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2 -SUPERMANIFOLDS Janusz Grabowski (Polish Academy of Sciences) 60 - PowerPoint PPT Presentation

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Z n 2 -SUPERMANIFOLDS Janusz Grabowski (Polish Academy of Sciences) 60 YEARS ALBERTO IBORT FEST Madrid, 5-9 March, 2018 Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 1 / 31 2 -supermanifolds Contents Batchelor-Gaw edzki theorem

  1. Z n 2 -SUPERMANIFOLDS Janusz Grabowski (Polish Academy of Sciences) 60 YEARS ALBERTO IBORT FEST Madrid, 5-9 March, 2018 Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 1 / 31 2 -supermanifolds

  2. Contents Batchelor-Gaw¸ edzki theorem Sign-rules and Z n 2 -graded algebras Z n 2 -supermanifolds n -fold vector bundles Superization of n -fold vector bundles Colored Batchelor-Gaw¸ edzki theorem Sketch of the proof: embedding C ∞ M ֒ → A M The talk is based on a joint work with Tiffany Covolo and Norbert Poncin: The category of Z n 2 -supermanifolds, J. Math. Phys. 57 (2016), 073503 (16pp). Splitting theorem for Z n 2 -supermanifolds, J. Geom. Phys. 110 (2016), 393-401. Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 2 / 31 2 -supermanifolds

  3. Batchelor-Gaw¸ edzki theorem A vector bundle is a locally trivial fibration τ : E → M which, locally over U ⊂ M , reads τ − 1 ( U ) ≃ U × R n and admits an atlas in which local trivializations transform linearly in fibers U ∩ V × R n ∋ ( x , y ) �→ ( x , A ( x ) y ) ∈ U ∩ V × R n , A ( x ) ∈ GL ( n , R ). The latter property can also be expressed in the terms of the gradation in which base coordinates x have degrees 0 and ‘linear coordinates’ y have degree 1. Linearity in y ′ s is now equivalent to the fact that changes of coordinates respect the degrees. This implies that there is a well-defined homogeneity structure, i.e. an action of the multiplicative monoid of reals, h : R × E ∋ ( t , v ) �→ h t ( v ) ∈ E , h t ( x , y ) = ( x , ty ) , (multiplication by reals) and its infinitesimal generator ∇ E = y a ∂ y a (the Euler/Liouville vector field). Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 3 / 31 2 -supermanifolds

  4. Batchelor-Gaw¸ edzki theorem Actually, one can prove (Grabowski-Rotkiewicz ’09) that a vector bundle structure is just a homogeneity structure h on a manifold E which is regular, i.e. d d t | t =0 h t ( v ) = 0 ⇔ v ∈ M = h 0 ( E ) . A homogeneity structure defines N -graded algebra generated by homogeneous functions: f ∈ C ∞ ( E ) is homogeneous of degree k ∈ N if f ◦ h t = t k f ( ∇ E ( f ) = kf ). If we replace the local fiber coordinates ( y i ) of degree 1 with coordinates ( ξ i ) which are not only of degree 1 but also odd, ξ i ξ j = − ξ j ξ i , then the coordinate transformations ( x , ξ ) �→ ( x , A ( x ) ξ ) remain consistent and define a supermanifold Π E = E [1] with M being its body. Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 4 / 31 2 -supermanifolds

  5. Batchelor-Gaw¸ edzki theorem Each coordinate neighbourhood U ⊂ M is then a ringed space with the sheaf O U of supercommutative rings O U ( V ) = C ∞ ( V )[ ξ 1 , . . . , ξ n ] of Grassmann polynomials in variables ( ξ i ) and coefficients in the algebra C ∞ ( V ) of smooth functions on V ⊂ U . Theorem (Gaw¸ edzki ’77, Batchelor ’79) Any supermanifold M with the body M is (non-canonically) diffeomorphic with a supermanifold Π E for a vector bundle τ : E → M. The superalgebra A ( M ) of smooth (super)functions on M is then isomorphic with the Grassmann algebra n A • ( E ) = � Sec (Λ j E ) , j =1 of multi-sections of E, where n is the rank of E. Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 5 / 31 2 -supermanifolds

  6. Z 2 -grading is not enough Supermanifolds Π E are special, because the Z 2 -grading in the structure sheaf comes from a Z -grading (actually, N -grading). Consider a supermanifold M with coordinates (even and odd) ( x a , ξ i ) and its tangent bundle T M with coordinates ( x a , ξ i , d x b , d ξ j ). We can consider T M as a supermanifold, viewing x a , d x b as even and ξ i , d ξ j as odd, or, closer to the standard convention, viewing x a , d ξ j as even and d x b , ξ i as odd. Much more natural is to take advantage with the additional N -grading on the vector bundle T M and to consider the algebra of functions as Z 2 × N (thus also Z 2 2 )-graded. Hence the sign convention for homogeneous elements z α , z β of bi-degrees α = ( α 1 , α 2 ) , β = ( β 1 , β 2 ) ∈ Z 2 2 is (Deligne convention) z α z β = ( − 1) � α,β � z β z α , where � α, β � = α 1 β 1 + α 2 β 2 is the ‘scalar product’. However, T M is not a standard supermanifold but rather Z 2 2 -supermanifold. Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 6 / 31 2 -supermanifolds

