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Graded-And-Microformal-Prop The University of Manchester Research Graded Geometry, Q-Manifolds, and Microformal Geometry DOI: 10.1002/prop.201910023 Document Version Accepted author manuscript Link to publication record in Manchester Research Explorer Citation for published version (APA): Voronov, T. (2019). Graded Geometry, Q-Manifolds, and Microformal Geometry: LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. Fortschritte der Physik. https://doi.org/10.1002/prop.201910023 Published in: Fortschritte der Physik Citing this paper Please note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscript or Proof version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version. General rights Copyright and moral rights for the publications made accessible in the Research Explorer are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Takedown policy If you believe that this document breaches copyright please refer to the University of Manchester’s Takedown Procedures [http://man.ac.uk/04Y6Bo] or contact [email protected] providing relevant details, so we can investigate your claim. Download date:06. Oct. 2021 Graded geometry, Q-manifolds, and microformal geometry 1,2 Theodore Th. Voronov ¤ below), to which we give a brief introduction as well. We give an exposition of graded and microformal ge- “Thick morphisms” (defined for ordinary manifolds, ometry, and the language of Q-manifolds. Q-manifolds supermanifolds or graded manifolds) generalize ordinary are supermanifolds endowed with an odd vector field of maps or supermanifold morphisms, but are not maps square zero. 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 ©: M M is a formal non-linear differ- ing pullbacks of functions are nonlinear. In particular, 1 ! 2 ential operator ©¤ : C1(M ) C1(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 1 “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 1 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- Æ 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 1 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 ¦T ¤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- 1 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.
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