Quantum LerayHirsch Chin-Lung Wang National Taiwan University - PowerPoint PPT Presentation

Quantum Leray–Hirsch Chin-Lung Wang National Taiwan University 2011, December 4 Pacific Rim Geometry Conference, Osaka 1 / 26

This is a joint work with Yuan-Pin Lee and Hui-Wen Lin. A general framework to determine g = 0 GW invariants: From I to J . Let τ = ∑ µ τ µ T µ ∈ H ( X ) , g µν = ( T µ , T ν ) , T µ = ∑ g µν T ν . q β T µ J X ( τ , z − 1 ) = 1 + τ � � ∑ z + z ( z − ψ ) , τ , · · · , τ n ! T µ 0, n + 1, β β ∈ NE ( X ) , n , µ q β T µ � � τ 1 z + ∑ τ z +( τ 1 . β ) T µ = e z ( z − ψ ) , τ 2 , · · · , τ 2 n ! e , 0, n + 1, β β � = 0, n , µ where τ = τ 1 + τ 2 with τ 1 ∈ H 2 ( X ) . Witten’s dilaton, string, and topological recursion relation in 2D gravity ⇐ ⇒ Givental’s symplectic space reformulation of GW theory. 2 / 26

Let H : = H ( X ) , H : = H [ z , z − 1 ]] , H + : = H [ z ] and H − : = z − 1 H [[ z − 1 ]] . H ∼ = T ∗ H + gives a canonical symplectic structure on H . ∞ k T µ z k ∈ H + . q µ q ( z ) = ∑ ∑ µ k = 0 The natural coordinates on H + are t ( z ) = q ( z ) + 1 z (dilaton shift), with t ( ψ ) = ∑ µ , k t µ k T µ ψ k ∈ H + the general descendent insertion. Let F 0 ( t ) be the generating function. The one form dF 0 gives a section of π : H → H + . Givental’s Lagrangian cone L = the graph of dF 0 . The existence of C ∗ action on L is due to the dilaton equation ∑ q µ k ∂ / ∂ q µ k F 0 = 2 F 0 . Thus L is a cone with vertex q = 0. − zJ : H → z H − is a section over τ ∈ H ∼ = − 1 z + H ⊂ H + . 3 / 26

Let L f = T f L be the tangent space of L at f ∈ L and L τ = L ( τ , dF 0 ( τ )) . (i) zL ⊂ L and so L / zL ∼ = H + / z H + ∼ = H has rank N : = dim H . (ii) L ∩ L = zL , considered as subspaces inside H . (iii) L is the tangent space at every f ∈ zL ⊂ L . Moreover, T f = L implies that f ∈ zL . Thus zL is the ruling of the cone. (iv) The intersection of L and the affine space − 1 z + z H − is parameterized by its image − 1 z + H ∼ = H ∋ τ under π . − zJ ( τ , − z − 1 ) = − 1 z + τ + O ( 1/ z ) is the function of τ whose graph is the intersection. (v) The set of all directional derivatives z ∂ µ J = T µ + O ( 1/ z ) spans L ∩ z H − ∼ = L / zL . 4 / 26

� Let R = C [ NE ( X )] be the ground (Novikov) ring. Denote a = ∑ q β a β ( z ) ∈ R { z } if a β ( z ) ∈ C [ z ] . All discussions are only as formal germs around the neighborhood of t = 0 ( q = − 1 z ). Lemma z ∇ J = ( z ∂ µ J ν ) forms a matrix whose column vectors z ∂ µ J ( τ ) generates the tangent space L τ of the Lagrangian cone L as an R { z } -module. In fact, by TRR, z ∇ J is the fundamental solution matrix of the Dubrovin connection on TH = H × H : ∇ z = d − 1 zd τ µ ⊗ ∑ T µ ∗ τ . µ Namely we have the quantum differential equation (QDE) z ∂ µ z ∂ ν J = ∑ ˜ C κ µν ( τ , q ) z ∂ κ J . 5 / 26