  7. Sign-rules and Z n 2 -graded algebras let K be a commutative unital ring, K × be the group of invertible elements of K , and let G be a commutative semigroup. A map ϕ : G × G → K × is called a commutation factor on G if ϕ ( g , h ) ϕ ( h , g ) = 1 , ϕ ( f , g + h ) = ϕ ( f , g ) ϕ ( f , h ) , ϕ ( g , g ) = ± 1 , for all f , g , h ∈ G . Note that these axioms imply that ϕ ( f + g , h ) = ϕ ( f , h ) ϕ ( g , h ) and that the condition ϕ ( g , g ) = ± 1 follows automatically from the other two axioms if K is a field. g ∈ G A g . Elements x from A g Let A be a G -graded K -algebra A = � are called G-homogeneous of degree or weight g =: deg( x ). The algebra A is said to be ϕ -commutative if ab = ϕ (deg( a ) , deg( b )) ba , for all G -homogeneous elements a , b ∈ A . Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 7 / 31 2 -supermanifolds

  8. Sign-rules and Z n 2 -graded algebras Homogeneous elements x with p (deg( x )) = p ( g ) := ϕ ( g , g ) = − 1 are odd, the other homogeneous elements are even. Graded algebras with commutation rules of this kind are known under the name color algebras. In this talk we will be interested in color associative algebras whose commutation factor is just a sign. In what follows, K will be R and ϕ will take the form ϕ ( g , h ) = ( − 1) � g , h � , for a ‘scalar product’ �− , −� : G × G → Z . This means that we use the commutation factor as the sign rule. In this note we confine ourselves to G = Z n 2 and the standard ‘scalar product’ of Z n 2 , what will lead to Z n 2 -Supergeometry with nicer categorical properties than the standard Supergeometry. More precisely, we propose a generalization of differential Z 2 -Supergeometry to the case of a Z n 2 -grading in the structure sheaf. Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 8 / 31 2 -supermanifolds

  9. Sign-rules and Z n 2 -graded algebras Example The real Clifford algebra Cl p , q ( R ) is the associative R -algebra generated by e i , where 1 ≤ i ≤ n and n = p + q , of R n , modulo the relations e i e j = − e j e i , i � = j , � +1 , i ≤ p e 2 = i − 1 , i > p . The pair of integers ( p , q ) is called the signature. Note that, as a vector space, Cl p , q ( R ) is isomorphic to the Grassmann algebra � � e 1 , . . . , e n � on the chosen generators. Cl p , q ( R ) is often understood as quantization of the Grassmann algebra (in the same sense as the Weyl algebra is a quantization of the symmetric algebra). The Clifford algebra Cl p , q ( R ) is a Z p + q +1 -commutative associative algebra 2 with the degree of e i being (0 , . . . , 0 , 1 , 0 , . . . , 0 , 1). Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 9 / 31 2 -supermanifolds

  10. Sign-rules and Z n 2 -graded algebras Actually, from the scalar product on Z n 2 we can obtain arbitrary sign rule. For, let S be a finite set, say S = { 1 , . . . , m } , and let ϕ : S × S → {± 1 } be any symmetric function. We can understand ϕ as a sign rule for the associative algebra A generated freely by elements y i , i = 1 , . . . , m , modulo the commutation identities y i y j = ϕ ( i , j ) y j y i . We then have the following. Theorem There is n ≤ 2 m and a map σ : S → Z n 2 , i �→ σ i , such that ϕ ( i , j ) = ( − 1) � σ i ,σ j � n . In other words, A can be made into a Z n 2 -commutative associative algebra. Note that there are non-nilpotent generators of non-zero degree. Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 10 / 31 2 -supermanifolds

  11. Z n 2 -supermanifolds The first idea is to define the function sheaf O U of a Z n 2 -superdomain U = ( U , O U ) , over any open V ⊂ U , as the Z n 2 -commutative associative unital R -algebra O U ( V ) = C ∞ U ( V )[ ξ 1 , . . . , ξ q ] of polynomials in the indeterminates ξ a of degrees deg( ξ a ) ∈ Z n 2 \ { 0 } with coefficients in smooth functions of V . However, for a proper development of differential calculus, we must be able to compose elements of degree 0 with smooth functions. But what is F ( x + ξ 2 ) for a 1-variable smooth function F , a variable x and a formal even variable ξ ? Since ξ is not nilpotent, the Taylor formula k ! F ( k ) ( x ) ξ 2 k leads to a formal power series. F ( x + ξ 2 ) = � 1 k Hence we are forced to take O U ( V ) = C ∞ U ( V )[[ ξ 1 , . . . , ξ q ]] . Z n J.Grabowski (IMPAN) Luxembourg, 5-9/3/2018 11 / 31 2 -supermanifolds

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