Geometric Recursion by Jrgen Ellegaard Andersen Center for Quantum - PowerPoint PPT Presentation

Geometric Recursion by Jørgen Ellegaard Andersen Center for Quantum Geometry of Moduli Spaces Aarhus University Denmark at the Inaugural Conference of the Institute of the mathematical Sciences of the Americas September 8, 2019. Most of the work presented is joint with Gaëtan Borot and Nicolas Orantin.

Setting for Geometric Recursion. Consider the following setting: S = Category of compact oriented surfaces (Morphisms are isotopy classes of diffeo’s). V = Category of vector spaces. A functor E : S → V A functorial assigment Ω Σ ∈ E (Σ) for every object Σ of S . We note that in fact Ω Σ ∈ E (Σ) Γ Σ , where Γ Σ is the mapping class group of Σ .

Setting for Geometric Recursion. Consider the following setting: S = Category of compact oriented surfaces (Morphisms are isotopy classes of diffeo’s). V = Category of vector spaces. A functor E : S → V A functorial assigment Ω Σ ∈ E (Σ) for every object Σ of S . We note that in fact Ω Σ ∈ E (Σ) Γ Σ , where Γ Σ is the mapping class group of Σ . Many construction in low dim. geometry and topology fit in this framework:

Setting for Geometric Recursion. Consider the following setting: S = Category of compact oriented surfaces (Morphisms are isotopy classes of diffeo’s). V = Category of vector spaces. A functor E : S → V A functorial assigment Ω Σ ∈ E (Σ) for every object Σ of S . We note that in fact Ω Σ ∈ E (Σ) Γ Σ , where Γ Σ is the mapping class group of Σ . Many construction in low dim. geometry and topology fit in this framework: Ex. 1. The constant function one on Teichmüller space T Σ : E (Σ) = C 0 ( T Σ ) , Ω Σ = 1 ∈ E (Σ) Γ Σ Ex. 2. Sums over all simple closed multi-curves as a functions on Teichmüller space: � � E (Σ) = C 0 ( T Σ ) , Ω Σ ( σ ) = f ( l σ ( γ c )) , σ ∈ T Σ . γ ∈ S Σ c ∈ π 0 ( γ ) • S Σ = multi-curves = the set of isotopy classes of embedded closed 1-dim. manifolds in Σ , such that no component is isotopic to a boundary component, nor are any two different components isotopic. • f : R + → C is decaying sufficiently fast at infinity.

Setting for Geometric Recursion. Ex. 3. Functions on Teichmüller space via spectral theory: E (Σ) = C 0 ( T Σ ) , Ω Σ ( σ ) = Tr ( f ( − ∆ σ )) • f : R → C is sufficiently fast decaying at infinity and ∆ σ Dirichlet-Laplace-Beltrami operator on the Riemann surface Σ σ , σ ∈ T Σ . Ex. 4. Weil-Petersson symplectic form on Teichmüller space: E (Σ) = Ω 2 ( T Σ ) , Ω Σ = ω WP . Ex. 5. Bers complex structure I Bers on Teichmüller space: E (Σ) = C ∞ ( T Σ , End ( T T Σ )) , Ω Σ = I Bers . Ex. 6. Closed form on Teichmüller space: E (Σ) = Ω ∗ ( T Σ ) , Ω Σ ∈ Ω ∗ ( T Σ ) Γ Σ , d Ω Σ = 0. • Representing non-trivial cohomology classes on moduli space of curves M (Σ) = T Σ / Γ Σ . Ex. 7. Fock-Rosly Poisson structure P FR on moduli spaces of flat connections M G (Σ) : E (Σ) = C ∞ ( M G (Σ) , Λ 2 TM G (Σ)) , Ω Σ = P FR ∈ E (Σ) Γ Σ . • G any semi-simple Lie group either complex or real. Ex. 8. Narasimhan-Seshadri complex structure on moduli spaces of flat connections M G (Σ , c ) : E (Σ) = C ∞ ( T Σ , C ∞ ( M G (Σ , c ) , End ( TM G (Σ , c ))) , Ω Σ = I NS ∈ E (Σ) Γ Σ . • G any real semi-simple Lie group and c is an assignment of conjugacy classes to each boundary components of Σ , in which we assume the holonomy around each boundary component is contained.

Setting for Geometric Recursion. Ex. 9. Ricci potentials on the moduli spaces of flat connections M G (Σ , c ) : E (Σ) = C ∞ ( T Σ , C ∞ ( M G (Σ , c ))) , Ω Σ = F Ricci ∈ E (Σ) Γ Σ . Ex. 10. Hitchin’s Hyper-Kähler structure on moduli spaces of parabolic Higgs bundles: E (Σ) = C ∞ ( T Σ , C ∞ ( M G (Σ , c ) , End ( TM G (Σ , c ))) × 3 , Ω Σ = ( I , J , K ) Hitchin ∈ E (Σ) Γ Σ . • G is a complex semi-simple Lie group and c is as before. Ex. 11. Representations of mapping class groups ρ : Γ Σ → Aut ( V ) : E (Σ) = Ω 1 ( T , T × End ( V )) , Ω Σ = u ρ ∈ E (Σ) Γ Σ . Ex. 12. Boundary vectors in TQFT Z : Ω Σ = Z ( X 3 ) ∈ E (Σ) Γ X , ∂ X = Σ . E (Σ) = Z (Σ) , Ex. 13. Any invariant I 3 of closed oriented 3-manifolds: E (Σ) = C [ Heegaard diagrams ( α, β ) on Σ] ∗ , Ω Σ = I 3 ∈ E (Σ) Γ Σ , I 3 ( α, β ) = I 3 ( X 3 ( α,β ) ) . Ex. 14. Any invariant I 4 of smooth closed oriented 4-manifolds: E (Σ) = C [ Tri-section diagrams ( α, β, γ ) on Σ] ∗ , Ω Σ = I 4 ∈ E (Σ) Γ Σ , I 4 ( α, β, γ ) = I 4 ( X 4 ( α,β,γ ) ) . Ex. 15. Closed forms representing cohomology classes from Gromov-Witten Theory: E (Σ) = Ω ∗ ( T Σ ) , Ω Σ = ϕ GW ∈ E (Σ) Γ Σ . Ex. 16. Amplitudes in closed string theory: E (Σ) = Ω top ( T Σ ) , Ω Σ = A Σ ∈ E (Σ) Γ Σ .

