of co- given by sending all odd variables and all even variables 1 1 1 ordinates ϕαβ : ϕα− ϕβ : ϕ− (Uα Uβ) ϕα− (Uα Uβ) of non-zero weight to 0. On the stalk Ex at each x00 U00 = ◦ β ∩ → ∩ 00 ∈ it gives a homomorphism ε: E R and the kernel are smooth maps of graded coordinate domains. We refer x00 → Kerε E is a unique maximal ideal. We shall consider to such atlases as smooth. Two smooth atlases for X are ⊂ x00 algebra homomorphisms E (W ) E (U) for graded coor- equivalent if their union is a smooth atlas. → dinate domains that can be expressed by substitutions Definition 2.1. A smooth graded manifold of a given in coordinates, yi ϕi (x), with the right-hand sides be- graded dimension is a local ringed space over R, X = = ing functions of the same class (i.e. smooth in the even (X00,AX ), with a Hausdorff second-countable underlying variables of zero weight and formal power series in the topological space X00, endowed with an equivalence class rest). (Possibly such are all the algebra homomorphisms of smooth atlases. The structure sheaf AX will be denoted for these algebras, but we do not want to dwell on that.) by EX or CX∞ and is called the sheaf of smooth functions. By augmentation, they induce algebra homomorphisms A smooth map f : X Y of smooth graded manifolds is → C∞(W ) C∞(U ) and hence the usual smooth maps of a morphism in the category of local ringed spaces over 00 → 00 the underlying coordinate domains ϕ : U W . We R represented in local charts by smooth maps of graded 00 00 → 00 define a morphism (also called a smooth map) between coordinate domains.

7 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

Speaking informally, a graded manifold is a super- tions on Sn(q) can be described as the “inverse limit”: manifold with a distinguished class of atlases where co- n(q) n n(q) n(q) ¯ ordinates are additionally assigned weights in Z and the C ∞(S ) f (fN , fS ) C ∞(R ) C ∞(R )¯ = = ∈ × ¯ transformations of coordinates preserve both weights and ¾ ¡ uN ¢ parities. Smooth maps between graded manifolds are ex- fN (uN ) fS . = u 2 pressed in coordinates in the same way as for ordinary | N | a manifolds and supermanifolds. They are formal power Note that the transformations of coordinates, uS a 2 2 2 2= u u − , p p u − , q q u − , where u − series in coordinates of non-zero weight. N | N | S = N | N | S = N | N | | N | = ¡ a 2¢ 1³ 2pN qN ´ Similarly defined are categories of graded manifolds in P (u ) − 1 ... , are formal power series a N − P a 2 + the mixed (real-complex) smooth and complex-analytic a(uN ) in the coordinates of non-zero weights p ,q . settings. N N One can construct more similar examples as “graded analogs” of classical (super)manifolds. It would be inter- 2.2.3 Simple examples. “Graded sphere”. “Graded esting to study them systematically together with graded groups” analogs of classical differential-geometric structures. Another collection of examples can be obtained from graded Lie algebras (not to be confused with Lie superal- Many examples of graded manifolds in applications arise gebras!). It is well known that Z-gradings play important from auxiliary constructions for ordinary manifolds. How- role in the theory of finite- and infinite-dimensional Lie ever, they can also arise in their own right as graded algebras. Such algebras can come with natural gradings, analogs of familiar differential-geometric objects. The fol- which are forgotten when the corresponding Lie groups lowing example is meant to illustrate this point. are constructed. By taking these gradings into account n(q) 1 1 one can obtain ‘graded versions’ of these groups. Example 2.4. Consider R + , where n(q) q− n q, 1 n n 1 = + a+ with coordinates x ,...,x ,x + , y, z, of weights w(x ) Example 2.5. The vector space Mat(n) of (real) square = 0, w(y) 1 and w(z) 1. Consider the equation n n matrices becomes Z-graded if we take the matrix = − = + × units Ei,i r , for a given r Z, is a basis of the subspace + ∈ 1 2 n 2 n 1 2 (x ) ... (x ) (x + ) 2yz 1 Mat(n)r (which consists of matrices with non-zero entries + + + + = only on the r th diagonal). Non-trivial graded components exist only for r n 1. For each r , there are exactly n r (the left-hand side is a quadratic form of weight 0). It spec- | | É − −| | ifies a graded sphere Sn(q) as a closed subspace Sn(q) elements on the r th diagonal, hence the graded dimen- n(q) 1 ⊂ sion R + . Acting as for the ordinary sphere, one can intro- n(q) n(q) n(q) n 1 n 1 duce two charts ϕN : R S \ N and ϕS : R X− r X− r r n(q) → → dimMat(n) (n r )q n (n r )(q q− ). S \ S, so that = r n 1 −| | = + r 1 − + =− + = 2 This is a graded Lie algebra with respect to the matrix 2uN n 1 uN 1 ϕN : x , x + | | − , commutator: = u 2 1 = u 2 1 | N | + | N | + [Mat(n)r ,Mat(n)s ] Mat(n)r s . 1 x ⊂ + ϕ−N : uN n 1 , (r ) = 1 x + By multiplying the generators e : Ei,i r of weight r by − ir = + ir a a parameters t of weight r (so to obtain an expression where x (x , y,z), uN (u ,pN ,qN ), where a 1,...,n, (r ) − = = N = of weight zero) and taking the exponential, we obtain an w(ua ) 0, w(p ) 1, w(q ) 1, and u 2 P (ua )2 N = N = − N = + | N | = a N + invertible matrix that can be regarded as a “point” of the 2pN qN , and similar formulas for ϕS (with the opposite n 1 graded group (a group object in the category of graded sign for x + ). This gives manifolds) GL(n)grad corresponding to the graded Lie al- uN gebra Mat(n), u S 2 X i (r ) = uN g exp t r e . | | = (r ) ir ir (exactly as for the ordinary sphere or the supersphere) Parameters t(r ) are global coordinates on this graded man- as the change of coordinates. This shows that Sn(q) is a ifold. Obviously, the underlying ordinary manifold is just 1 smooth graded manifold of dimension n(q) q− n q. the group of diagonal matrices with positive entries and n(q) = + + The underlying topological space of S is the ordinary the graded group GL(n)grad can be regarded as its formal sphere Sn of dimension n. The algebra of smooth func- neighborhood in the Lie group GL(n).

8 2.2.4 Constructions with graded manifolds Another example is provided by differential forms on a vector bundle. Recall that in supergeometry, pseudodiffer- There are obvious analogs of constructions for ordinary ential forms, which we with an abuse of language will call manifolds and supermanifolds, such as submanifolds, simply “forms”, are functions on the antitangent bundle. products, etc. Closed submanifolds are locally specified by systems of equations of constant rank. It is required that Example 2.3. Consider for a vector bundle E, its antitan- the equations be homogeneous both in parity and weight, gent (parity reverses tangent) ΠTE. It is again a double where the notion of rank is understood as ‘graded rank’. vector bundle Then the dimension of S X is dimS dim X r , where ΠTE ΠTM 1 ⊂ = − r r (q) Zˆ [q,q− ] is the rank of the system of equations. −−−−−→ = ∈   As for the product X Y of graded manifolds, it is most   × y y natural to consider it as bi-graded. (“Bi-grading” refers to E M two Z gradings, with a single parity.)) −−−−−→ Every vector bundle by default can be considered as It has two weights corresponding to the two vector bundle a graded manifold so that linear coordinates in the fibers structures: w #dxa #dui and w #ui #dui . Here 1 = + 2 = + are assigned weight 1. Then fiberwise linear maps are induced weight is w #ui #dui w and degree is deg + = + = 2 = the same as weight-preserving. #dxa #dui w . In [72], [61] we discovered and used + = 1 All objects on a graded manifold assume weights, grading w w #dxa #ui on forms on E. Note that 1 − 2 = − e.g. tangent vectors, covectors, vector fields, etc. Tangent the de Rham differential has degree 1 in this grading. and cotangent bundles for graded manifolds carry a bi- + 0 grading. One Z-grading (actually, ZÊ ) is the vector bun- Besides grading, manifolds can be endowed with a fil- dle grading by degree in fiber coordinates. Another is the tration. For example, for a bi-graded manifold, one of the induced weight. gradings can become a filtration if more general transfor- mation are considered. Such is the case of “resolvent de- Example 2.1. If xa are coordinates of weights w a, the par- gree” in [14], which is preserved only as a filtration under tial derivatives ∂/∂xa have weights w a. Hence we assign a − canonical transformations. We do not formalize filtered weights w to the momentum variables pa canonically − a manifolds here, since this notion should be clear. conjugate to x . We arrive at the cotangent bundle T ∗M for a graded manifold M as a bi-graded manifold. The first grading induced from M we continue to call weight and 2.2.5 Structure of a graded manifold it is given by w(xa) w a,w(p ) w a. The second grad- = a = − ing we call degree and it just expresses the vector bundle structure: degxa 0,degp 1. Let X (X00,EX ) be a graded manifold. Denote by JX : = a = + = 2 = (EX ) 0 (EX ) 0 the ideal generated by all functions of non- Example 2.2. For a vector bundle E M regarded as a 6= + 6= → zero weight. Its zero locus is a closed submanifold X0 graded manifold in the usual way, its cotangent bundle = (X00,EX0 ), EX0 EX /JX , with the same underlying topo- T ∗E is a double vector bundle [38], with the side bundles = logical space X00. Note that in general X0 is a supermani- E M and E ∗ M: → → fold. It should not be confused with X00, which has a nat- T ∗E E ∗ ural structure of an ordinary manifold, X00 (X00,EX ), −−−−−→ = 00   where EX00 EX /((EX ) 0 (EX )odd) . In general, X00 X0,   = 6= + 6= y y only X X . Only if there are no odd coordinates of zero 00 ⊂ 0 weight, then X is an ordinary manifold and X X . We E M 0 0 = 00 −−−−−→ will be more concerned with X0. Powers of the ideal JX (this is related with the Mackenzie–Xu theorem, see 4.3.1). k 1 define infinitesimal neighborhoods Xk (X00,EX /J + ), The double vector bundle structure gives two gradings on = X i of the closed submanifold X0 X . This is an infinite se- T ∗E : w #p #p and w #p #u . Here we denote ⊂ 1 = a + i 2 = a + quence and X is its direct limit: by xa coordinates on the base, by ui coordinates in the fibers of E, and by pa, pi the conjugate momenta. Com- X0 X1 ... Xk Xk 1 ...... X . pared to out previous analysis, we have w #ui #p ⊂ ⊂ ⊂ + ⊂ = − i = w2 w1 as induced weight and deg #pa #pi w1 as − = + = Consider the normal bundle to X0 in X , defined as degree. usual as the quotient (TX )/TX . Denote it N. To see |X0 0 (In physics, the above grading w #ui #p appearÑ ˛A its structure, denote local coordinates of weight zero on = − i under the name “ghost number”, see [14].) X by xa and local coordinates of non-zero weights, by yi .

9 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

Transformation of coordinates has the form graded manifold itself, it has the same graded dimension as X . a a x x (x0, y0), = Note now that the graded manifold X is formal in the i i directions normal to the submanifold X . (Changes of y y (x0, y0), 0 = coordinates are given by formal power series in coordi- where x0 and y0 denote ‘new’ coordinates of weights 0 nates of non-zero weights.) In the same way as for super- and 0, respectively, xa0 and yi 0 . The right-hand sides are manifolds and their underlying ordinary manifolds, there 6= formal power series in coordinates of non-zero weights. are no actual (non-infinitesimal) ‘intermediate’ neighbor- For the induced transformation of fiber coordinates in the hoods between X0 and X . Therefore, in the smooth case, tangent bundle we obtain exactly as for smooth supermanifolds, an analog of the tubular neighborhood theorem gives a (non-canonical) a a a a ∂x i ∂x x˙ x˙ 0 (x0, y0) y˙ 0 (x0, y0), diffeomorphism = ∂xa0 + ∂yi 0 i i X N i a ∂y i ∂y =∼ y˙ x˙ 0 (x0, y0) y˙ 0 (x0, y0). = a0 + i 0 ∂x ∂y as graded manifolds. This is the classification theorem Note that the Jacobi matrix for transformation of coor- for smooth graded manifolds. dinates on X is a block matrix with blocks numbered by In the complex-analytic case, one should expect an a weights. In the formulas above, the matrix ( ∂x ) is the analog of Vaintrob’s theorem [55] for complex-analytic ∂xa0 i supermanifolds: namely, that a complex-analytic graded zero-zero block of the Jacobi matrix, the matrix ( ∂y ) con- ∂yi0 manifold X is a deformation of the respective normal sists of possibly several other diagonal blocks, while the bundle N. a i matrices ( ∂x ) and ( ∂y ) consist of off-diagonal blocks. In the same way as for smooth supermanifolds, the ∂yi0 ∂xa0 The entries in the diagonal blocks have weight zero; possibility to describe a smooth graded manifold as the entries in the off-diagonal blocks are of correspond- a graded vector bundle over an ordinary base (non- ing non-zero weights. The diagonal blocks are invertible canonically), does not make smooth graded manifold and, as formal power expansions in variables of no-zero not interesting. The key difference between graded vec- weights, will remain invertible is all such variables are set tor bundles and graded manifolds is that the latter have to zero. (One may say that the Jacobi matrix for transfor- more morphisms as transformations mixing variables of mation of coordinates on a graded manifold takes values different weights with the only condition that the total in the graded general linear group.) Upon restriction to weight — as well as parity — be preserved . Note also that, i even more, such morphisms can themselves depend on X0, all coordinates of non-zero weights y become zero and, in particular, all elements of off-diagonal blocks of parameters of non-zero weight leading to graded man- the Jacobi matrix will vanish. Hence the transformation of ifolds of maps, in particular already mentioned graded coordinates on the normal bundle N as a vector bundle groups, etc. etc. A remark giving a different perspective is that, in some over X0 will be cases, the formal power series defining transformations a a x x (x0 ,0) of variables in a graded manifold X can happen to be 0 = 0 the Taylor series of genuine smooth transformations of for the coordinates on the base X0, which we have marked even coordinates in an ordinary manifold X˜ . (Such are with the subscript 0 to distinguish them from those on X , the above examples of the “graded sphere” Sn(q) and the and graded group GL(n)grad.) In general, one case see a graded ∂yi manifold as a formal germ of an ordinary (super)manifold i i 0 y˙0 y˙0 (x0,0) of the “ordinary” dimension n obtained as n n(1) for = ∂yi 0 = dim X n(q). = for the fiber coordinates, where likewise we have attached the subscript to distinguish them from (a part of) coordi- nates on the tangent bundle TX . The normal bundle N is 2.2.6 Graded manifolds of maps. Functor of points a Z-graded vector bundle over a non-graded base X0, so it is a direct sum of ordinary vector bundles (with assigned One may wish to consider “graded manifolds of map- weights). (Everything is in the category of supermanifolds, pings”. First of all, they have to be (in general) infinite- which makes no real difference here.) If we treat N as a dimensional, hence strictly speaking outside the scope

