On holonomy of supermanifolds arXiv:math/0703679 v3 1 ( Z 2 = { 0 , - PDF document

Anton Galaev (Masaryk University, Brno, Czech Republic) On holonomy of supermanifolds arXiv:math/0703679 v3 1 ( Z 2 = { ¯ 0 , ¯ Vector superspace : V = V ¯ 0 ⊕ V ¯ 1 } ) Homogeneous elements: x ∈ V ¯ 0 ∪ V ¯ 1 0 is called even, | x | = ¯ x ∈ V ¯ 0; 1 \{ 0 } is called odd, | x | = ¯ x ∈ V ¯ 1; V and W are vector superspaces ⇒ V ⊗ W and Hom( V, W ) are vector superspaces: ( V ⊗ W ) ¯ 0 = ( V ¯ 0 ⊗ W ¯ 0 ) ⊕ ( V ¯ 1 ⊗ W ¯ 1 ) ( V ⊗ W ) ¯ 1 = ( V ¯ 0 ⊗ W ¯ 1 ) ⊕ ( V ¯ 1 ⊗ W ¯ 0 ) 0 ) ⊕ Hom( V ¯ Hom( V, W ) ¯ 0 = Hom( V ¯ 0 , W ¯ 1 , W ¯ 1 ) � = { f ∈ Hom( V, W ) | f ( x ) | = | x |} (morphisms) � Hom( V, W ) ¯ 1 = Hom( V ¯ 0 , W ¯ 1 ) ⊕ Hom( V ¯ 1 , W ¯ 0 ) | f ( x ) | = | x | + ¯ � = { f ∈ Hom( V, W ) 1 , x � = 0 } � 1

0 ⊕ A ¯ 1 , · : A ⊗ A → A , | xy | = | x | + | y | Superalgebra: A = A ¯ A is called commutative if xy = ( − 1) | x || y | yx Example. The Grassmann superalgebra i =0 Λ i R n = Λ even ⊕ Λ odd is commutative Λ( n ) = ⊕ n Lie superalgebra: g = g ¯ 0 ⊕ g ¯ 1 , [ · , · ] : g ⊗ g → g , | [ x, y ] | = | x | + | y | 1) [ x, y ] = ( − 1) | x || y | [ y, x ] 2) [[ x, y ] , z ] + ( − 1) | x | ( | y | + | z | ) [[ y, z ] , x ] + ( − 1) | z | ( | x | + | y | ) [[ z, x ] , y ] = 0 ⇒ g ¯ 0 is a Lie algebra and g ¯ 1 is a g ¯ 0 -module gl ( n | m, K ) = { ( A B C D ) } Example. K = R or C 0 = { ( A 0 gl ( n | m, K ) ¯ 0 D ) } ≃ gl ( n, K ) ⊕ gl ( m, K ) C 0 ) } ≃ ( K n ⊗ ( K m ) ∗ ) ⊕ (( K n ) ∗ ⊗ K m ) 1 = { ( 0 B gl ( n | m ) ¯ [ X, Y ] = XY − ( − 1) | X || Y | Y X 2

Supermanifold: M n | m = ( M, O M ) M is a smooth n -dim. manifold, O M is a sheaf of superalgebras over R such that locally O M ( U ) ≃ O M ( U ) ⊗ Λ( m ) ( x i ) ( i = 1 , ..., n ) coordinates on M , ( ξ α ) ( α = 1 , ..., m ) a basis of R m ⇒ ( x i , ξ α ) = ( x a ) are called coordinates on M (put x n + α = ξ α and assume a = 1 , ..., n + m ) f ∈ O M ( U ) ⇒ m f α 1 ...α r ξ α 1 · · · ξ α r , f = ˜ � � ˜ f + f, f α 1 ...α r ∈ O M ( U ) r =1 α 1 < ··· <α r f ( x ) := ˜ x ∈ U ⇒ f ( x ) ⇒ f is not determined by its values at all points of U !!! The tangent sheaf: T M = ( T M ) ¯ 0 ⊕ ( T M ) ¯ 1 ,  �  | X | = ¯ � i, X is R -linear    �  ( T M ) ¯ X : O M ( U ) → O M ( U ) i ( U ) = � � X ( fg ) = X ( f ) g + ( − 1) | f || g | fX ( g )  �    � The vector fields ∂ i = ∂ x i , ∂ α = ∂ ξ α form a local basis of T M ( U ) ⇒ T M is a locally free sheaf of supermodules over O M Example. E → M a vector bundle ⇒ O M ( U ) := Λ(Γ( U, E )) defines a supermanifold M . 3

