Harmonic maps, Toda frames and extended Dynkin diagrams Emma Carberry Katharine Turner University of Sydney University of Chicago 3rd of December, 2011 Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Coxeter automorphism on G C / T C and conditions for it to preserve a real form de Sitter spheres S 2 n 1 and isotropic flag bundles Toda integrable system and relationship to cyclic primitive maps from a surface into G / T Solution in terms of ODEs (finite type) Applications to superconformal tori in S 2 n 1 Applications to Willmore tori in S 3 . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Coxeter automorphism on G C / T C Let G C be a simple complex Lie group and T C a Cartan subgroup. The homogeneous space G C / T C is naturally a k -symmetric space. That is, we have an automorphism σ : G C → G C with σ k = 1 and σ ) id ⊂ T C ⊂ G C ( G C σ . Recall that a non-zero α ∈ ( t C ) ∗ is a root with root space G α ⊂ g C if ∀ H ∈ t , R α ∈ G α . [ H , R α ] = α ( H ) R α Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Choose a set of simple roots , that is roots { α 1 , . . . , α N } such that every root can be written uniquely as N � α = m j α j , j = 1 where all m j ∈ Z + or all m j ∈ Z − . The height of α is h ( α ) = � N j = 1 m j and the root of minimal height is called the lowest root . Let η 1 , . . . , η N ∈ t C be the dual basis to α 1 , . . . , α N and σ : G C → G C be conjugation by N exp ( 2 π i � η j ) (Coxeter automorphism) . k j = 1 Then σ has order k , where k − 1 is the maximal height of a root of g C . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Choose a set of simple roots , that is roots { α 1 , . . . , α N } such that every root can be written uniquely as N � α = m j α j , j = 1 where all m j ∈ Z + or all m j ∈ Z − . The height of α is h ( α ) = � N j = 1 m j and the root of minimal height is called the lowest root . Let η 1 , . . . , η N ∈ t C be the dual basis to α 1 , . . . , α N and σ : G C → G C be conjugation by N exp ( 2 π i � η j ) (Coxeter automorphism) . k j = 1 Then σ has order k , where k − 1 is the maximal height of a root of g C . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Let G be a real simple Lie group with Cartan subgroup T and assume that the Coxeter automorphism preserves the real form G . I will describe class of harmonic maps from the surface into G / T which are given simply by solving ordinary differential equations and give a relationship between these maps and the Toda equations. This will generalise work of Bolton, Pedit and Woodward for the case when G is compact. Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Example: SO ( 2 n , 1 ) Let R 2 n , 1 denote R 2 n + 1 with the Minkowski inner product x 1 y 1 + x 2 y 2 + · · · + x 2 n y 2 n − x 2 n + 1 y 2 n + 1 Consider the de Sitter group SO ( 2 n , 1 ) of orientation preserving isometries of R 2 n , 1 . Take as Cartan subgroup T = diag ( 1 , SO ( 2 ) , . . . , SO ( 2 ) , SO ( 1 , 1 )) . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Define ˜ a k ∈ t ∗ , k = 1 , . . . , n by � 0 � � � � ��� 0 a 1 a n ˜ a k diag 0 , , . . . = a k . − a 1 0 a n 0 Take as simple roots of so ( 2 n , 1 , C ) the roots α 1 = i ˜ a 1 , α k = i ˜ a k − i ˜ a k − 1 for 1 < k < n and α n = ˜ a n − i ˜ a n − 1 . The lowest root is then α 0 = − ˜ a n − i ˜ a n − 1 , which is of height − 2 n + 1. Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Then writing η j for the dual basis of t C , conjugation by n � π i � � Q = exp η j n j = 1 � � 2 π � � � π � r π � � = diag 1 , R , R , . . . , R , − I 2 n n n is an automorphism of order 2 n . It is not hard to prove directly in this case that the real form SO ( 2 n , 1 ) is preserved by the Coxeter automorphism. Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Let �· , ·� denote the complex bilinear form � z , w � = z 1 w 1 + z 2 w 2 + · · · + z 2 n w 2 n − z 2 n + 1 w 2 n + 1 on C 2 n + 1 . A subspace V ⊂ C 2 n + 1 is isotropic if � u , v � = 0 for all u , v ∈ V . Geometrically, SO ( 2 n , 1 ) / T = SO ( 2 n , 1 ) / ( 1 × SO ( 2 ) ×· · ·× SO ( 2 ) × SO ( 1 , 1 )) is the full isotropic flag bundle Fl ( S 2 n 1 ) = { V 1 ⊂ V 2 ⊂ · · · ⊂ V n − 1 ⊂ T C S 2 n 1 | V j is an isotropic sub-bundle of dimension j } Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
