Proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian $LG(2,4)$
Fangze Sheng


Refined Dubrovin conjecture for the Lagrangian Grassmanian LG(2,4)]Proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian LG(2,4)

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China
fzsheng2000@mail.ustc.edu.cn

§ ABSTRACT  
The Dubrovin conjecture predicts a relationship between the monodromy data of the Frobenius manifold associated to the quantum cohomology of a smooth projective variety and the bounded derived category of the same variety. A refinement of this conjecture was given by Cotti, Dubrovin and Guzzetti, which is equivalent to the Gamma conjecture II proposed by Galkin, Golyshev and Iritani. The Gamma conjecture II for quadrics was proved by Hu and Ke. The Lagrangian Grassmanian LG(2,4) is isomorphic to the quadric in ℙ^4. In this paper, we give a new proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian LG(2,4) by explicit computation. 

[
    Fangze Sheng
    September 9, 2024
=====================










§ INTRODUCTION



The monodromy data of a semisimple Dubrovin–Frobenius manifold M <cit.> consist of four constant matrices (μ, R, S, C). Here, μ and R are given by monodromy data of the ODE system associated to the extended Dubrovin connection on M ×ℂ^* at z=0, the matrix S, called the Stokes matrix, describes asymptotic behaviours of solutions to the same system as |z| →∞, and the matrix C, called the central connection matrix, relates two canonically-constructed analytic solutions. Dubrovin proved that one can reconstruct the Dubrovin–Frobenius structure of a semisimple Dubrovin–Frobenius manifold from its monodromy data (μ, R, S, C) <cit.>. 

Let X be a D-dimensional smooth projective variety with vanishing odd cohomology, and X_0,k,β the moduli space of stable maps of degree β∈ H_2(X;ℤ)/ torsion with target X from curves of genus 0 with k distinct marked points.  Choose a homogeneous basis ϕ_1 = 1,ϕ_2,…,ϕ_n of the cohomology ring H^*(X;ℂ) such that ϕ_α∈ H^2q_α(X;ℂ), α=1,…,n,  0=q_1<q_2 ≤…≤ q_n-1 < q_n = D. The Gromov–Witten (GW) invariants of X of genus 0 and degree β are the integrals <cit.>

    ∫_[X_0,k,β]^ virt ev_1^*(ϕ_α_1) ⋯ ev_k^*(ϕ_α_k),   α_1,…,α_k=1,…,n.
 
Here, ev_a, a=1,…,k, are the evaluation maps, and [X_0,k,β]^ virt is the virtual fundamental class. The genus 0 primary free energy F_0( v;Q) for the GW invariants of X is defined by

    F_0( v;Q) = ∑_k≥0∑_α_1,…,α_k ≥0∑_βQ^β v^α_1⋯ v^α_k/k!∫_[X_0,k,β]^ virt ev_1^*(ϕ_α_1) ⋯ ev_k^*(ϕ_α_k),

where (v^1,…,v^n) are indeterminates, and 

    Q^β=Q_1^m_1⋯ Q_r^m_r    ( for β=m_1β_1+…+m_rβ_r)

is an element of the Novikov ring ℂ [[Q_1,…,Q_r]]. 
Here, (β_1,…,β_r) is a basis of H_2(X;ℤ)/ torsion. When X is Fano, it is known that F_0( v;Q) is a power series of v^1, Q_1 e^v^2, …, Q_r e^v^r+1, v^r+2, …, v^n and one can 
take Q_1=⋯=Q_r=1 (cf. <cit.>). If F_0( v):=F_0( v; Q)|_Q_1=⋯=Q_r=1 is convergent, then an analytic Dubrovin–Frobenius structure can be defined on its convergence domain. The quantum cohomology of X is the commutative ring given by flat vector fields on this Dubrovin–Frobenius manifold, and its restriction to H^2(X;ℂ) is called the small quantum cohomology of X. The quantum cohomology of X carries a canonical calibration <cit.> obtained from genus 0 GW invariants, which can also be fixed by using an analytic criterion <cit.> (cf. <cit.>). 









 
In his 1998 ICM talk <cit.>, Dubrovin proposed a conjecture on the relationship between monodromy data of the quantum cohomology of a smooth projective variety and the bounded derived category of the same variety. We refer the readers to Section <ref> for a precise statement of the conjecture (see Conjecture <ref>) and for a refinement proposed by Cotti, Dubrovin and Guzzetti <cit.> (see Conjecture <ref>). Note that part 3 of the refined Dubrovin conjecture <ref> is equivalent <cit.> to the Gamma conjecture II proposed by Galkin, Golyshev and Iritani <cit.>. Recently, some new significance of the Dubrovin conjecture (and its refined version) is given in <cit.>. 

