Holonomy of supermanifolds Anton Galaev Masaryk University (Brno, - PowerPoint PPT Presentation

Holonomy of supermanifolds Anton Galaev Masaryk University (Brno, Czech Republic) Anton Galaev Holonomy of supermanifolds

The case of smooth manifolds Let E → M be a vector bundle over a smooth manifold M , ∇ a connection on E . γ : [ a , b ] → M a curve in M τ γ : E γ ( a ) → E γ ( b ) the parallel transport along γ x ∈ M , τ pt x = id E x , τ γ⋆µ = τ µ ◦ τ γ , τ γ − 1 = ( τ γ ) − 1 . Anton Galaev Holonomy of supermanifolds

The holonomy group at the point x : � Hol x ( ∇ ) := { τ γ � γ is a loop at x } ⊂ GL ( E x ) ≃ GL ( m , R ) . � The restricted holonomy group at the point x : � Hol 0 x ( ∇ ) := { τ γ � γ is a loop at x , γ ∼ pt x } ⊂ Hol x ( ∇ ) . � Fact: Hol x ( ∇ ) ⊂ GL ( E x ) is a Lie subgroup, Hol 0 x ( ∇ ) is the identity component of Hol x ( ∇ ) . The holonomy algebra at the point x : hol x ( ∇ ) := LA Hol x ( ∇ ) = LA Hol 0 x ( ∇ ) ⊂ gl ( E x ) ≃ gl ( m , R ) . Anton Galaev Holonomy of supermanifolds

Theorem. (Ambrose, Singer, 1952) � hol x ( ∇ ) = { ( τ γ ) − 1 ◦ R γ ( b ) ( τ γ ( X ) , τ γ ( Y )) ◦ τ γ � γ ( a ) = x , X , Y ∈ T x M } . � Anton Galaev Holonomy of supermanifolds

The fundamental principle: { parallel sections X ∈ Γ( E ) } ← → { X x ∈ E x | Hol x X x = X x } ( X ∈ Γ( E ) is parallel if ∇ X = 0, or for any γ : [ a , b ] → M , τ g ammaX γ ( a ) = X γ ( a ) ) Anton Galaev Holonomy of supermanifolds

Holonomy of Riemannian manifolds ∇ = ∇ g , ( M , g ) , E = TM , Hol ( ∇ ) ⊂ O ( n ), hol ( ∇ ) ⊂ so ( n ) Consider two Riemannian manifolds ( M , g ) , ( N , h ) , then ( M × N , g + h ) is also a Riemannian manifold and Hol ( M × N ) = Hol ( M ) × Hol ( N ) . Conversely: Theorem (De Rham) If ( M , g ) is complete and simply connected, then M = N 0 × N 1 × · · · × N r , Hol ( M ) = { id } × Hol ( N 1 ) × · · · × Hol ( N r ) , Hol ( N i ) are irreducible. In general exists local decomposition of ( M , g ) . Anton Galaev Holonomy of supermanifolds

If ( M , g ) is an indecomposable simply connected symmetric Riemannian space, then M = G / H , where G is the group of transvections, then Hol coincides with the isotropy representation of H . Simply connected symmetric Riemannian spaces are classified, hence all possible Hol are known. Anton Galaev Holonomy of supermanifolds

Connected irreducible holonomy groups of non-locally symmetric Riemannian manifolds (M. Berger 1953): � n � n � n � n � � � � SO ( n ) , U , SU , Sp , Sp · Sp (1) , 2 2 4 4 Spin(7) ⊂ SO (8) , G 2 ⊂ SO (7) . Anton Galaev Holonomy of supermanifolds

Special geometries: SO ( n ): ”general” Riemannian manifolds; U ( n 2 ): K¨ ahlerian manifolds; SU ( n 2 ): Calabi-Yau manifolds or special K¨ ahlerian manifolds, Ric = 0, parallel spinors; Sp ( n 4 ): hyper-K¨ ahlerian manifolds, Ric = 0, parallel spinors; Sp ( n 4 ) · Sp (1): quaternionic-K¨ ahlerian manifolds, Einstein; Spin (7): 8-dimensional manifolds with a parallel 4-form, Ric = 0, parallel spinors; G 2 : 7-dimensional manifolds with a parallel 3-form, Ric = 0, parallel spinors. Anton Galaev Holonomy of supermanifolds

Supermanifolds 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 Anton Galaev Holonomy of supermanifolds

Let E be a locally free sheaf of supermodules over O M of rank p | q . 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 X ∈ E ( U ) is not defined by its values! Anton Galaev Holonomy of supermanifolds

