Imaginary time Hamiltonian flows and applications to Quantization, - PowerPoint PPT Presentation

Imaginary time Hamiltonian flows and applications to Quantization, Kahler geometry and representation theory Jos´ e Mour˜ ao CAMGSD, Mathematics Department, IST XXXVIII Workshop on Geometry Methods in Physics Bialowieza, June 30 – July 6 On work in collaboration with T. Baier, J. Hilgert, O. Kaya & J. P. Nunes

Index 1. K¨ ahler manifolds and space of K¨ ahler metrics . . . . . . . . . . . . . . . . 3 2. The geometry of the space of K¨ ahler metrics on M and HCMA 6 3. Explicit “rotation” of hamiltonian flows to imaginary time . . . .18 4. Infinite dimensional spaces of new solutions of the HCMA on elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5. Applications of special geodesics in the space of K¨ ahler metrics 23 5.1. Geometric quantization 24 5.2. Links with representation theory 34 2

1. K¨ ahler manifolds and space of K¨ ahler metrics K¨ ahler manifolds ( M, ω, J ) are symplectic manifolds ( M, ω ) with a compatible complex structure J , ie such that the bilinear form γ ( X, Y ) := ω ( X, JY ) is a Riemannian metric, so that we get 3 structures, ( M, ω, J, γ ). A symplectic manifold may not have compatible complex structures but if it has one it has an infinite dimensional space of them. The symplectic form is automatically of type (1 , 1) for any compatible complex structure and has a locally defined J -dependent K¨ ahler potential k J , ω = i 2 ∂ J ∂ J k J Example - CP n The Fubini-Study K¨ ahler form reads ω FS = i ∂ k FS = i ∂ log(1 + | z 1 | 2 + · · · + | z n | 2 ) 2 ∂ ¯ 2 ∂ ¯ 3

On the other hand, fixing J , on a compact manifold M , two J –compatible closed 2–forms ω and ω ′ are in the same cohomology class iff their K¨ ahler potentials k, k ′ can be chosen to differ by a global function k ′ = k + φ, φ ∈ C ∞ ( M ) Then, the space of K¨ ahler forms compatible with J , in the given cohomology class, is naturally given by � � φ ∈ C ∞ ( M ) : ω φ = ω + i H ( ω, J ) ∼ 2 ∂ J ¯ = H 0 ( ω, J ) / R := ∂ J φ > 0 / R This (infinite dimensional) manifold (convex open nbd of 0 in C ∞ ( M )) has a natural metric introduced by Mabuchi, � ω n where ω φ = ω + i φ 2 ∂ J ¯ G φ ( h 1 , h 2 ) = (1) h 1 h 2 n ! , ∂ J φ M 4

Example - CP n ahler potentials on CP n with fixed cohomology class is then The space of K¨ given by the following open convex subset of C ∞ ( CP n ): � � � � φ ∈ C ∞ ( CP n ) : ω φ = i log(1 + | z 1 | 2 + · · · + | z n | 2 ) + φ 2 ∂ ¯ H 0 ( ω FS , J ) = > 0 ∂ C ∞ ( C P n ) ⊂ (2) So H 0 ( ω, J ) has trivial topology but a very interesting metric. As showed by Donaldson, the Mabuchi metric is the metric associated with the realization of H ( ω, J ) as the symmetric space Ham C ( M, ω ) / Ham( M, ω ). 5

2. Geometry on the space of K¨ ahler metrics on M and HCMA Let M be compact and simply connected. Theorem 1 (Mabuchi/Semmes/Donaldson) The geodesics for the me- tric (1) are the stationary points of the energy functional � � n � 1 � ω + i 2 ∂ ¯ ∂φ t φ 2 ˙ E ( φ ) = t dt . n ! 0 M Donaldson further shows that H with the Mabuchi metric is an infinite dimen- sional analogue of the symmetric spaces of non–compact type of the form PSL ( N, C ) /PSU ( N ) , with PSL ( N, C )–invariant metric. 6

(I) First argument supporting H ( ω, J ) ∼ = Ham C ( M, ω ) /Ham ( M, ω ): H as a quotient Let � � ψ − 1 � ∗ ( ω ) ∈ H � Ham C ( M, ω ) := ψ ∈ Diff ( M ) : (3) not a subgr ⊂ Diff( M ) we obtain, from Moser theorem, that the following map is a bijection ∼ Ham C ( M, ω ) /Ham ( M, ω ) = H ( ω, J ) � ψ − 1 � ∗ ( ω ) . [ ψ ] �→ 7

(II) Second argument supporting H ∼ = Ham C ( M, ω ) /Ham ( M, ω ): Tangent space at a K¨ ahler potential We have T ω φ H ∼ = C ∞ ( M ) / R and � � = − i 2 ∂ ¯ L JX ω φ ∂H, ωφ H 8

(III) Third argument supporting H ∼ = Ham C ( M, ω ) /Ham ( M, ω ): Curvature formulas Theorem 2 (Donaldson) The curvature of the Mabuchi me- tric (1) and the sectional curvature read R φ ( f 1 , f 2 ) f 3 = − 1 4 {{ f 1 , f 2 } φ , f 3 } φ , K φ ( f 1 , f 2 ) = − 1 4 ||{ f 1 , f 2 } φ || 2 φ . for all f 1 , f 2 , f 3 ∈ T φ H , where � � � ∼ f ∈ C ∞ ( M ) : M f ω n T φ H = φ = 0 = Lie ( Ham ( M, ω φ )) . 9