10 a a a of the definition above. Difficulty with introducing such immediately identify the variables ϕ0 and ϕ1 with x and objects comes from two separate but entangled causes. dxa, respectively, the latter considered as coordinates on One is their infinite-dimensionality, and fundamentally ΠTM. this is the same difficulty that we have for ordinary man- Example 2.7. Consider an even variable t of weight ifolds when we want to define a manifold of maps. The 1 1 as a coordinate on Rq− . Find the graded manifold other cause of the difficulty is of ‘graded’ nature and has − 1 q− to be overcome already for supermanifolds. It is resolved Map(R , X ), for an arbitrary graded manifold X . The q 1 by allowing for odd as well as non-zero-weight parame- “coordinates” on Map(R − , X ) will be the power series ters in the formulas for the mappings (possibly, infinite a a +∞X 1 n a number of them). In brief, the graded manifold of maps x ϕ (t) t ϕn , = = n 0 n! Map(X ,Y ) for finite-dimensional graded manifolds X and = Y is defined “in the weak sense” by the formula with indeterminate coefficients ϕa , where ε(ϕa ) ε(xa) n n = and w(ϕa ) w(xa) n, for all n 0,1,2,... . Although Map(Z ,Map(X ,Y )) Map(Z X ,Y ), n = + = = × in this case the graded manifold of maps is infinite- where Map stands for the set of morphisms in the category dimensional (unlike the previous example), its infinite- of graded manifolds and Z is an arbitrary graded manifold. dimensionality is easily controllable. The transformation a The equality should be understood as an isomorphism of law for the variables ϕn follows from the expansion in t a a a a functors. The right-hand-side serves as the definition of of x x (ϕ0 tϕ0 ...), where x x (x0) is a change of = 0 + 1 + = the left-hand-side, i.e., Map(X ,Y ) is defined as the repre- coordinates on X . One gets senting object for the functor Z Map(Z X ,Y ) (if ex- a a 7→ × ϕ x (ϕ0 ) isted). In other words, the functor is known and we work 0 = 0 with it as if it were representable (this is what is meant ∂xa a a0 ϕ1 ϕ1 (ϕ0) by “weak sense”). The meaning of the above formula is = ∂xa0 that, for a given graded manifold Z , we consider all maps 2 a a a a b ∂ x a ∂x X Y depending on coordinates on Z as external pa- ϕ ϕ 0 ϕ 0 (ϕ0 ) ϕ 0 (ϕ0 ) → 2 1 1 b a 0 2 a 0 rameters; then Map(X ,Y ), if one can define it, serves as = ∂x 0 ∂x 0 + ∂x 0 the “universal family” of maps and coordinates on it are ... “universal” parameters. Acting naively, we can describe the graded manifold The transformation law for the variables of weight n in- Map(X ,Y ) as follows. If xa and yi are, respectively, local volves only variables of weights n, so can be truncated É 1 coordinates on X and Y , then “coordinates” on Map(X ,Y ) at any n. The graded manifold Map(Rq− , X ) is the inverse are functions yi ϕi (x), x (xa), defined by expansions limit of finite-dimensional graded manifolds. We recog- = = 1 over odd variables and variables of non-zero weights, nize in Map(Rq− , X ) the infinite-order tangent bundle ( ) (N) where the coefficients of the expansions, which should T ∞ X , which is the limit of higher tangent bundles T X , be ordinary smooth functions of the coordinates xa of taken with their natural graded structures. (Or spaces of weight zero, are treated formally as having the required jets of parameterized curves in X .) parities and weights (possibly, non-zero). What about functions on a graded manifold X ? Can In the following two examples we can avoid, or partly they be fit into the above? avoid, the problem arising from infinite-dimensionality. The graded manifold Map(X ,R) is the manifold of Example 2.6. For any graded manifold X , all even functions of weight zero. To odd functions or functions of non-zero weight, one needs to consider 0 1 Map(R | , X ) ΠTX . = Map(X ,ΠR), Map(X ,R[n]) or Map(X ,ΠR[n]). These are linear graded manifolds and we have Map(X ,ΠR) This is well known (at least in the non-graded case). In- = ΠMap(X ,R), etc. deed, if xa are local coordinates on X and τ is the single 0 1 The way how graded manifold of maps is introduced coordinate on R | , ε(τ) 1, w(τ) 0, then “coordinates” = = is an example of the idea the “functor of points”. Its ori- on Map(R0 1, X ) are functions of τ, | gins are in algebraic geometry and it is well known for a a a supermanifolds. Namely, every graded manifold X de- x ϕ(τ) ϕ0 τϕ1 , = = + fines a contravariant functor on the category of graded where w(ϕa) w(ϕa) w(xa), ε(ϕa) ε(xa), and ε(ϕa) manifolds, Z Map(Z , X ) (to the category of sets). Ele- 0 = 1 = 0 = 1 = 7→ ε(xa) 1 . By checking the transformation law, one can ments of the set Map(Z , X ) are called Z -points of X . In +

11 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

particular, the points of the underlying topological space 2.3.2 Canonical linear model 0 X00 are exactly R -points of X . A graded manifold is com- pletely defined by its functor of points. Hence the general Non-negatively graded manifolds are the most direct gen- idea when a graded manifold is being looked for in some eralization of vector bundles. There is one problem re- problem, to find it first “in the weak sense”, i.e. introduce lated with the fact that, unlike vector bundles, sections of first a functor that should serve as it functor of points for such a nonlinear bundle cannot be added or multiplied graded manifold in question and then see if it is indeed by numbers, so we seem to lose an algebraic arena where representable. (The analogy with a “weak solution” of a algebraic structures such as brackets can be defined. We differential equation is due to K. Fukaya.) shall show here that for a non-negatively graded manifold E regarded as a fiber bundle E M, its structure group → GG(V ) possesses a natural faithful finite-dimensional lin- 2.3 Non-negatively graded manifolds ear representation ρ. It plays the role of the standard rep- resentation of the general linear group, and reduces to it in the linear case. The associated vector bundle ρ(E) M 2.3.1 Non-negatively graded manifold as a fiber bundle → can be seen as a canonical “linearization” of the graded manifold E. It can be defined directly as corresponding to a - Suppose all local coordinates x for a graded manifold the projective module Vect (E) over C ∞(M) consisting of E have non-negative weights. Then: transformations of vector fields on E of negative weight. coordinates cannot have infinite power series and are nec- Consider the standard fiber V , which is positively essarily polynomial (see Example 2.3). We can arrange co- graded, and the space of vector fields. It naturally expands - ordinates by increasing weights. Then the coordinates of by weights as Vect(V ) Vect (V ) Vect+(V ), zero weight transform between themselves, coordinates = ⊕ - of weight 1 undergo linear transformation, coordinates Vect (V ) Vect N (V ) ... Vect 1(V ). + = − ⊕ ⊕ − of weight 2 transform linearly between themselves but + - can have a term quadratic in coordinates of weight 1, The group GG(V ) acts on Vect (V ). + etc. We arrive at a canonical tower of fibrations: Theorem 1. The representation of GG(V ) on Vect-(V ) is faithful. E EN EN 1 ...E2 E1 E0 M (1) = → − → → → = Proof. On the infinitesimal level, if a vector field of zero weight commutes with all vector fields of negative weights, Here N is the top weight of local coordinates on E. The then in particular it commutes with all partial derivatives subscript for Ek means the top weight for Ek . The first fi- ∂/∂yi , therefore it has constant coefficients, hence is zero. bration E E M is a vector bundle, the rest are affine 1 → 0 = bundles. Altogether this assembles into a fiber bundle E M with special form polynomial transition functions. - → We call the representation of GG(V ) on Vect (V ) the This picture was introduced in [62]. fundamental representation. D The standard fiber for E M is some R where It is finite-dimensional. In the case of GL(n) it is the P w → n D w 0 nw q , i.e., an affine space with coordinates standard representation on R . = > assigned with some positive weights. Denote this model The associated bundle F M corresponding to the → graded space by V . Denote by GG(V ) the group of graded fundamental representation of the group GG(V ) is called polynomial transformations of V (“general graded”). It the fundamental vector bundle of the graded manifold E. depends only on dimension of V . Its sections can be identified with Vect-(E). Since the Non-negatively graded manifolds because of restric- representation ρ is faithful, the bundle E M (its transi- → tions posed by their bundle structure are particularly use- tion functions) can be recovered from the vector bundle ful for encoding various differential-geometric informa- F M. We shall use this vector bundle when considering → tion. The method is placing a bound on “height” (top “non-linear Lie algebroids” in 3.4.5. weight of local coordinates) combined with “component analysis” of some graded quantity. The simplest but still very useful case is to treat vector bundles as graded mani- 2.4 Historical remarks about graded notions folds. This helps e.g for description of Lie algebroids and multiple Lie algebroids (see below in 3.3). (Also the de- Graded notions have long played important role in dif- scription of Courant algebroids by Roytenberg [46], [47].) ferent areas of mathematics, from gradings appearing in

12 the theory of Lie algebras where it was used as a tool in Z-grading in the structure sheaf with the same restric- classification and e.g. for measuring growth of infinite- tion. (He commented also that many of his constructions dimensional algebras, to graded objects in topology and are valid just for supermanifolds and do not require Z- 0 differential geometry, where grading was used for induc- grading.) Ševera [51] introduces a version of ZÊ -graded tion and “dimensional” arguments and as a source of “sign manifolds where parity equals degree mod 2 and they be- rule”. Nijenhuis–Richardson [42] developed the basics of come popular especially combined with a Q-structure graded algebras using grading by an arbitrary abelian under the name NQ-manifolds (N presumably for N). group endowed with a parity homomorphism. (They also Graded manifolds as defined here (with Z Z -grading) × 2 anticipated Lie supergroups.) were introduced and studied in [62]. In particular, the Looking at other important works, we may notice that tower of fibrations (1) for non-negatively graded mani- probably until the physics works related with BRST quan- folds appeared there. We have used them as a standard tization, Z-grading for algebras was almost always either language ever since, see e.g. [65], [25], [67], [66], [68]. 0 0 ZÊ or ZÉ . Tate [54] uses non-negatively graded alge- bras with homological differential. Milnor and Moore [41] 0 by “graded” mean ZÊ -graded. Deligne–Griffiths–Morgan– 3 Language of Q-manifolds. Description Sullivan [9] and Sullivan [53] saw graded algebras as non- of algebraic and geometric structures negatively Z-graded. Quillen [45] uses non-negative grad- ing. Boardman says quite explicitly that “graded” means In this section we will introduce the language of Q- Z 0-graded [3] and proceeds to establishing the sign rule Ê manifolds, which are supermanifolds endowed with an as a precise theorem. odd vector field of square zero. They provide a powerful Berezin, working on implementation of his program tool for describing differential-geometric and algebraic of supermathematics (before the name) made a decisive structures. From the viewpoint of algebraic-geometric step in separating Z -grading responsible for signs from Z- 2 duality, Q-manifolds are the geometric counterpart of dif- grading. In particular, he studied automorphisms of Grass- ferential Z -graded algebras and can be seen as a basis of mann algebra as a Z -graded algebra [2] and on these 2 2 a “non-linear homological algebra”. paths discovered Berezinian. Without that, there would be no supermanifolds. (Supermanifolds were briefly known for some as “graded manifolds” following Kostant, but this 3.1 Definition of a -manifold. Main notions usage has now gone.) Re-introduction of Z-grading into Q supergeometry, in a different way, is a new turn of Hegel’s dialectic spiral. 3.1.1 Definition and model examples Schlessinger and Stasheff in their famous long-secret work [48] use graded as Z-graded while noting that in Definition 3.1. A Q-manifold is a supermanifold en- 2 many cases it will be either 0 (as cochains in topology) dowed with an odd vector field Q such that Q 0. Such Ê = or 0 (as in algebraic geometry); they however consider a vector field is called homological. A homological vector É a bi-graded case for the “Tate–Józefiak resolution”. (Józe- field is also referred to as a Q-structure. fiak [18] generalized Tate’s resolution to graded case.) Note that for an odd Q, Q2 1 [Q,Q]. We may some- = 2 (Working in a different area, the present author found times write a Q-manifold as a pair (M,Q). a non-standard Z-grading for pseudodifferential forms If M is a Q-manifold and xa are local coordinates on on a vector bundle and used it for a study of integral trans- M, so that Q Qa(x)∂/∂xa, the condition Q2 0 is ex- = = forms [72], [61].) pressed by Supermanifolds graded additionally by Z (and some- a b times endowed with several gradings and/or filtration) Q ∂aQ 0. (2) appeared without any particular name or mathematical = formalization in Henneaux–Teitelboim [14]. They how- Remark. The notion of a Q-manifold was introduced by ever were quite explicit that parity and Z-grading (e.g. A. S. Schwarz, see [50]. The notation Q, can be traced “ghost number”) are independent and the latter can be back to the earlier study of supersymmetry in physics, positive and negative. where the letter Q was a standard notation for a super- Kontsevich in [29] introduces the tensor category of charge, i.e. an odd operator such that Q2 H, where = “graded vector spaces” as a full subcategory of Z-graded H is the (quantum) Hamiltonian or more generally an super vector spaces for which parity equals degree mod 2 even symmetry generator. If such an even symmetry van- and also “graded manifolds” as supermanifolds with extra ishes for whatever reason, we arrive at the situation when