In the past two decades, the Dubrovin conjecture has been proved for some cases. Dubrovin proposed a method to directly compute the monodromy data in <cit.>, where the Stokes matrix for the quantum cohomology of ℙ^2 was computed. In <cit.>, Dubrovin computed the central connection matrix for ℙ^2, and proved the Dubrovin conjecture for this case. The Stokes matrix for an arbitrary complex projective space was computed by Guzzetti in <cit.>, and part 2 of Conjecture <ref> was proved (part 1 of Conjecture <ref> for this case was known). The same results were also obtained by Tanabé in <cit.>. 

Recently, the refined Dubrovin conjecture for an arbitrary Grassmanian was proved by Cotti, Dubrovin and Guzzetti <cit.> (cf. <cit.>). 

In <cit.>, Cotti computed the monodromy data and then proved the refined conjecture for Hirzebruch surfaces. Part 2 of Conjecture <ref> has been verified for other cases, including cubic surfaces by Ueda <cit.>, minimal Fano threefolds by Golyshev <cit.>, a class of orbifold projective lines by Iwaki and Takahashi <cit.>, weighted projective spaces and certain smooth Fano hypersurfaces by Cruz Morales and van der Put <cit.>. The Gamma conjecture II has been proved for arbitrary Grassmanians in <cit.> (cf. <cit.>), for quadrics of arbitrary dimensions by Hu and Ke <cit.> and for certain toric Fano manifolds by Fang and Zhou <cit.>.









Let V be the vector space ℂ^2n, equipped with a symplectic form. A subspace Σ of V is called isotropic if the restriction of the symplectic form to Σ vanishes. The maximal possible dimension of an isotropic subspace is n, and in this case Σ is called a Lagrangian subspace. The Lagrangian Grassmanian LG(n,2n) is the projective complex manifold which parameterizes Lagrangian subspaces Σ in V. A description of the small quantum cohomology of LG(n,2n) was formulated by Kresch and Tamvakis by quantum Schubert calculus <cit.> (cf. <cit.>). The existence of a full exceptional collection in the bounded derived category of coherent sheaves on a Lagrangian Grassmanian was proved for LG(3,6) by Samokhin <cit.>, for LG(4,8) by Polishchuk and Samokhin <cit.>, and for LG(n,2n), n ≥ 1 by Fonarev <cit.>. 


The main result of this paper is the following theorem. 

    The refined Dubrovin conjecture holds for the Lagrangian Grassmanian LG(2,4). 




Since LG(2,4) is isomorphic to the quadric in ℙ^4, Theorem <ref> can also be deduced from the corollary of Hu-Ke <cit.>. In this paper, we will give an explicit proof by computing the monodromy data for the quantum cohomology of LG(2,4). 

The rest of the paper is organized as follows. In Section <ref>, we review the monodromy data of semisimple Dubrovin–Frobenius manifolds. In Section <ref>, we recall some basic notions of derived category, and state the Dubrovin conjecture and a refined version. In Section <ref>, we briefly review some results of the small quantum cohomology of LG(2,4) and then give our proof of Theorem <ref>.




§ REVIEW ON MONODROMY DATA OF SEMISIMPLE DUBROVIN–FROBENIUS MANIFOLDS


A Frobenius algebra is a triple (A,   e,  ⟨ , ⟩), where A is a commutative and associative algebra over ℂ with unity e, and ⟨ , ⟩: A× A→ℂ is a symmetric, non-degenerate and bilinear product satisfying 
⟨ x· y , z⟩ = ⟨ x, y· z⟩, ∀ x,y,z ∈ A. 
A Dubrovin–Frobenius structure of charge D <cit.> on a complex manifold M is a family of 
Frobenius algebras (T_p M,   e_p,  ⟨ , ⟩)_p∈ M, 
depending holomorphically on p and satisfying the following three axioms:


  1  The metric ⟨ , ⟩ is flat; moreover, 
denote by ∇ the Levi–Civita connection of ⟨ , ⟩, then it is required that ∇ e = 0. 

  2 Define a 3-tensor field c by 
c(X,Y,Z):=⟨ X· Y,Z⟩, for X,Y,Z being holomorphic vector fields on M. 
Then the 4-tensor field ∇ c is required to be 
symmetric. 

  3
There exists a holomorphic vector field E on M satisfying 

    [E, X· Y]-[E,X]· Y - X· [E,Y] = X· Y, 
         E ⟨ X,Y ⟩ - ⟨ [E,X],Y⟩ - ⟨ X, [E,Y]⟩ 
    = (2-D) ⟨ X,Y⟩.


