Planar maps, circle patterns, conformal point processes and 2D gravity

Transcription

1 Planar maps, circle patterns, conformal point processes and 2D gravity François David*, Institut de Physique Théorique, Saclay joint work with Bertrand Eynard Direction des Sciences de la Matière, CEA & * Institut National de Physique, CNRS arxiv: x to appear in AIHPD ANNALES DE L INSTITUT HENRI POINCARÉ D, Combinatorics, Physics and their Interactions and work in progress

2 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

3 2D continuum quantum gravity Feynman path integral over 2D Riemannian metrics (+ matter fields) Z Z D[g ab ]exp µ d 2 p Z x g D[X] e S[X,g] 0 This is a fairly well understood theory

4 Polyakov functional integral, use the conformal gauge g ab =ĝ ab e A. Polyakov 1981 The Faddeev-Popov ghost systems leads to the effective action for the remaining Z conformal Z factor. D[g ab ]= The effective theory is the Liouville theory (conformal anomaly consistency condition), a CFT and integrable quantum theory S L ['] = 1 Z pĝ ( ˆr') 2 + Q ˆR ' + µ R e ' 2 For pure gravity Q = c L =1+6Q 2 = 26 c M Real positive action for Dĝe [ ] det(r FP ĝ )= c M =0 1 <c M apple 1 Z Dĝ['] e S L[']

5 2D discrete quantum gravity * Discretize metric Random planar lattices (maps) surfaces aléatoires Courtesy Nicolas Curien et al. F. D., J. Fröhlich, V. Kazakov & A. Migdal (circa ) Also fairly well understood

6 2 Discrete 2d gravity Random matrices recursion relations integrability Continuous 2d gravity Topological gravity QFT CFT integrability combinatorics Cori-Vauquelin- Schaeffer-... bijections combinatorics probabilities Planar maps (diagrams) Brownian maps & well labelled trees KPZ relations Liouville theory & conformal field theories conformal invariance probabilities Gaussian free field, SLE Still lacking: a constructive & geometrical link between the discrete and the continuous formulations

7 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

8 The Koebe-Andreev-Thurston theorem There is a bijection between triangulations and circle packings, modulo SL(2,C) Möbius transformations v 1 v3 v 3 v 2 v3 Illustrations borrowed to Schramm & Mishenko

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10 A generalisation of circle packings: circle patterns Circles meeting at common points. The angles of intersection of the circles are given. Find the planar pattern and the radii of the circles. theorem of existence and unicity (Igor Rivin 1994, Ann. of Math.) circle packing = circle patterns with angles 0 or /2 only

11 F. David, June 16, 2014 Wien, Erwin Schrödinger Institute

12 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

13 T =(T,{ }) { } angles 1) the angles are positive 2) around each vertex 3) around each closed cycle «Dressed» planar maps T an abstract triangulation of the sphere, with e attached to the edges e such that 0 < e < X e =2 e!v X e2c e Sphere Sphere

14 Delaunay triangulations and planar maps There is a bijection* between such «dressed» planar maps and Delaunay triangulations in the plane, such that the circle intersection angles are e = * modulo global SL(2, C) transformations (as usual) e Sphere Plane

15 Voronoï tessellation and Delaunay triangulation in the plane Delaunay condition: no vertex of a triangle must be inside the circumscribed circle to any another triangle Local flatness (p.l. manifold) X e!v e =2 >0 From Wikimedia Commons Last condition on cycles no change of orientations or foldings (to be discussed later) This excludes some triangulations but it is conjectured that one keeps generic ones (universality)

16 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

17 ( ) = ( )) T + Take as initial measure on triangulations the uniform measure on ( ) triangulations and the flat measure on the angles (+ inequalities) µ( T ) = µ(t,d )) = uniform(t ) d (e) e E(T ) v V(T ) (e) 2 ev Question: which measure does this induce on Delaunay triangulations? For T this consider N+3 points, 3 fixed by ( ) D N+3 = C N+3 /SL(2, C) ' C N dµ(z 4,,z N+3 ) =? T + SL(2, C) ( ) = E( ) E( ) F. David, June 16, 2014 Wien, Erwin Schrödinger Institute

18 Transition between Delaunay triangulations by edge flips the flip of link e occurs when (e) =0 Moving the points allows to explore the whole space of Delaunay triangulations and of dressed abstract triangulations

19 Elementary example: the hexahedron (5 points) moving the 5 th point

20 1 st question: Which sets of edges form independent basis for the angles? Answer: The sets whose complementary form a cycle-rootedspanning-tree of the triangulation with odd length cycle