13 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

Q2 0 (see e.g. [74]). Homological vector fields were Remark. The previous example is classical. Functions = studied by Vaintrob [56, 58]. Seminal role was played by on Πg can be identified with the “standard cochain com- the work of Alexandrov–Kontsevich–Schwarz–Zaboronsky plex” C ∗(g) of a Lie algebra g and the vector field Q is the (AKSZ) [1] and the application of Q-manifolds by Kontse- Chevalley–Eilenberg differential in this complex. We shall vich in [29,30]. But before Q-manifolds were formalized use this example as a model for describing other struc- as a mathematical notion, homological vector fields had tures. One should also compare the formula for the vector existed in physics as BRST symmetries (for Becchi–Rouet– field Q on Πg with the formulas {y , y } ck y for the i j = i j k Stora and I. Tyutin), see monograph [14]. The physicists’ Lie–Poisson bracket (=Berezin–Kirillov bracket) on g∗ and approach for a long time was only half-geometrical, as {η ,η } ck η for the Lie–Schouten bracket (odd ana- i j = i j k they were main drawing from known algebraic methods log of Lie–Poisson) on Πg∗. For a Lie algebra, these three of homological algebra (e.g. Tate resolution). structures are different equivalent manifestations of a Lie Example 3.1. For any (super)manifold M, the superman- algebra structure itself. If we drop the restrictions e.g. the ifold ΠTM is a Q-manifold. The Q-structure is given by linearity for the brackets, we will arrive at Q-manifolds, the de Rham differential: Poisson manifolds and odd Poisson manifolds as three different non-linear generalizations of Lie algebras. ∂ Q d dxa . = = ∂xa The general philosophy is that a Q-manifold is a non- linear analog of a (co)chain complex. Respectively, we This example from many viewpoints plays the same role will introduce now the analogs for chain maps and for for Q-manifolds as T M with the canonical symplectic ∗ cohomology. We will also give the analog of the complex structure plays for symplectic manifolds. (Also, we shall of homomorphisms. see that from some abstract viewpoint, the Q-structure on ΠTM as well as the even and odd symplectic structures on T M and ΠT M are manifestations of “one and the ∗ ∗ 3.1.2 Q-morphisms same structure”.)

Example 3.2. Let V be a Z2-graded vector space, which Definition 3.2. A morphism of Q-manifolds (M1,Q1) to we treat as a supermanifold and actually as a graded man- (M2,Q2) or a Q-morphism or a Q-map is a supermanifold ifold (in the usual way). An odd differential on V , i.e. an map ϕ: M M that intertwines Q and Q , i.e. such 1 → 2 1 2 odd linear operator d : V V such that d 2 0 defines a that the vector fields Q and Q are ϕ-related. → = 1 2 “linear vector field” Recall that in general for vector fields Q1 and Q2 (that a b ∂ for this purpose do not have to be homological) the con- Q x da , = ∂xb dition of being intertwined by a map ϕ or being ϕ-related can be formulated in two equivalent ways: either as the (where (d b) is the matrix of the linear operator d), which a commutativity of the diagram is a Q-structure. Note that w(Q) 0 for the natural Z- = T ϕ grading. If V is a cochain complex, i.e. is itself endowed (Π)TM (Π)TM 1 −−−−−→ 2 with a Z-grading so that degd 1, the corresponding x x = +   (3) supermanifold becomes bi-graded (by weight and by de- Q1 Q2 gree), and degQ 1. ϕ = + M1 M2 −−−−−→ Example 3.3. Let g be a Lie algebra (we will shortly gen- (if vector fields are seen as sections of the tangent bundles; eralize to Lie superalgebras). Consider the supermanifold if a vector field is even, it is a section of TN M and if i → Πg. Let ξ be linear coordinates on Πg corresponding to a it is odd, it is a section of ΠTM M, so we have or not i → basis ei in g. (Because g is purely even, all coordinates ξ have Π in the above diagram). Or as the equality are odd.) Consider a vector field on Πg Q ϕ∗ ϕ∗ Q , (4) 1 ◦ = ◦ 2 1 i j k ∂ Q ξ ξ ci j , where ϕ is the pullback of functions and vector fields are = 2 ∂ξk ∗ regarded as operators on functions. k a i where ci j are the structure constants of g in the basis ei . If x and y are local coordinates on Q-manifolds M1 The vector field Q is odd and w(Q) 1 w.r.t. grading and M2, the condition that a map ϕ: M1 M2 is a Q- = + → given by the linear structure. One can check that Q2 0 morphism is expressed by k = due to the Jacobi identity for ci j . Moreover, the condition i a ∂ϕ i 2 Q 0 is exactly equivalent to the Jacobi identity in g. Q1 (x) a (x) Q2(ϕ(x)), (5) = ∂x =

14 i i where ϕ∗(y ) ϕ (x). Hence one may wish to explore the space of leaves = Z(M)/B, which is a kind of “non-linear homology” [1]. Proposition 3.1. In the examples above, i.e. ΠTM, Πg for Functions on Z(M)/B are those functions on Z(M) that a Lie algebra g, and the Q-manifold corresponding to a are constant in the directions of B. complex (V,d),— Q-morphisms preserving grading are In the model examples we obtain the following. For a equivalent to, respectively: arbitrary maps M M ; Lie 1 → 2 Q-manifold corresponding to a complex (V,d), Z(M)/B algebra homomorphisms g h; chain maps V W. → → coincides with the usual cohomology Z (V,d)/B(V,d). For We see that for maps ΠTM ΠTM the condition the rest, the answers are less obvious. One can see that 1 → 2 that a map is a Q-morphism is an “integrability condition”. for ΠTM and for any x M Z(ΠTM), Bx Imdx ∈ = = = If we relax preservation of grading, more Q-maps appear. Kerdx Tx M. (There is no homology in the tangent = For example, general Q-maps ΠTM ΠTM in local co- spaces, which is in a certain sense the condition of non- 1 → 2 ordinates are specified by formulas yi ϕi (x,dx) (where degeneracy of a Q-structure, see [50].) Hence functions = the r.h.s. is arbitrary), instead of yi ϕi (x). on Z(ΠTM)/B are locally constant functions on M, i.e. = 0 H (M), and Z(ΠTM)/B π0(M). This does not feel very =∼ satisfying and one may wish to modify the interpretation 3.1.3 Zero locus and its involutive distribution. of Z(M)/B (e.g. by considering “points” that are more gen- “Non-linear homological algebra” eral than ordinary R-valued points, so to be able to detect more information such as the whole cohomology algebra Definition 3.3. The zero locus of a Q-manifold M is the H ∗(M)). zero locus (the set of zeros) of the vector field Q. Notation: For the case of Πg, one can identify Z(Πg)/B with the Z(M) or Z(Q). space of orbits of the adjoint action of a Lie group associ- ated with g (note that the adjoint action preserves Z(Πg)). In coordinates, if Q Qa(x)∂/∂xa, then Z(M) is speci- = We can elaborate this as follows. It makes sense to con- fied by the equation sider a slightly more generalize setting.

Qa(x) 0. (6) Example 3.4. Let g be a differential Lie superalgebra, i.e. = besides the Lie bracket it is equipped also with an odd If ϕ: M M is a Q-map, it maps Z(M ) to Z(M ). operator d such that d is a derivation of the bracket and 1 → 2 1 2 In the above examples, we obtain the following subsets d 2 0. This is described by a field Q on Πg of the form = as zero loci. µ 1 ¶ ∂ For ΠTM with d, it is specified by the equation dxa Q ξi Qk ξi ξj Qk (7) = i ji k 0, hence Z(ΠTM) M. = + 2 ∂ξ = For a complex V (V,d), we obtain Z(V ) Kerd (the = = (the first term is responsible for d, the second for the usual subspace Z (V,d) of cocycles). bracket). In a coordinate-free form, For a Lie algebra g, the zero locus Z(Πg) Πg is a conic ⊂k i j 1 subspace given by the quadric equations c ξ ξ 0. Q(ξ) dξ [ξ,ξ] (8) i j = = − 2 The zero locus Z(M) comes equipped with a canoni- cally defined distribution, as follows. The vector field Q (the minus sign has some explanation, compare 3.4.1). induces a linear transformation Q : TQ(x) in the tan- Hence the equation of the zero locus is x = gent space T M for all x Z(M), and Q2 0. It is easy to x ∈ x = 1 see (e.g. by using local coordinates) that KerQx Tx Z(M). dξ [ξ,ξ] 0. = − 2 = Hence ImQ KerQ give a distribution on Z(M). Denote x ⊂ x it B, so that B ImQ . Note that points of Πg are the same as odd elements of x = x g. We look for solutions of (??) that can depend on ar- Proposition 3.2 ( [50], [1]). The distribution B on Z(M) is bitrary external parameters some of which may be odd. involutive, [B,B] B. ⊂ The infinitesimal transformation defined by Q on Πg is ξ ξ εQ(ξ) (where ε is odd). It lifts to the action on Proof. Observe (e.g. in local coordinates) that the vector 7→ + arbitrary tangent vectors ξ˙ by fields tangent to Z(M) can be described as elements of the Lie subalgebra Ker(adQ) Vect(M) restricted to Z(M), ˙ ˙ ˙ ˙ ⊂ ξ ξ ε(dξ [ξ,ξ]) and the distribution B can be similarly described by the 7→ + − ideal Im(adQ) Ker(adQ), and thus the involutivity fol- (where ξ˙ can be of any parity). In particular, if ξ Z(Q), ⊂ ∈ lows. the action preserves TξΠg. So the linear operator Qξ is the

15 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

“covariant derivative”: we shall define in 3.4.1). Basically, with any such an alge- bra is associated a deformation functor which is roughly Qξ(η) dη [ξ,η], = − Z(M)/B for the corresponding graded Q-manifold M. where η g (of arbitrary parity). The equation of the zero (Note that we did not consider a Z-grading in the exam- ∈ 2 locus is the “zero curvature” condition (d adξ) 0. ple; but in concrete situations it plays important role.) It is − = Hence the tangent space TξZ(Πg) KerQξ consists of roughly a set whose points are moduli or deformations of = “covariantly constant” vectors. The infinitesimal shift of structures of a considered type. “Functor” refers to depen- ξ Z(Πg), by η TξZ(Πg), ξ ξ εη, in the case of dence on a base of deformations, i.e. a choice of an algebra ∈ ∈ 7→ + η ImQξ is ξ ξ ε(dη [ξ,η]). This can be viewed as from which parameters are taken. The idea that deforma- ∈ 7→ + − an ”infinitesimal gauge transformation” of ξ, i.e. the in- tions of geometric and algebraic structures are controlled 1 1 finitesimal form of a transformation ξ dg g − gξg − by graded Lie algebras was put forward by Nijenhuis 7→ − + elements of a differential group integrating g. This is a (see Nijenhuis–Richardson [42]), as an abstract frame- usual Lie group with a Q-structure coming from d on g. work modeled on the previous work on deformations of Hence at least locally the leaves of the distribution B are complex structures (Frölicher–Nijenhuis [11], Kodaira– the same as gauge orbits. Spencer [28] and Kuranishi [35]) and associative algebras Remark. Introducing the zero locus of a given homologi- (Gerstenhaber [12]). It was Nijenhuis who brought to the cal vector field Q and then taking quotient of it by a distri- fore the “deformation equation” dξ 1 [ξ,ξ] 0, called ± 2 = bution on it can be compared with the logic of BRST the- also the Maurer–Cartan equation or master equation. (Ni- ory [14] (see also [37], [21]). In BRST theory it goes in the jenhuis was very much ahead of his time, he possessed opposite direction: a given “constraint surface” or “shell”, for example a working replacement of Lie supergroups which has to be factorized by symmetries generating an under the name “analytic graded Lie algebras”.) Then involutive distribution, is first “resolved” by a version of there followed the work of Schlessinger–Stasheff [48] of Tate [54] or Tyurina [43] resolutions, which means effec- 1979 and the work of Goldman–Millson [13], who used tively replacing a submanifold by a non-positively graded Deligne’s ideas that quasi-isomorphic DG Lie algebras Q-manifold which as a fiber bundle over the original am- define the same deformation theory and that instead of bient manifold, then it is further enlarged to a Z-graded Q- taking the deformation functor as a set of equivalence manifold (with negative and positive weights) where the classes, one should consider it as the corresponding ac- vector field Q (denoted traditionally as s and called “BRST tion groupoid (“Deligne’s groupoid”). Then it was Kont- differential”) incorporates information about constraints sevich [29,30] who formulated everything in terms of for- and symmetries. The procedure is non-unique and the mal Q-manifolds, identifying solutions of the deformation correct picture should take care of this non-uniqueness. equation with points of the zero locus Z(Q), and estab- If one recalls that in standard homological algebra com- lished invariance of the deformation functor under L ∞ plexes are taken up to quasi-isomorphism to get the de- quasi-isomorphisms (much more than DG Lie!), which rived category, then analogously one should expect ap- was the crucial step for formulating and proving his cele- pearance of some “derived Q-manifolds”. Investigations brated formality theorem. in this direction coming from the side of derived alge- braic geometry (which has been around for some time) are already on the way, see e.g. Pridham [44]. (See also [5], 3.1.5 Q-structure on the space of maps. [4].) The future theory should need to be able to incor- porate also microformal morphisms introduced below in This is an analog of the complex of homomorphisms. We Sections 4,5. will not use this construction in the rest of the paper, but wanted to include it because of its importance. Sup- pose M1 and M2 are Q-manifolds. Consider the infinite- 3.1.4 Remark: on deformation of structures using dimensional supermanifold (or graded manifold, if M1 Q-manifolds. and M2 are graded) of ll maps Map(M1,M2). Claim: it has a natural Q-structure (defined first in [1]). It has numer- Example 3.4 above was a glance into the apparatus of de- ous applications, in original paper [1] as well as in many formation theory (which will remain outside the scope of others, e.g. [7]. this text). The modern viewpoint is that every algebraic The construction is as follows (we use the exposition or geometric structure or, better, type of structure, is con- given in [73]). trolled by a particular differential graded Lie superalge- Consider first arbitrary vector fields Qi on Mi (not as- bra (or its generalization such as an L -algebra, which suming them homological). To them corresponds a vector ∞

