Higher Supergeometry Revisited

Transcription

1 Higher Supergeometry Revisited Norbert Poncin University of Luxembourg Supergeometry and applications December 14-15, 2017 N. Poncin Higher SG University of Luxembourg 1 / 42

2 Outline * Motivations and applications Z n 2 Berezinian Z n 2 Manifolds and Z n 2 morphisms Z n 2 Batchelor-Gawedzki theorem Z n 2 integral calculus Outlook * Joint with T. Covolo, J. Grabowski, V. Ovsienko, S. Kwok,... N. Poncin Higher SG University of Luxembourg 2 / 42

3 Motivations and Applications MOTIVATIONS AND APPLICATIONS N. Poncin Higher SG University of Luxembourg 3 / 42

4 Motivations and Applications New sign rule Higher gradings and modified sign rule Supergeometry: coordinates x of degree 0 ξ of degree 1 N. Poncin Higher SG University of Luxembourg 3 / 42

5 Motivations and Applications New sign rule Higher gradings and modified sign rule Z 2 2-Supergeometry: coordinates x of degree (0, 0) y of degree (1, 1) ξ of degree (0, 1) η of degree (1, 0) N. Poncin Higher SG University of Luxembourg 3 / 42

6 Motivations and Applications New sign rule Higher gradings and modified sign rule Z 3 2-Supergeometry: coordinates x of degree (0, 0, 0) y of degree (0, 1, 1)... ξ of degree (0, 0, 1) η of degree (0, 1, 0)... y η = ( 1) (0,1,1),(0,1,0) η y N. Poncin Higher SG University of Luxembourg 3 / 42

7 Motivations and Applications New sign rule Higher gradings and modified sign rule Z 3 2-Supergeometry: coordinates x of degree (0, 0, 0) y of degree (0, 1, 1)... ξ of degree (0, 0, 1) η of degree (0, 1, 0)... y η = ( 1) (0,1,1),(0,1,0) η y New features: Even coordinates may anticommute: ( 1) (1,1,0),(1,0,1) = 1 Odd coordinates may commute: ( 1) (1,0,0),(0,1,0) = +1 Non-zero degree coordinates may not be nilpotent: ( 1) (1,1,0),(1,1,0) = +1 N. Poncin Higher SG University of Luxembourg 3 / 42

8 Motivations and Applications New sign rule Examples Physics: - Parastatistical supersymmetry - String orbifolds - Anyons Algebra: - Super differential forms (n = 2) α β = ( 1) deg(α) deg(β)+p(α) p(β) β α - Quaternions H (n = 3) - Clifford algebras Cl p,q (n = p + q + 1) Geometry: - Superized higher vector bundles, e.g., T T M, T T M... - Tangent bundle of a supermanifold N. Poncin Higher SG University of Luxembourg 4 / 42

9 Motivations and Applications New sign rule Tangent bundle to a supermanifold Supermanifold M : (x, ξ) (x, ξ,dx,dξ) (0, 1, 1 + 0, 1 + 1) + usual sign rule T M : supermanifold C (x,d ξ)[ξ,d x] (x, ξ,dx,dξ) ((0, 0), (0, 1), (1, 0), (1, 1)) + new sign rule T M : Z 2 2-manifold C (x)[[ξ,d x,d ξ]] N. Poncin Higher SG University of Luxembourg 5 / 42

10 Motivations and Applications Local model Coherent differential calculus Z 2 2-manifold: (0, 0), (1, 1), (0, 1), (1, 0) φ : {x, y, ξ, η} {x, y, ξ, η } x = x + y 2, y = y, ξ = ξ, η = η Local model: F (x ) = F (x + y 2 ) = α (R p, C (U)[[y, ξ, η]]) 1 α! ( α x F )(x)y2α x = r f r x (x)y 2r + r gx r (x)y 2r+1 ξη y = r f r y (x)y 2r+1 + r gy r (x)y 2r ξη ξ = r f r ξ (x)y 2r ξ + r gξ r (x)y 2r+1 η η = r f r η (x)y 2r η + r gη r (x)y 2r+1 ξ N. Poncin Higher SG University of Luxembourg 6 / 42