We now give conditions under which a choice of real form g of a simple complex Lie algebra g C , Cartan subalgebra t C and simple roots α j yield a Coxeter automorphism σ = Ad exp ( 2 π i j = 1 η j ) which � N k preserves the real Lie algebra g . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
The condition for the Coxeter automorphism σ to preserve g is that for the simple roots α 1 , . . . , α N we have α j ∈ {− α 0 , . . . , − α N } , ¯ α ( X ) = α (¯ where ¯ X ) and α 0 is the lowest root. We will now use a Cartan involution to express this reality condition in terms of the extended Dynkin diagram. Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
A Cartan involution for g is an involution Θ of g C such that � X , Y � Θ = −� X , Θ( Y ) � is positive definite on g , where �· , ·� denotes the Killing form. Alternatively, it is an involution for which k ⊕ i m is compact, where k = + 1-eigenspace of Θ m = − 1-eigenspace of Θ . We may choose a Cartan involution which preserves the given Cartan subalgebra t Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Proposition Let g be a real simple Lie algebra, t a Cartan subalgebra and Θ be a Cartan involution preserving t . Choose simple roots α 1 , . . . , α N and let σ = Ad exp ( 2 π i j = 1 η j ) be the corresponding � N k Coxeter automorphism of g C . Then the following are equivalent: σ preserves the real form g , 1 σ commutes with Θ , 2 Θ defines an involution of the extended Dynkin diagram for 3 g C consisting of the usual Dynkin diagram augmented with the lowest root α 0 . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
For a Θ -stable Cartan subalgebra t , t is maximally Θ defines a permutation ⇔ of the Dynkin diagram for g C compact and so when t is maximally compact (e.g. g is compact), the real form g is preserved by any Coxeter automorphism defined by simple roots for t . The more interesting case is when we have an involution of the extended Dynkin diagram which does not restrict to an involution of the Dynkin diagram (i.e. t is not maximally compact). Call these non-trivial involutions. Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
α 0 α 1 α N A N α 2 α N − 1 α 2 . . . E 7 α 7 α 6 α 5 α 4 α 3 α 1 α 0 α 0 α 0 . . . B N α 2 α 1 α 2 α N − 1 α N E 6 α 6 α 5 α 4 α 3 α 1 . . . C N α 0 α 1 α 2 α N − 1 α N F 4 α 0 α 0 α N − 1 α 1 α 2 α 3 α 4 . . . D N α 1 α 2 α N − 2 α N G 2 α 0 α 2 α 1 α 2 E 8 α 0 α 8 α 6 α 3 α 7 α 5 α 4 α 1 There are nontrivial involutions for all root systems except E 8 , F 4 and G 2 . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Theorem Every involution of the extended Dynkin diagram for a simple complex Lie algebra g C is induced by a Cartan involution of a real form of g C . More precisely, let g C be a simple complex Lie algebra with Cartan subalgebra t C and choose simple roots α 1 , . . . , α N for the root system ∆( g C , t C ) . Given an involution π of the extended Dynkin diagram for ∆ , there exists a real form g of g C and a Cartan involution Θ of g preserving t = g ∩ t C such that Θ induces π and t is a real form of t C . The Coxeter automorphism σ determined by α 1 , . . . , α N preserves the real form g . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
Primitive Maps and Loop Groups The Coxeter automorphism σ : g → g of order k induces a Z k -grading k − 1 g C = � g σ [ g σ j , g σ l ] ⊂ g σ j , j + l , j = 0 j denotes the e j 2 π i where g σ k -eigenspace of σ . We have the reductive splitting g = t ⊕ p with k − 1 p C = t C = g σ � g σ j , 0 , j = 1 and if ϕ is a g -valued form we may decompose it as ϕ = ϕ t + ϕ p . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
A smooth map f of a surface into a symmetric space ( G / K , σ ) is harmonic if and only if for a smooth lift F : U → G of f : U → G / K , the form ϕ = F − 1 dF has the property that for each λ ∈ S 1 ϕ λ = λϕ ′ p + ϕ k + λ − 1 ϕ ′′ p satisfies the Maurer-Cartan equation d ϕ λ + 1 2 [ ϕ λ ∧ ϕ λ ] = 0 . Moreover given a family of flat connections as above, we can recover a harmonic map f : U → G / K on any simply connected U . Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams
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