16 field on Map(M1,M2) that we call the difference construc- 3.2 Digression: derived brackets tion: Recall (for reference purposes) the definition of a Lie su- d(Q1,Q2)[ϕ]: ϕ∗Q2 ϕ Q1 , peralgebra (we prefer not to use “graded Lie algebras” to = − ∗ avoid contradiction with the Lie algebras that are graded). where the “pullback” ϕ∗Q2 of a vector field Q2 on the tar- A Z2-graded vector space L L0 L1 with an even bi- get M2 and the “pushforward” ϕ Q1 of a vector field Q1 on = ⊕ ∗ linear operation which we denote by [ , ] is a Lie super- the source M are defined respectively as ϕ∗Q : Q ϕ − − 1 2 = 2 ◦ algebra (and the operation is referred to as ‘Lie bracket’) if and ϕ Q1 T ϕ Q1. Here we treat vector fields as sections ∗ = ◦ antisymmetry of the tangent bundles (not as operators on functions). u˜v˜ Both ϕ∗Q2 and ϕ Q1 are vector fields along ϕ, i.e. can be [u,v] ( 1) [v,u] (11) ∗ perceived as infinitesimal variations of ϕ or elements of = − − the tangent space Tϕ Map(M1,M2). So is the difference and Jacobi identity (which we write in the Leibniz form) d(Q1,Q2)[ϕ] for each ϕ. Hence we have vector fields on [u,[v,w]] [[u,v],w] ( 1)u˜v˜[v,[u,w]] (12) Map(M1,M2), in particular, the vector field d(Q1,Q2). The = + − zeros of the vector field d(Q1,Q2) are precisely such ϕ that are satisfied. Q1 and Q2 are ϕ-related. If only (12) is satisfied (no antisymmetry assumed), It is convenient to use the notation Q2 and Q∗ for ∗ 1 then L is called a Loday or Leibniz algebra and the bracket the vector fields induced on the space of maps, so that is referred to as ‘Loday bracket’. Q2 [ϕ] ϕ∗Q2 and Q∗[ϕ] ϕ Q1. (The position of the ∗ = 1 = ∗ One can modify these notions by shifting parity so that star corresponds to post- or pre-composition with the in- the bracket becomes odd (with respect to the new parity). finitesimal diffeomorphism generated by the vector field.) Its properties differ by the shift of parities in all the signs. In this notation, Such structures are called an odd Lie superalgebra or an odd Loday algebra. d(Q1,Q2) Q2 Q1∗ . = ∗ − Fix an odd linear operator D on a Loday algebra L It immediately follows that under both “star” operations which is a derivation of the bracket (for example, D ad∆ = map commutator on M1 or M2 is mapped to commutator for an odd element ∆). Define a new operation of the on Map(M1,M2), and that any two vector fields with the opposite parity to the original: lower star and the upper star automatically commute. [u,v] : [D(u),v] (13) Hence the main result: D = ±

Proposition 3.3. For homological vector fields Qi on Mi , (sign not essential and can be properly chosen). the difference construction d(Q ,Q ) is a homological vec- 1 2 Theorem 3.1 ( [31]). Suppose D2 0. Then the operation tor field on Map(M1,M2) . = [u,v]D defines on L a new Loday algebra structure (of the Explicit formula: opposite parity). See also [32]. Operation (13) is called derived bracket. d(Q1,Q2) Q2 Q1∗ It has many applications. Note that even if the original = ∗ − = algebra is a Lie superalgebra, the new algebra does not Z µ ∂ϕi ¶ δ Dx Qi (ϕ(x)) Qa(x) (x) . (9) generally satisfy antisymmetry. However, it may be satis- 2 − 1 a i M1 ∂x δϕ (x) fied (for some elements) for an additional reason. (up to common sign depending on conventions for There is a related construction of “higher derived Berezin integral). brackets” that we will introduce shortly. They automati- cally satisfy (anti)symmetry, but a price that one has to For three Q-manifolds and a composition of maps consider an infinite sequence of brackets instead of one. ϕ : M M and ϕ : M M 2 there is a for- 21 1 → 2 32 2 → 3 mula [73] :

3.3 Lie algebroids and multiple Lie algebroids d(Q ,Q )[ϕ ϕ ] 1 3 32 ◦ 21 = d(Q2,Q3)[ϕ32] ϕ21 T ϕ32 d(Q1,Q2)[ϕ21] (10) Recall that a Lie algebroid over a manifold M is a vector ◦ + ◦ bundle E M with a structure of a Lie (super)algebra on → (it is an analog of the Leibniz formula). the space of sections and a fiberwise linear map a : E →

17 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

TM over M called anchor, so that the Leibniz rule is satis- to the situation for Lie (super)algebras. Later we shall fied: show constructively how all structures on ΠE, E ∗ and ˜ ΠE ∗ correspond to each other. (This will be done for the [u, f v] a(u)(f )v ( 1)f u˜ f [u,v] (14) = + − homotopy case, see 4.4.1.) There is a multiple analog of Lie algebroids: double where u,v are sections and f a function on M. (We formu- Lie algebroids, triple Lie algebroids, etc. Double Lie alge- late everything in the super setting.) See [38] as a general broids were first introduced by K. Mackenzie (see [39]) by source on Lie algebroids and Lie groupoids. using some nontrivial dualization process and then an Consider the parity reversed vector bundle ΠE M. → equivalent simplifying formulation was found in [68]. It Let Q Vect(ΠE) of weight 1. If xa,ξi are local coordi- ∈ + can be described as follows. Multiple Lie algebroids live nates on ΠE so that ξi of parity i˜ 1 are linear coordinates + on multiple vector bundles. The simplest way to define on the fibers, the general form of Q is then a k-fold vector bundle is to say that it is a k-fold graded ∂ 1 ∂ manifold (e.g. bi-graded for double vector bundle) such Q ξi Qa(x) ξi ξj Q( x) . = i ∂xa + 2 ji ∂ξk that each of the weights of local coordinates is 0 or 1. This leads to a fiber bundle structure with multilinear transi- Theorem 3.2 (Vaintrob [57]). The structure of a Lie alge- tion functions (see [68]). In particular, a double vector broid in E is equivalent to the Q-structure on ΠE of weight bundle is a commutative square of ordinary vector bun- 1. + dles (plus some extra conditions). Similarly fir the k-fold In other words, if Q as above is odd (this is automatic if case. There are commuting parity reversions in each of E is purely even) and satisfies Q2 0, it defines a Lie alge- the k directions, and one can consider the total parity = broid structure in E, and conversely. We can give explicit reversion. Then a k-fold Lie algebroid is specified by k formulas: commuting homological vector fields Q1,..., Qk such that wi (Q j ) δi j for the k weights w1,...,wk . See [68]. Double u˜ = ι ( 1) [[Q,ι ],ι ] (15) Lie algebroids in particular arise as Drinfeld doubles of [u,v] = − u v Lie bialgebroids introduced in [40]. See [39]. and a(u)(f ) [Q,ιu](f ). (16) = 3.4 L -algebras. L -morphisms. L -algebroids. ∞ ∞ ∞ Here ιu is a vector field on ΠE of weight 1 defined by “Non-linear Lie algebroids” u˜ i − i a section u C ∞(M,E) by ιu ( 1) u (x)∂/∂ξ if u i ∈ = − = u (x)ei . This is an odd isomorphism between Vect 1(ΠE) 3.4.1 L -algebras − ∞ and C ∞(M,E). (Note that there are no vector fields of weights less than 1 on ΠE.) L -algebras or “strongly homotopy Lie algebras” (SHLA) − ∞ If E1 and E2 are Lie algebroids over the same base M, originated in physics and were mathematically first de- it is not a problem to define a (fixed base) Lie algebroid fined by Lada and Stasheff [36]. They exist in two parallel morphism E E . This is just a fiberwise linear map over equivalent versions: “symmetric” and “antisymmetric”. 1 → 2 M preserving brackets and anchors. In particular, a : E We shall define both. Below we work with Z2-grading only. → TM is itself a Lie algebroid morphism. However, there is If necessary, a Z-grading can also be taken into account no obvious way of defining a Lie algebroid morphism over (but it does not affect identities). different bases (because there is no mapping of sections). Definition 3.4 (L -algebra: antisymmetric version). A ∞ I highly non-trivial definition was found in [15]. vector space L L L with a collection of multilinear = 0 ⊕ 1 Theorem 3.3 ( [57]). A fiberwise linear map E E over operations called brackets 1 → 2 a map of bases M1 M2 is a Lie algebroid morphism if [ ,..., ]: L ... L L (for k 0,1,2,...) → and only if the induced fiberwise linear map ΠE ΠE is − − | × {z × } → = 1 → 2 k times a Q-morphism. such that This is the most efficient way of dealing with mor- – the parity of the kth bracket is k mod 2; phisms of Lie algebroids. – all brackets are antisymmetric (in Z2-graded sense); P P β Let us mention that a Lie algebroid structure in E is – r s n shuffles( 1) [[xσ(1),...,xσ(r )],...,xσ(r s)] 0, for + = − + = also equivalent to a Poisson bracket on E ∗ and a Schouten all n 0,1,2,3,... = β r s α α (= odd Poisson, Gerstenhaber) bracket on ΠE ∗, both (here ( 1) ( 1) sgnσ( 1) and ( 1) is the Koszul − = − − − brackets having to be of weights 1. This is analogous sign). −

18 A parallel notion is as follows. A proof of the theorem follows from a more general construction producing L -algebras that we will consider Definition 3.5 (L -algebra: symmetric version). A vector ∞ ∞ in 3.4.3. space V V V with a collection of multilinear opera- = 0 ⊕ 1 The homological vector field Q has an expansion tions called brackets

∂ { ,..., }: V ... V V (for k 0,1,2,...) Q Qk (ξ) − − | × {z × } → = k k times = ∂ξ = µ 1 1 ¶ ∂ such that Qk ξi Qk ξi ξj Qk ξi ξj ξl Qk ... . (21) 0 + i + 2 ji + 3! l ji + ∂ξk – all brackets are odd; – all brackets are symmetric (in Z2-graded sense); k k k k P P α Up to signs, the Taylor coefficients Q0 , Qi , Q ji , Ql ji , etc., – r s n shuffles( 1) {{vσ(1),...,vσ(r )},...,vσ(r s)} 0, for + = − + = are structure constants of the 0-ary, unary, binary, ternary, all n 0,1,2,3,... = etc., brackets in V (or L). Interpretation of the “higher Ja- (here ( 1)α is the Koszul sign). − cobi identities” in the definition of L -algebras is simpli- ∞ (Note that here signs come from parities only!) fied if Q(0) is assumed to be zero. (In general, it is known The two variants of an L -algebra are related by a as “curvature” and the L -algebras that we defined are ∞ ∞ change of parity. Let V ΠL. Then the relation between called “curved”.) Then the first identity says that the unary = brackets in L and V ΠL is given by the formula bracket (which is a linear operator) is a differential; the = second identity says that it is a derivation of the binary {Πx ,...,Πx } ( 1)εΠ[x ,...,x ], . (17) bracket; the third identity says that the “usual” Jacobi 1 n = − 1 n identity for the binary bracket is satisfied up to a chain ho- where ε Px˜ (n k). Hence it is sufficient to consider motopy, the operator of chain homotopy being the ternary = k − just one variant, though in examples both can appear. bracket. And then there is an infinite sequence of further It is more convenient to analyze the symmetric version identities satisfied by the ternary bracket and the “higher (with all odd brackets). Let V be equipped a symmetric homotopies” that arise. (This explains “strongly homo- L -algebra structure. Because of symmetry, all operations topy”, not just “homotopy” in the name.) ∞ are determined by their values on coinciding even argu- ments: {ξ,...,ξ} for even ξ V . (We use the letter ξ for an ∈ even vector in V as a reminder of V being ΠL.) They can 3.4.2 L -morphisms be assembled into a formal odd vector field Q on V : ∞ Here again (after morphisms of Lie algebroids) the supe- X 1 Q(ξ) {ξ,...,ξ} . (18) riority of the Q-manifold language becomes compelling. = n! | {z } Suppose L and K are L -algebras in the antisymmetric n times ∞ version, and V ΠL and W ΠK are L -algebras in the = = ∞ We can express back the bracket operations in V and L in symmetric version. What should be the “correct” notion terms of Q, as follows: of a morphism? Denote it by a special arrow, L K . We have to establish what L K should be. {u1,...,un} [...[Q,u1],...,un](0) (19) If we start from a linear map L K and require it be a = → chain map (commute with the differentials), what should and be required from it with respect to the binary brackets? It would be too restrictive (and in hindsight, of little use) to ι([x ,...,x ]) ( 1)ε[...[Q,ι(x )],...,ι(x )](0). (20) 1 n = − 1 n require that the binary bracket in L is precisely mapped to the binary bracket in K . In view of the homotopy nature For elements of L, we use the operation ι similar to x˜ i i of an L -structure, it is natural to expect preservation of that used above for Lie algebroids, ι(x): ( 1) x ∂/∂ξ ∞ i =i − ∈ binary brackets only up to homotopy, which should be Vect(ΠL) if x x ei L. We denote by ξ linear coordi- = ∈ considered part of structure. Hence there is an algebraic nates on V and identify vectors from V with vector fields homotopy operator Λ2L K (equivalently, S2V W ). with constant coefficients. → → By analogy with the brackets, one expects to have an Theorem 3.4. Formulas above define L -algebra struc- infinite sequence of such “higher homotopies” Λk L ∞ → tures in V and L (in the respective version) if and only if Q K or Sk (ΠL) ΠK that should be subject to an infinite → is homological, Q2 0. sequence of identities involving the higher brackets in L =