11 Motivations and Applications Higher Berezinian Key-concept (0, 0) (1, 1) (0, 1) (1, 0) (x,y,ξ,η) (x, y, ξ, η ) = (1, 1) (0, 0) (1, 0) (0, 1) (1, 0) (0, 1) (0, 0) (1, 1) (0, 1) (1, 0) (1, 1) (0, 0) Big diagonal blocks: even Z n 2 -degrees Small diagonal blocks: degree zero N. Poncin Higher SG University of Luxembourg 7 / 42

12 -Berezinian: direct approach HIGHER TRACE AND BEREZINIAN OF MATRICES OVER A CLIFFORD ALGEBRA Journal of Geometry and Physics (2012), 62(11), N. Poncin Higher SG University of Luxembourg 8 / 42

13 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Berezinian A: supercommutative algebra Theorem! group morphism such that Ber A D Ber I B I Ber : GL 0 (A) (A 0 ) = det A det -1 D I = 1 = Ber C I It is defined by Ber A B C D = det(a BD 1 C) det -1 (D) N. Poncin Higher SG University of Luxembourg 8 / 42

14 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Z n 2-Berezinian A: Z n 2 -commutative algebra Theorem! group morphism such that Z n 2Ber : GL 0 (A) (A 0 ) Z n 2Ber = det? det -1? Z n 2Ber I I = 1 = Z n 2Ber I I It is defined by Z n 2Ber A B C D =? N. Poncin Higher SG University of Luxembourg 9 / 42

15 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Z n 2-Determinant A: (Z n 2 ) even -commutative algebra Theorem 1.! algebra morphism such that Z n 2det : gl 0 (A) A 0 Z n 2det Z n 2det I I I I = det = 1 = Z n 2det I I I I 2. Z n 2det(X) is linear in the entries of X N. Poncin Higher SG University of Luxembourg 10 / 42

16 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Quasideterminants (I.Gelfand and V.Retakh) Ber A B C D = det(a BD 1 C) det -1 (D) N. Poncin Higher SG University of Luxembourg 11 / 42

17 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Quasideterminants (I.Gelfand and V.Retakh) Ber A B C D = det(a BD 1 C) det -1 (D) ( ) A B = A BD 1 C C D 11 N. Poncin Higher SG University of Luxembourg 11 / 42

18 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Quasideterminants (I.Gelfand and V.Retakh) Ber A Definition B C D = det(a BD 1 C) det -1 (D) ( ) A B = A BD 1 C C D 11 For a square matrix X with entries in a ring R, j-th column X ij := x ij r j i (Xij ) 1 c i j R i-th row N. Poncin Higher SG University of Luxembourg 11 / 42

19 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Examples X = x a b c y d e f z X 11 = x bz 1 e (a bz 1 f)(y dz 1 f) 1 (c dz 1 e) X 11 = X = x a b c y d e f z ( x bz 1 e a bz 1 f c dz 1 e y dz 1 f ) Quasi-determinants are rational functions N. Poncin Higher SG University of Luxembourg 12 / 42

20 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Z n 2-Determinant A: (Z n 2 ) even -commutative algebra Theorem 1.! algebra morphism such that Z n 2det : gl 0 (A) A 0 Z n 2det Z n 2det I I I I = det = 1 = Z n 2det I I I I 2. Z n 2det(X) is linear in the entries of X N. Poncin Higher SG University of Luxembourg 13 / 42

21 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace UDL decomposition X = UDL N. Poncin Higher SG University of Luxembourg 14 / 42

22 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace UDL decomposition X = UDL D = X 11 X 1: X qq N. Poncin Higher SG University of Luxembourg 14 / 42

23 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace UDL decomposition X = UDL D = X 11 X 1: X qq N. Poncin Higher SG University of Luxembourg 14 / 42

24 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace UDL decomposition X = UDL D = X 11 X 1: X qq N. Poncin Higher SG University of Luxembourg 14 / 42

25 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace UDL decomposition X = UDL D = X 11 X 1: X qq N. Poncin Higher SG University of Luxembourg 14 / 42

26 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace UDL decomposition X = UDL D = X 11 X 1: X qq Theorem Z n 2det(X) = det( X 11 ) det( X 1:1 22 )... det(x rr ) N. Poncin Higher SG University of Luxembourg 14 / 42