A complex manifold endowed with a Dubrovin–Frobenius structure of charge D is  
called a Dubrovin–Frobenius manifold of charge D, with ⟨  , ⟩ 
being called the invariant flat metric and E the Euler vector field.
We denote the Dubrovin–Frobenius manifold by (M, ·,  e, ⟨ , ⟩,  E). 

A point p on a Dubrovin–Frobenius manifold M is called a semisimple point, if T_pM is semisimple. 
A Dubrovin–Frobenius manifold is called semisimple, if its generic points are semisimple points.

For an n-dimensional Dubrovin–Frobenius manifold (M,  ·,   e,  ⟨ , ⟩, E), an flat affine connection ∇̃, called extended Dubrovin connection, can be defined <cit.> on M×ℂ^* through 
    
    ∇̃_X Y   :=∇_X Y + z X · Y,
            
    ∇̃_∂_z Y   := ∂ Y∂ z + E · Y- 1z(2-D/2 id - ∇ E)Y,
            
    ∇̃_∂_z∂_z   :=0,     ∇̃_X∂_z :=0.

Here, X,Y are arbitrary holomorphic vector fields on M ×ℂ^* with 
zero component along ∂/∂ z.
Choose flat coordinates v=(v^1,…,v^n) such that E=∑_β=1^n ((1-D/2-μ_β) v^β + r^β) ∂/∂ v^β. The differential system for a flat section y=(y^1,…,y^n)^T reads (see <cit.>)

    ∂ y/∂ v^α = z C_α y,   α=1,…,n, 
       d y/d z = (𝒰+μ/z) y,

where C_α is the matrix of multiplication by ∂/∂ v^α, 𝒰 is the matrix of multiplication by the Euler vector field and μ = diag(μ_1,…,μ_n).

Fix v. Then there exists a fundamental matrix solution to (<ref>) of the form <cit.> 

    𝒴_0(z) = Φ(z) z^μ z^R,

where R is a constant matrix, Φ(z) is an analytic matrix-valued function on ℂ satisfying 

    Φ(0) = I,   Φ(-z)^TηΦ(z)= η.

The matrices μ,R are the monodromy data at z=0 of the Dubrovin–Frobenius manifold <cit.>. 
 

Now we restrict our consideration to a semisimple point p∈ M.  Denote the restriction of coordinate vector fields (∂/∂ v^1,…,∂/∂ v^n) to this point by (e_1,…,e_n). Let u_1,…, u_n be eigenvalues to 𝒰 and assume they are pairwise distinct. Let f_1,…,f_n be the normalized eigenvectors such that

    f_i · f_j = δ_ij f_j and ⟨  f_i ,  f_j ⟩ = δ_ij,      i,j=1,…,n.

Let Ψ=(ψ_iα) be the transition matrix from the orthonormal basis (f_1,…,f_n) to the basis (e_1,…,e_n), i.e.,

    e_α=∑_i=1^n ψ_iα f_i ,   α=1,…,n.
 
For any fundamental matrix solution 𝒴(z) to (<ref>), let Y(z)=Ψ𝒴(z). Then Y=Y(z) satisfies the equation 

    dY/dz = (U+V/z) Y ,

where U = Ψ𝒰Ψ^-1 =  diag(u_1,…,u_n) and V=ΨμΨ^-1. 

Note that z=∞ is an irregular singularity of (<ref>) 
of Poincaré rank 1. Then <cit.> (cf. <cit.>) 
 equation (<ref>) has a unique formal solution Y_ formal(z) of the form

    Y_ formal(z) = G (z) e^z U,

where G (z) is a formal power series of z^-1 with G(∞)=I and satisfies the orthogonality condition G (-z)^T G (z) = I. 

A line ℓ through the origin in the complex z-plane is called admissible if

    Re  z(u_i- u_j)|_z ∈ℓ∖ 0≠ 0,   ∀  i≠ j.

Fix any admissible line ℓ with an orientation. Denote by ϕ∈[0,2π) the angle from the positive real axis and to the positive direction of ℓ. 
Construct two sectors

    Λ_ left: ϕ-ε< arg  z< ϕ + π+ ε, 
    Λ_ right: ϕ-π-ε< arg  z< ϕ + ε,

where ε is a sufficiently small positive number. From <cit.>(cf. <cit.>)
there exist unique fundamental matrix solutions Y_L/R(z) to the ODE system (<ref>), analytic in Π_ left/ right, such that 

    Y_L/R(z) ∼ Y_ formal(z)

as |z|→∞ within the sectors. 
We will consider the behaviour of the matrix solutions on the narrow sectors

    Λ_+: ϕ-ε <  arg z < ϕ+ε,

and 

    Λ_-: ϕ-π-ε <  arg z < ϕ-π+ε.