19 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

and K . Handling such a sequence directly would be very be obtained by linearity. (For any multilinear expression, complicated. It is convenient to turn to the symmetric by using auxiliary odd factors, one can make all arguments description. A sequence of linear maps SkV W meant even and then take the auxiliary constants out using the → to be “higher homotopies” (one can note that they all linearity, and this would give the desired formula for argu- have to be even) assemble similarly with what we did for ments with arbitrary parities.) the brackets into one formal non-linear map ϕ: V W . Let ϕ: V W be a formal map of vector spaces en- → → (One cannot do the same directly in terms of L and K .) dowed with structures of L -algebras. Define its Taylor ∞ The language of Q-manifolds provides now a one-line components (symmetric multilinear maps) by the formu- solution. las

Definition 3.6. An L -morphism V W is an infinite ϕn(u1,...,un): ∂u ...∂u ϕ(0), (22) ∞ 1 n sequence of even linear maps SkV W which are the = → Taylor coefficients of a formal non-linear Q-morphism where ∂u means the usual derivative along a vector. (We ϕ: V W , where V and W are regarded as formal Q- substantially use here the linear structure of W , otherwise → manifolds. it would make no invariant sense.) We shall also need the notion of an L -structure “shifted” by a constant vector. An L -morphism L K is an infinite sequence of ∞ ∞ ξ0 linear maps Λk L K (of alternating parities; for k 1, If ξ0 is such a vector, we consider a vector field Q (ξ) → = = even) such that the corresponding sequence Sk (ΠL) Q(ξ ξ0). (Such shifts or “twistings” under more abstract → + ΠK is an L -morphism ΠL ΠK . guise were considered in [8].) Clearly, if Q is a homological ∞ → vector field, its shift Qξ0 is again homological vector field. Shortly: an L -morphism V W is just a Q-morphism We denote the brackets generated by Qξ0 as { ,..., }ξ0 . ∞ − − V W ; an L -morphism L K is a Q-morphism ΠL They effectively correspond to expanding Q not at 0, but → ∞ → ΠK . (We do not have to use a special arrow for V and at ξ0. W , since it is an ordinary map.) i µ Proposition 3.4. The condition that ϕ: V W is an L - If ξ and η are linear coordinates on V and W respec- → ∞ µ µ morphism is equivalent to the following infinite sequence tively, we can write ϕ∗(η ) ϕ (ξ) and expand as = of identities, for n 0,1,2,3,... with arguments u ,...,u = 1 n ∈ µ µ i µ 1 i j µ V which are assumed to be even: ϕ (ξ) ϕ0 ξ ϕi ξ ξ ϕji ... = + + 2 + n X X ¡ ¢ µ ϕk 1 {uσ(1),...,uσ(n k)},uσ(n k 1),...,uσ(n) For simplicity assume that ϕ 0, i.e. the origin is pre- + − − + = 0 = k 0 (n k,k)- served, and that both algebras have no curvature. Then = shuffles− by expanding the equation of a Q-morphism n X X X © ϕi1 (uτ(1),...,uτ(i1)),..., µ ∂ϕ µ r 1 i1 ... ir n combinations τ i = + + = of i ,...,i Q1(ξ) Q (ϕ(ξ)) i1 0,...,ir 0 1 r ∂ξi = 2 > > out of n ªϕ0 ϕir (uτ(ir 1 1),...,uτ(ir )) . (23) we obtain, in the first order: − + j µ µ Here combinations of i ,...,i out of n mean symmet- Q ϕ ϕλQ , 1 r i j = i λ ric combinations in each group (order unimportant, e.g. increasing) and in the second order: For example, we can write down the identities for n = k µ λ ν µ k µ k µ λ µ 0,1,2. Qi j ϕk ϕi ϕj Qνλ Qi ϕjk Q j ϕik ϕi j Qλ . ± ± = ± ± ± For n 0: = up to signs. The first order condition means that the lin- ϕ (Ω) Ωϕ0 . ear term ϕ : V W is a chain map. The second order 1 = 1 → condition means that ϕ1 preserves the binary brackets up For n 1: = to a chain homotopy given by ϕ2. (This is what we have ϕ ({u}) ϕ (Ω,u) {ϕ (u)}ϕ0 . started from heuristically above.) One can obtain in this 1 + 2 = 1 way the full set of identities that should be satisfied by For n 2 the Taylor components ϕ : SkV W (with the proper = k → signs). ϕ1({u1,u2}) ϕ2({u1},u2) ϕ2({u2},u1) ϕ3(Ω,u1,u2) We shall give the general formula for even arguments + + + = {f (u ,u )}ϕ0 {ϕ (u ),ϕ (u )}ϕ0 . only, hence without signs, but so that the correct signs can 2 1 2 + 1 1 1 2

20 By Ω we denote curvature, i.e. {∅}, in any L -algebra C ∞(M,E) defining in it an (antisymmetric) L -algebra ∞ ∞ Compare with the identities obtain above under the sim- structure and a sequence of fiberwise multilinear maps plifying assumptions that the origin is fixed and there are E ... E TM called anchors, so that the Leibniz ×M ×M → no curvature. identities hold:

[u1,...,un 1, f un] 3.4.3 Higher derived brackets − = α a(u1,...,un 1)(f )un ( 1) f [u1,...,un], (25) − + − We want to explain why the condition Q2 0 for a formal = α (u˜1 ... u˜n 1 n)f˜ homological vector field on a vector space V encodes where ( 1) ( 1) + + − + . − = − the higher Jacobi identities of an L -algebra. This can be ∞ Consider the parity reversed vector bundle ΠE M. shown directly, but we will give a more general framework. → We can treat the total space ΠE as a formal neighborhood An abstract setup is as follows. Let L be a Lie superalgebra of the zero section. with a direct sum decomposition into two subalgebras: – An L -algebroid structure on E M is equivalent to a L K V . Assume that V is abelian (all brackets are zero). ∞ → = ⊕ formal homological vector field on the supermanifold Let ∆ be an odd element of L. Define a sequence of new ΠE. odd brackets on V by the formula: – An L -morphism of L -algebroids Φ: E1 E2 is spec- ∞ ∞ ified by a formal (in general, nonlinear) Q-morphism {u1,...,uk }: P[...[∆,u1],...,uk ], (24) = Φ: ΠE ΠE . 1 → 2 where P is the projection on V parallel to K . They are (We refer to Φ: ΠE1 ΠE2 also as L -morphism.) → ∞ called the higher derived brackets generated by ∆. One Example 3.6. The collection of all anchors assembles into can see that they are symmetric (by the Jacobi identity in an L -morphism ΠE ΠTM . L and the commutativity of V ). ∞ → Theorem 3.5 ( [63]). If ∆2 0, then the higher derived = brackets generated by ∆ define on V a structure of an L - 3.4.5 “Non-linear Lie algebroids” ∞ algebra. Consider a non-negatively graded manifold E. As we know, (There are also generalizations to arbitrary derivations it is a fiber bundle E M, where M E0 (see 2.3.1) with and a homotopical algebra interpretation, see [64].) → = polynomial transition functions preserving weights. Sup- Example 3.5 (universal). Take L : Vect(V ) for a vector pose N is the top weight of local coordinates. (If N 1, we = = space V regarded as a supermanifold. Then L K V come back to vector bundles.) It is possible to develop in = ⊕ where elements of V are treated as constant vector fields such a setup an analog of the Lie algebroid theory [66]. and K is the space of vector fields vanishing at the origin. Definition 3.7. A structure of a non-linear Lie algebroid Clearly, these are subalgebras and V is abelian. Projection on the graded manifold E is defined by a (formal) homo- P is evaluation at zero. Then an arbitrary homological logical vector field Q Vect(E) of weight 1. vector field Q Vect(V ) defines on V a structure of an ∈ + ∈ L -algebra and we arrive at the formulas (19). What is an algebraic structure associated with such an ∞ object? Note that unlike vector bundles, sections here are This example is universal, i.e. all L -algebras arise this ∞ not additive, so not suitable for algebraic operations. As way and are specified by some Q. However, the advantage we have found, the “correct” vector space is the space of the general construction is that L -algebras can also - ∞ of all vector fields of negative weight Vect (E). It is a arise from different (not necessarily universal) data (L = nilpotent (but in general not abelian) Lie subalgebra in K V,∆). We will meet many examples later. ⊕ Vect(E). Higher derived brackets can be defined, but are not (anti)symmetric. Because of grading, everything re- duces to a differential and a binary derived bracket, on 3.4.4 L -algebroids ∞ top of the original commutator of vector fields. In [66] we have presented a list of identities satisfied by such a struc- This notion combines properties of L -algebras and Lie ∞ ture. Note that one can non-canonically identify E with algebroids. Let E M be a (super) vector bundle. → a graded vector bundle (the normal bundle to the zero Definition 3.1. An L -algebroid structure in E M con- section). Then the field Q induces an L -algebroid struc- ∞ → ∞ sists of a sequence of brackets on the space of sections ture in this normal bundle (with an extra grading). It is

21 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

non-canonical and is defined up to an L -isomorphism. Definition 4.1 ( [70], [71]). A microformal (aka thick) mor- ∞ The algebraic structure in Vect-(E) is, on the other hand, phism Φ: M M is a formal Lagrangian submanifold 1 → 2 canonical. In a sense, both structures contain the same Φ T ∗M T ∗M w.r.t. ω ω specified locally by a gen- ⊂ 2 × 1 2 − 1 information. erating function of the form S(x,q): Note that for an non-linear Lie algebroid E there is q d yi p dxa d(yi q S) on Φ, an anchor a : E ΠTM defined as the composition i − a = i − → T p Q, where Q is regarded as a map E ΠTE and where S(x,q), regarded as part of the structure, is a formal ◦ → T p : ΠTE ΠTM is tangent to the projection E M. power series in the momentum variables on the target → → If a is a fibration, we call E a transitive non-linear Lie manifold M2 : algebroid. (This generalizes transitivity for ordinary Lie al- i 1 i j 1 i jk gebroids [38].) Note that the anchor is always a Q-map, so S(x,q) S0(x) S (x)qi S (x)q j qi S (x)q q j qi ... = + +2 +3! k + intertwines Q on E with d on ΠTM. Hence for a transitive non-linear Lie algebroid we can introduce local coordi- We refer to S as the generating function of a thick mor- nates as xa,dxa, yi , where yi are fiber coordinates over phism Φ. the base ΠTM, and the homological vector field Q takes Remark. There is close similarity between our notion the form of a microformal (thick) morphism between two man- ∂ ∂ ifolds and the notion of a symplectic micromorphism Q dxa Qi (x,dx, y) , (26) = ∂xa + ∂yi between symplectic micromanifolds of Cattaneo–Dherin– Weinstein [6]. A “symplectic micromanifold” is defined as the first term being de Rham differential on ΠTM. This the germ of a symplectic manifold at a Lagrangian sub- can be compared with the “Q-bundles” considered by Ko- manifold and a “symplectic micromorphism” between tov and Strobl [34], [52]. They assumed a bundle structure such germs is defined as the germ of a canonical relation over ΠTM with the extra restriction that Q in a local trivi- between symplectic manifolds representing the germs. alization splits into d and a homological vector field on Since by the symplectic tubular neighborhood theorem the standard fiber. Compared with (26) this would mean every symplectic manifold near a Lagrangian submani- no dependence on x,dx in the second term. As we showed fold looks like its cotangent bundle, symplectic microman- in [67], the possibility of such a gauge follows from the ifolds can be represented by cotangent bundles and every “non-abelian Poincaré lemma”. This covers some part of symplectic micromorphism defines a thick morphism be- transitive Lie algebroid theory [38]. An interesting ques- tween the Lagrangian manifolds by “passing from germs tion would be to consider integration of such non-linear to (infinite) jets”. The big difference is in “the morphisms Lie algebroids in an analogy with Mackenzie’s theory (for of what” are the corresponding notions. For symplectic transitive Lie algebroids). micromorphisms, these are (the germs of) the cotangent bundles. For thick or microformal morphisms, these are the manifolds themselves. Hence, we look for an action of 4 Microformal geometry. Classical thick such morphisms on functions by analog of pullbacks by morphisms smooth maps. From the viewpoint of symplectic geome- try, this is an action on functions on Lagrangian submani- In this and the next section, we give a concise introduction folds. to microformal geometry. The key references are: [70], [69], Now we introduce these pullbacks. [71] for main ideas and constructions, also [59]; and [60] and [22] for further development and applications. 4.1.2 Pullback by a microformal morphism

4.1 Main constructions Let Φ: M M be a thick morphism with a generating 1 → 2 function S. 4.1.1 Definition of a microformal (thick) morphism Definition 4.2. The pullback Φ∗ is a formal mapping a Φ∗ : C∞(M2) C∞(M1) of functional supermanifolds de- Let M1, M2 be supermanifolds with local coordinates x , → i fined by the formula (see [70]) y . Let pa and qi be the corresponding conjugate mo- menta (fiber coordinates in T ∗M1 and T ∗M2) and let a i i ω1 dpadx and ω2 dqi d y be the symplectic forms Φ∗[g](x) g(y) S(x,q) y q , (27) = = = + − i on T ∗M1 and T ∗M2.

