Special geometry Simon G. Chiossi Special geometry with solvable - PowerPoint PPT Presentation

Special geometry with solvable Lie groups Special geometry Simon G. Chiossi Special geometry with solvable Lie groups Lie groups’ actions Six dimensions Seven dimensions Nilpotent/Solvable Lie groups Nilmanifolds Holonomy Groups and Applications in String Theory – Prototypical example Solvmanifolds Universität Hamburg, July 2008 Non-compact homogeneous Einstein spaces Half-flatness Geometry with torsion Spinors Strings attached ‘Simultaneous’ structures End Simon G. Chiossi Polytechnic of Turin 1

Special geometry with solvable Lie groups Simon G. Chiossi Under the auspices of a famous scientist from Hamburg Special geometry Lie groups’ actions Six dimensions Seven dimensions Nilpotent/Solvable Lie groups Nilmanifolds Prototypical example Solvmanifolds Non-compact homogeneous Einstein spaces Half-flatness Geometry with torsion Spinors Strings attached ‘Simultaneous’ structures End 2

Special geometry with solvable Lie groups Special geometry 1 Simon G. Chiossi Lie groups’ actions Six dimensions Special geometry Seven dimensions Lie groups’ actions Six dimensions Seven dimensions Nilpotent/Solvable Lie Nilpotent/Solvable Lie groups 2 groups Nilmanifolds Nilmanifolds Prototypical example Prototypical example Solvmanifolds Non-compact Solvmanifolds homogeneous Einstein spaces Non-compact homogeneous Einstein spaces Half-flatness Half-flatness Geometry with torsion Spinors Strings attached ‘Simultaneous’ structures 3 Geometry with torsion End Spinors Strings attached ‘Simultaneous’ structures End 4 3

Special geometry Holonomy groups of Ricci-flat metrics with solvable Lie groups Simon G. Chiossi Special geometry dim 6 7 8 Lie groups’ actions group SU ( 3 ) G 2 Spin ( 7 ) Six dimensions Seven dimensions Nilpotent/Solvable Lie groups local examples: easy(-ish) to find Nilmanifolds Prototypical example complete/compact examples: harder, but fortunately the Solvmanifolds explicit knowledge of the metric is often unnecessary Non-compact homogeneous Einstein spaces Half-flatness In dims 6, 7, 8 interesting structures are determined by Geometry with torsion differential forms lying in open orbits under the action of Spinors Strings attached GL ( n , R ) ‘Simultaneous’ structures For instance, in the intermediate dimension a certain 3-form End determines the whole geometry Maximal subgroups of G 2 : SO ( 3 ) , SO ( 4 ) , SU ( 3 ) And G 2 ⊂ SO ( 8 ) � Spin ( 7 ) -, PSU ( 3 ) -, Sp ( 2 ) Sp ( 1 ) -geometry. 4

Special geometry G-structures with solvable Lie groups Simon G. Chiossi Special geometry Spin ( 6 ) = SU ( 4 ) acts transitively on S 7 Lie groups’ actions Six dimensions Seven dimensions SU ( 3 ) = SO ( 7 ) SO ( 6 ) = SO ( 8 ) Nilpotent/Solvable Lie Spin ( 7 ) = RP 7 groups G 2 Nilmanifolds Prototypical example Solvmanifolds Non-compact Different sets of reductions are parametrised by the same homogeneous Einstein spaces space, which by the way admits G 2 structures Half-flatness Geometry with torsion Related to this Spinors • S 6 = G 2 / SU ( 3 ) Strings attached ‘Simultaneous’ structures • ( S 6 , g round ) ⊂ R 7 has an almost complex structure J inherited End from the vector cross product on R 7 • J is nearly Kähler 5