In fact, the sectors Λ_ left/ right where the asymptotic behaviour (<ref>) holds can be extended. 
We introduce Stokes rays, which are oriented rays R_ij in ℂ defined by

    R_ij:={-i(u_i-u_j)ρρ≥ 0}.

By definition, an admissible line ℓ must contain no Stokes rays. The sectors Λ_ left/ right can be extended up to the first nearest Stokes ray (see <cit.>). We denote the extended sectors by Π_ left/ right and the corresponding extended narrow sectors by Π_+ and Π_-. 
Within the narrow sector Π_+, the two analytic fundamental matrix solutions in (<ref>)  is related by a constant matrix S called the Stokes matrix with respect to the 
admissible line ℓ (cf. <cit.>):

    Y_L(z) =Y_R(z) S,    z∈Π_+.
 
Similarly, according to <cit.> (cf. <cit.>), in the narrow sector Π_-, we have

    Y_L(z) =Y_R(z) S^T,    z∈Π_-.
 
Within the narrow sector Π_+, the fundamental matrix solutions 
Y_0(z):=Ψ𝒴(z) and Y_R(z) 
to the ODE system (<ref>) also be related by a constant matrix C called the central connection matrix with respect to the 
admissible line ℓ: 

    Y_R(z) = Y_0(z) C.

When M is semisimple, the quadruple (μ, R, S, C) is called the monodromy data of M <cit.>.






S and C satisfy the following restrictions:

    C S^T S^-1 C^-1 = e^2π i μ e^2π i R,


    S = C^-1 e^-π i R e^-π i μη^-1 (C^T)^-1.
  



Following Dubrovin <cit.>, we consider an action of the braid group ℬ_n on the set of monodromy data (S, C). Denote by β_1,2,…, β_n-1,n the generators of ℬ_n, satisfying the relations 

    β_i,i+1β_i+1,i+2β_i,i+1=β_i+1,i+2β_i,i+1β_i+1,i+2,
    β_i,i+1β_j,j+1=β_j,j+1β_i,i+1,  if  |i-j|>1.

Let ℓ be an admissible line and (S, C) be the monodromy data with respect to ℓ. Choose the order of canonical coordinates (u_1,…,u_n) such that S is an upper triangular matrix. The action of the elementary braid β_i,i+1∈ℬ_N, 1≤ i ≤ n-1 has the form <cit.>

    β_i,i+1(S):= K^β_i,i+1(S)  S  K^β_i,i+1(S),
      β_i,i+1(C):= C  (K^β_i,i+1(S))^-1,

where

    (K^β_i,i+1(S))_kk=1,     k=1,…, N   k≠ i,i+1,


    (K^β_i,i+1(S))_i+1,i+1=-s_i,i+1,


    (K^β_i,i+1(S))_i,i+1=(K^β_i,i+1(S))_i+1,i=1,

and all other entries of the matrix K^β_i,i+1 are equal to zero. 



An important class of Dubrovin–Frobenius manifolds come from quantum cohomology. To be precise, let X be a D-dimensional smooth projective variety with vanishing odd cohomology, and X_g,k,β the moduli space of stable maps of degree β∈ H_2(X;ℤ)/ torsion with target X from curves of genus g with k distinct marked points. Here, g,k≥0. We denote the Poincaré pairing on 
H^*(X;ℂ) by ⟨ , ⟩. Choose a homogeneous basis ϕ_1 = 1,ϕ_2,…,ϕ_n of the cohomology ring H^*(X;ℂ) such that ϕ_α∈ H^2q_α(X;ℂ), α=1,…,n, 
 0=q_1<q_2 ≤…≤ q_n-1 < q_n = d. 
 
The integrals

    ∫_[X_g,k,β]^ virt c_1(ℒ_1)^i_1 ev_1^*(ϕ_α_1) ⋯ c_1(ℒ_k)^i_k ev_k^*(ϕ_α_k),   α_1,…,α_k=1,…,l,   i_1,…,i_k≥0,

are called Gromov–Witten (GW) invariants of X of genus g and degree β <cit.>. Here, ev_a, a=1,…,k are the evaluation maps, ℒ_a is the ath tautological line bundle on X_g,k,β, 
and [X_g,k,β]^ virt is the virtual fundamental class.
The genus g primary free energy F_g( v;Q) for the GW invariants of X is defined by 

    F_g( v;Q) = ∑_k≥0∑_α_1,…,α_k ≥0∑_βQ^β v^α_1⋯ v^α_k/k!∫_[X_g,k,β]^ virt ev_1^*(ϕ_α_1) ⋯ ev_k^*(ϕ_α_k),

where (v^1,…,v^n) are indeterminates, and 

    Q^β=Q_1^m_1⋯ Q_r^m_r   ( for β=m_1β_1+…+m_rβ_r)

is an element of the Novikov ring ℂ [[Q_1,…,Q_r]]. Here, (β_1,…,β_r) is a basis of H_2(X;ℤ)/ torsion. 