22 i for g C∞(M ), where q and y are determined from the defines a map ϕ : M M (depending on a function ∈ 2 i g 1 → 2 equations g !) as a formal perturbation of the map ϕ ϕ : M M = 0 1 → 2 given by the linear term in S(x,q): ∂g i i˜ ∂S qi (y), y ( 1) (x,q) (28) = ∂yi = − ∂q yi ϕi (x) ϕi (x) Si j (x)∂ g(ϕ(x)) ... , (30) i = g = + j + ˜ (giving yi ( 1)i ∂S (x, ∂g (y)) solvable by iterations). and therefore the formula for the pullback becomes = − ∂qi ∂y ³ i ´¯ Remark. For ordinary manifolds, we do not have to Φ∗[g](x) g(y) S(x,q) y q ¯ . = + − i ¯ ∂g think about parity of functions. (Though we can con- y ϕg (x),q (ϕg (x)) = = ∂y sider odd functions on purely even manifolds if needed, but they will be families incorporating odd parameters Note that the function g enters in two ways: explicitly as a i and even “universal” such families, rather than individ- summand in g(y) S(x,q) y qi and implicitly through i + − ual functions.) For supermanifolds, we have to distin- y and qi . This is the source of the non-linearity (expect guish between even and odd functions (or ‘bosonic’ and for the case of a function S linear in the momenta, where ‘fermionic’ fields in physical parlance) because they sat- the equations for y and q decouple and the dependence isfy different commutation rules. So above C∞(M) stands on g disappears). for the supermanifold of even (bosonic) functions. Un- Therefore, for a general thick morphism Φ: M M 1 → 2 like the familiar case, when pullbacks are linear and can the pullback Φ∗ : C∞(M ) C∞(M ) is a formal non- 2 → 1 be applied to functions regardless of their parity, the linear differential operator : pullbacks defined above work only for even functions. (For odd functions, see 4.1.7.) We use C (M) with bold- 0 1 i j ∞ Φ∗[g](x) S (x) g(ϕ(x)) S (x)∂i g(ϕ(x))∂j g(ϕ(x)) ... face for a supermanifold of even functions (rather than = + +2 + a set) and distinguish it from the Z2-graded vector space (Higher order terms can also be calculated [70], but their C ∞(M) C ∞(M) C ∞(M) . = 0 ⊕ 1 form is not very elucidating.) As we shall see, these non-linear differential opera- Heuristically, if f Φ∗[g], then = tors possess special properties, so they are far from being Λ Λ Φ (29) arbitrary. f = g ◦ (composition of relations), where Λ graph(d f ) . Note f = that equation (27) contains more information than (29) 4.1.4 Coordinate invariance because (27) is an equality for functions themselves, not the derivatives. More important is that (27) and (28) give Generating functions of thick morphisms are not scalar a constructive procedure for calculating pullbacks. functions, in the sense that they are geometric objects whose representations depend on coordinate systems. They possess the following non-trivial transformation law. 4.1.3 Description of pullbacks Transformation Law (for generating functions). A gener-

0 i ating function S(x,q) as a geometric object on M1 M2 Example 4.1. Let S(x,q) S (x) ϕ (x)qi . Then: Φ∗[g ] × 0 = + = transforms by S ϕ∗g (an ordinary pullback with a shift by a fixed func- + tion). i i 0 S0(x0,q0) S(x,q) y q y q . (31) = − i + i 0 Remark. Ordinary maps M M can be identified with 1 → 2 Here S(x,q) is the expression for S in ‘old’ coordinates thick morphism that have generating functions of the and S (x ,q ) is the expression for S in ‘new’ coordinates. form S ϕi (x)q , i.e. linear in momenta. 0 0 0 = i At the r.h.s., the variables xa and yi 0 are given by substi- a a i i i Write a general generating function as tutions: x x (x0) and y 0 y 0 (y), while q and y are = = i determined from 0 i S(x,q) S (x) ϕ (x)qi ... i = + + ∂y 0 ˜ ∂S q (y)q , yi ( 1)i (x,q). (32) i i i 0 (note the notation for the linear term). Then the equation = ∂y = − ∂qi

˜ ∂S ∂g One can see that the cocycle condition is satisfied by yi ( 1)i (x, (y)) = − ∂qi ∂y this formula (because it has a “coboundary” form). This

23 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

µ transformation law can either be postulated as part of S32(y,r ). Here z are local coordinates on M3 and by rµ the definition of thick morphisms or deduced from the we denoted the corresponding conjugate momenta. requirement that the corresponding formal canonical re- Theorem 4.2. The composition Φ32 Φ21 is well-defined lation have the same expression in terms of the generating ◦ as a thick morphism Φ31 : M1 M3 with the generating function in all coordinate systems. In all cases, we have → function S31 S31(x,r ), where the crucial proposition: = i S31(x,r ) S32(y,r ) S21(x,q) y qi (35) Proposition 4.1. If a generating function S transforms ac- = + − i a cording to the transformation law given by (31), the canon- and y and qi are expressed through (x ,rµ) from the sys- ical relation Φ T ∗M ( T ∗M ) specified by S and the tem ⊂ 2 × − 1 operation of pullback Φ∗C∞(M2) C∞(M1) do not de- ∂S32 ˜ ∂S21 → q (y,r ), yi ( 1)i (x,q), (36) i i pend on a choice of coordinates. = ∂y = − ∂qi

which has a unique solution as a power series in rµ and a functional power series in S . 4.1.5 Key fact: derivative of pullback 32 If we think about thick morphisms as of (formal) As the pullback by a thick morphism is a non-linear canonical relations, there is their composition as “set- mapping of vector spaces of functions (more accurately, theoretic” relations. The point of the above statement is we have to speak about the corresponding infinite- that set-theoretic composition leads actually to a relation dimensional supermanifolds), it is natural to ask about of the same type, i.e. a thick morphism, and we are able to give a formula for its generating function. its derivative or variation for a given function g C∞(M ). ∈ 2 The answer is remarkable. Theorem 4.1. Let Φ: M M be a thick morphism. Con- 1 → 2 4.1.7 Further facts sider the pullback

Φ∗ : C∞(M ) C∞(M ). (33) (A) Formal category. Composition of thick morphisms is 2 → 1 associative and (Φ Φ )∗ Φ∗ Φ∗ . In the lowest order, 32◦ 21 = 21◦ 32 Then for every g C∞(M ), the derivative T Φ∗[g] is given ∈ 2 the composition is as in SManoC∞, whose arrows are by pairs (ϕ , f ) with the composition (ϕ , f ) (ϕ , f ) 21 21 32 32 ◦ 21 21 = (ϕ32 ϕ21,ϕ21∗ f32 f21). Thick morphisms form a formal T Φ∗[g] ϕ∗g , (34) ◦ + = category (“formal thickening” of the category SManoC∞). where ϕ∗ : C ∞(M ) C ∞(M ) is the usual pullback Notation: EThick. g 2 → 1 with respect to the map ϕ : M M defined by yi (B) Relation with gradings. The notion of a thick mor- g 1 → 2 = i˜ ∂S ∂g phism as such and the construction of the pullback of ( 1) (x, (y)) (depending perturbatively on g, ϕg − ∂qi ∂y = functions by a thick morphism do not require any super ϕ0 ϕ1 ϕ2 ...). + + + or graded structure. We formulated them for superman- (Explanation of notation: C∞(M) is the supermanifold ifolds with an eye on applications. If necessary, an extra of functions, whose ‘points’ are even functions; C ∞(M) Z-grading can be included. One only needs to assume is a Z2-graded vector space, which can identified with that a generating function S has weight 0. This would give the tangent space Tg C∞(M) to C∞(M), for an arbitrary correct weights for all other quantities in our formulas. g C∞(M).) ∈ (C)“Fermionic version”. There is a fermionic version A direct proof of Theorem 4.1 was given in [70]. An al- based on anticotangent bundles ΠT ∗M and odd generat- ternative proof can be obtained by consideration of quan- ing functions S(x, y∗): “odd thick morphisms” tum thick morphisms (see the next Section 3). (This was suggested by H. Khudaverdian, whom I thank.) Ψ: M1 M2 ⇒ Corollary. For every g, the derivative T Φ∗[g] of Φ∗ is an induce nonlinear pullbacks algebra homomorphism C ∞(M2) C ∞(M1). → Ψ∗ : ΠC∞(M ) ΠC∞(M ) 2 → 1 on odd functions (“fermionic fields”), and their compo- 4.1.6 Composition law sition gives another formal category, OThick, which is a formal thickening of SManoΠC∞. What is said above Consider thick morphisms Φ : M M and Φ : M M about the possibility of introduction of an extra Z-grading 21 1 → 2 31 2 → 3 with generating functions S S (x,q) and S applies in the fermionic case as well. 21 = 21 32 =

24 4.2 Application to homotopy Poisson brackets Remark. “Hamilton–Jacobi” vector fields such as QH live on spaces of functions. They should not be confused with 4.2.1 P - and S -structures (homotopy Poisson and Hamilton vector fields. One can write such a vector field ∞ ∞ Schouten) QH on C∞(M) (the supermanifold of even functions on M) for any Hamiltonian H C ∞(T ∗M) irrespective of ∈ Definition 4.3. A P - (resp., S -) structure on a super- its parity. As it was shown in [70], [QH ,QH ] Q(H ,H ) ∞ ∞ 1 2 = 1 2 manifold M is an antisymmetric (resp., symmetric) L - (i.e. the canonical Poisson bracket maps to the commu- ∞ structure on C ∞(M) such that the brackets are multi- tator of vector fields on C∞(M)). The same is true for the derivations of the associative product. A supermanifold fermionic case, i.e. for functions P on ΠT ∗M and vector with a P -structure (resp., an S -structure) is called a fields QP on ΠC∞(M) (the supermanifold of odd func- ∞ ∞ P -manifold (resp., an S -manifold). tions on M). ∞ ∞ 1. A P -structure on M is specified by an even function ∞ P C ∞(ΠT ∗M) satisfying [P,P] 0, by the formula ∈ = 4.2.2 Key theorem: pullback as an L -morphism {f ,..., f } : [...[P, f ],..., f ] . (37) ∞ 1 k P = 1 k |M Let M1 and M2 be S -manifolds, with Hi C ∞(T ∗Mi ), 2. An S -structure on M is specified by an odd function ∞ ∈ ∞ i 1,2. H C ∞(T ∗M) satisfying (H, H) 0, by the formula ∈ = = Definition 4.4 (S or “homotopy Schouten” thick mor- {f1,..., fk }H : (...(H, f1),..., fk ) M . (38) ∞ = | phism). A thick morphism Here [ , ] stands for the canonical odd Schouten bracket − − (canonical antibracket) on functions on ΠT ∗M (which Φ: M M 1 → 2 can be identified with multivector fields on M) and ( , ) − − stands for the canonical even Poisson bracket on func- is homotopy Schouten or an S thick morphism if ∞ tions on T ∗M (i.e. Hamiltonians on M). Sometimes we π∗H π∗H . (41) refer uniformly to H or P as to the master Hamiltonian of 1 1 = 2 2 the corresponding S - or P -structure. ∞ ∞ It follows that P -brackets have alternating parities: Here πi are the restrictions on Φ of the projections of ∞ T ∗M T ∗M on T ∗M . the binary bracket is even, the unary bracket and ternary 2 × 1 i bracket are odd, etc. A P -structure on M is a homotopy ∞ Note: condition (41) is expressed by the Hamilton– analog of an ordinary (even) Poisson bracket. Jacobi equation for S(x,q) As for S -brackets, they are all odd and a S -structure ∞ ∞ on M is a homotopy analog of an odd Poisson (or ³ ∂S ´ ³ q ∂S ´ H1 x, H2 ( 1) ,q . (42) Schouten or Gerstenhaber) bracket. ∂x = − ∂q Formulas (37) and (38) are particular cases of “higher derived brackets” [63, 64], and the fact that the “master Theorem 4.4. If a thick morphism of S -manifolds ∞ equations” [P,P] 0 and (H, H) 0 imply higher Jacobi Φ: M1 M2 is S , then the pullback = = → ∞ identities of L -algebras follows from a general theorem ∞ Φ∗ : C∞(M ) C∞(M ) from [63]. On the other hand, the universal description of 2 → 1 an L -algebra is given by a (formal) homological vector ∞ is an L -morphism of the homotopy Schouten brackets. field. What are the homological vector fields correspond- ∞ ing to P - and S -structures? ∞ ∞ Explicitly: if the Hamilton–Jacobi equation (42) holds, Theorem 4.3 ( [70]). The homological vector fields corre- then Φ∗ intertwines the homological vector fields Q H2 ∈ sponding to P - and S -structures with “master Hamil- Vect(C∞(M2)) and QH Vect(C∞(M1)). ∞ ∞ 1 ∈ tonians” P C ∞(ΠT ∗M) (even) and H C ∞(T ∗M) (odd) ∈ ∈ have the Hamilton–Jacobi form: Z 4.2.3 Analog for P -structures ³ ∂ψ ´ δ ∞ QP Dx P x, Vect(ΠC∞(M)) (39) = M ∂x δψ(x) ∈ Let M1 and M2 be P -manifolds, with Pi C ∞(ΠT ∗Mi ), ∞ ∈ and i 1,2. = Z ³ ∂f ´ δ We have to use the fermionic version of thick mor- QH Dx H x, Vect(C∞(M)). (40) = M ∂x δf (x) ∈ phisms now.