27 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Z n 2-Berezinian Theorem! group morphism Z n 2Ber : GL 0 (A) (A 0 ) such that Z n 2Ber = 1 Z n 2Ber I I = 1 = Z n 2Ber I I It is defined by Z n 2Ber A C B D = N. Poncin Higher SG University of Luxembourg 15 / 42

28 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Z n 2-Berezinian Theorem! group morphism Z n 2Ber : GL 0 (A) (A 0 ) such that Z n 2Ber = Z n 2det Z n 2det 1 Z n 2Ber I I = 1 = Z n 2Ber I I It is defined by Z n 2Ber A C B D = N. Poncin Higher SG University of Luxembourg 15 / 42

29 -Berezinian: direct approach Berezinian, Zn 2 -Berezinian, Zn 2 -Determinant, Zn 2 -Trace Z n 2-Berezinian Theorem! group morphism such that Z n 2Ber : GL 0 (A) (A 0 ) Z n 2Ber = Z n 2det Z n 2det 1 Z n 2Ber I I = 1 = Z n 2Ber I I It is defined by Z n 2Ber A B C D = Zn 2det(A BD 1 C) Z n 2det -1 (D) For n = 1, Z n 2Ber = Ber; for A = H, Z n 2Ber = Ddet ; Liouville formula N. Poncin Higher SG University of Luxembourg 15 / 42

30 -Geometry I: manifolds and morphisms THE CATEGORY OF Z n 2 -SUPERMANIFOLDS JOURNAL OF MATHEMATICAL PHYSICS (2016), 57(7) N. Poncin Higher SG University of Luxembourg 16 / 42

31 -Geometry I: manifolds and morphisms Zn 2 -manifolds Functor of points approach P C[C n ], V = {z C n : P (z) = 0} Aff, C[V ] = C[C n ]/(P ) CA Sol P : CA A Sol P (A) = {a A n : P (a) = 0} Set Sol P = Hom CA (C[V ], ) Fun(CA, Set) Hom Aff (, V ) Fun(Aff op, Set) : C c c := Hom C (, c) FunSh(C op, Set) Lim c Lim c Colim c Colim c N. Poncin Higher SG University of Luxembourg 16 / 42

32 -Geometry I: manifolds and morphisms Zn 2 -manifolds Functor of points approach P C[C n ], V = {z C n : P (z) = 0} Aff, C[V ] = C[C n ]/(P ) CA Sol P : CA A Sol P (A) = {a A n : P (a) = 0} Set Sol P = Hom CA (C[V ], ) Fun(CA, Set) Hom Aff (, V ) Fun(Aff op, Set) : C c c := Hom C (, c) FunSh(C op, Set) Lim c Lim c Colim c Colim c N. Poncin Higher SG University of Luxembourg 16 / 42

33 -Geometry I: manifolds and morphisms Zn 2 -manifolds Functor of points approach Representable Sh(C): trivial spaces Sh(C): spaces Locally representable Sh(C): varieties or manifolds Var(C) Examples: Var(Aff): schemes Var(Z n 2 -Domain): Z n 2 -manifolds N. Poncin Higher SG University of Luxembourg 17 / 42

34 -Geometry I: manifolds and morphisms Zn 2 -manifolds Locally ringed space approach Definition A Z n 2 -manifold is a Z n 2 -graded locally ringed space (M, A M ) that is locally modeled on (R p, C R p( )[[ξ1,..., ξ q ]]), where the ξ a are Z n 2 -commutative. N. Poncin Higher SG University of Luxembourg 18 / 42

35 -Geometry I: manifolds and morphisms Zn 2 -manifolds Reconstruction theorem Proposition A topological space that is covered by Z n 2 -graded Z n 2 - commutative coordinate systems (x, y,..., ξ, η,...) and is endowed with Z n 2 -degree preserving coordinate transformations ϕ βα : (x, y,..., ξ, η,...) (x, y,..., ξ, η,...) that satisfy the cocycle condition defines a Z n 2 -manifold. ϕ γβ ϕ βα = ϕ γα, N. Poncin Higher SG University of Luxembourg 19 / 42

