Introduction to the BKMP conjecture
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1 I P h T SGCSC, 3rd November Institut Henri Poincaré s a c l a y Introduction to the BKMP conjecture Gaëtan Borot
2 Introduction to the BKMP conjecture I Topological string theories A Gromov-Witten invariants II Calabi-Yau 3-folds and mirror symmetry in open sector : state of knowledge 1
3 String theory : Kähler target Topological if observables ind. of worldsheet metric Topological string theory SUSY sigma models with a twist Anomaly free string theory Superstring theories compactified on (no fluxes) SUSY Calabi-Yau Many observables vanish if we do not have Two types of theories twist A type IIA Kähler parameters of twist B type IIB periods 2
4 Topological string theory SUSY sigma models with a twist Twist A genus # branes homology class branes windings Amplitudes = Gromov-Witten invariants Closed sector Kähler parameters of couplings for type IIA Open sector branes end on a lagrangian submanifold 3
5 Topological string theory SUSY sigma models with a twist Twist B Amplitudes = encoding deformation of complex structure of X Other choice of moduli Genus 0 closed sector Genus 1 closed sector symplectic basis of related to Ray-Singer torsion of BRST symmetry Ward id. Holomorphic anomaly eqn. for Special geometry couplings for type IIB 4
6 Topological string theory SUSY sigma models with a twist Twist B Amplitudes = encoding deformation of complex structure of (2006) No general definition for stable closed sector open sector A model B model if exists mirror symmetry (2007) Remodeling the B model Bouchard, Klemm, Mariño, Pasquetti intrinsic definition of when mirror of a toric CY 5
7 Mathematical definition of GW invariants/a model Closed GW defined when Kähler Kontsevich (96), Li, Tian (96) ; etc. where Localization techniques when toric CY Graber, Pandharipande (97) where Hodge integrals Topological vertex (when toric CY) Aganagic, Klemm Mariño, Vafa (03) from = algorithm to compute open and closed GW Mathematical theory of the topological vertex Li, Liu, Liu, Zhou (04) 6 Other methods to compute GW
8 Definition A complex manifold is Calabi-Yau if it admits a nowhere vanishing volume form Theorem When is a compact Calabi-Yau (Yau) Definition Kähler parameters = real coordinates on All toric CY 3-folds can be constructed as : identification under action are the Kähler parameters Theorem is Calabi-Yau iff 7
9 All toric CY 3-folds can be constructed as : with Choose one triplet which is a coordinate patch We may realize as a fibration over fibre angles 2D picture (projection at ) 8
10 All toric CY 3-folds can be constructed as : with characterized by the degeneration locus of the fibre Example and act trivially acts trivially acts trivially acts trivially 9
11 All toric CY 3-folds can be constructed as : with characterized by the degeneration locus of the fibre Example 9
12 All toric CY 3-folds can be constructed as : with characterized by the degeneration locus of the fiber In each coordinate patch we may work out the action of toric diagram trivalent graph edges acts trivially edge length = Kähler parameter 10
13 A model B model When with Mirror defined by with id. under action are complexified Kähler parameters The mirror is a conic fibration over Locus of fibre degeneration defines the mirror curve such that Eqn. of 11
14 A model B model The mirror is a conic fibration over Locus of fiber degeneration = mirror curve Eqn. always of the form thickening of the toric diagram of 12
15 in open sector A model? B model? = We have to specify for which brane configuration In the A model construction A brane is supported by an edge of the toric diagram winding around the edge : momentum slope in fibre : framing generated by 13
16 in open sector A model? B model? = We have to specify for which brane configuration In the mirror theory (B model) A brane configuration is determined by position a choice of parametrization such that framing fixing 14 Beware : has nothing to do with the worldsheet
17 Prediction of mirror symmetry in open sector in the B model depends in fact on the choice of framings expansion near contains all information on open GW Schematically For a specific choice a brane edge & framings in A model is some Laplace transform of computed with same framings in B model 15
