Deformation Theory. I Maxim Kontsevich Yan Soibelman Current address:InstitutdesHautesEtudes´ Scientifiques, Bures-sur-Yvette, F-91440, France E-mail address: [email protected] Current address:DepartmentofMathematics,KansasStateUniversity,Man- hattan, KS 66506, USA E-mail address: [email protected] 1991 Mathematics Subject Classification. 3 THIS IS A VERY PRELIMINARY DRAFT OF THE FIRST VOLUME OF THE BOOK. IT IS INCOMPLETE AND CONTAINS MANY MISPRINTS AND, PROBABLY, MISTAKES. IT IS PLACED ON MY HOMEPAGE BECAUSE OF THE HIGH DEMAND EVEN ON SUCH AN INCOMPLETE WORK. USE IT AT YOUR OWN RISK. Plan INTRODUCTION. CHAPTER 1. Elementary deformation theory 1.1. Algebraic examples. 1.1.1. Associative algebras. 1.1.2. Lie algebras. 1.1.3. Commutative algebras. 1.1.4. Example: limitations of the formal deformation theory. 1.2. Geometric examples. 1.2.1. Local systems. 1.2.2. Holomorphic vector bundles. 1.2.3. Complex structures. 1.3. Schlessinger axioms. 1.4. DGLAs and Deligne’s groupoids. CHAPTER 2. Tensor categories 2.1. Language of linear algebra. 2.1.1. From spaces to algebras. 2.1.2. Linearization of differential geometry. 2.1.3. Less trivial example: space of maps. 2.1.4. Avoiding individual vectors. 2.1.5. Graphs and acyclic tensor calculus. 2.1.6. Why it is useful to speak algebraically? 2.2. Definition of tensor category. 2.2.1. Facultative properties of tensor categories. 2.3. Examples of tensor categories. 2.3.1. Classical examples. 2.3.2. Supervector spaces. 2.3.3. Z-graded spaces, complexes. 2.3.4. Tensor category GLt. 2.4. Signs and orientations. 2.5. Applications of supermathematics. 2.5.1. Idenitification of symplectic and orthogonal geometry. 2.5.2. Where does de Rham complex come from? 5 2.6. Pseudo-tensor categories, operads, PROPs. 2.7. Supermanifolds. 2.7.1. Definition. 2.7.2. Superschemes. 2.7.3. Diffeomorphisms of 0 1-dimensional vector space. | 2.7.4. Representation of the group scheme S∗. 2.7.5. One more remark on the origin of de Rham complex. 2.8. Appendix: Symmetric monoidal categories. CHAPTER 3. dg-manifolds 3.1. Formal manifolds in tensor categories. 3.1.1. Formal manifolds and coalgebras. 3.1.2. Vector fields, tangent spaces. 3.1.3. Inverse function theorem. 3.2. Formal pointed dg-manifolds. 3.2.1. Main definition. 3.2.2. Remark about non-formal dg-manifolds. 3.2.3. L -algebras. 3.2.4. Morphisms∞ of L -algebras. 3.2.5. Pre-L -morphisms.∞ 3.2.6. L -algebras∞ and formal pointed dg-manifolds. 3.2.7. Tangent∞ complex. 3.3. Homotopy classification of formal pointed dg-manifolds and L - algebras. ∞ 3.3.1. Main examples: Lie algebras and DGLAs. 3.3.2. Proof of the minimal model theorem. 3.3.3. Massey operations. 3.3.4. Cofibrant dg-manifolds and homotopy equivalence of dg-algebras. 3.4. Deformation functor. 3.4.1. Groupoid and the foliation in the case of supermanifolds. 3.4.2. Deformation functor associated with a formal pointed dg-manifold. CHAPTER 4. Examples 4.1. DGLAs and dg-manifolds associated with deformation complexes of algebras. 4.1.1. Associative, Lie and commutative algebras. Truncated Hochschild and Chevalley complex. 4.2. DGLAs structures on geometric deformation complexes. 4.2.1. Systems of polynomial equations. 4.2.2. Koszul complex gives rise to a dg-manifold. 4.2.3. Polynomial systems with a group action. 4.2.4. dg-manifolds with homotopy actions of Lie algebras. 4.2.5. Formal differential geometry. 4.2.6. Systems of differential equations with local group of symmetries (gauge groups). 6 4.2.7. BRST. 4.2.8. DGLAs and dg-manifolds associated with deformations of local systems, complex vector bundles, complex manifolds. 4.2.9. dg-manifolds associated with the space of holomorphic maps. Maps(Y, X)= Maps(Y, ΠT 0,1X).Thelatterisanon-formaldg-manifold.Butwecantakeafor- mal neighborhood of a point. Then we get the deformation theory of a morphism f : Y X. 4.2.10.→ Lie algebroids= global dg-manifolds with coordinates in degrees 0 and 1.Note:thecaseofcoordinatesindegrees0 and +1 corresponds to systems of equations− from 4.2.1. 4.2.11. Deformations of representations of the fundamental group of K¨ahler manifold: Goldman and Millson, Simpson. CHAPTER 5. Operads and algebras over operads 5.1. Generalities on operads. 5.1.1. Polynomial functors, operads, algebras. 5.1.2. Examples of operads. 5.1.3. Colored operads. 5.1.4. Non-linear operads. 5.1.5. Pseudo-tensor categories and colored operads. 5.2. Trees. 5.3. Resolutions of operads. 5.3.1. Topological motivation. 5.4. Resolutions of linear operads. 5.4.1. Filtered resolutions of algebras. 5.4.2. Boardman-Vogt resolution of a linear operad. 5.4.3. Example: BV-resolution of an associative operad. 5.4.4. Cofibrant BV-resolution. 