36 -Geometry I: manifolds and morphisms Zn 2 -manifolds Nilpotency Formal series Invertibility of superfunctions: f C (U)[ξ 1,..., ξ q ] invertible f 0 C (U) invertible Proof: Nilpotency N. Poncin Higher SG University of Luxembourg 20 / 42

37 -Geometry I: manifolds and morphisms Zn 2 -manifolds Nilpotency Formal series Invertibility of Z n 2 -functions: f C (U)[[ξ 1,..., ξ q ]] invertible f 0 C (U) invertible Proof: Formal power series N. Poncin Higher SG University of Luxembourg 20 / 42

38 -Geometry I: manifolds and morphisms Zn 2 -morphisms Z n 2-morphisms Z n 2 -morphism: ringed space morphism (ψ, ψ ) : (M, A) (N, B) Commutation with base projections: B(V ) ψ A(ψ 1 (V )) ε V ε ψ 1 (V ) C N (V ) ψ C M (ψ 1 (V )) ψ is C 0 with respect to the J -adic topology, J = ker ε A is Hausdorff-complete for the J -adic topology N. Poncin Higher SG University of Luxembourg 21 / 42

39 -Geometry I: manifolds and morphisms Zn 2 -morphisms Fundamental Z n 2-Morphism Theorem Theorem A Z n 2 -morphism (ψ, ψ ) : (M, A M ) (V, CV [[ξ 1,..., ξ q ]]) is completely and uniquely defined by the pullbacks ψ x i and ψ ξ a of the base coordinates x i and the formal coordinates ξ a The result that makes Z n 2 -Geometry a reasonable theory N. Poncin Higher SG University of Luxembourg 22 / 42

40 -Geometry II : Batchelor-Gawedzki Theorem SPLITTING THEOREM FOR Z n 2 -MANIFOLDS JOURNAL OF GEOMETRY AND PHYSICS (2016), 110 N. Poncin Higher SG University of Luxembourg 23 / 42

41 Z n 2 -Geometry II : Batchelor-Gawedzki Theorem Double vector bundles Double vector bundles Definition 1: E 01 τ E E 01 E E 11 τ E E 10 E 10 τ E 01 M M τ E 10 M Trivial example: E 11 = ker τ E E 01 ker τ E E 10 E 01 E 10 E 11 E E 01 E 10 E 11 and Γ(E 01 E 10 E 11 ) N. Poncin Higher SG University of Luxembourg 23 / 42

42 -Geometry II : Batchelor-Gawedzki Theorem Double vector bundles Double vector bundles Definition 2: Pair of commuting Euler vector fields on a manifold. Definition 3: Locally trivial fiber bundle with standard fiber V 01 V 10 V 11 s.th. ξ a = fu(x)ξ a u η b = gv(x)η b v y c = h c w(x)y w + ku,v(x)ξ c u η v N. Poncin Higher SG University of Luxembourg 24 / 42

43 -Geometry II : Batchelor-Gawedzki Theorem Zn 2 -graded vector bundles, higher vector bundles, Zn 2 -manifolds Split Z n 2-manifolds Vector bundle: E M ΠE := E[1] A(ΠE) = Γ( E ) = r k=0 Γ( k (ΠE) ) Supermanifold: M = (M, A(ΠE)) Graded vector bundle: E = E 01 E 10 E 11 M ΠE := E 01 [01] E 10 [10] E 11 [11] A(ΠE) = k 0 Γ( k (ΠE) ) Z 2 2-manifold: M = (M, A(ΠE)) N. Poncin Higher SG University of Luxembourg 25 / 42

44 -Geometry II : Batchelor-Gawedzki Theorem Zn 2 -graded vector bundles, higher vector bundles, Zn 2 -manifolds Non-split Z n 2-manifold Double vector bundle: E ξ a η b y c = fu(x)ξ a u = gv(x)η b v = h c w(x)y w + ku,v(x)ξ c u η v > Superization, coherence, cocycle condition Z 2 2-manifold NOT canonically split N. Poncin Higher SG University of Luxembourg 26 / 42