25 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

Definition 4.5 (P or “homotopy Poisson” odd thick mor- defined canonically up to some choice of signs. Depending ∞ phism). An odd thick morphism on this choice, it is either symplectomorphism or antisym- plectomorphism. Ψ: M M 1 ⇒ 2 The Mackenzie–Xu diffeomorphism κ: T ∗E T ∗E ∗ is homotopy Poisson or a P thick morphism if → ∞ plays the key role for our construction of adjoint as a thick morphism. π∗P π∗P . (43) 1 1 = 2 2 Theorem 4.6 (“adjoint”). For any fiberwise, in general Now πi are the restrictions on Ψ of the projections of nonlinear, map of vector bundles Φ: E1 E2, there is a ΠT ∗M ΠT ∗M on ΠT ∗M . → 2 × 1 i thick morphism, which we call the (fiberwise) adjoint, This is expressed by the Hamilton–Jacobi equation for an odd generating function S(x, y∗) Φ∗ : E ∗ E ∗ . (46) 2 → 1 ³ ∂S ´ ³ ∂S ´ P1 x, P2 , y∗ . (44) The thick morphism Φ∗ is fiberwise in the natural sense. It ∂x = ∂y∗ is an ordinary map and coincides with the usual adjoint if Theorem 4.5. If an odd thick morphism of P -manifolds the map Φ is fiberwise-linear, and has the same functorial ∞ property (Φ Φ ) Φ Φ . Construction: Φ: M1 M2 is P , then the pullback 1 2 ∗ 2∗ 1∗ ⇒ ∞ ◦ = ◦ ¡ ¢op Ψ∗ : ΠC∞(M ) ΠC∞(M ) Φ∗ : κ κ)(Φ T ∗E ∗ ( T ∗E ∗), (47) 2 → 1 = × ⊂ 1 × − 2 is an L -morphism of the homotopy Poisson brackets. where κ: T ∗E T ∗E ∗ is the Mackenzie–Xu diffeomor- ∞ → That is: if (44) holds, then Ψ∗ intertwines the ho- phism. mological vector fields Q Vect(ΠC∞(M )) and Q P2 ∈ 2 P1 ∈ Corollary (pushforward). There is a pushforward of func- Vect(ΠC (M )). ∞ 1 tions on the dual bundles (as the pullback by the adjoint) Further application that we have in mind (to L - ∞ algebroids) requires first a short digression. This is an- other application of the language of thick morphisms, this Φ : (Φ∗)∗ : C∞(E1∗) C∞(E2∗) (48) time having nothing to do in principle with homotopy ∗ = → brackets, but simply to maps of vector spaces or vector maps the subspace of sections C∞(M,E ) C∞(E ∗) to 1 ⊂ 1 bundles. It is as follows. C∞(M,E2). It coincides on sections with the obvious push- forward v Φ v. 7→ ◦ 4.3 “Non-linear adjoint” 4.3.2 The fermionic analog: parity reversed adjoint We shall show that the notion of the adjoint of a linear transformation has an analog for non-linear transforma- For the fermionic version (for adjoint combined with par- tions, but now as a thick morphism rather than an ordi- ity reversion in vector bundles), we need the following nary map. (Again, there are parallel bosonic and fermionic analog of the Mackenzie–Xu theorem: versions.) We work in the setting of vector bundles over a fixed base to avoid complications for different bases. Theorem 4.7 ( [62]). For a vector bundle E, there is a dif- (See [16] for duality for vector bundles by using two cate- feomorphism gories, with “morphisms” and “comorphisms”.) ΠT ∗E ΠT ∗(ΠE ∗), (49) =∼ defined canonically up to a choice of signs, and which is 4.3.1 The adjoint for a nonlinear transformation an (anti)symplectomorphism.

The construction is based on the following fundamental Theorem 4.8 (“antiadjoint”). For any fiberwise, in gen- fact. eral nonlinear, map of vector bundles Φ: E E , there 1 → 2 Theorem (Mackenzie–Xu [40]). For dual vector bundles E is an odd thick morphism, which we call the (fiberwise) and E ∗, there is a diffeomorphism antiadjoint,

Π T ∗E T ∗E ∗ , (45) Φ∗ : ΠE ∗ ΠE ∗ . (50) =∼ 2 ⇒ 1

26 It is an ordinary map and coincides with the usual adjoint E combined with parity reversion if the map Φ is fiberwise- Π Π Π ¡ A linear. The equality (Φ Φ )∗ Φ∗ Φ∗ holds. Con- 1 ◦ 2 = 2 ◦ 1 ¡ A struction: ¡ A ¡ A Π ¡ ¢op ¡ A Φ∗ : χ χ)(Φ ΠT ∗(ΠE ∗) ( ΠT ∗(ΠE ∗)), (51) = × ⊂ 1 × − 2 ΠE ΠE ∗ @ where χ: ΠT ∗E ΠT ∗(ΠE ∗) is the odd analog (49) of the → @ Mackenzie–Xu diffeomorphism. E ∗

Corollary (pushforward of functions on the antidual bun- Theorem 4.9. The following structures are equivalent: – L -algebroid structure in vector bundle E M dles). For the antidual bundles, there is a pushforward ∞ → – P -structure on supermanifold E ∗ ∞ – S -structure on supermanifold ΠE ∗ ∞ Π Π – Q-structure (homological vector field) on supermanifold Φ : (Φ∗ )∗ : ΠC∞(ΠE1∗) ΠC∞(ΠE2∗). (52) ∗ = → ΠL It maps the subspace of sections C∞(M,E ) ΠC∞(ΠE ∗) 1 ⊂ 1 The quickest way to see that and to obtain explicit to C∞(M,E ). It coincides on sections with v Φ v. 2 7→ ◦ formulas is to use the Mackenzie–Xu theorem and its odd analog discussed above. The cotangent bundle T ∗(ΠE) and the anticotangent bundle ΠT ∗(ΠE) are double vector 4.4 Application to L -algebroids bundles: ∞ T ∗(ΠE) ΠE ∗ In this subsection, we construct a homotopy analog of the −−−−−→     familiar relation between Lie algebras and linear Poisson y y brackets. Recall that a Lie algebra structure for a vector ΠE M space (i.e. a Lie bracket defined for its elements) is equiv- −−−−−→ alent to a linear Poisson structure on the dual space (i.e. a (and similarly for ΠT ∗(ΠE)) and hence are naturally bi- Poisson bracket on functions on the dual space), known graded. If xa,ξi are local coordinates on ΠE (where ξi variably as “Lie–Poisson” or “Berezin–Kirillov” bracket. are linear coordinates in the fibers), natural coordinates a i Also, a linear map of vector spaces is a Lie algebra homo- on T ∗(ΠE) will be x ,ξ ,pa,πi (here pa,πi are the corre- a morphism if and only if its adjoint (which is the map of the sponding conjugate momenta). Similarly for ΠE ∗: x ,ηi a i dual spaces in the opposite direction) is a Poisson map. In and x ,ηi ,pa,π . Up to signs, the Mackenzie–Xu transfor- i i the supercase, to that one can add the similar statements mation is the exchange of ξ ,πi with π ,ηi . The bi-grading for odd Poisson bracket on the antidual space. What we is given by the two non-negative weights: w1 #pa #πi i i = + = will do, we will give an analog for L -algebroids. This will #pa #ηi and w2 #pa #ξ #pa #π . (Incidentally, ∞ + = + i = + use P - and S -structures and require the language of their difference w2 w1 #ξ #πi is the physicists’ “ghost ∞ ∞ − = − thick morphisms. number” gh.) A formal homological vector field

a ∂ i ∂ 4.4.1 Recollection: manifestations of an L -algebroid Q Q (x,ξ) a Q (x,ξ) ∞ = ∂x + ∂ξi structure on ΠE lifts to an odd Hamiltonian H C ∞(T ∗(ΠE)), ∈ Let a (super) vector bundle E M have a structure of where H Q p, i.e. → = · a L -algebroid. Recall from 3.4.4 that this means a se- a i ∞ H Q (x,ξ)pa Q (x,ξ)πi . quence of brackets and a sequence of anchors satisfying = + certain properties, namely that the brackets define in the It has weights w (H) 1 and w (H) 0. The applica- 1 = + 2 Ê space of sections an L -algebra structure and the an- tion of the Mackenzie–Xu transformation turns H into ∞ chors appear in the Leibniz type formulas for the brackets H ∗C ∞(T ∗(ΠE ∗)) of the same weights. Now w2 is the (with respect to multiplication by functions). We have grading by the degrees of the momenta on ΠE ∗, so H ∗ seen in 3.4.4 that such a structure is encoded by a formal generates an infinite number of odd brackets. Lifting of homological vector field on the supermanifold ΠE. We vector fields maps commutator to the canonical Pois- can now include into consideration also the bundles E ∗ son bracket and the Mackenzie–Xu transformation pre- and ΠE ∗: serves the Poisson brackets (possibly up to a sign). Hence

27 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

[Q,Q] 0 is equivalent to (H, H) 0 and (H ∗, H ∗) 0. for an L -algebroid E M assemble into a single (non- = = = ∞ → So we get an S -structure on ΠE ∗, which is the homo- linear, in general) bundle map ∞ topy analog of the familiar Lie–Schouten bracket for Lie a : ΠE ΠTM algebras or Lie algebroids. A function f lifted from ΠE ∗ → to T ∗(ΠE ∗) will have weights w (f ) w(f ) (its degree in over M (which we also refer to as anchor). 1 = ηi ) and w2(f ) 0. Note the canonical Poisson bracket on Corollary. The anchor for an L -algebroid E M in- = ∞ → T ∗(ΠE ∗) has bi-weight ( 1, 1). From this we can see that duces L -morphisms − − ∞ S -brackets on ΠE ∗ induced by an L -algebroid struc- ∞ ∞ C∞(ΠE ∗) C∞(ΠT ∗M) ture in E will have weights n 1 for an n-ary bracket, → − + i.e. 1 for the 0-bracket, 0 for the 1-bracket, 1 for the of the homotopy Schouten brackets, and + − 2-bracket, 2 for a 3-bracket, and so on. (So actually we − ΠC∞(E ∗) ΠC∞(T ∗M). need to include these weights into the theorem.) → of the homotopy Poisson brackets. Similar argument applies to E ∗, where a P -structure, ∞ which is the homotopy analog of the usual Lie–Poisson bracket, is obtained. Now there is a sequence of brackets 4.4.4 Application to higher Koszul brackets for a of alternating parities, but they again will have weights homotopy Poisson manifold n 1 for a bracket with n arguments. − + We observe that all these equivalent structures on the Let M be a P -manifold. By applying the above to the L - three neighbors, ΠE, ΠE and E are described basically ∞ ∞ ∗ ∗ algebroid structure induced in T M (see [25]) we arrive by one geometric object, which only gets different mani- ∗ at the following statement. festations. Corollary. On a homotopy Poisson manifold M, there is an L -morphism ∞ 4.4.2 L -morphisms of Lie-Poisson and Lie-Schouten Ω(M) C∞(ΠTM) C∞(ΠT ∗M) A(M), ∞ = → = brackets between the higher Koszul brackets on forms induced by the homotopy Poisson structure and the canonical Schouten Consider an L -morphism Φ: E1 E2 of L -algebroids ∞ ∞ bracket on multivector fields. over a base M. It is given by a Q-map Φ: ΠE ΠE . To 1 → 2 This gives solution for the problem posed in [25] simplify notation, we shall suppress indications on parity (where higher Koszul brackets were introduced) and reversion, i.e. write Φ instead of ΦΠ and Π instead of ∗ which was the initial motivation that led us to thick mor- Φ Π. ∗ phisms. See more details in [71] and [22].