Domain and target categories for Geometric Recursion. The category of surfaces we consider S : Objects: Compact oriented surfaces Σ of negative Euler characteristic with a marked point on each boundary component together with an orientation of the boundary, such that ∂ Σ = ∂ − Σ ∪ ∂ + Σ , and such that the inclusion map ∂ − Σ ⊂ Σ induces π 0 ( ∂ − Σ) ∼ = π 0 (Σ) . Morphisms: Isotopy classes of orientation preserving diffeomorphisms which preserves marked points and orientations on the boundary modulo isotopies which also preserves all this structure.

Domain and target categories for Geometric Recursion. The category of surfaces we consider S : Objects: Compact oriented surfaces Σ of negative Euler characteristic with a marked point on each boundary component together with an orientation of the boundary, such that ∂ Σ = ∂ − Σ ∪ ∂ + Σ , and such that the inclusion map ∂ − Σ ⊂ Σ induces π 0 ( ∂ − Σ) ∼ = π 0 (Σ) . Morphisms: Isotopy classes of orientation preserving diffeomorphisms which preserves marked points and orientations on the boundary modulo isotopies which also preserves all this structure. The category of vector spaces V : Objects: Hausdorff, complete, locally convex topological vector spaces over C . Morphisms: Morphisms of locally convex topological vector spaces.

Main Idea behind Geometric Recursion Suppose now we have a functor b 1 E : S → C . γ c c We want to recursively define for every object Σ of S b i Ω Σ c B b 1 .b i P c Ω Σ ∈ E (Σ) Γ Σ The B case. recursing in the Euler characteristic χ = χ (Σ) . The basic idea is b 1 γ 2 C b 1 c P c c to recursively remove pairs of pants which are embedded around the γ 1 c Ω Σ c components of ∂ − Σ , so that χ goes up by one in each step ending with χ = − 1 which is a pair of pants P or a one holed torus T . The C case. This will require: γ 2 Ω Σ c (2) b 1 C b 1 • Disjoint union morphisms: ⊔ : E (Σ 1 ) × E (Σ 2 ) → E (Σ 1 ⊔ Σ 2 ) c P c c γ 1 • Glueing morphisms: Θ β : E (Σ 1 ) × E (Σ 2 ) → E (Σ 1 ∪ β Σ 2 ) c Ω Σ c (1) for subset β ⊂ π 0 ( ∂ + Σ 1 ) × π 0 ( ∂ − Σ 2 ) consisting of disjoint pairs. • Starting data A ∈ E ( P ) Γ P , D ∈ E ( T ) Γ T giving Ω P = A , Ω T = D . • Recursion data B b ( b ∈ π 0 ( ∂ + P ) ), C ∈ E ( P ) . The C case. But in order to have mapping class group invariance persist through the recursion, we will also need to be able to make sense of the following infinite sum � � Θ b ′ ( B b , Ω Σ c ) + Ω Σ = Θ b , b ′ ( C , Ω Σ c ) . P ∈P B (Σ) P ∈P C (Σ) where P B (Σ) and P C (Σ) are the sets of isotopy classes of embeddings of pair of pants into Σ of type B and C respectively and ∂ + P = b ∪ b ′ .

Admissible initial data Definition Initial data for a given target theory E are assignments • A , C ∈ E ( P ) Γ P . • B b ∈ E ( P ) for b ∈ π 0 ( ∂ + P ) such that ϕ ( B b ) = B ϕ ( b ) for all ϕ ∈ Γ( P ) . • D ∈ E ( T ) Γ T .

Admissible initial data Definition Initial data for a given target theory E are assignments • A , C ∈ E ( P ) Γ P . • B b ∈ E ( P ) for b ∈ π 0 ( ∂ + P ) such that ϕ ( B b ) = B ϕ ( b ) for all ϕ ∈ Γ( P ) . • D ∈ E ( T ) Γ T . Definition The initial data is called admissible if A , B , C , D satisfies certain decay properties.

The recursion and the main existence theorem Let ( A , B , C , D ) be an admissible initial data for a target theory E . Definition • Ω ∅ := 1 ∈ E ( ∅ ) = K , • Ω P := A , • Ω T := D . For Σ a connected object of S with Euler characteristic χ (Σ) ≤ − 2 we seek to inductively define � � Ω Σ := 1 Θ b ′ ( B b • Θ b , b ′ ( C , Ω Σ c ) + P c , Ω Σ c ) 2 P ∈P C (Σ) P ∈P B (Σ) as an element of E (Σ) . For disconnected objects Σ , we declare � Ω Σ := Ω Σ( a ) . a ∈ π 0 (Σ) Theorem (Andersen, Borot and Orantin) The assignment Σ �→ Ω Σ is well-defined. More precisely, the above series defining Ω Σ converges absolutely for any of the seminorms of E (Σ) , and it is functorial. In particular, Ω Σ ∈ E (Σ) Γ Σ .