Contact manifolds Anna Fino Contact manifolds and SU ( 2 ) -structures in 5-dimensions SU ( n ) -structures Sasaki-Einstein structures Hypo structures Hypo evolution equations η -Einstein structures Hypo-contact structures Classification “Holonomy Groups and Applications in String Theory”, New metrics with holonomy SU ( 3 ) Hamburg – 14 - 18 July 2008 Sasakian structures Link with half-flat structures From hypo to half-flat From half-flat to hypo New metrics with holonomy G 2 SU ( n ) -structures in ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures Examples Contact reduction Anna Fino 1 Dipartimento di Matematica Università di Torino
Contact manifolds SU ( 2 ) -structures in 5 -dimensions Anna Fino SU ( 2 ) -structures in Definition 5-dimensions Sasaki-Einstein structures An SU ( 2 ) -structure ( η, ω 1 , ω 2 , ω 3 ) on N 5 is given by a 1-form η Hypo structures Hypo evolution equations and by three 2-forms ω i such that η -Einstein structures Hypo-contact structures ω i ∧ ω j = δ ij v , v ∧ η � = 0 , Classification i X ω 3 = i Y ω 1 ⇒ ω 2 ( X , Y ) ≥ 0 , New metrics with holonomy SU ( 3 ) Sasakian structures where i X denotes the contraction by X . Link with half-flat structures From hypo to half-flat From half-flat to hypo Remark New metrics with holonomy G 2 The pair ( η, ω 3 ) defines a U ( 2 ) -structure or an almost contact SU ( n ) -structures in metric structure on N 5 , i.e. ( η, ξ, ϕ, g ) such that ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures ϕ 2 = − Id + ξ ⊗ η, η ( ξ ) = 1 , Examples Contact reduction g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . 2
Contact manifolds SU ( 2 ) -structures in 5 -dimensions Anna Fino SU ( 2 ) -structures in Definition 5-dimensions Sasaki-Einstein structures An SU ( 2 ) -structure ( η, ω 1 , ω 2 , ω 3 ) on N 5 is given by a 1-form η Hypo structures Hypo evolution equations and by three 2-forms ω i such that η -Einstein structures Hypo-contact structures ω i ∧ ω j = δ ij v , v ∧ η � = 0 , Classification i X ω 3 = i Y ω 1 ⇒ ω 2 ( X , Y ) ≥ 0 , New metrics with holonomy SU ( 3 ) Sasakian structures where i X denotes the contraction by X . Link with half-flat structures From hypo to half-flat From half-flat to hypo Remark New metrics with holonomy G 2 The pair ( η, ω 3 ) defines a U ( 2 ) -structure or an almost contact SU ( n ) -structures in metric structure on N 5 , i.e. ( η, ξ, ϕ, g ) such that ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures ϕ 2 = − Id + ξ ⊗ η, η ( ξ ) = 1 , Examples Contact reduction g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . 2
Contact manifolds SU ( 2 ) -structures in 5 -dimensions Anna Fino SU ( 2 ) -structures in Definition 5-dimensions Sasaki-Einstein structures An SU ( 2 ) -structure ( η, ω 1 , ω 2 , ω 3 ) on N 5 is given by a 1-form η Hypo structures Hypo evolution equations and by three 2-forms ω i such that η -Einstein structures Hypo-contact structures ω i ∧ ω j = δ ij v , v ∧ η � = 0 , Classification i X ω 3 = i Y ω 1 ⇒ ω 2 ( X , Y ) ≥ 0 , New metrics with holonomy SU ( 3 ) Sasakian structures where i X denotes the contraction by X . Link with half-flat structures From hypo to half-flat From half-flat to hypo Remark New metrics with holonomy G 2 The pair ( η, ω 3 ) defines a U ( 2 ) -structure or an almost contact SU ( n ) -structures in metric structure on N 5 , i.e. ( η, ξ, ϕ, g ) such that ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures ϕ 2 = − Id + ξ ⊗ η, η ( ξ ) = 1 , Examples Contact reduction g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . 2
Contact manifolds Sasaki-Einstein structures Anna Fino Example SU ( 2 ) -structures in Sasaki-Einstein structure 5-dimensions Sasaki-Einstein structures Hypo structures d η = − 2 ω 3 , d ω 1 = 3 η ∧ ω 2 , d ω 2 = − 3 η ∧ ω 1 . Hypo evolution equations η -Einstein structures Hypo-contact On S 2 × S 3 there exist an infinite family of explicit structures Classification Sasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram, New metrics with holonomy SU ( 3 ) ...]. Sasakian structures Link with half-flat structures Definition (Boyer, Galicki) From hypo to half-flat From half-flat to hypo g = dr 2 + r 2 g ( N 2 n + 1 , g , η ) is Sasaki-Einstein if the conic metric ˜ New metrics with holonomy G 2 on the symplectic cone N 5 × R + is Kähler and Ricci- flat (CY). SU ( n ) -structures in ( 2 n + 1 ) -dimensions • N 2 n + 1 × R + has an integrable SU ( n + 1 ) -structure, i.e. an Generalized Killing spinors Contact SU ( n ) -structures Hermitian structure ( J , ˜ g ) , with F = d ( r 2 η ) , and a Examples Contact reduction ( n + 1 , 0 ) -form Ψ = Ψ + + i Ψ − of lenght 1 such that dF = d Ψ = 0 ⇒ ˜ g has holonomy in SU ( n + 1 ) . 3 • N 2 n + 1 has a real Killing spinor, i.e. the restriction of a parallel spinor on the Riemannian cone [Friedrich, Kath].