Remark The above expressions are in full agreement with the formulas for the curvature of the finite dimensional symmetric spaces K C /K , R ( X, Y ) Z = − 1 4[[ X, Y ] , Z ] and K ( X, Y ) = − 1 4 || [ X, Y ] || 2 . for all X, Y, Z ∈ T 0 K C /K ∼ = iLie ( K ) ∼ = Lie ( K ) and the Lie brac- kets are calculated in Lie ( K ). 10

(IV) Fourth argument supporting H ∼ = Ham C ( M, ω ) /Ham ( M, ω ): Limit of spaces of Bergman metrics H = lim N →∞ PSL ( N, C ) /PSU ( N ) 1 Let L → M be a very ample holomorphic line bundle with c 1 ( L ) = 2 π [ ω ] and dim H 0 ( M, L p ) = d p + 1. Every ordered basis s = ( s 0 , . . . , s d p ) defines an embedding i s : M → CP d p and the p –th root of the pullback of the Fubini-Study hermitian structure defines an hermitian structure on L − → M , � � 1 /k 1 i ∗ FS p ( s ) = s h FS = ( � d p j =0 | s j ( z ) | 2 ) 1 /p � � ∼ k ( s ) = − log( FS p ( s )) : s a basis of H 0 ( M, L p ) B p = = GL ( d p + 1) /U ( d p + 1) . 11

Every k ∈ H 0 ( ω, J ) defines an inner product on H 0 ( M, L p ) via the Hermi- tean structure h p ( k ) = e − pk � s ) ω n k � s, ˜ s � k = h p ( k )( s, ˜ n ! M Let s p ( k ) be an orthonormal basis for �· , ·� k and let B p ∼ H 0 ( ω, J ) − → = GL ( d p + 1) /U ( d p + 1) k �→ k p = − log ( FS p ( s p ( k ))) . Then, we have: Theorem 3 (Tian, 1990) k = lim p →∞ k p . 12

(V) Fifth argument supporting H ∼ = Ham C ( M, ω ) /Ham ( M, ω ): Geodesic equations on H and imaginary time Hamiltonian flows The Homogeneuous Complex Monge–Amp` ere (HCMA) equation is the following nonlinear equation on a complex ( n + 1)–dimensional manifold N � ∂ 2 K � MA ( K ) := det = 0 , ∂z j ∂ ¯ z l or, equivalently, � � n +1 = 0 . ∂ ¯ ∂K (4) It is a very difficult equation with very few (genuinly complex) rank n solutions known. 13

Even for n = 1 the HCMA equation is very nontrivial. Relation with geodesics on H Let us for simplicity consider the case n = 1. Functions K on (open subsets of) N = [0 , T ] × S 1 × M , which are ( a ) S 1 –invariant and 1 = ∂ 2 K ( b ) such that g 1¯ z ( t, z, ¯ z ) > 0 ∂z∂ ¯ so that k t = K ( t, · ) is a path of K¨ ahler potentials on M ). The HMA equation for these functions coincides with the geodesic equations for k t . � ∂ N K � 2 = 0 ∂ N ¯ ⇔ ∂ 2 K ∂ 2 K z − | ∂ 2 K z | 2 = 0 ⇔ ∂t 2 ∂z∂ ¯ ∂t∂ ¯ ∂ 2 K 1 | ∂ 2 K ∂t 2 = g 1¯ z | 2 ⇔ ∂t∂ ¯ k t || 2 ¨ k t = ||∇ ˙ (5) k t Analogously in higher dimensions 14

Ellaborating on an idea of Semmes and Donaldson we will show how to reduce the Cauchy problem for the Mabuchi geodesics, with k t = k + φ t .  k t || 2 ¨ ||∇ ˙ k t =   k t k t ∈ C ∞ ( U ) , H ∈ C ∞ ( M ) . (6) k 0 = k,   ˙ k 0 = − H, to the problem of finding the integral curves of the Hamiltonian H , where ω = i vector field X ω 2 ∂ ¯ ∂k , followed by “rotating” t to the imaginary axis (in the complex t –plane) √ ?? exp( sX ω − 1 tX ω H ) � exp( H ) ∈ Ham C ( M, ω ) ⊂ Diff ( M ) , (7) in a certain way. 15

To make sense of (7) we will be working on the symplectic pic- ture (see section 3 below) in which ω is fixed and the complex structure J t changes. Then the imaginary time integral curves in (7) are solutions of the following coupled system � ˙ J t X ω H = ∇ γ t H = x t � � ∗ ( J ) . exp( √− 1 tX ω (8) J t = H ) A solution of (6) is given formally by the K¨ ahler potential φ t of ω t in �� � − 1 � ∗ √ − 1 tX ω ω t = exp( H ) (9) ( ω ) . This is the so called Donaldson formal solution of the CHMA. 16

The problem is that to find the imaginary time flow exp( √− 1 tX ω H ) with (8) is equivalent to solving a complicated system of PDE (see [Burns–Lupercio–Uribe, 2013]). So it is not clear what have we gaigned in going from the original HCMA (6) to the coupled system (8). NO PDE needed! 17

3. Explicit “rotation”of hamiltonian flows to imaginary time The missing step to transform Donaldson formal solution of the Cauchy problem (6) for the HCMA given by (9) into an actual solution is the rotation √ exp( sX ω − 1 tX ω H ) � exp( H ) . In the present section we will describe our solution to this pro- blem obtained in [M-Nunes, IMNR2015]. One key technical tool to rotate the flow is the Gr¨ obner theory of Lie series of vec- tor fields (which is still very popular in numerical methods in astronomy – satelite motion, exoplanets, etc). 18