45 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Z n 2-Batchelor-Gawedzki Theorem Batchelor, Gawedzki, Kirillov and Rudakov Smooth and real analytic, but not holomorphic Theorem Any Z n 2 -manifold (M, A) is non-canonically split, i.e., there exists a non-canonical isomorphism where E is a Z n 2 \ {0}-vector bundle. A A(ΠE), N. Poncin Higher SG University of Luxembourg 27 / 42

46 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) )? N. Poncin Higher SG University of Luxembourg 28 / 42

47 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) )? J /J 2 0 J A C 0 N. Poncin Higher SG University of Luxembourg 28 / 42

48 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) ) J /J 2 = Γ((ΠE) ) k 0 k (J /J 2 ) Sh A? 0 J A C 0 N. Poncin Higher SG University of Luxembourg 28 / 42

49 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) ) 0 J A C 0 J /J 2 = Γ((ΠE) ) k 0 k (J /J 2 ) Stalks A N. Poncin Higher SG University of Luxembourg 28 / 42

50 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) ) J /J 2 = Γ((ΠE) ) k 0 k (J /J 2 ) Sh A? 0 J A C 0 N. Poncin Higher SG University of Luxembourg 28 / 42

51 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) ) J /J 2 = Γ((ΠE) ) k 0 k (J /J 2 ) A J /J 2 A? 0 J A C 0 N. Poncin Higher SG University of Luxembourg 28 / 42

52 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) ) J /J 2 = Γ((ΠE) ) k 0 k (J /J 2 ) A 0 J A C 0 J /J 2 A 0 J 2 J J /J 2 0? N. Poncin Higher SG University of Luxembourg 28 / 42

53 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (I) Step 1: A Sh A(ΠE) = k 0 k Γ((ΠE) ) J /J 2 = Γ((ΠE) ) k 0 k (J /J 2 ) A 0 J A C 0 J /J 2 A 0 J 2 J J /J 2 0 C ϕ A? N. Poncin Higher SG University of Luxembourg 28 / 42

54 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (II) Step 2: C ϕ ϕ k A = lim A/J k ϕ k+1 k π k π k+1... A/J k A/J k+1... f k,k+1 ϕ k+1,ω : C (Ω) A(Ω)/J k+1 (Ω) ϕ k+1,u : C (U) A(U)/J k+1 (U) ϕ k+1,u U V = ϕ k+1,v U V N. Poncin Higher SG University of Luxembourg 29 / 42

55 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (III) ω k+1,uv := ϕ k+1,u U V ϕ k+1,v U V ω k+1,uv Der(C (U V ), Γ(U V, k (ΠE) )) N. Poncin Higher SG University of Luxembourg 30 / 42

56 -Geometry II : Batchelor-Gawedzki Theorem Proof of the generalized Batchelor-Gawedzki Theorem Sketch of proof (III) ω k+1,uv := ϕ k+1,u U V ϕ k+1,v U V ω k+1,uv Γ(U V, T M k (ΠE) ) ω k+1 Ž1 = ˇB 1 ϕ k+1,u U V ϕ k+1,v U V = η k+1,v U V η k+1,u U V ϕ k+1,u + η k+1,u consistent (correction of ϕ k+1,u ) N. Poncin Higher SG University of Luxembourg 30 / 42

57 -Berezinian: cohomological approach COHOMOLOGICAL APPROACH TO THE GRADED BEREZINIAN Journal of Noncommutative Geometry, 9 (2015), N. Poncin Higher SG University of Luxembourg 31 / 42

58 -Berezinian: cohomological approach Pre integral calculus Determinant module A a commutative algebra M a free A-module, rank r, bases (e i ), (e i ), e j = e i Bi j Det(M) = r M : rank 1 A-module with e 1... e r = e 1... e r det(b) N. Poncin Higher SG University of Luxembourg 31 / 42

59 -Berezinian: cohomological approach Pre integral calculus Z n 2-Berezinian module A a Z n 2 -commutative algebra M a free Z n 2 -graded A-module, total rank r, bases (e i ), (e i ), e j = e i Bi j Definition Z n 2Ber(M) is a rank 1 A-module on which B GL 0 (A) acts as Z n 2Ber(B) N. Poncin Higher SG University of Luxembourg 32 / 42