Now we restrict our consideration to the case that X is Fano. Then for any fixed k≥0 and α_1,…,α_k ∈{1,…,n}, 
the sum ∑_β in (<ref>) is a finite sum. Moreover, by the divisor equation <cit.>

    ∫_[X_g,k+1,β]^virev_1^*(ϕ_α_1)⋯ev_k^*(ϕ_α_k)ϕ=(∫_βϕ) ∫_[X_g,k,β]^virev_1^*(ϕ_α_1)⋯ev_k^*(ϕ_α_k).
 
we know that 
F_0( v;Q) is a power series of v^1, Q_1 e^v^2, …, Q_r e^v^r+1, v^r+2, …, v^n. Take Q_1=⋯=Q_r=1. 
Denote 

    F_0( v):=F_0( v; Q)|_Q_1=⋯=Q_r=1.

Assume that the power series F_0( v) has a convergence domain Ω. Then F_0( v) leads to an analytic Dubrovin–Frobenius structure of charge D on Ω <cit.>, with the invariant flat metric ⟨ , ⟩, the unity vector field e=∂/∂ v^1 and the Euler vector field E given by 

    E=c_1(X)+∑_α=1^n(1-12degϕ_α)v^αϕ_α,

where c_1(X) is the first Chern class of X. We call the commutative ring given by flat vector fields on the Dubrovin–Frobenius manifold by the quantum cohomology of X, denoted by QH(X), and the restriction of this ring to Ω∩ H^2(X;ℂ) (i.e. points with coordinates v^i=0 unless i=2,…, r+1) the small quantum cohomology of X. Note that the constant matrix R in (<ref>) can be chosen to correspond to  c_1(X) ∪ H^*(X;ℂ)→ H^*(X;ℂ) <cit.>. 


For the quantum cohomology of a smooth projective variety X, a canonical calibration <cit.>  obtained from genus 0 GW invariants can be chosen. 



Fix a point whose coordinates are v=(v^1,…,v^n) on the quantum cohomoloy of a smooth projective variety X. Then a calibration at this point is given by

    𝒴_ top(z):=Φ_ top(z) z^μ z^R,
    Φ_ top(z)_λ^γ:   =δ_λ^γ+∑_k,l≥ 0∑_β∑_α_1,…,α_kh_λ,k,l,β,α^γ/k! v^α_1… v^α_k z^l+1.

Here

    h_λ,k,l,β,α^γ:=∑_ϵη^ϵγ∫_[X_0,k+2,β]^virtc_1(ℒ_1)^l  ev_1^*(ϕ_λ)  ev_2^*(ϕ_ϵ)∏_j=1^k ev^*_j+2(ϕ_α_j).


We call (<ref>) the topological-enumerative solution of (<ref>) .


In <cit.>, Galkin, Golyshev and Iritani gave an analytic criterion to obtain the above calibration when X is Fano. 

Let X be a Fano manifold and fix a point v on the small quantum cohomology of X (i.e. point with coordinates v^i=0 unless i=2,…, r+1). Then among all solutions of the form

    Φ(z)z^μ z^R
 to (<ref>) around z=0, the topological-enumerative solution (<ref>) is the unique one for which the product

    H(z)=z^-μΦ(z)z^μ,
 
is holomorphic at z=0 and satisfies

    H(0)=exp((∑_i=2^r+1 v^i ϕ_i)∪(-)).










§ EXCEPTIONAL COLLECTIONS, GAMMA CLASSES AND A REFINED VERSION OF THE DUBROVIN CONJECTURE
 Let X be a smooth projective variety of complex dimension D with odd-vanishing cohomology, and denote by 𝒟^b(X) the bounded derived category of coherent sheaves on X.  Given E,F∈ Ob(𝒟^b(X)), define Hom^*(E,F) as the ℂ-vector space

    Hom^*(E,F):=⊕_k∈ℤHom(E,F[k]).