5.5. Operads as algebras over the colored operad .Freeoperads. OP 5.6. Resolutions of classical operads. 5.6.1. Generators. 5.6.2. Differential. 5.7. Deformation theory of algebras over operads. 5.7.1. Statement of the problem and two approaches. 5.7.2. Deformations and free operads. 5.7.3. Example: A -operad and A -algebras. 5.7.4. Checking the∞ conjecture for a∞ free operad. 5.7.5. Homotopical actions of the Lie algebras of derivations. 5.7.6. Independence of a choice of resolution of operad. 5.7.7. Theorem: deformation theory of operads As, Lie, Comm are trivial, i.e. they are rigid. Colored operad is rigid. The operad Pois of Poisson algebras is not rigid. OP 5.8. Koszul duality for operads and algebras. 5.9. Hochschild complex and Deligne’s conjecture. 7 5.9.1. Operad of little discs and Fulton-Macpherson operad. 5.9.2. Higher-dimensional generalization of Deligne’s conjecture. 5.9.3. Digression: homotopy theory and deformation theory. 5.9.4. Universal Hochschild complex and deformation theory. 5.9.5. Formality of the operad of little discs. 5.10. Deformation theory of algebras over PROPs. 5.10.1. PROP of bialgebras. 5.10.2. PROP of Lie bialgebras. CHAPTER 6. A -algebras and non-commutative geometry ∞ 6.1. Motivations. 6.1.1. From associative algebras to abelian categories. 6.1.2. Triangulated categories. 6.1.3. Digression about non-commutative geometries. 6.1.4. Why should one generalize triangulated categories? 6.2. Non-commuttaive formal schemes. 6.2.1. Reminder on coalgebras. 6.2.2. Representability theorem and non-commutative small schemes. 6.2.3. Non-commutative smooth schemes Formal and partially formal mani- folds. Theorem: formal manifold=smooth functor Artink Sets. 6.2.4. Free product of smooth schemes. → 6.2.5. Small scheme Maps(X, Y ). 6.3. Formal neighborhood of a small subscheme. 6.4. Elements of non-commutative differential geometry. 6.4.1. Implicit and inverse function theorems. 6.5. A -algebras. 6.5.1. Non-commutative∞ dg-manifolds. 6.5.2. Non-unital A -algebras. 6.5.3. A -algebra Centr∞ (f). 6.5.4. Remark∞ about the unit. 6.5.5. Non-commutative dg-line L and weakly unital A -algebras. 6.5.6. A -structure on a subcomplex. ∞ 6.5.7. Yoneda∞ lemma. 6.5.8. Odd line and double Maps construction as smoothening. 6.5.9. Two constructions of unital A -algebras: via Yoneda and double Maps construction. ∞ 6.5.10. Generalized (with m0)andfilteredA algebras. Minimal model theorem in this case. ∞ 6.6. Examples of A -algebras. 6.6.1. A -algebras and∞ A -modules. ∞ ∞ 6.7. Elliptic spaces. 6.8. Hochschild complex of an A -algebra and non-commutative ge- ometry. ∞ 6.8.1. Hochschild complex again. 8 6.8.2. Comparison with the deformation theory via Boardman-Vogt resolution. 6.8.3. Geometry of homological Hochschild complex. 6.9. Additional structures in finite-dimensional case. 6.10. Non-commutative symplectic manifolds. 6.10.1. Volume forms. 6.10.2. Digression about matrix integrals. 6.11. dg-case and A -algebras with scalar product. ∞ CHAPTER 7. Appendix BIBLIOGRAPHY 9 Introduction 0.1. Subject of the deformation theory can be defined as “study of moduli spaces of structures”. This general definition includes large part of mathematics. For example one can speak about the “moduli space” of objects or morphisms of a category, as long as one can give a meaning to a “variation” of either of them. Homotopy theory is a deformation theory, since we study “variations” of topo- logical spaces under homotopies. Moduli of smooth structures, complex structures, etc. are well-known examples of the problems of deformation theory. In algebra one can study moduli spaces of algebraic structures: associative multiplications on a given vector space, homomorphisms between two given groups, etc. Rephrasing the well-known quote of I. Gelfand one can say that any area of mathematics is a kind of deformation theory. In addition, a typical theory in physics depends on parameters (masses, charges, coupling constants). This leads physicists to believe that the concept of the “moduli space of theories” can be useful. For example, correlators can be computed as integrals over the moduli space of fields, dualities in the string theory can be explained in terms of special points (“cusps”) of the compactified moduli spaces of certain theories, etc. In other words, deformation theory can have applications in theoretical physics. It is quite surprising that despite the importance of the subject, there is no “general” deformation theory. At the same time it is clear that there is a need in such a theory. This feeling was supported by a number of important examples which has been worked out in algebra and geometry starting from the end of 50’s.