Let E be a locally free sheaf of supermodules over O M of rank p | q . x ∈ M consider the fiber at x : E x := E ( U ) / ( O M ( U )) x E ( U ), where x ∈ U and ( O M ( U )) x ⊂ O M ( U ) are functions vanishing at x . For X ∈ E ( U ) consider the value X x ∈ E x Example. E = T M ⇒ ( T M ) x = T x M and ( T x M ) ¯ 0 = T x M Consider the vector bundle E = ∪ x ∈ M E x → M . X �→ ˜ ˜ We get the projection ∼ : E ( U ) → Γ( U, E ) , X, X x = X x Let ( e A ) A = 1 , ..., p + q be a basis of E ( U ) X ∈ E ( U ) ⇒ X = X A e A ( X A ∈ O M ( U )) ⇒ ˜ X = ˜ X A ˜ e A Connection on E : ∇ : T M ⊗ R E → E |∇ X Y | = | X | + | Y | , ∇ Y fX = ( Y f ) X + ( − 1) | Y || f | f ∇ Y X ∇ fY X = f ∇ Y X and Locally: ∇ ∂ a e B = Γ A Γ A aB e A , aB ∈ O M ( U ) ∇ = ( ∇| Γ( TM ) ⊗ Γ( E ) ) ∼ : Γ( TM ) ⊗ Γ( E ) → Γ( E ) is a connection on E ˜ ˜ iB are Cristoffel symbols of ˜ Γ A ∇ γ : [ a, b ] ⊂ R → M τ γ : E γ ( a ) → E γ ( b ) the parallel displacement along γ . τ γ : E γ ( a ) → E γ ( b ) is an isomorphism of vector superspaces. Problem: Define holonomy of ∇ (it must give information about all parallel sections of E !) 4

Parallel sections X ∈ E ( M ) is called parallel if ∇ X = 0 ∇ X = 0 ⇒ ˜ ∇ ˜ X = 0 ( � !!!) Locally:  ∂ i X A + X B Γ A iB = 0 ,   ∇ X = 0 ⇔ ∂ γ X A + ( − 1) | X B | X B Γ A γB = 0    ( ∂ γ r ...∂ γ 1 ( ∂ i X A + X B Γ A iB )) ∼ = 0 , ( ∗ )   ⇔ r = 0 , ..., m ( ∂ γ r ...∂ γ 1 ( ∂ γ X A + ( − 1) | X B | X B Γ A γB )) ∼ = 0 ( ∗∗ )   X A + ˜ ∇ ˜ ˜ X = 0 ⇔ ∂ i ˜ X B ˜ Γ A iB = 0 Prop. A parallel section X ∈ E ( M ) is uniquely defined by its value at any point x ∈ M . Proof. ∇ X = 0 ⇒ ˜ ∇ ˜ X = 0; ˜ X x = X x uniquely determine ˜ X , i.e. we know the functions ˜ X A . γ = − ˜ X B ˜ Further, use ( ∗∗ ): X A Γ A γB , γγ 1 = − ˜ γ 1 ˜ X A X B Γ A γBγ 1 + X B Γ A γB ... ⇒ we know the functions X A . � 5

Def. (holonomy algebra) hol ( ∇ ) x := � � � � r ≥ 0 , Y, Z, Y i ∈ T y M � ◦ ¯ τ − 1 ∇ r Y r ,...,Y 1 R y ( Y, Z ) ◦ τ γ ⊂ gl ( E x ) ≃ gl ( p | q, R ) � γ � ¯ ∇ : connect on T M | U � � Note: hol ( ˜ ∇ ) x ⊂ ( hol ( ∇ ) x ) ¯ ( � = !) 0 Lie supergroup G = ( G, O G ) is a group object in the category of super- manifolds; G is uniquely given by the Harish-Chandra pair ( G, g ), where g = g ¯ 0 ⊕ g ¯ 1 is a Lie superalgebra, g ¯ 0 is the Lie algebra of G . Denote by Hol( ∇ ) 0 x the connected Lie subgroup of GL(( E x ) ¯ 0 ) × GL(( E x ) ¯ 1 ) corresponding to ( hol ( ∇ ) x ) ¯ 0 ⊂ gl (( E x ) ¯ 0 ) ⊕ gl (( E x ) ¯ 1 ) ⊂ gl ( E x ); x · Hol( ˜ Hol( ∇ ) x := Hol( ∇ ) 0 ∇ ) x ⊂ GL(( E x ) ¯ 0 ) × GL(( E x ) ¯ 1 ). Def. Holonomy group: H ol ( ∇ ) x := (Hol( ∇ ) x , hol ( ∇ ) x ); the restricted holonomy group: H ol ( ∇ ) 0 x := (Hol( ∇ ) 0 x , hol ( ∇ ) x ). Def. (infinitesimal holonomy algebra) hol ( ∇ ) inf := x ◦ ¯ < τ − 1 ∇ r Y r ,...,Y 1 R x ( Y, Z ) ◦ τ γ | r ≥ 0 , Y, Z, Y 1 , ..., Y r ∈ T x M > ⊂ hol ( ∇ ) x γ Theorem. If M , E and ∇ are analytic, then hol ( ∇ ) x = hol ( ∇ ) inf x . 6