A collection (E_1,…, E_n) of objects of 𝒟^b(X) is called an exceptional collection if 


  * Hom^*(E_k,E_k) is a one dimensional ℂ-algebra for each 1≤ k ≤ n, generated by the identity morphism,

  * we have Hom^*(E_j,E_k)=0 for j>k.

An exceptional collection is full if any triangular subcategory containing all objects of this collection is equivalent to 𝒟^b(X) via the inclusion functor.


Now let TX be the tangent bundle of X, and let δ_1,…,δ_D be the Chern roots of TX, so that 

    c_k(TX)=σ_k(δ_1,…,δ_r),   1≤ k≤ D,
where σ_k is the k-th elementary symmetric polynomial.
The Gamma classes of X are the characteristic classes (cf. <cit.>)

    Γ_X^±:=∏_j=1^DΓ(1±δ_j),

where γ is the Euler-Mascheroni constant and ζ is the Riemann zeta function.

Consider an arbitrary vector bundle V on X, and let τ_1,…,τ_r be the Chern roots of V. 
The Chern character of V is the characteristic class Ch(V)∈ H^*(X) defined by Ch(V):=∑_j=1^r e^2π i τ_j.


The following conjecture was proposed by Dubrovin in his 1998 ICM talk.

 Assume that X is Fano and that the power series F_0(v) has a convergence domain Ω. Then

    
  1. the quantum cohomology of X is semisimple if and only if 𝒟^b(X) admits a full exceptional collection. 

If the quantum cohomology of X is semisimple, then there exists a full exceptional collection (E_1,…, E_n) of 𝒟^b(X) such that: 

    
    
  2. the Stokes matrix S is equal to the inverse of the Euler matrix (χ(E_j,E_k))_1≤ j,k ≤ n;
    
  3. the central connection matrix has the form C=C'C” where the columns of C” are coordinates of Ch(E_i), and C'  H^*(X;ℂ)→ H^*(X;ℂ) is an operator commuting with c_1(X)∪ H^*(X;ℂ)→ H^*(X;ℂ).



 

In <cit.>, Dubrovin suggested that the matrix C' should be given by using the Gamma class of X. In <cit.>, Cotti, Dubrovin and Guzzetti proposed a refinement of Conjecture <ref>. 


    Assume that X is a Fano variety of dimension D with odd-vanishing cohomology and that the power series F_0(v) has a convergence domain Ω. Then

    
  1. the quantum cohomology of X is semisimple if and only if 𝒟^b(X) admits a full exceptional collection. 

If the quantum cohomology of X is semisimple, then there exists a full exceptional collection (E_1,…, E_n) of 𝒟^b(X) such that: 

     
    
  2. the Stokes matrix S is equal to the inverse of the Euler matrix (χ(E_j,E_k))_1≤ j,k ≤ n;
    
  3. the central connection matrix C, connecting the solution Y_R (see (<ref>)) with the topological-enumerative solution Y_ top=Ψ·𝒴_ top of Proposition <ref>, is equal to the matrix whose columns are given by the coordinates of 
    
    i^D̅/(2π)^D/2Γ^-_X∪exp(-π i c_1(X))∪ Ch(E_k),    1≤ k ≤ n.

    Here D̅ = D mod 2. 




      In <cit.>, Cotti, Dubrovin and Guzzetti proved that part 3 of Conjecture <ref> is equivalent to the Gamma conjecture II proposed by Galkin, Golyshev and Iritani <cit.>.


    In several occasions, the convergence assumption in Conjecture <ref> and <ref> is true. In <cit.>, Dubrovin suggested that the convergence can be proved when the quantum cohomology of X is semisimple. Cotti then proved that if the small quantum cohomology of X is semisimple, then F_0 has a convergence domain <cit.>. 














§ PROOF OF THE REFINED DUBROVIN CONJECTRUE FOR LG(2,4)






 §.§ Small quantum cohomology of LG(2,4)

It is known that the Lagrangian Grassmanian LG(2,4) is isomorphic to the 3-dimensional quadric. 
The cohomology ring H^*(LG(2,4);ℂ) has the following ring presentation (cf. <cit.>)

    H^*(LG(2,4);ℂ) ≅ℂ[x_1,x_2]/⟨ x_1^2 - 2 x_2,  x_2^2 ⟩.


From the general theory of Schubert calculus <cit.>, it is known that H^*(LG(2,4);ℂ) is a complex linear space of dimension 4 and a basis is given by Schubert classes: 

    σ_0:=1,  σ_1,  σ_2,  σ_2,1.

We denote them by e_1:=σ_0, e_2:=σ_1, e_3:=σ_2, e_4:=σ_2,1, and denote by v^i the coordinate with respect to e_i. Then the coordinates in the small quantum cohomology are

    v=(0,v^2,0,0).

