The Global Geometry of SLE Roland Friedrich MPI Roma, 10.09.2008 - PowerPoint PPT Presentation

The Global Geometry of SLE Roland Friedrich MPI Roma, 10.09.2008

What is SLE? The correlator � O � γ [0 ,t ] can be recognised as a section of a certain bundle L h over the moduli space of Riemann surfaces. 6 In this context it is clear F. & Kalkkinen (2003) that P X ( γ ) is consequently a holonomy, or a Wilson line, of this section when parallel transported from the fibre at X to the fibre over X \ γ with respect to the connection d + T . Recall that if the correlator � O � satisfies ˆ H � O � = 0, the operator creates a state in the Verma module V 2 , 1 . This module is closed under Virasoro action, which in turn is generated by the stress-energy tensor T . Since the Loewner process involves only insertions of the stress-energy tensor in the correlator, the final correlator � O � γ [0 ,t ] has to be that of an operator belonging to the same Verma module and satisfying the same di ff erential equation. This is true irrespective of the moduli of the Riemann surface, and provides indeed an independent analytic characterisation of the correlators � O � γ [0 ,t ] as those sections of L h that are annihilated by ˆ H . ˆ � ....... The CFT analysis leads us to consider sections of the line bundle L h , which is a twisted version of the standard determinant bundle defined on the moduli space M g, 1 . By using the above defined projection π , we can construct the pull-back bundle π ∗ L h on � M g, 1 . This bundle carries now a transitive Virasoro action, and can be equipped with a flat connection ∇ = d + L − 1 . In this way the Der K action is lifted to a Virasoro action in the quantum theory. The generator of the Loewner process H := κ ˆ 2 L 2 − 1 − 2 L − 2 (71) should therefore be seen naturally as a map ˆ H : Γ ( π ∗ L h ) − → Γ ( π ∗ L h +2 ) . (72)

The relevant example for „standard“ SLE • Let X be the Riemann sphere ˆ C = P 1 ( C ) with x ∈ X , a marked point. Let O x be the completion of the local ring at x , (the stalk of the structure sheaf at x ). • D x := Spec O x non-canonically isomorphic to D • choose a formal co-ordinate t x at x , i.e., a topological generator of the maximal ideal m x of O x . Interesting point: infinity ∞ Generalisation: abstract “half-disc”, i.e., disc with an involution.

The group Aut(O) • O : completed topological C -algebra C [[ z ]], with resp. to the natural filtration (Krull topology) • Aut( O ): the group of continuous automorphisms of O . • Aut( O ) ≃ a 1 z + a 2 z 2 + . . . , with a 1 ∈ C ∗ (formal power series). Der + ( O ) = z 2 C [[ z ]] ∂ z Aut + ( O ) ∩ ∩ Aut( O ) Der 0 ( O ) = z C [[ z ]] ∂ z ∩ Der( O ) = C [[ z ]] ∂ z Associated spaces: power series development at infinity: { bz + b 0 + b 1 Aut( O ∞ ) := z + · · · , b � = 0 } . { z + b 0 + b 1 Aut + ( O ∞ ) := z + · · · , } .

Proposition 1 (TUYKN). 1. Aut( O ) = C ∗ ⋉ Aut + ( O ) , semi-direct product of C ∗ and Aut + ( O ) . Aut( O ) acts on itself by composition. 2. Aut( O ) + is a pro-algebraic group, i.e. Aut( O ) + = lim − Aut + ( O / m n ) ← 3. The exponential map exp : Der + ( O ) → Aut + ( O ) , is an isomorphism.

The infinite Kähler manifold of univalent functions Definition 1. M := { f ∈ O ( D ) | conformal, with f (0) = 0 and f ′ (0) = 1 } , the set of univalent functions . M is a subset of the semi-direct product R + ⋉ Aut + ( O ), where � � ∞ � c k z k Aut + ( O ) := z 1 + , k =1 and it is enough to study the traces in Aut + ( O ). Now, this space has a natural affine structure with co-ordinates { c k } , and the identity map corresponding to the origin 0. By the De Branges-Bieberbach theorem one has | a n | ≤ n and therefore M can be identified with an open subset of � M ⊂ Ball C (0 , n + 1) n ≥ 1

Stochastic Löwner Equation Definition 1 (Stochastic Lœwner Equation). For z ∈ H , t ≥ 0 define g t ( z ) by g 0 ( z ) = z and ∂g t ( z ) 2 = (1) . g t ( z ) − W t ∂t The maps g t are normalised such that g t ( z ) = z + o (1) when z → ∞ and W t := √ κ B t where B t ( ω ) is the standard one-dimensional Brownian motion, starting at 0 and with variance κ > 0. Itˆ o form: f t ( z ) := g t ( z ) − W t , 2 f t ( z ) = d f t ( z ) dt − dW t . For a non-singular boundary point x ∈ R , the generator A for the Itˆ o-diffusion X t := f t ( x ) is d 2 A = 21 dx − κ d dx 2 . 2 x

Witt algebra / Virasoro algebra Define first order differential operators: ℓ n := − x n +1 d n ∈ Z , dx yields A = κ 2 ℓ 2 − 2 − 2 ℓ − 1 . [ L n , L m ] = ( n − m ) L n + m + ˜ c 12( n 3 − n ) δ n + m, 0 ,

