Counting surfaces of any topology, with Topological Recursion

Counting surfaces of any topology, with Topological Recursion 1... g 2... 3 n+1 = 1 g h h I 1 + J/I g 1 2 n+1 Quatum Gravity, Orsay, March 2013

Contents Outline 1. Introduction counting surfaces, discrete surfaces and Riemann surfaces 2. Topological Recursion - Definition - Interpretation - Application 3. Examples - Weil-Petersson volumes - Discrete surfaces, and continuous limit 4. General properties 5. Conclusion

1. Introduction: Counting surfaces

Counting Discrete surfaces In Combinatorics or Statistical Physics: count discrete surfaces ("maps") made of polygons, possibly carrying a "color": L 1 Generating series for counting: W g,n = v t v Σ M g,n(v) L 2 1 t #3 gons 3 t #4 gons 4 t #5 gons 5... #Aut x L 1+1 1 x L 2+1 2... xn Ln+1 g = genus, n = #boundaries, v = #vertices. Link with Tutte s equations [Tutte 60], with Random matrices [BIPZ 78], with trees [Schaeffer, BDG,...]

Counting Riemann surfaces In (topological) String theory or in Enumerative Geometry: "count" Riemann surfaces ("world-sheets") conformally embedded into a "target space" X: X conformal map world sheet Generating series for counting: W g,n = β e t.β Σ M g,n(x,β) Λ(Σ) #Aut(Σ) n i=1 e x i.#winding( i Σ) g = genus, n = #boundaries, β H 2 (X, Z) = #degree. Λ = cohomology class, e.g. Chern class, Hodge class,...

Continuum limit?? Question: limit of large discrete surfaces Riemann surfaces? 2 ways of viewing continuous surfaces: Quantun gravity CFT: Liouville CFT, KPZ,... Topological gravity: Witten Kontsevich, Intersection numbers, Gromov-Witten invariants, Chern classes, Chern-Simons,... in 1991, Kontsevich proved Witten s claim: generating series of topological gravity = KdV Tau-function (= scaling limit of random matrices = scaling limits of numbers of larges maps).

2. Topological Recursion

Topological Recursion Assume you know the "Disc" generating function W 0,1 (x): Disc : genus =0, # boundary=1. Assume you know the "Cylinder" generating function W 0,2 (x 1, x 2 ): Cylinder : genus=0, # boundary=2. 1 2 Out of them, define: Definition (the "Recursion kernel") K a ( 1, ) = 1 2 s a() W 0,2( 1,.) W 0,1 () W 0,1 (s a ()) 1

Topological Recursion Definition (Topological Recursion) The W g,n are said to satisfy the "topological recursion" iff: 1... g 2... 3 n+1 I h 1 = 1 + g 1 g h J/I 2 n+1 J {}}{ W g,n+1 (x 0, x 1,..., x n ) = [ Res K a(x 0, x) W g 1,n+2 (x, s a (x), J) x a a ] + W h,1+ I (x, I)W h,1+ I (s a (x), I ) dx h+h =g, I I =J

Example: Topological Recursion Example: W 1,2 2 1 Example: W 2,1 + + +

Interpretation: Topological Recursion 1... g 2 3 = 1... n+1 h g h I 1 + J/I g 1 2 n+1 Recursion kernel = "short": K ( 1, ) = 1 2 W 0,2( 1,.) W 0,1 () W 0,1 ( ) = 1 =

3. Examples of surfaces which satisfy the TR

Many surface counting problems do satisfy the TR Counting Discrete surfaces L 1 Counting Riemann surfaces with Weil-Petersson metrics Counting Riemann surfaces with products of Chern classes of cotangent bundles (Intersection numbers, Kontsevich) Counting Riemann surfaces with Hodge class (Hurwit numbers) = counting ramified coverings of the sphere Gromov-Witten Invariants = counting holomorphic maps of Riemann surfaces into a Calabi-Yau target space Many more... (random matrices, 2d, 3d Young tableaux, crystal growth models, knot theory,...) L2

Many surface counting problems do satisfy the TR Many more... (random matrices, 2d, 3d Young tableaux, crystal growth models, knot theory,...)