The corresponding Schubert polynomials are
    
    𝔖_0 = 1 , 
         
       𝔖_1 = x_1 , 𝔖_2= 1/2x_1^2 , 
         
       𝔖_2,1=1/2x_1^3,

which are images of the Schubert classes under the isomorphism (<ref>). 
The first Chern class of LG(2,4) is c_1=3σ_1. The matrix of the multiplication c_1∪ under the basis (<ref>) is

    R = (
    [ 0 0 0 0; 3 0 0 0; 0 6 0 0; 0 0 3 0;   ]).

The matrix of Poincaré pairing 

    η(α,β):=∫_LG(2,4)α∧β

is 

    η = (
    [ 0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0;   ]).


 
The small quantum cohomology ring qH^*(LG(2,4);ℂ) has the following ring presentation <cit.>: 

    qH^*(LG(2,4);ℂ) ≅ℂ[x_1,x_2,q]/⟨ x_1^2 - 2 x_2, -q x_1 + x_2^2 ⟩,

where q=e^v^2. 
Images of the Schubert classes under the isomorphism (<ref>) are <cit.>
    
    𝔖_0^q = 1 , 
         
       𝔖_1^q = x_1 , 𝔖_2^q= 1/2x_1^2 , 
         
       𝔖_2,1^q=1/2x_1^3-q.


The multiplication table of the small quantum cohomology of LG(2,4) is shown in <ref>. 




It is known that q=1 (i.e. v^i=0, 1≤ i ≤ 4) is a semisimple point of the small quantum cohomology of LG(2,4) <cit.>. Then according to a result of Cotti <cit.> we know that F_0( v) (see (<ref>)) has a convergence domain. 











 §.§ Deformed flat connection





 At the point q=1, equation (<ref>) becomes

    dydz=(𝒰+μz)y,

where

    𝒰 = (
    [ 0 0 3 0; 3 0 0 3; 0 6 0 0; 0 0 3 0;   ])
 










and 

    μ = (
    [ -3/2    0    0    0;    0 -1/2    0    0;    0    0  1/2    0;    0    0    0  3/2;      ]).




    The system (<ref>) can be reduced to an equivalent scalar ODE
    
    D^4φ-108  z^3 Dφ-162 z^3φ=0,

    where D = z d/dz.


    Let y=(y_1,…,y_4)^T and y_4(z)=z^3/2φ(z). From the system (<ref>), y_1,y_2,y_3 are uniquely determined by φ through the following equations: 
    
    y_1(z)   =z^2 φ^(3)(z)+φ'(z)+3 z φ”(z)-54 z^2 φ(z)/54 √(z),
    
        y_2(z)   =1/18(z^3/2φ”(z)+√(z)φ'(z)),
    
        y_3(z)   =1/3 z^3/2φ'(z),

where φ satisfies the scalar ODE (<ref>). 



The indicial equation of (<ref>) at z=0 reads

    r^4=0.

A solution of (<ref>) is (cf. <cit.>)

    φ̃(z)= ∑^∞_d=0(2d)!(d!)^5z^3d=1 + 2 z^3 + 3/4z^6 + 5 /54z^9 + 35 /6912z^12 + 7 /48000z^15+O(z^18).






By Proposition <ref>, the topological-enumerative solution of (<ref>) is 

    𝒴_top(z) = Φ_top(z) z^μz^R,

where 

    μ = (
    [ -3/2    0    0    0;    0 -1/2    0    0;    0    0  1/2    0;    0    0    0  3/2;      ]),  R=(
    [ 0 0 0 0; 3 0 0 0; 0 6 0 0; 0 0 3 0;   ]),

and 

    Φ_top(z)    = I +  (
    [ 0 0 1 0; 0 0 0 1; 0 0 0 0; 0 0 0 0;   ]) z+ (
    [  0 -2  0  0;  0  0  0  0;  0  0  0  2;  0  0  0  0;    ]) z^2+(
    [  2  0  0  1;  0 -2  0  0;  0  0 -2  0;  0  0  0  2;    ]) z^3
    
       +(
    [    0    0 -3/2    0;    4    0    0  3/2;    0    0    0    0;    0    0   -4    0;      ])z^4
    +(
    [    0  3/2    0    0;    0    0 -7/2    0;    8    0    0  3/2;    0    8    0    0;      ])z^5+(
    [  13/4     0     0   1/2;     0  33/4     0     0;     0     0 -17/4     0;     0     0     0   3/4;       ])z^6
    
                   +(
    [      0      0 -19/12      0;   -5/2      0      0   5/12;      0   25/2      0      0;      0      0   -5/2      0;        ])z^7+O(z^8).

