p : X → S be a smooth toric bundle with fiber divisor D = ∑ t i D i . Let ¯ H ( X ) is a free over H ( S ) with finite generators { D e : = ∏ i D e i i } e ∈ Λ . Let t s ¯ T s ∈ H ( S ) . H ( X ) has basis { T e = T ( s , e ) = ¯ ¯ t : = ∑ s ¯ T s D e } e ∈ Λ + . t s the ¯ Denote by ∂ ¯ T s ≡ ∂ ¯ T s directional derivative on H ( S ) , ∂ e = ∂ ( s , e ) : = ∂ ¯ t s ∏ ∂ e i t i , i and the naive quantization T e ≡ ∂ z e ≡ ∂ z ( s , e ) : = z ∂ ¯ t s ∏ z ∂ e i t i = z | e | + 1 ∂ ( s , e ) . ˆ i As usual, the T e directional derivative on H ( X ) is denoted by ∂ e = ∂ T e . This is a special choice of basis T µ (and ∂ µ ) of H ( X ) . ∂ z e and z ∂ e are very different, but they are also closely related. 6 / 26

p : X → S be a split toric bundle quotient from � L ρ → S . The Let ¯ hypergeometric modification of J S by the ¯ p -fibration takes the form D I X ( ¯ t , D , z , z − 1 ) : = q β e z +( D . β ) I X / S ( z , z − 1 ) J S t , z − 1 ) , ∑ β S ( ¯ β β ∈ NE ( X ) ( D ρ + L ρ ) . β where I X / S = ∏ ρ ∈△ 1 1/ ∏ ( D ρ + L ρ + mz ) comes from fiber β m = 1 localization, and the product is directed when ( D ρ + L ρ ) . β ≤ − 1. In general positive z powers may occur in I X . Nevertheless for each β ∈ NE ( X ) , the power of z in I X / S ( z , z − 1 ) is bounded above by a β constant depending only on β . I is defined only on the subspace ˆ t : = ¯ � t + D ∈ H ( S ) ⊕ i C D i ⊂ H ( X ) . Theorem (J. Brown 2009) ( − z ) I X ( ˆ t , − z ) lies in the Lagrangian cone L of X. 7 / 26

Definition (GMT) For each ˆ t , say zI ( ˆ t ) lies in L τ of L . The correspondence ˆ t �→ τ ( ˆ t ) ∈ H ( X ) ⊗ R is called the generalized mirror transformation . Proposition (BF) (1) The GMT: τ = τ ( ˆ t ) satisfies τ ( ˆ t , q = 0 ) = ˆ t. (2) Under the basis { T e } e ∈ Λ + , there exists an invertible N × N matrix-valued formal series B ( τ , z ) , the Birkhoff factorization, such that � � � � t , z , z − 1 ) z ∇ J ( τ , z − 1 ) ∂ z e I ( ˆ = B ( τ , z ) , where ( ∂ z e I ) is the N × N matrix with ∂ z e I as column vectors. The first column vectors are I and J respectively (string equation). 8 / 26

Proof zI ∈ L ⇒ z ∂ I ∈ T L = L . Then z ( z ∂ ) I ∈ zL ⊂ L and so z ∂ ( z ∂ ) I ∈ L . Inductively, ∂ z e I ∈ L . The factorization ( ∂ z e I ) = ( z ∇ J ) B ( z ) follows. t s ¯ From ˆ t = ∑ ¯ T s + ∑ t i D i , it is easy to see that t / z = T e e ˆ z ∂ e e t / z = T e e t / z . ∂ z e e ˆ t / z , t ) ≡ T e e ˆ Hence, modulo NE ( X ) , ∂ z e I ( ˆ t / z , z ∂ e J ( τ ) ≡ T e e τ / z . To prove (1), modulo all q β ’s we have t / z ≡ ∑ e ˆ B e ,1 ( z ) T e e τ ( ˆ t ) / z . e ∈ Λ + Thus e ( ˆ t − τ ( ˆ t )) / z ≡ ∑ B e ,1 ( z ) T e , e which forces that τ ( ˆ t ) ≡ ˆ t and B e ,1 ( z ) ≡ δ T e ,1 . Then we also have B ( τ , z ) ≡ I N × N . In particular B is invertible. This proves (2). 9 / 26