21 2 nd question: What is the Jacobian of the change of angle variables between two basis of edges? Answer: Jacobian = 1! Indeed... G T + { } = = ( µ(t,d ) = 1 2 uniform(t + ) ) ( ) = e E 0 (T ) So, the measure over the points is given locally (for a given = E ( ) E( ) Delaunay triangulation) by a simple Jacobian E µ(t,d ) = dµ(z) = ( ) ( ) N+3 v=4 d 2 z v + { ( ) d (e) det J T (z) {1,2,3} Ē e J T (z) v, z v ) e E(T E ) v V(T )( ) ciated to the three fixed vertices ( ) { } E

22 = ( ( ) + ) The matrix elements = of the ( ) Jacobian are made of simple poles J v,e e, J v,e e z v tex is = a vertex of one of the tw J v1,e = i z v4 z v1 z v3 z v1 J v3,e = i 2 1 z v3 z v1 ( ) = ( ) 1 z v3 z v2 J v2,e = i 2 1 z v3 z v2 D( ) J v4,e = i 2 1 z v4 z v2 1 z v4 z v2 1 z v4 z v1 ( D The determinant ({ ) } of the Jacobian matrix is locally a rational function of the s and z v s z v D T (z) {1,2,3} = det J T (z) {1,2,3} Ē 0

23 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

24 Definition 2.3 (triangle rooted spanning 3-tree) Let T be a planar triangulation with N + 3 vertices, and = f 0 be a face (triangle) of T,with3verticesV( ) = (v 1,v 2,v 3 ) and 3 edges E( ) = (e 1,e 2,e 3 ). Let E(T ) = E(T ) E( ) be the set of 3N edges of T not in. We call a -rooted 3-tree of T ( R3T )afamilyf of three disjoint subsets (I, I, I ) of edges of E(T ) such that: 1. (I, I, I ) are disjoint and disjoint of E( ) 2. Each I E( ), I E( ), I E( ) is a cycle rooted spanning tree of T with cycle. I I I ( ) 5: 2 inequivalent triangle-rooted spanning 3-trees of a = planar triang nb: -rooted spanning 3-trees are NOT Schnyder woods! F. David, June 16, 2014I I I Wien, Erwin Schrödinger Institute

25 Theorem 2.3 Let T be a planar triangulation of the plane with N +3 vertices. If the 3 fixed points (v 1,v 2,v 3 ) belong to a triangle (face) of T,themeasuredeterminanttakes the form D T (z) {1,2,3}) = 4 N F=(I,I,I ) R3T of T (F) e=(v v ) I 1 z v z v e=(v v ) I 1 z v z v (2.10) where (F) = ±1 is a sign factor, coming from the topology of T and of F, thatwillbe defined later. This is a non trivial extension of the spanning tree representation ( ) of the determinant of scalar Laplacians. I I I This representation is specific to this Jacobian matrix D. D It is useful for proof of convergence, factorization properties, etc. Can we use it for more?

26 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

27 Hyperbolic volume of triangle = volume of ideal tetrahedron above the triangle in hyperbolic Poincaré half-space ( ) ) = ( ) Vol(f) = L( 1 ) + L( 2 ) + L( 3 ) ki-milnor = Im(Li function 2 (z)) + ln(z)arg (1 z) z = z 3 z 1 z 2 z1 ( ) = ( ( )) ( Action of a triangulation = sum over volumes A T = triangles f F(T ) Vol(f) A

28 = ( + ) Define the N x N matrix D u v @ z v A T (z) 1. D u v is a Kähler form on D N+3 i.e. D>0 2. is countinuous (no discontinuity when a flip occurs) D u v 3. The measure determinant is the Kähler volume form The (2N x 2N) Jacobian has been reduced to a N x N Kähler determinant! D T (z) {1,2,3}) = det (D u, v ) u,v {1,2,3} D ( ) { }) But it is not a determinantal process!

29 It is clear that 1 2 surprising, theuinitial measure d v,f independent = 2 v,f+ = 1 v,f+ over v = 2 4 f v 4 f v Sv Sv ˆ ˆ This is not too angles can be written as a combination of Chern classes In other words, uv is a connection globally defined on the bundle Lv along the fiber is 1, its curvature duv is then the Chern class: 1 n f 1 c1 (Lv )apparently = duv = depend d v,f Notice that the coefficient Cv T=might triangulation T, on the + d v,f+ 2 4 f =2 f =1 2 ever, it was proved by Kontsevich that it doesn t 4 v N 2 v = ± N! 2, and we have: 22 2N +1 d e e This shows that our measure DT (z) {1,2,3} is also the measure of topological gr and the 4 angleproofs measure isthe a measure on (a subset of) the moduli of results fiber of the circle bundle L T, is a circle centered at v. A point space of the punctured Riemann sphere M0,N +3 a point on the circle, and a coordinate for that point is the angle the segment [v, f ] where f is a face adjacent to v.a basis of edges 4.1 Chosing So this measure is related (and possibly equivalent) to the WeilProofover of Th.M 2.10,N +3 and our model is related to Peterson4.1.1 measure of the bundle L is a 2-form on T, notice that it is independent s topological 2Dv. defined gravitythe measure on T N +3 to be the uniform measure on triangula Wearound have rigin of labelling of faces v v v N +3 f,v N +3 ds, if we label the edges around v in the trigonometric order e1,..., en, tensored with the flat Lebesgue measure on angles e s (constrained by (2.1)): F. David, June 16, n e 1 d e d e v = (3.6) Wien, Erwin Schrödinger Institute

