Solvable Lie groups and the shear construction

Solvable Lie groups and the shear construction Marco Freibert jt. with Andrew Swann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel 19.05.2016

1 Swann s twist 2 The shear construction The left-invariant shear The shear in general 3 Examples 4 Summary and Outlook

Setting of the twist Double fibration picture: Two principal A-bundles (A connected Abelian Lie group) P π π W M < A =AP A M =A > W endowed with a principal A P = A-connection 1-form θ Ω 1 (P, a P ) of π such that (i) the principal actions commute and (ii) H := ker θ is a connection also for π W. Call α Ω k M, α W Ω k W H-related (α H α W ) iff π α H = π W α W H. H for arb. tensor fields using T π(p) M H p T πw (p)w.

Pushing data down to M Idea: Construct W, θ, P from twist data on M. Twist data + properties of α Ω k M determine properties of H-related α W Ω k W. We have W = P/A M for the principal A M = A-action of πw. Induced curvature form ω Ω 2 (M, a P ) on M. A M -action on P descends to A M action on M. ξ a M X(P) inf. princ. action of πw is lift of induced inf. ξ action a M X(M) and (i) and (ii) from last slide L ξ θ = 0. (if ξ(a M ) ker θ = TP)

Lifting Abelian actions Proposition (Swann) An A M -action on M lifts to R n -action on P preserving θ iff (i) a C (M, a P a M ) with da = ξ ω and (ii) ξ ω = 0.

Lifting Abelian actions Proposition (Swann) An A M -action on M lifts to R n -action on P preserving θ iff (i) a C (M, a P a M ) with da = ξ ω and (ii) ξ ω = 0. Then L ξ ω = 0 and ξ = ξ + ρ π a with ξ : a M X(P) horizontal lift of ξ, ρ : a P X(P) inf. princ. A P -action of π. Definition We call (a, ξ, ω) C (M, a P a M ) X(M) a M Ω2 (M, a P ) with invertible a fulfilling dω = 0 and (i) and (ii) from above twist data on M and W := P/ ξ(a M ) the twist of (M, a, ξ, F ) if smooth.

Some results If A M = T n and ω has integral periods, one can construct principal T n -bundle P and lift A M = T n -action on M to a T n -action on P preserving princ. conn. θ with curv. ω. For α Ω k M (!)α W Ω k W with α H α W L ξ α = 0. If this is the case, then: dα ( ξ a 1 α ) ω H d W α W. Differential on W equals twisted differential d ξ a 1 ω on M. Twist is invertible.

Examples HKT/SKT twists which are not hyperkähler/kähler from easy hyperkähler/kähler manifolds. E.g. HKT metrics on torus bundles over K3-surfaces Σ by twisting Σ T 4k with T 4k acting on right factor as usual (Swann). Twists of certain hyperkähler manifolds (after elementary deformations ) to hyperkähler manifolds (generalizing hyperkähler modifications) (Swann) quaternionic Kähler manifolds (reinterpretation and uniqueness of hk/qk correspondence of Haydys) (Maciá, Swann) Any compact nilmanifold by several twists from T n, see next slide.

The left-invariant twist G := M, P, H := W Lie groups, etc., (a, ξ, F ) left-invariant twist data ξ : a G g LA hom., a const., ω Λ 2 g a P closed with ξ ω = da = 0. Want to be able to twist every left-invariant form L ξ α = 0 α g ξ(a G ) is a central ideal in g. a P = ρ(a P ) p g is a central extension determined by ω and ξ(a G ) p h is a central extension as well. n nilpotent LA with LCS n = n 0, n 1,..., n r = {0} a := n r 1 central and n can be twisted to n/a a as LAs. n can be transformed to R N by r 1 successive twists.

The left-invariant shear Generalizing the twist Problem: Find a generalization such that all 1-connected solvable Lie groups can be obtained from R n, such that the space of invariant forms is not that restrictive (e.g., if g LA and X g not central α g with L X α 0) and such that the set of shear data is more general (e.g. dω 0 or ξ ω = 0 does not imply a constant).