Special geometry Examples of interaction with solvable Lie groups Simon G. Chiossi • Hypersurface theory X n ֒ → Y n + 1 , quotients X / S 1 , and the Special geometry Lie groups’ actions like Six dimensions Seven dimensions • conical singularities constructed from NK structures: Nilpotent/Solvable Lie groups the cone of SU ( 2 ) 3 / SU ( 2 ) deforms to a complete smooth Nilmanifolds = R 4 × S 3 [Bryant-Salamon] Prototypical example holonomy metric on Y ∼ Solvmanifolds Non-compact Similarly for Ber = SO ( 5 ) / SO ( 3 ) , AW = SU ( 3 ) / U ( 1 ) homogeneous Einstein spaces Half-flatness ⇒ the sine cone dt 2 + ( sin 2 t ) g has weak • ( M 6 , g ) NK = Geometry with torsion holonomy G 2 (so Einstein). Its singularities at t = 0 , π Spinors Strings attached approximate G 2 -holonomy cones [Acharya & al], see ‘Simultaneous’ structures End [Fernández & al] too This example has the flavour of Killing spinors • ALC singularities of [Gibbons–Lü–Pope–Stelle] 6

Special geometry Tensors and representations with solvable Lie groups Let ( X d , g ) be Riemannian and φ a tensor, define Simon G. Chiossi G = { a ∈ SO ( d ) : a ∗ φ = φ } Special geometry so Λ 2 T ∗ X = so ( d ) = g ⊕ g ⊥ and Hol ( g ) ⊆ G ⇐ ⇒ ∇ φ = 0 Lie groups’ actions Six dimensions By analogy with the complex case, these are often referred to Seven dimensions Nilpotent/Solvable Lie as integrable G-geometries groups Nilmanifolds • ∇ φ is identified with the intrinsic torsion, an element in Prototypical example Solvmanifolds T ∗ ⊗ g ⊥ ∼ Non-compact = W 1 ⊕ W 2 ⊕ . . . ⊕ W N homogeneous Einstein spaces = R 7 when with N irreducible components. Notice so ( d ) Half-flatness g Geometry with torsion d = 6 , 7 , 8 Spinors Strings attached ‘Simultaneous’ structures d φ G N End 2 m almost complex structure J U (m) 4 2 m non-degenerate 2-form σ U (m) 4 7 positive generic 3-form G 2 4 8 positive generic 4-form Spin (7) 2 4 k quaternionic 4-form Sp (k) Sp (1) 6 7

Special geometry Six dimensions with solvable Lie groups Simon G. Chiossi • g Riemannian metric Special geometry • J orthogonal almost complex structure Lie groups’ actions Six dimensions J ∈ End TM : J 2 = − 1 , g ( JX , JY ) = g ( X , Y ) Seven dimensions Nilpotent/Solvable Lie groups • σ non-degenerate 2-form Nilmanifolds Prototypical example σ ( X , Y ) = g ( JX , Y ) Solvmanifolds Non-compact • Ψ ∈ Λ 3 , 0 T ∗ M a complex volume form homogeneous Einstein spaces Ψ ∧ ¯ Ψ = 4 3 i σ 3 Half-flatness σ ∧ Ψ = 0 , Geometry with torsion • ψ + = Re Ψ with open orbit in Λ 3 R 6 Spinors Strings attached (determines J , hence ψ − = J ψ + = Im Ψ ) ‘Simultaneous’ structures End = ⇒ Complex and symplectic aspects are linked: the structure is determined by choosing ψ + , σ only, for SL ( 3 , C ) ∩ Sp ( 6 , R ) = SU ( 3 ) 8

Special geometry SU ( 3 ) -intrinsic torsion with solvable Lie groups Simon G. Chiossi Special geometry The holonomy group Hol ( g ) is contained in SU ( 3 ) iff all forms Lie groups’ actions are constant for the Levi–Civita connection Six dimensions Seven dimensions ∇ ψ ± = 0 ∇ σ = 0 , Nilpotent/Solvable Lie groups Obstruction: Nilmanifolds Prototypical example Solvmanifolds ∇ J ∈ T ∗ ⊗ su ( 3 ) ⊥ ∼ W ± 1 ⊕ W ± 2 ⊕ W 3 ⊕ W 4 ⊕ W 5 Non-compact = homogeneous Einstein spaces Half-flatness where W j are the so-called ‘Gray–Hervella classes’ Geometry with torsion Spinors The intrinsic torsion is completely determined by the exterior Strings attached derivatives of σ, ψ + and ψ − ( n > 3 only σ, ψ + !) ‘Simultaneous’ structures End ⇒ all forms are closed: d σ = 0 , d ψ ± = 0 ∇ J = 0 ⇐ � M is a Calabi–Yau manifold 9