Theorem (BF/GMT) There is a unique, recursively determined, scalar-valued differential operator C e ∂ z e = 1 + P ( z ) = ∑ q β P β ( t i , ¯ t s , z ; z ∂ t i , z ∂ ¯ ∑ t s ) , e ∈ Λ + β ∈ NE ( X ) \{ 0 } with P β polynomial in z, such that P ( z ) I = 1 + O ( 1/ z ) . Moreover, t ) , z − 1 ) = P ( z ) I ( ˆ t , z , z − 1 ) , J ( τ ( ˆ with τ ( ˆ t ) being determined by the 1/ z coefficient of the right-hand side. Proof . We construct P ( z ) by induction on β ∈ NE ( X ) . We set P β = 1 for β = 0. Suppose that P β ′ has been constructed for all β ′ < β . We set P < β ( z ) = ∑ β ′ < β q β ′ P β ′ . Let A 1 = z k 1 q β ∑ e ∈ Λ + f e ( t i , ¯ t s ) T e be the top z -power term in P < β ( z ) I . If k 1 < 0 then we are done. 10 / 26

Otherwise we remove it via the “naive quantization” A 1 : = z k 1 q β ∑ e ∈ Λ + f e ( t i , ¯ ˆ t s ) ∂ z e . In ( P < β ( z ) − ˆ A 1 ) I = P < β ( z ) I − ˆ A 1 I , the term A 1 is removed since t / z = A 1 e ˆ t / z = A 1 + A 1 O ( 1/ z ) . A 1 e ˆ A 1 I ( q = 0 ) = ˆ ˆ All the newly created terms have curve degree q β ′′ with β ′′ > β in NE ( X ) . Thus we keep on removing the new top z -power term A 2 , which has k 2 < k 1 . The process stops in k 1 steps and we define P β by q β P β = − ∑ 1 ≤ j ≤ k 1 ˆ A j . By induction we get P ( z ) = ∑ β ∈ NE ( X ) q β P β as expected. Q: Is it possible to get explicit forms/analytic properties of P or B ? 11 / 26

From now on we work with the projective local model of a split P r flop f : X ��� X ′ with bundle data ( S , F , F ′ ) , where r r F ′ = L ′ � � F = L i and i . i = 0 i = 0 The contraction ψ : X → ¯ X has exceptional loci ¯ ψ : Z = P S ( F ) → S ψ ∗ F ′ ⊗ O Z ( − 1 ) . Similarly we have Z ′ ⊂ X ′ , N ′ . with N = N Z / X = ¯ ¯ p ψ p : X = P Z ( N ⊕ O ) → Z → S is a double projective The local model ¯ bundle . Leray–Hirsch = ⇒ for h , ξ being the relative hyperplane classes , H ( X ) = H ( S )[ h , ξ ] / ( f F , f N ⊕ O ) , where the Chern polynomials take the form (we identify L with c 1 ( L ) ) r r a i : = ∏ ( h + L i ) , b i : = ξ ∏ ( ξ − h + L ′ ∏ ∏ f F = f N ⊕ O = b r + 1 i ) . i = 0 i = 0 12 / 26

Γ f ] ∈ A ( X × X ′ ) induces an The graph correspondence F = [ ¯ isomorphism F : H ( X ) ∼ = H ( X ′ ) in the group level: for ¯ t ∈ H ( S ) , th i ξ j = ¯ t ( F h ) i ( F ξ ) j = ¯ t ( ξ ′ − h ′ ) i ξ ′ j , F ¯ i ≤ r . F also preserves the Poincar´ e pairing, but not the ring structure. Theorem (LLW 2010) F induces an isomorphism of quantum rings QH ( X ) ∼ = QH ( X ′ ) under analytic continuations in the K¨ ahler moduli formally defined by F q β = q F β , β ∈ NE ( X ) . Let γ , ℓ be the fiber line class in X → Z → S . Then F γ = γ ′ + ℓ ′ , but F ℓ = − ℓ ′ �∈ NE ( X ′ ) . So analytic continuations are necessary. Li–Ruan 2000 ( r = 1, dim X = 3), LLW 2006 (simple P r flop in any dimension, S = pt), LLW 2008 (simple flop, any g ≥ 0). 13 / 26