Determinant line bundles (regularised determinants) To every Jordan domain one can associate the determinant of the Laplacian (with respect to the Euclidean metric and Dirichlet boundary conditions) det(∆ D ) := det(∆ g Eucl . ) trivial bundle over M , where f ∈ M denotes the uniformising map from the unit disc D onto the domain D , containing the origin. det(∆ f ( D ) ) f  π  � M

Polyakov`s conformal anomaly Consider the space F of all flat metrics on D which are conformal to the Euclidean metric, obtained by pull-back. For f : D → D a conformal equivalence, define φ := log | f ′ | . which gives a correspondence of harmonic functions on D with with F via ds = | f ′ || dz | = e φ | dz | . To fix the SU(1 , 1)-freedom, we divide by it. We work with the equivalence classes, so, e.g. 0 corresponds to the orbit of the Euclidean metric under SU(1 , 1). det(∆ D ) = e − 1 S 1 ( 1 H 2 φ ∗ dφ + φ | dz | ) · det(∆ D ) 6 π

Semi-group property Consider the sequence of conformal maps between domains D , D, G : f g − − − → D − − − → G . D d The relation of det(∆ G ) and det(∆ D ) is obtained via dz g ( f ( z )) = g ′ ( f ( z )) · f ′ ( z ) , and log | g ′ ( f ( z )) · f ′ ( z ) | = log | g ′ ( f ( z )) | + log | f ′ ( z ) | . � �� � � �� � =: ψ ( z ) =: φ ( z ) � 1 The property of harmonic functions: 2 ( φ∂ n ψ + ψ∂ n φ ) = 0 , gives S 1 2 φ ∗ dφ + φ | dz | ) · det(∆ D ) det(∆ G ) = e − 1 S 1 ( 1 1 S 1 ( 1 H H 2 ψ ( f ( z )) ∗ dψ ( f ( z ))+ ψ ( f ( z )) | dz | ) · e 6 π 6 π � �� � � �� � I. II. where � (1 ψ ∗ d ˜ ˜ ψ + ˜ ˜ ψ ( w ) := log | g ′ ( w ) | , = ψ | dw | ) with I. 2 ∂D = det(∆ D ) II.

model 1 probability density (model & time dependent) id model 2 Lie vector fields {L n } M ⊂ Aut( O ) model 3

Virasoro algebra, Gelfand-Fuks and Weil-Petersson The Virasoro algebra Vir C is spanned polynomial vector fields e n = − ie inθ d dθ , n ∈ Z , and c , with commutation relations [ c , e n ] = 0 and [ e m , e n ] = [ e m , e n ] + ω c,h ( e m , e n ) · c , with the extended Gelfand-Fuks cocycle � 2 π ω c,h ( v 1 , v 2 ) := 1 (2 h − c 1 ( θ ) − c � � 12) v ′ 12 v ′′′ 1 ( θ ) v 2 ( θ ) dθ , 2 π 0 and v 1 , v 2 being complex valued vector fields on S 1 . There exists a two-parameter family of K¨ ahler metrics on this space, with the form at the origin ∞ 2 hk + c � � � 12( k 3 − k ) w c,h := dc k ∧ d ¯ c k , k =1

Analytic line bundles • The Witt algebra has a representation in terms of the Lie fields L e n which act transitively on Aut + ( O ). • To have an action of Vir C , one has to introduce a determinant line bundle. • The line bundle E c,h is trivial, with total space E c,h = Aut + ( O ) × C . It is parametrised by pairs ( f, λ ), where f is a univalent function and λ ∈ C . It carries the following action L v + τ c ( f, λ ) = ( L v f, λ · Ψ( f, v + τ c )) , where � 2 � � wf ′ ( w ) � v ( w ) dw w + c w 2 S ( f, w ) dw Ψ c,h ( f, v + τ c ) := h w + iτc , f ( w ) 12 and where � 2 S ( f, w ) := { f ; w } := f ′′′ ( w ) f ′ ( w ) − 3 � f ′′ ( w ) , 2 f ′ ( w ) The central element c acts fibre-wise linearly by multiplication with ic .

Transitive action 0 − − − → C − − − → Vir C − − − → Witt − − − → 0           � � � � � 0 − − − → C − − − → Θ E c,h − − − → Θ M − − − → 0           � � � � � 0 − − − → C − − − → E c,h − − − → M − − − → 0 An action of a Lie algebra is a morphism from the Lie algebra g to the tangent sheaf, and it is transitive if the map g ⊗ O X → Θ X is surjective. point-wise Algebraically, the situation corresponds to so-called Harish-Chandra pairs surjective. ⊗ O → ( g , K ). Algebraically, the situation corresponds to so-called Harish-Chandra pairs ( g , K ).

Virasoro and Verma modules 1 • The space of holomorphic sections | σ � ∈ O ( E c,h ) ≡ Γ(Aut + ( O ) , E c,h ) of the line bundle E c,h carries a Vir C -module structure. • Let P be the set of (co-ordinate dependent) polynomials on M , defined by P ( c 1 , . . . , c N ) : Aut( O ) / m N +1 → C , with m the unique maximal ideal. • P corresponds to the sections O ( M ) of the structure sheaf O M of M and it carries an action of the representation of the Witt algebra in terms of the Lie fields L n ≡ L e n . • In affine co-ordinates { c n } , e.g. ∞ ∂ ∂ � L n = + ( k + 1) c k n ≥ 1 . ∂c k ∂c n + k k =1