30 Conformal invariance dµ(z) =d 2 z det(d) is a conformal point process Independence of the 3 fixed points and H = det D \ a,b,c (z) 3(z a,z b,z c ) 2 SL(2, C) invariance 3(z a,z b,z c )=(z a z b )(z a z c )(z b z c ) is independent of the choice of points z! w = az + b cz + d H(z) = N+3 Y i=1 w 0 (z i ) with ad bc =1 2 H(w) = N+3 Y i=1 1 cz i + d 2 H(w)

31 Consequence: D is a very singular but integrable measure One expects large fluctuations of the density of points at all scales, consequence of conformal invariance Poisson process on the sphere collapse of half of the points X The sum of angles around the collapsed points e! 2 + e!v

32 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

33 Local geometrical representation of D as a sum over triangles D u, v = f D u, v (f), D u, v (f) z v Vol(f) ( ) D(f) = 1 cot( 2 ) + cot( 3 ) cot( 3 ) i cot( 2 ) + i cot( 8 R(f) 2 3 ) + i cot( 3 ) + cot( 1 ) cot( 1 ) i cot( 2 ) i cot( 1 ) + i cot( 1 ) + cot( 2 ) (

34 D as a discretized V( ) Fadeev-Popov determinant! The Hermitean form D can be written as D(f) = i,j vertices of f V( ) F( ) Local derivative operator from (f) = 1 2i (f) = 1 2i ( ) (v i )D i (f) (v j ) = Area(f) R(f) 2 V( ) = ( = ) ( ) (v 1 )( z 3 z 2 ) + (v 2 )( z 1 z 3 ) + (v 3 )( z 2 z 1 ) Area(f) (v 1 )(z 3 z 2 ) + (v 2 )(z 1 z 3 ) + (v 3 )(z 2 z 1 ) Area(f) (f) (f) (vertices)! (faces) = Geometric characterization of r r f,v v = 1 z v z v + log(radius(f)) z v 0 z v

35 ( ) = ( ) ( ) = If complex functions are identified with real vector fields = z, = z ( ) = ( ) Area(f) = d 2 w f 1 R(f) 2 = e (w f ) One gets with D(f) X Area(f) = r ˆ R(f) 2 (f) r (f) = ( ) = ( ) f D = d 2 w e z z z z (w) (f) = 2log(R(f)) ( One can rewrite the Kähler form as a discretized version of the Faddeev-Popov determinant in Polyakov s formulation of two dimensional gravity and of non-crititical string theory! Indeed...

36 Functional integral over 2d Riemannian metrics, conformal gauge ˆ ˆ Faddeev-Popov ghost systems Integrating over the b s only = ( one gets ) = + The Kähler D form operator that appears is nothing but Therefore the discretised the FP determinant and ( ) g ab (z) = ab e (z) Faddev-Popov determin D[ ] = D[ ] ( field on the Voronoï lattice D[g ab ] = ( ) = ) = ( ) = ( ) = ( ) det( FP ) = ˆ = ( ) = ( ) = ( ) = ( ) = D[ ] ( ( ) + ( ) ˆ D[c, D[ b] ] exp = ( ) = ( ) ˆ ˆ det( FP ) = D[ ] det( FP ) d 2 z e (b zz ( c) zz + b z z ( c) z z ) ( ) D[c] exp d 2 z c z z c z = = = ( ) = ( ) = D = FP (f) = 2log(R(f)) ) = ( ( (b, c) plays the role of a discretized Liouville (

37 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Conclusion, open questions

38 We have an explicit quasi-conformal embedding of planar «dressed admissible» 2-dimensional maps onto the complex plane This point process is well defined for finite number of points It has many nice and interesting properties It is conformally invariant Its form is what is expected from continuum 2d quantum gravity But we would like to be able to characterise its «continuum limit», namely the limit when the density of points become infinite, and the corresponding statistical system This is much more difficult... renormalization group methods needed (work in progress) Any help and ideas from mathematicians is welcome Thank you for your interest!