Special geometry almost Hermitian taxonomy with solvable Lie groups Simon G. Chiossi Special geometry comp dim R U ( 3 ) -module SU ( 3 ) -module Lie groups’ actions Six dimensions W ± [Λ 3 , 0 ] 1+1 [ ] R R Seven dimensions 1 W ± 8 + 8 [ [ V ] ] su ( 3 ) su ( 3 ) Nilpotent/Solvable Lie 2 groups [Λ 2 , 1 S 2 , 0 W 3 12 [ 0 ] ] Nilmanifolds Prototypical example Λ 1 Λ 1 W 4 6 Solvmanifolds Non-compact Λ 1 Λ 1 W 5 6 homogeneous Einstein spaces Half-flatness For instance Geometry with torsion Spinors Strings attached C 3 , G × T m • ∇ J ∈ W 3 ⊕ W 4 ⇐ ⇒ N J = 0 e.g. ‘Simultaneous’ structures Z ( S 4 ) End • ∇ J ∈ W 1 ⇐ ⇒ M is nearly Kähler KT = S 1 × H 3 / Γ • ∇ J ∈ W 2 ⇐ ⇒ d σ = 0 • ∇ J ∈ W 4 ⇐ ⇒ loc. conformally Kähler SU ( 2 ) × U ( 1 ) You name it . . . 10

Special geometry G 2 structures with solvable Lie groups Simon G. Chiossi On a 7-manifold Y with tangent spaces T y Y = R 6 ⊕ R and Special geometry SU ( 3 ) × { 1 } structure, define Lie groups’ actions Six dimensions Seven dimensions ϕ = σ ∧ e 7 + ψ + Nilpotent/Solvable Lie groups Nilmanifolds ∗ ϕ = ψ − ∧ e 7 + 1 2 σ 2 Prototypical example Solvmanifolds Non-compact In terms of an ON basis homogeneous Einstein spaces Half-flatness ϕ = e 127 + e 347 + e 135 + e 425 + e 146 + e 236 + e 567 Geometry with torsion Spinors Strings attached [Engel, Reichel] Stab ( ϕ ) = G 2 ‘Simultaneous’ structures End ⇒ open GL ( 7 , R ) -orbit in Λ 3 T ∗ Y = [Bryant] Such a ϕ determines the metric g and ∗ ϕ [Fernández–Gray] Hol ( g ) ⊆ G 2 ⇐ ⇒ d ϕ = 0 , d ∗ ϕ = 0 11

Special geometry with solvable Lie groups Simon G. Chiossi The intrinsic torsion of a G 2 structure ∇ ϕ ∈ Λ 1 ⊗ g ⊥ Special geometry 2 = X 1 ⊕ X 2 ⊕ X 3 ⊕ X 4 Lie groups’ actions Six dimensions is encoded into the exterior derivatives d ϕ, d ∗ ϕ Seven dimensions Nilpotent/Solvable Lie groups class type conditions Nilmanifolds Prototypical example Solvmanifolds — G 2 holonomy d ϕ = 0 = d ∗ ϕ Non-compact homogeneous Einstein X 1 weak holonomy d ϕ = λ ∗ ϕ spaces � Half-flatness d ∗ ϕ = 4 θ ∧ ∗ ϕ Geometry with torsion X 4 conformally G 2 Spinors d ϕ = 3 θ ∧ ϕ Strings attached ‘Simultaneous’ structures X 2 calibrated d ϕ = 0 End X 1 ⊕ X 3 d ∗ ϕ = 0 cocalibrated X 1 ⊕ X 3 ⊕ X 4 G 2 T d ∗ ϕ = ϑ ∧ ∗ ϕ 12