18 BKMP proposition in the B model, when are given by the topological recursion applied to The topological recursion Algorithm to associate to any plane curve differential forms numbers Bouchard, Klemm, Mariño, Pasquetti (07) Eynard, Orantin (05-...) Properties first, definition after Symplectic invariance under inv. inv. inv. inv. inv. inv. 16 Nice properties under variation of Integrability
19 Recipe for the topological recursion Data : and a heat kernel on For instance, when : Branchpoints : Recursion kernel : which are simple zeroes of Recursion on ( - Euler characteristic) Initialization Recursive def. by a residue formula To find the free energies 17
20 Geometrical interpretation of the topological recursion Initialization : disk amplitude cylinder amplitude Recursion kernel renormalized propagator Recursive def. by a residue formula 18
21 Geometrical interpretation of the topological recursion Initialization : disk amplitude cylinder amplitude Recursion kernel renormalized propagator Recursive def. by a residue formula 18
22 Geometrical interpretation of the topological recursion Initialization : disk amplitude cylinder amplitude Recursion kernel renormalized propagator Recursive def. by a residue formula 18
23 Geometrical interpretation of the topological recursion Initialization : disk amplitude cylinder amplitude Recursion kernel renormalized propagator Recursive def. by a residue formula 18
24 Geometrical interpretation of the topological recursion Initialization : disk amplitude cylinder amplitude Recursion kernel renormalized propagator Recursive def. by a residue formula 18
25 Geometrical interpretation of the topological recursion Initialization : disk amplitude cylinder amplitude Recursion kernel renormalized propagator To find the free energies 18
26 Geometrical interpretation of the topological recursion Initialization : disk amplitude cylinder amplitude Recursion kernel renormalized propagator To find the free energies combinatorial factor 18
27 BKMP proposition Bouchard, Klemm, Mariño, Pasquetti (07) constructed from the mirror curve Example (no Kähler parameter) at canonical framing at framing 1 branchpoint, at at zoom at branchpoint mirror symmetry Predicts that GW invariants of are encoded in of the curve of eqn. 19
28 Example at From Hodge integrals theory Open GW( ) at are related to simple Hurwitz numbers Ekedahl, Lando, Shapiro, Vainshtein (99) Coverings Riemann surface of genus Arbitrary ramification over Simple ramification elsewhere Predicts that simple Hurwitz numbers are encoded in for the curve of eqn. 20 Bouchard, Mariño (07)
29 Proofs available Simple Hurwitz numbers are encoded in for the curve of equation by matrix model representation properties of the topological recursion Borot, Eynard, Mulase, Safnuk (09) by combinatorics (Hodge integrals, cut-and-join) Eynard, Mulase, Safnuk (09) GW invariants of are encoded in for the curve of equation by combinatorics (Hodge integrals, cut-and-join) Zhou (09) we understand the basic block in the topological vertex ( = algorithm to compute all GW( ) for toric CY) 21
30 Towards a full proof Matrix model representation of GW( ) for any toric CY Eynard, Marchal, Kashani-Poor (10) But But - not easy to prove that the matrix model curve is equivalent to the mirror curve - not easy to establish to which brane configuration should correspond Can we prove the topological vertex from the topological recursion? i.e can we understand how and give 22
31 General references «Enumerative invariants in alg. geom. & string theory» Abramovich, Mariño, Thaddeus, Vakil Lect. Notes in Maths, 1947 (Springer, 2005) Review on the topological recursion Eynard, Orantin BKMP conjecture math-ph/ «Remodeling the B model» Bouchard, Klemm, Mariño, Pasquetti BKMP conjecture for Hurwitz numbers Bouchard, Mariño hep-th/ math.ag/ Towards a proof 23 Proof of BKMP for Hurwitz numbers Borot, Eynard, Mulase, Safnuk Proof of BKMP for Zhou «A matrix model for the topological string» Eynard, Marchal, Kashani-Poor math-ph/ math.ag/ hep-th/ hep-th/
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