The left-invariant shear Abelian extensions An Abelian extension a P p g is determined by ω Λ 2 g a P, η g gl(a P ) with dω = η ω and η being a representation of g on a P. Reinterpretation of the data: η defines a flat connection on the trivial vector bundle F := G a P ω Ω 2 (G, F ) and dω = η ω is equivalent to d ω = 0. Using p = a P g as vector spaces, let θ : p a P p be the projection. Then θ Ω 1 (P, π F ) with d θ = π ω.

The left-invariant shear Lifting LA homomorphisms Given ξ : a G g injective LA homomorphism, make ansatz ξ = ξ + ρ a : a G p with a : a G a P, ρ : a P p. Question: When is ξ(a G ) an Abelian ideal in p? For the answer, set γ := a 1 ξ ω + a 1 ηa g gl(a G ) connection on trivial vector bundle E := G a G. Induced connections on E r F s, a Ω 0 (G, E F ), ξ : E TG, ρ : π F TP, ξ : π E TP bundle maps.

The left-invariant shear The left-invariant shear Proposition (F., Swann) (a) Let ξ : a G g as before. Then ξ(a G ) is an Abelian ideal in p iff (i) (, ξ) is torsion-free, i.e. [ξ(e 1 ), ξ(e 2 )] = ξ( ξ(e1)e 2 ξ(e2)e 1 ) e 1, e 2 Γ(E), (ii) ξ ω = 0 and (iii) L ξ α = 0 α g. If this is the case, all connections are flat and so ( d ) 2 = 0, L ξ ω = 0, L ξ θ = 0 and we call h := p/ ξ(a G ) the shear of g. (b) Any two solvable Lie algebras of the same dimension are related by a sequence of shears.

The shear in general Lifting certain bundle maps (I) Setting: (E, ), (F, ) flat vector bundles of the same rank k over M, ξ : E TM vb map such that (, ξ) is torsion-free, ω Ω 2 (M, F ) with d ω = 0, π : P M surjective submersion with k-dimensional fibres, θ Ω 1 (P, π F ), ρ : π F TP vb map with θ ρ = id π F, dπ ρ = 0 and d θ = π ω. Question: Can one lift ξ to ξ : π E TP with L ξ θ = 0 and (, ξ) being torsion-free?

The shear in general Lifting certain bundle maps (II) Proposition (F., Swann) A lift ξ as on the last slide exists iff (i) a Ω 0 (M, E F ) = Hom(E, F ) with d a = ξ ω and (ii) ξ ω = 0. Then ξ = ξ + ρ π a and L ξ ω = 0. Definition Shear data on M is given by (a, ξ, ω) Hom(E, F ) Hom(E, TM) Ω 2 (M, F ) with invertible a such that (ξ, ) is torsion-free, d ω = 0 and (i) and (ii) from the above proposition hold for (E, ), (F, ) flat vbs of the same rank. If there (P, θ, ρ) as before and ξ : π E TP is build as above, then the leaf space W := P/ ξ(π E) is called the shear of (M, a, ξ, ω) if it is smooth.

The shear in general Shearing differential forms Using H := ker θ, define the relation α H α W for α Ω k M and α W Ω k W as in the twist case. Then: For α Ω k M (!)α W Ω k W with α H α W iff L ξ α = 0 (new invariance condition). If α H α W, then dα ( ξ a 1 α ) ω H d W α W ( same relation for differentials).

A toy example (I) (N 4, σ 0 ) symplectic manifold (M, σ) := (N R t S 1 ϕ, σ 0 + dt dϕ) symplectic. Vbs E = M R = F, F can. flat conn., E flat conn. defined by closed γ Ω 1 (M), ξ : M R TM, ξ(1) = t. ω Ω 2 Z (N) principal S 1 -bdle π : P N, θ Ω 1 (P) princ. conn. with d θ = dθ = π ω, ρ inf. princ. action (extend everything trivially to M = N R S 1 /P R S 1 ).