Theorem 4.10. An L -morphism Φ: E1 E2 over a base ∞ M induces morphisms of the homotopy structures: – S thick morphism Φ∗ : ΠE ∗ ΠE ∗ 5 Quantum thick morphisms ∞ 2 → 2 – P odd thick morphism Φ∗ : E ∗ E ∗ ∞ 2 ⇒ 2 This gives L -morphisms of the homotopy Lie–Schouten So far, the statements about bosonic and fermionic thick ∞ and homotopy Lie–Poisson brackets, respectively (by push- morphisms were completely parallel to each other. This forward): cannot remain always the case because of the substantial difference between even and odd symplectic geometry

Φ : C∞(ΠE1∗) C∞(ΠE2∗) (see e.g. [26], [27] and [49]; also [23] [24]). In this section we ∗ → will see that bosonic thick morphisms governed by even and generating functions S(x,q) have quantum counterparts which are special type Fourier integral operators speci- Φ : ΠC∞(E1∗) ΠC∞(E2∗). fied by certain “quantum generating functions” S (x,q). ∗ → ħ In the same way as (classical) bosonic thick morphisms give L -morphisms for S -brackets, there is a construc- ∞ ∞ 4.4.3 Example: L -morphisms induced by the anchor tion of “quantum” S , -brackets (equivalent to a higher ∞ ∞ ħ order quantum “BV operator”), which are not S , but ∞ Recall that the “higher anchors” tend to S when 0, and we show how to obtain L - ∞ ħ → ∞ morphisms for S , -brackets using quantum thick mor- ∞ ħ E ... E TM phisms. The main references here are [69] and [71]. ×M ×M →

28 5.1 Main construction S0(x,q): lim S (x,q) as the (classical) generating func- = 0 ħ ħ→ tion of a (classical) thick morphism Φ: M1 M2. Then for We treat Planck’s constant as a formal parameter. → i g(y) ħ any oscillatory wave function of the form w(y) e on = ħ M2, the quantum pullback is given by

5.1.1 Quantum pullbacks and quantum thick morphisms £ i g ¤ i f (x) Φˆ ∗ e e ħ , (55) ħ = ħ

We need first to introduce suitable classes of functions. where f Φ∗[g](1 O( )) and Φ∗ is the pullback by the ħ = + ħ Besides C ∞(M)[[ ]], smooth functions on a manifold classical microformal morphism Φ: M M defined by ħ 1 → 2 M which are formal power series in (“formal power S (x,q). ħ 0 series” for us always means non-negative powers), we We say that Φ lim Φˆ . introduce the algebra of (formal) oscillatory wave func- = 0 ħ→ To be able to regard legitimately the limit S0(x,q) tions, which we denote OC∞(M), obtained by adjoining to = ħ i f (x) lim S (x,q) as a classical generating function, we need to 0 ħ C ∞(M)[[ ]] formal oscillating exponentials e ħ , where ħ knowħ→ of course that it possesses the required transforma- f C ∞(M)[[ ]]. The usual rules of manipulating with ex- ∈ ħ ponentials are assumed to hold. tion law. We will see that shortly.

Definition 5.1. Consider supermanifolds M1 and M2.A ˆ ˆ quantum pullback Φ∗ is a linear operator Φ∗ : OC∞(M2) 5.1.3 Explicit formula for quantum pullbacks ħ → OC∞(M1) defined by the integral formula ħ Z i i Suppose ˆ (S (x,q) y qi ) (Φ∗[w])(x) D y—Dq e ħ ħ − w(y), (53) = T M2 0 i ∗ S (x,q) S (x) ϕ (x)qi S+(x,q), (56) ħ = + + where a function S (x,q) is called quantum generating ħ ħ ħ ħ where S+(x,q) is the sum of all terms of order 2 in qi . function. It is a formal power series in momentum vari- ħ Ê ables on M2 : Theorem 5.2. The action of Φˆ ∗ defined by S (x,q) can be ħ expressed as follows: 0 i S (x,q) S (x) ϕ (x)qi µ ³ ´ ¶ ħ = + i 0 i ∂ ħ ħ ¡ ¢ S (x) S+ x, ħi ∂y Φˆ ∗w (x) e e ħ w(y) ¯ . (57) 1 i j 1 i jk ħ ħ ħ ¯ = ¯ yi ϕi (x) S (x)q j qi S (x)qk q j qi ... (54) ¯ = + 2 ħ + 3! ħ + ¯ ħ with coefficients formal power series in .A quantum Hence the quantum pullback Φˆ ∗ is a special type ˆ ħ thick (or microformal) morphism Φ: M1 M2 is defined formal linear differential operator over the ‘quantum- →ħ as the corresponding arrow in the dual category. 0 perturbed’ map ϕ : M1 M2. Here S (x) gives the phase ħ n m i → ħ (In the integral, —Dq : (2π )− (i ) Dq in dimension factor, ϕ (x)qi gives the map, and the term S+(x,q) is re- = ħ ħ n m.) sponsibleħ for “quantum corrections”. ħ | There is a question, in which sense to understand os- cillatory integrals such as (53). This is achieved by a for- mal version of the stationary phase formula [71]. An ax- 5.1.4 Further facts iomatic theory of formal oscillatory integrals is developed by A. Karabegov in [20]. (A) Transformation law. Quantum generating functions The outward appearance (54) of a quantum generating transform under changes of coordinates by the following function S (x,q) seems the same as our previous func- formula: ħ tions S(x,q) apart from a dependence on . We shall see i Z i ¡ ¢ ħ S0 (x0,q0) S (x(x0),q) yq y0(y)q0 e D y —Dq e ħ − + . (58) however, that there is a difference: namely, in the respec- ħ ħ = ħ tive transformation laws. a a a a i i i i Here x x (x0), x 0 x 0 (x) and y y (y0), y 0 y 0 (y) = = = = are mutually inverse changes of local coordinates on M1 5.1.2 Classical limit and M2 respectively. In particular, a corollary is that in the limit 0, the classical transformation law from 4.1.4 is ħ → Theorem 5.1. Let Φˆ : M1 M2 be a quantum thick mor- recovered. This justifies taking the classical limit in Theo- →ħ phism with a quantum generating function S . Consider rem 5.1. ħ

29 Th. Th. Voronov: Graded geometry, Q-manifolds, and microformal geometry

(B) Composition. Quantum thick morphisms can be Example 5.1. On a supermanifold M, an arbitrary - ħ composed. The composition is given by an integral for- differential operator of order n has in local coordinates mula similar to that defining quantum pullbacks. the form More details see in [71]. n a1...an ∆ ( i ) A (x)∂a1 ...∂an = − ħ ħ + n 1 a1...an 1 0 ( i ) A − (x) ...... A (x). 5.2 Higher BV-structures − ∂a1 ∂an 1 − ħ ħ − + + ħ For such operators, the principal symbol is 5.2.1 Digression: brackets generated by an operator

a1...an a1...an 1 σ(∆) A0 (x)pa1 ...pan A0 − (x)pa1 ...pan 1 Let A be a commutative algebra with 1 over C[[ ]]. Let = + − + ħ ∆ be a linear operator on A. Consider two sequences of multilinear operations (of parity ∆˜ and symmetric in the ... A0(x) supersense): + 0 Definition 5.2 (a modification of Koszul’s [33]; see [63]). (subscript 0 means substituting 0), which is a well- ħ = Quantum brackets generated by ∆ : defined inhomogeneous fiberwise polynomial function k on T ∗M. It is the master Hamiltonian for the classical {a1,...,ak }∆, : ( i )− [...[∆,a1],...,ak ](1); ħ = − ħ brackets generated by ∆. (For a formal -differential oper- ħ classical brackets generated by ∆ : ator, such a master Hamiltonian is a formal power series k in momenta.) {a1,...,ak }∆,0 : lim ( i )− [...[∆,a1],...,ak ](1) = 0 − ħ ħ→ We say that: – ∆ is a formal -differential operator if all quantum 5.2.3 S , -algebras ħ ∞ ħ brackets are defined; – ∆ is an -differential operator of order n if all quantum Let an operator ∆ on A be odd. (We assume it is formal ħ É brackets vanish for k n. -differential.) > ħ Proposition 5.2. If ∆2 0, then the quantum brackets = define an L -algebra (in the odd symmetric version). 5.2.2 More on brackets generated by ∆ ∞ This follows from general theory [63]. The quantum brackets additionally satisfy the modi- Proposition 5.1 (Explicit formulas for quantum brackets). fied Leibniz identity We have: – For k 0, {∅}∆, ∆(1) ; = ħ = 1¡ ¢ {a1,...,ak 1,ab}∆, {a1,...,ak 1,a}∆, b – for k 1, {a}∆, ( i )− ∆(a) ∆(1)a ; − ħ = − ħ + ħ = = − ħ 2¡ − a˜b˜ α˜ – for k 2, {a,b}∆, ( i )− ∆(ab) ∆(a)b ( 1) ∆(b)a ( 1) a{a1,...,ak 1,b}∆, ( i ){a1,...,ak 1,a,b}∆, = ¢ ħ = − ħ − − − + − − ħ + |− ħ {z − ħ} ∆(1)ab ; extra term – for general k, the expression for the kth bracket generated where α˜ a˜(1 a˜1 ... a˜k 1). by ∆ is = + + + − We call such an algebraic structure an S , -algebra. ∞ ħ Note that an operator ∆ and the S , -brackets gener- {a1,...,ak }∆, ∞ ħ ħ = ated by it contain the same data, and they both are fully k k X s X α defined by the 0-bracket and 1-bracket. ( i )− ( 1) ( 1) ∆(aτ(1) ... − ħ − − Since an S , -algebra is in particular an L -algebra, s 0 (k s,s)-shuffles ∞ ħ ∞ = − we may ask about the corresponding homological vec- aτ(k s))aτ(k s 1) ...aτ(k) , tor field (which should live on A). The answer is in the − − + following statement. where ( 1)α ( 1)α(τ;a˜1,...,a˜k ) is the Koszul sign for per- − = − mutation of commuting factors of given parities. Lemma 5.1. The quantum brackets generated by ∆ corre- Remark. The notion of an -differential operator can be spond to the “Batalin-Vilkovisky homological vector field” ħ defined by induction: ord ∆ k if for all a A,[∆,a] on A (regarded as a supermanifold) ħ É ∈ = i B, where ord B k 1 (and ord ∆ 0 if ∆ commutes i i δ ħ ħ É − ħ = a ¡ a¢ with multiplication by all a A). Q e− ħ ∆ e ħ . (59) ∈ = δa

30 5.3 BV-manifolds and BV quantum morphisms 5.3.3 From a quantum BV morphism to a classical S ∞ thick morphism 5.3.1 Definitions Let M be a BV-manifold with a BV-operator ∆. In the limit We introduce the following terminology which may be 0, ∆ gives an S -structure. ħ → ∞ non-standard, but is convenient for our present purpose. Lemma 5.2. The master Hamiltonian of the S -structure Definition 5.3. (1) A BV-manifold is a supermanifold M ∞ generated by ∆ is equipped with an odd formal -differential operator ∆, 2 ħ i a i a ∆ 0. The operator ∆ is called the BV-operator. x pa x pa = H(x,p) lim e− ħ ∆(e ħ ). (62) (2) A (quantum) BV-morphism of BV-manifolds (M ,∆ ) = 0 1 1 ħ→ and (M2,∆2) is a quantum thick morphism Φˆ : M1 M2 →ħ Theorem 5.4 (“analog of Egorov’s theorem”). Let M and such that 1 M2 be BV-manifolds and let Φˆ : M1 M2 be a BV quan- →ħ ∆1 Φˆ ∗ Φˆ ∗ ∆2 . tum thick morphism. Then its classical limit Φ: M M ◦ = ◦ 1 → 2 is an S thick morphism for the induced S -structures. Since a BV-operator ∆ induces a sequence of quan- ∞ ∞ tum brackets, and is defined by the 0- and 1-brackets, Explicitly: the intertwining relation ∆ Φˆ ∗ Φˆ ∗ ∆ 1 ◦ = ◦ 2 a BV-structure and an S , -structure on a manifold M implies the Hamilton-Jacobi equation for the classical ∞ ħ are equivalent. In particular, the space of functions on a thick morphism Φ lim Φˆ : = 0 BV-manifold is an L -algebra with respect to quantum ħ→ ∞ brackets generated by ∆. ³ ∂S ´ ³ ∂S ´ A natural question: how to obtain an L -morphism of H1 x, H2 ,q . (63) ∞ ∂x = ∂q quantum brackets from a quantum BV-morphism? Note that unlike the classical case, the quantum pullback oper- Note: that Φ∗ is an L -morphism of classical brackets ∞ ator Φˆ ∗ is linear, so cannot be the answer. It turns out that follows per se from the statement for quantum brackets the solution is given by a formula motivated by the sta- and Φˆ ∗. However Theorem 5.4 is a subtler statement: that tionary phase method but without passing to the classical a BV quantum thick morphism (the intertwining condi- limit! tion for ∆-operators) induces a classical S thick mor- ∞ phism (the condition expressed by the Hamilton–Jacobi equation (63). One can see an analogy with the famous 5.3.2 L -morphism of quantum brackets induced by a Egorov theorem [10], which was one of motivating ex- ∞ quantum BV-morphism amples for Hörmander’s theory of Fourier integral oper- ators [17]. This poses the question about a possibility of ! Define a non-linear transformation Φˆ : C∞(M2) C∞(M1) quantization for the whole picture: i.e. lifting of an S - ħ → ħ ∞ by the formula structure to a S , - (= quantum BV) structure and lifting ∞ ħ of a classical S thick morphism to a BV quantum thick ! i Φˆ : ħ ln Φˆ ∗ exp , (60) ∞ = i ◦ ◦ morphism. See more in [71]. ħ ! ¡ i g ¢ or Φˆ (g) ħ ln Φˆ ∗ e , for a g C∞(M ). = i ħ ∈ 2 ˆ ħ Theorem 5.3. If Φ: M1 M2 is a BV quantum mor- 6 Potential further development. Some ! →ħ phism, then Φˆ is an L -morphism of the S , -algebras of ∞ ∞ ħ problems and open questions functions. In greater detail: Φˆ ! is a morphism of infinite- dimensional Q-manifolds C∞(M2) C∞(M1) with the ħ → ħ 6.1 “Non-linear algebra-geometry duality” homological vector fields Q∆1 and Q∆2 , where Z i f ¡ i f ¢ δ – Define a non-linear homomorphism of (super)algebras Q∆ Dx e− ħ ∆ e ħ . (61) = δf (x) to be a non-linear map A A (variant: formal map) 1 → 2 Since in the limit 0, quantum brackets generated such that its derivative at every element a A1 is an al- ħ → ∈ by ∆ become classical brackets, and the transformation gebra homomorphism. Question: how to describe such ! Φ in the classical limit gives the pullback Φ∗ by the cor- maps? responding classical thick morphism, as a corollary we – In particular, is it true that all such non-linear homo- obtain that Φ∗ gives an L -morphism of the classical S - morphisms between algebras C ∞(M) are pullbacks by ∞ ∞ brackets. In fact, we can prove more than that. thick morphisms?

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