Weil-Petersson volumes M g,n = moduli space of Riemann surfaces of genus g, with n boundaries. decompose into 2g 2 + n "pairs of pants" 3g 3 + 2n geodesic lengths l i, twist angles θ i = Fenchel-Nielsen coordinates on M g,n. W g,n = M g,n i dl i dθ i j=boundaries e x j l j. l 1 l2 l 3 l 1 l2 l 3 Theorem W g,n s Weil-Petersson volumes satisfy the TR with W 0,1 (x) = 1 π sin 2π, W 0,2 (x 1, x 2 ) = 1 4 x 1 x 2 ( x 1 x 2 ) 2 Proof: TR recursion for W g,n Laplace transform of Mirakhani s recursions.

Counting Discrete surfaces L 1 L2 W g,n = v t v Σ M g,n(v) 1 t #3 gons 3 t #4 gons 4 t #5 gons 5... #Aut x L 1+1 1 x L 2+1 2... xn Ln+1 Theorem (E2004) Generating functions of discrete surfaces satisfy the topological recursion J {}}{ W g,n+1 (x 0, x 1,..., x n ) = Res K (x 0, x) x a + h+h =g, I I =J [ W g 1,n+2 (x, x, J) ] W h,1+ I (x, I)W h,1+ I (x, I )

Counting Discrete surfaces Proof: by Tutte s recursion = edge removal, then, use Cauchy residue formula and move integration contours. Remark: Tutte s equation = bijective, but moving integration contours bijective.,

Limit of Large Discrete surfaces W g,n = v t v Σ M g,n(v) 1 t #3 gons 3 t #4 gons 4 t #5 gons 5... #Aut x L 1+1 1 x L 2+1 2... xn Ln+1 Take a scaling limit t t c, t k t kc, x i a c, then: Theorem (Bergère, E 2009) W g,n (x 1,..., x n ; t 3, t 4,... ; t) (t t c ) (2 2g n)µ+nν Wg,n ( x i a c (t t c ) ν ; t k t kc (t t c ) ν k ) (1 + o(t t c)) where exponents ν, µ, ν k given by KPZ, and W g,n satisfy the TR, and W g,n s are determinants of KdV Baker-Akhieer kernel. In other words: asymptotic number of large maps topological gravity (Witten s conjecture).

4. General Properties of TR

General properties of the F g and W (g) n For arbitrary W 0,1 (and W 0,2 =fund. 2nd kind form of W 0,1 ), the W g,n have many interesting properties: Integrability (formal) Hirota equation, MKP hierarchy Symplectic invariance F g = W g,0 is left invariant under y = W 0,1 (x) ỹ = W 0,1 ( x) if they ate related by transformations which conserves dx dy = d x dỹ. Modular properties Holomorphic anomaly equation Singular limits if the curve E becomes singular, then scaling lim W g,n satisfy TR. Dilaton equation. W g,n =homogeneous of deg 2 2g n. Diagramatic representation: Givental s formalism, trees... g-trees map" bijection? and many other properties... etc

5. Conclusion

Conclusion: The TR is a universal recursion relation satisfied by many enumeration problems (also appears in random matrices, enumeration of partitions, 3d partitions,...) Once we know W 0,1 and W 0,2, the TR is very efficient at actually computing the higher genus generating functions. Question: find geometric (bijective) proofs of TR for various models? (for the moment all known proofs use non-bijective steps). Further prospects: other generaliations of the topological recursion with boundary operators. What does it compute on the A-model side? link with AdS/CFT? recent new conjecture: topological recursion with S =A-polynomial, computes asymptotics of Jones polynomial of knots? [Dijkgraaf-Fuji-Manabe 2010, Borot-E 2012]. Find a proof??? categorification refinement, AGT

The end Thank you for your attention Acknowledgments: This work was supported by: by the ANR project GranMa ANR-08-BLAN-0311-01, by the ANR project GIMP Géométrie et intégrabilité en physique mathématique ANR-05-BLAN-0029-01, by the European Science Foundation through the Misgam program, by the Quebec government with the FQRNT.