Connection on E : ∇ : T M ⊗ R E → E |∇ ξ X | = | ξ | + | X | , ∇ ξ fX = ( ξ f ) X + ( − 1) | ξ || f | f ∇ ξ X ∇ f ξ X = f ∇ ξ X and Locally: ∇ ∂ a e B = Γ A Γ A aB ∈ O M ( U ) aB e A , ∇ = ( ∇| 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 displac. along γ (defined by ˜ ∇ ). τ γ : E γ ( a ) → E γ ( b ) is an isomorphism of vector superspaces. Anton Galaev Holonomy of supermanifolds

Problem: Define holonomy of ∇ (it must give information about all parallel sections of E !) Anton Galaev Holonomy of supermanifolds

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 = 0 γ B ( ∂ 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 γ B )) ∼ = 0 ( ∗∗ ) X A + ˜ ∇ ˜ ˜ X = 0 ⇔ ∂ i ˜ X B ˜ Γ A iB = 0 Anton Galaev Holonomy of supermanifolds

Proposition. 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 γ B , γγ 1 = − ˜ γ 1 ˜ X A X B Γ A γ B γ 1 + X B Γ A γ B ... ⇒ we know the functions X A . � Anton Galaev Holonomy of supermanifolds

Definition (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 ) � ¯ γ ∇ : connect on T M | U � Note: hol ( ˜ ∇ ) x ⊂ ( hol ( ∇ ) x ) ¯ ( � = !) 0 Anton Galaev Holonomy of supermanifolds

Lie supergroup G = ( G , O G ) is a group object in the category of supermanifolds; 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 ); Hol ( ∇ ) x := Hol ( ∇ ) 0 x · Hol ( ˜ ∇ ) 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 ). Anton Galaev Holonomy of supermanifolds

Definition (Infinitesimal holonomy algebra). hol ( ∇ ) inf := x < ¯ ∇ r Y r ,..., Y 1 R x ( Y , Z ) | r ≥ 0 , Y , Z , Y 1 , ..., Y r ∈ T x M > ⊂ hol ( ∇ ) x Theorem. If M , E and ∇ are analytic, then hol ( ∇ ) x = hol ( ∇ ) inf x . Anton Galaev Holonomy of supermanifolds

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 Anton Galaev Holonomy of supermanifolds

← − : Hol ( ˜ ∇ ) x preserves X x ∈ E x ⇒ ∃ X 0 ∈ Γ( E ) , ˜ = ∇ X 0 = 0 , ( X 0 ) x = X x X 0 = X A e A , X A 0 ∈ O M ( U ) 0 ˜ ( ∗∗ ) 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 iB )) ∼ = ( ∂ γ r ...∂ γ 2 (( − 1) ( | A | + | B | ) | X B | R A ( ∂ γ r ...∂ γ 1 ( ∂ i X A + X B Γ A B γ 1 i X B )) ∼ = ( ∂ γ r ...∂ γ 3 (( − 1) ( | A | + | B | ) | X B | ¯ ∇ γ 2 R A B γ 1 i X B )) ∼ = · · · = (( − 1) ( | A | + | B | ) | X B | ¯ B γ 1 i X B ) ∼ = 0 , ∇ r − 1 γ r ,...,γ 2 R A this proves ( ∗ ) Anton Galaev Holonomy of supermanifolds

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. � Parallel tensor fields � A x ∈ T p , q � M annihilated by hol ( ∇ ) x � x ← → and preserved by Hol ( ˜ of type ( p , q ) on M ∇ ) x Anton Galaev Holonomy of supermanifolds

Examples of parallel structures on ( M , ∇ ) Hol ( ˜ parallel structure on M hol ( ∇ ) is ∇ ) is contained in contained in complex structure gl ( k | l , C ) GL ( k , C ) × GL ( l , C ) �� A 0 �� � odd complex structure, q ( n , R ) � A ∈ GL ( n , R ) 0 A i.e. odd automorphism (queer Lie J of T M with J 2 = − id superalgebra) Riemannian supermetric, osp ( p 0 , q 0 | 2 k ) O ( p 0 , q 0 ) × Sp (2 k , R ) i.e. even non-degenerate supersymmetric metric osp sk (2 k | p , q ) even non-degenerate Sp (2 k , R ) × O ( p , q ) super skew-symmetr. metric �� A 0 �� � A ∈ GL ( n , R ) � odd non-degenerate pe ( n , R ) 0 A supersymmetric metric (periplectic Lie superalgebra) �� A 0 pe sk ( n , R ) �� � odd non-degenerate super � A ∈ GL ( n , R ) 0 A skew-symmetric metric Anton Galaev Holonomy of supermanifolds