Contact manifolds Sasaki-Einstein structures Anna Fino Example SU ( 2 ) -structures in Sasaki-Einstein structure 5-dimensions Sasaki-Einstein structures Hypo structures d η = − 2 ω 3 , d ω 1 = 3 η ∧ ω 2 , d ω 2 = − 3 η ∧ ω 1 . Hypo evolution equations η -Einstein structures Hypo-contact On S 2 × S 3 there exist an infinite family of explicit structures Classification Sasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram, New metrics with holonomy SU ( 3 ) ...]. Sasakian structures Link with half-flat structures Definition (Boyer, Galicki) From hypo to half-flat From half-flat to hypo g = dr 2 + r 2 g ( N 2 n + 1 , g , η ) is Sasaki-Einstein if the conic metric ˜ New metrics with holonomy G 2 on the symplectic cone N 5 × R + is Kähler and Ricci- flat (CY). SU ( n ) -structures in ( 2 n + 1 ) -dimensions • N 2 n + 1 × R + has an integrable SU ( n + 1 ) -structure, i.e. an Generalized Killing spinors Contact SU ( n ) -structures Hermitian structure ( J , ˜ g ) , with F = d ( r 2 η ) , and a Examples Contact reduction ( n + 1 , 0 ) -form Ψ = Ψ + + i Ψ − of lenght 1 such that dF = d Ψ = 0 ⇒ ˜ g has holonomy in SU ( n + 1 ) . 3 • N 2 n + 1 has a real Killing spinor, i.e. the restriction of a parallel spinor on the Riemannian cone [Friedrich, Kath].
Contact manifolds Sasaki-Einstein structures Anna Fino Example SU ( 2 ) -structures in Sasaki-Einstein structure 5-dimensions Sasaki-Einstein structures Hypo structures d η = − 2 ω 3 , d ω 1 = 3 η ∧ ω 2 , d ω 2 = − 3 η ∧ ω 1 . Hypo evolution equations η -Einstein structures Hypo-contact On S 2 × S 3 there exist an infinite family of explicit structures Classification Sasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram, New metrics with holonomy SU ( 3 ) ...]. Sasakian structures Link with half-flat structures Definition (Boyer, Galicki) From hypo to half-flat From half-flat to hypo g = dr 2 + r 2 g ( N 2 n + 1 , g , η ) is Sasaki-Einstein if the conic metric ˜ New metrics with holonomy G 2 on the symplectic cone N 5 × R + is Kähler and Ricci- flat (CY). SU ( n ) -structures in ( 2 n + 1 ) -dimensions • N 2 n + 1 × R + has an integrable SU ( n + 1 ) -structure, i.e. an Generalized Killing spinors Contact SU ( n ) -structures Hermitian structure ( J , ˜ g ) , with F = d ( r 2 η ) , and a Examples Contact reduction ( n + 1 , 0 ) -form Ψ = Ψ + + i Ψ − of lenght 1 such that dF = d Ψ = 0 ⇒ ˜ g has holonomy in SU ( n + 1 ) . 3 • N 2 n + 1 has a real Killing spinor, i.e. the restriction of a parallel spinor on the Riemannian cone [Friedrich, Kath].
Contact manifolds Sasaki-Einstein structures Anna Fino Example SU ( 2 ) -structures in Sasaki-Einstein structure 5-dimensions Sasaki-Einstein structures Hypo structures d η = − 2 ω 3 , d ω 1 = 3 η ∧ ω 2 , d ω 2 = − 3 η ∧ ω 1 . Hypo evolution equations η -Einstein structures Hypo-contact On S 2 × S 3 there exist an infinite family of explicit structures Classification Sasaki-Einstein metrics [Gauntlett, Martelli, Sparks, Waldram, New metrics with holonomy SU ( 3 ) ...]. Sasakian structures Link with half-flat structures Definition (Boyer, Galicki) From hypo to half-flat From half-flat to hypo g = dr 2 + r 2 g ( N 2 n + 1 , g , η ) is Sasaki-Einstein if the conic metric ˜ New metrics with holonomy G 2 on the symplectic cone N 5 × R + is Kähler and Ricci- flat (CY). SU ( n ) -structures in ( 2 n + 1 ) -dimensions • N 2 n + 1 × R + has an integrable SU ( n + 1 ) -structure, i.e. an Generalized Killing spinors Contact SU ( n ) -structures Hermitian structure ( J , ˜ g ) , with F = d ( r 2 η ) , and a Examples Contact reduction ( n + 1 , 0 ) -form Ψ = Ψ + + i Ψ − of lenght 1 such that dF = d Ψ = 0 ⇒ ˜ g has holonomy in SU ( n + 1 ) . 3 • N 2 n + 1 has a real Killing spinor, i.e. the restriction of a parallel spinor on the Riemannian cone [Friedrich, Kath].
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