A toy example (II) Investigate when a C (M) nowhere vanishing defines shear data (a, ξ, ω) s.t. we can shear σ 2 to σ 2 W with d W σ 2 W = 0: (a, ξ, ω) shear data da aγ = d a = t ω = 0 d log(a) = γ.

A toy example (II) Investigate when a C (M) nowhere vanishing defines shear data (a, ξ, ω) s.t. we can shear σ 2 to σ 2 W with d W σ 2 W = 0: (a, ξ, ω) shear data da aγ = d a = t ω = 0 d log(a) = γ. σ 2 = σ 2 0 + σ 0 dt dϕ is t -invariant but still L ξ σ2 = L t σ 2 + γ ( t σ 2) = γ dϕ σ 0 = 0 γ = fdϕ for f C (M). 0 = d W σ 2 W = dσ2 a 1 ( t σ 2) ω = a 1 dϕ σ 0 ω σ 0 ω = 0. Hence, (a, ξ, ω) shear data s.t. we can shear σ 2 to σ 2 W with d W σ 2 W = 0 a C (S 1 ) nowhere vanishing with d log(a) = γ and σ 0 ω = 0. In the twist case: a C R \ {0}. So more flexibility here. However, the shear W is always diffeomorphic to P S 1 and σ 2 W = π σ 2 0 1 a π σ 0 θ dϕ.

Shears of calibrated G 2 -structures on almost Abelian Lie algebras A G 2 -structure ϕ Ω 3 M is called calibrated iff dϕ = 0. Calibrated G 2 -structures on 7d almost Abelian Lie algebras (aalas) g, i.e. g = R 6 R, have been clasified by F. Classification of all calibrated G 2 -structures on LAs of the form ( h 3 R 3) R (Exact result, see next slide): (1) Determination of all shears of calibrated G 2 -structures on aalas to calibrated G 2 -structures on LAs of the form ( h3 R 3) R (and more, e.g. also for cocalibrated). (2) Proof that any calibrated G 2 -structure on LA of the form ( h3 R 3) R can be sheared to one on an aala. (3) Left-invariant shear is invertible Classification.

Calibrated G 2 -structures on (h 3 R 3 ) R Theorem (F., Swann) g 7d LA with 6d ideal u = h 3 R 3, X u with X = 1, ϕ Λ 3 g G 2 -structure, U 1 := [u, u] J[u, u], U 2 := U 1 z(u), U 3 := z(u) u. Then ϕ is calibrated iff J[u, u] z(u) and (i) z(u) J-invariant, λ, µ R, lin. cpx. h ij : U i U j s. t. ( 2λI2 ) h 21 h 31 ad(x ) u = 0 3λI 2+µJ h 32+ h 0 0 λi 2 µj w.r.t. u = U 1 U 2 U 3, h : U 3 U 2 certain non-cpx. lin. map or (ii) z(u) not J-invariant, ρ [u,u] Λ 2 z(u) = 0 and h 1 sl(u, ρ) with h 1 U1 = 0, h 1 (z(u)) z(u) s. t. ad(x ) u = diag( 2λ, 6λ, 3λI 2, λi 2 ) + h 1 w.r.t. u = [u, u] J[u, u] U 2 U 3 for λ R specified by ρ and g.

Summary The shear naturally popes up in the left-invariant setting as a generalization of the twist by replacing central with Abelian extensions. The general shear is obtained by replacing principal bundles with Abelian structure group by foliated manifolds with tangent spaces of leaves being pullbacks of flat vector bundles. Analogous results for the shear when replacing exterior derivatives by covariant exterior derivatives and introducing covariant Lie derivatives.

Some open problems Find interesting non-invariant examples! Can one obtain (P, θ, ρ) under certain circumstances from data on M? Can one invert the shear?