Theorem.   X x ∈ E x annihilated by hol ( ∇ ) x     { X ∈ E ( M ) , ∇ X = 0 } ← → and preserved by Hol( ˜ ∇ ) x     ¯ ∇ r Proof. − → : ∇ X = 0 ⇒ Y r ,...,Y 1 R ( Y, Z ) X = 0 ∇ ˜ ˜ X is preserved by Hol( ˜ ˜ ∇ X = 0 ⇒ ⇒ ∇ ) x X = 0 ¯ ∇ r = ⇒ Y r ,...,Y 1 R y ( Y, Z ) ◦ τ γ X x = 0 ⇒ X x is annihilated by hol ( ∇ ) x ← − : Hol( ˜ ⇒ ∃ X 0 ∈ Γ( E ) , ˜ ∇ ) x preserves X x ∈ E x = ∇ X 0 = 0 , ( X 0 ) x = X x X 0 = X A e A , X A 0 ˜ 0 ∈ O M ( U ) ( ∗∗ ) defines X A γγ 1 ...γ r ∈ O M ( U ) for all γ < γ 1 < · · · < γ r , 0 ≤ r ≤ m − 1. We get X A ∈ O M ( U ) , consider X = X A e A ∈ E ( U ). Claim: ∇ X = 0 . To prove (by induction over r ): X A satisfy ( ∗ ) and ( ∗∗ ) for all γ 1 < · · · < γ r , 0 ≤ r ≤ m ( ∂ γ r ...∂ γ 1 ( ∂ i X A + X B Γ A iB )) ∼ = ( ∂ γ r ...∂ γ 2 (( − 1) ( | A | + | B | ) | X B | R A Bγ 1 i X B )) ∼ = ( ∂ γ r ...∂ γ 3 (( − 1) ( | A | + | B | ) | X B | ¯ ∇ γ 2 R A Bγ 1 i X B )) ∼ = · · · = (( − 1) ( | A | + | B | ) | X B | ¯ Bγ 1 i X B ) ∼ = 0 , ∇ r − 1 γ r ,...,γ 2 R A this proves ( ∗ ) 7

Parallel subsheaves A subsheaf F ⊂ E of O M -supermodules is called a locally direct if locally there exists a basis of E ( U ) some elements of which form a basis of F ( U ) A distribution on M is a locally direct subsheaf of T M F ⊂ E is parallel if ∇ Y X ∈ F ( U ) for all Y ∈ T M ( U ) and X ∈ F ( U ) Theorem. { parallel locally direct subsheaves F ⊂ E of rank p 1 | q 1 } {F x ⊂ E x of dimension p 1 | q 1 preserved by hol ( ∇ ) x and Hol( ˜ ← → ∇ ) x } 8

Linear connections ∇ a connection on E = T M , E = ∪ y ∈ M T y M = T M , E ¯ 0 = TM Hol( ˜ hol ( ∇ ) ⊂ gl ( n | m, R ) , ∇ ) ⊂ GL( n, R ) × GL( m, R ) Theorem.     A x ∈ T p,q x M annihilated by hol ( ∇ ) x Parallel tensor fields         ← → and preserved by Hol( ˜ of type ( p, q ) on M ∇ ) x         Let g be a bilinear form on a vector superspace V . g is even if g ( V ¯ 0 , V ¯ 1 ) = g ( V ¯ 1 , V ¯ 0 ) = 0 g is odd if g ( V ¯ 0 , V ¯ 0 ) = g ( V ¯ 1 , V ¯ 1 ) = 0 g is supersymmetric if g ( x, y ) = ( − 1) | x || y | g ( y, x ) g is super skew-symmetric if g ( x, y ) = − ( − 1) | x || y | g ( y, x ) Example. Let g be non-degenerate even and supersymmetric ⇒ g | V ¯ 0 is a usual non-degenerate symmetric bilinear form (of sign. ( p, q )) 0 × V ¯ and g | V ¯ 1 is a usual non-degenerate skew-symmetric bilinear form 1 × V ¯ so ( p, q | 2 k, R ) is a subalgebra of gl ( p + q | 2 k, R ) preserving g A B 1 B 2 �� � � � � A ∈ so ( p, q ) , C t 2 = C 2 , C t − B t so ( p, q | 2 k, R ) = 3 = C 3 2 C 1 C 2 � B t C 3 − C t 1 1 1 ≃ R p + q ⊗ R 2 k so ( p, q | 2 k, R ) ¯ 0 ≃ so ( p, q ) ⊕ sp (2 k, R ) , so ( p, q | 2 k, R ) ¯ 9