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Graded Manifolds: Some Issues

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