60 -Berezinian: cohomological approach Pre integral calculus Z n 2-Berezinian module K = A ΠM A M, H(K) = H r (K) = [ω] A d = i Πe i ε i Φ : B GL 0 (A) Φ B (B, Zn 2 t B 1 ) Aut 0 (H(K)) (A 0 ) Φ B = Z n 2Ber(B) ω = ω Z n 2Ber(B) ω : algebraic Z n 2 Berezinian volume N. Poncin Higher SG University of Luxembourg 33 / 42

61 -Supergeometry III: integration theory INTEGRATION ON Z n 2 -MANIFOLDS IN PROGRESS ORBILU N. Poncin Higher SG University of Luxembourg 34 / 42

62 -Supergeometry III: integration theory Local Zn 2 -Berezinian section Local Berezinian section (U, X = (x, y, ξ, η)) : ((0, 0), (1, 1), (0, 1), (1, 0)) M = Ω 1 (M)(U) : (d x,d y,d ξ,d η) M = T (M)(U) : ( x, y, ξ, η ) N. Poncin Higher SG University of Luxembourg 34 / 42

63 -Supergeometry III: integration theory Local Zn 2 -Berezinian section Local Berezinian section (U, X = (x, y, ξ, η)) : ((0, 0), (1, 1), (0, 1), (1, 0)) M = Ω 1 (M)(U) : (d x,d y,d ξ,d η) M = T (M)(U) : ( x, y, ξ, η ) ω = d xd y ξ η N. Poncin Higher SG University of Luxembourg 34 / 42

64 -Supergeometry III: integration theory Local Zn 2 -Berezinian section Local Berezinian section (U, X = (x, y, ξ, η)) : ((0, 0), (1, 1), (0, 1), (1, 0)) M = Ω 1 (M)(U) : (d x,d y,d ξ,d η) M = T (M)(U) : ( x, y, ξ, η ) ω = d xd y ξ η ω(x) = ω(x ) Z n 2Ber( X X) ω(x) f(x) = ω(x ) f(x(x )) Z n 2Ber( X X) =: ω(x ) f (X ) N. Poncin Higher SG University of Luxembourg 34 / 42

65 -Supergeometry III: integration theory Integral of an integrable Zn 2 -Berezinian section Global Berezinian section Definition A Berezinian section of a Z 2 2-manifold with an oriented base is a family ω(x)f(x), ω(x )f (X ),..., whose components transform according to the rule f (X ) = f(x(x )) Z n 2Ber( X X) N. Poncin Higher SG University of Luxembourg 35 / 42

66 -Supergeometry III: integration theory Integral of an integrable Zn 2 -Berezinian section Z 2 2-integral I β: Berezinian section supported in a Z 2 2-domain X = (x, y, ξ, η) 1 1 β = ω(x)f(x) = (d xd y ξ η ) f kab (x)y k ξ a η b k=0 a=0 b=0 Definition d x β := d y k=0 d x d y ξ η f(x, y, ξ, η) = f k11 (x)y k := d x f 011 (x) R β: arbitrary Berezinian section partition of unity N. Poncin Higher SG University of Luxembourg 36 / 42

67 -Supergeometry III: integration theory Integral of a generalized Zn 2 -Berezinian section Z 2 2-integral II σ: generalized Berezinian section supported in a Z 2 2-domain X = (x, y, ξ, η) 1 1 σ = ω(x) L(X) = (d xd y ξ η ) f kab (x)y k ξ a η b k= N a=0 b=0 Definition σ = d x d y k= N d x f 111 (x) R f k11 (x)y k := σ: arbitrary generalized Berezinian section partition of unity N. Poncin Higher SG University of Luxembourg 37 / 42

68 -Supergeometry III: integration theory Integral of a generalized Zn 2 -Berezinian section Change-of-variables formula σ: generalized Berezinian section supported in two domains U, U (X, X ) σ = ω(x) L(X) = ω(x ) L (X ) N. Poncin Higher SG University of Luxembourg 38 / 42

69 -Supergeometry III: integration theory Integral of a generalized Zn 2 -Berezinian section Change-of-variables formula σ: generalized Berezinian section supported in two domains U, U (X, X ) σ = ω(x) L(X) = ω(x ) L (X ) = ω(x ) L(X(X )) Z n 2Ber( X X) N. Poncin Higher SG University of Luxembourg 38 / 42

70 -Supergeometry III: integration theory Integral of a generalized Zn 2 -Berezinian section Change-of-variables formula σ: generalized Berezinian section supported in two domains U, U (X, X ) σ = ω(x) L(X) = ω(x ) L (X ) = ω(x ) L(X(X )) Z n 2Ber( X X) Theorem ω(x) L(X) = ω(x ) L(X(X )) Z n 2Ber( X X) U U N. Poncin Higher SG University of Luxembourg 38 / 42

71 -Supergeometry III: integration theory Integral of a distributional Zn 2 -Berezinian section Z 2 2-integral III : distributional Berezinian section supported in a Z 2 2-domain X = (x, y, ξ, η) = ω(x) l Nf l (X)δ (l) (y) = (d xd y ξ η ) l N ( ) 1 1 f kab;l (x)y k ξ a η b δ (l) (y) k=0 a=0 b=0 Definition = d x d y l N ( ) f k11;l (x)y k δ (l) (y) = k=0 d x d y l N( 1) ( ) l y l f k11;l (x)y k δ(y) = k=0 d x l N( 1) l l!f l11;l (x) R N. Poncin Higher SG University of Luxembourg 39 / 42

72 -Supergeometry III: integration theory Integral of a distributional Zn 2 -Berezinian section ω(x) f l (X)δ (l) (y) =? l N ω(x ) Z n 2Ber( Φ) Φ l N f l (X)δ (l) (y) N. Poncin Higher SG University of Luxembourg 40 / 42

73 -Supergeometry III: integration theory Integral of a distributional Zn 2 -Berezinian section ω(x) f l (X)δ (l) (y) =? l N ω(x ) Z n 2Ber( Φ) Φ l N f l (X)δ (l) (y) = ω(x ) Z n 2Ber( (Φ 2 Φ 1 )) (Φ 2 Φ 1 ) l N f l (X)δ (l) (y) N. Poncin Higher SG University of Luxembourg 40 / 42

74 -Supergeometry III: integration theory Integral of a distributional Zn 2 -Berezinian section ω(x) f l (X)δ (l) (y) =? l N ω(x ) Z n 2Ber( Φ) Φ l N f l (X)δ (l) (y) = ω(x ) Z n 2Ber( (Φ 2 Φ 1 )) (Φ 2 Φ 1 ) l N f l (X)δ (l) (y) = ω(x ) Z n 2Ber( Φ 2 ) Z n 2Ber( Φ 1 ) (Φ 2 Φ 1 ) l N f l (X)δ (l) (y) N. Poncin Higher SG University of Luxembourg 40 / 42

75 -Supergeometry III: integration theory Integral of a distributional Zn 2 -Berezinian section ω(x) f l (X)δ (l) (y) =? l N ω(x ) Z n 2Ber( Φ) Φ l N f l (X)δ (l) (y) = ω(x ) Z n 2Ber( (Φ 2 Φ 1 )) (Φ 2 Φ 1 ) l N f l (X)δ (l) (y) = ω(x ) Z n 2Ber( Φ 2 ) Z n 2Ber( Φ 1 ) (Φ 2 Φ 1 ) l N f l (X)δ (l) (y) = ω(x ) Z n 2Ber( Φ 2 ) Z n 2Ber( Φ 1 ) Φ 1 Φ 2 f l (X)δ (l) (y) l N N. Poncin Higher SG University of Luxembourg 40 / 42

76 OUTLOOK OUTLOOK N. Poncin Higher SG University of Luxembourg 41 / 42

77 OUTLOOK Current and upcoming work Z n 2 -differential calculus arxiv: Z n 2 -versions of inverse & implicit function, constant rank, Frobenius arxiv: Z n 2 -integral calculus ORBilu: Categorical Z n 2 -Geometry and Molotkov s work Functional analytic issues in Z n 2 -Geometry Applications in Physics N. Poncin Higher SG University of Luxembourg 41 / 42

78 O UTLOOK T HANK YOU FOR YOUR ATTENTION N. Poncin Higher SG University of Luxembourg 42 / 42