 §.§ Computation of the Stokes matrix and the central connection matrix

It suffices to compute the monodromy data at q=1 due to the simisimplicity. 
Let ϵ=e^2/3iπ. The canonical coordinates at q=1 are 

    u_1=0, u_2=3· 2^2/3,  u_3=3· 2^2/3ϵ^2,  u_4=3· 2^2/3ϵ.

The Stokes rays (see (<ref>)) are 

    R_12   ={iρ: ρ≥0}, R_13={-ρ e^i π/6: ρ≥0}, R_14={ρ e^-i π/6: ρ≥0}, 
        
      R_23   ={-ρ e^i π/3: ρ≥0}, R_24={ρ e^-i π/3: ρ≥0}, 
      R_34   ={ρ: ρ≥0}.




We fix the admissible line ℓ: 

    ℓ={ρ e^i π/4: ρ∈ℝ}.

Let 

    Π_left={z:π/6 < argz < 4π/3},  Π_right={z:-5π/6< argz < π/3}.

The narrow sectors are then given by

    Π_+={z:π/6< argz <π/3}, 
          Π_-={z:-π/6< argz <π/3}.





  §.§.§ Solutions to the scalar ODE (<ref>)


    The following functions are solutions of the ODE (<ref>): 

    
  * the function

    φ_1(z)=∫_Λ_1Γ (s)^4/Γ(s+1/2)2^-2 s e^π  i s z^-3 s ds,    -π/2<  z<π/2,

where Λ_1 is any line in the complex plane from the point κ - i ∞ to κ + i ∞ for any κ>0; 

    
  * the function 

    φ_2(z)=∫_Λ_2Γ (s)^4 Γ(1/2-s) 2^-2 s z^-3 s ds,    -5π/6<  z<5π/6,

where Λ_2 is any line in the complex plane from the point κ - i ∞ to κ + i ∞ for any 0<κ<1/2. 



    This Lemma can be obtained via a direct verification (cf. <cit.>). 


Now we consider asymptotic behaviours of φ_1 and φ_2 as |z| →∞ and several identities. 

   The following asymptotic behaviours hold as |z| →∞ 

    -z^3/2/2√(2)π^2φ_1(z)   = 1/√(6) e^u_4 z(1+O(1z)),    -π/2<  z<π/2,
        
       -z^3/2/√(2)π^3φ_2(z)   = -i/√(2) e^u_1 z(1+O(1z)),    -π/6<  z<π/2.
 


   The proof of this Lemma is an elementary exercise by using Laplace method (cf. <cit.>).



   Let ϵ=e^2/3iπ. The functions φ_1, φ_2 satisfy the following identity
    
    φ_2(zϵ^-1)= 2πφ_1(z)-φ_2(z),    -π/6<  z<π/2.



    This Lemma can be obtained via a direct verification (cf. <cit.>). 



    Any solution φ to (<ref>) satisfies the following identity
    
    φ(zϵ^4)-4φ(zϵ^3)+6φ(zϵ^2)-4φ(zϵ)+φ(z)=0.


 
(cf. <cit.>) Note that if φ(z) is a solution to (<ref>) then φ(zϵ) also is. Then the map z↦ zϵ defines a linear map on the solution space of (<ref>) 
, where the solutions have the form

    φ(z)=∑_n≥ 0z^3n(a_n+b_nlog z+c_n(log z)^2+d_n(log z)^3 ),

where a_0, b_0, c_0, d_0 are arbitrary constants, and a_n, b_n, c_n, d_n, n≥ 1 can be obtained recursively. Choosing a basis of solutions with coefficients (a_0,b_0,c_0,d_0)=(1,0,0,0), (0,1,0,0) and (0,0,1,0), (0,0,0,1), the matrix of the operator

    (Aφ)(z):=φ(zϵ)

is triangular whose diagonal entires are all 1. By Cayley–Hamilton theorem we then deduce that
(A- I)^5=0, namely

    A^4-4A^3+6A^2-4A+I=0.








  §.§.§ Asymptotic behaviour of fundamental matrix solutions






Let Ψ be the transition matrix from (f_1,f_2,f_3,f_4)  to (e_1,e_2,e_3,e_4) at q=1, i.e. (see (<ref>))

    e_α=∑_j=1^4 ψ_j α f_j ,     α = 1,2,3,4.

We can compute directly 

    f_1    = i/√(2)e_1-i/√(2)e_4,
    
      f_2    = 1/√(6)e_1+1/√(2)√(3)e_2+√(2)/√(3)e_3+1/√(6)e_4,
    
      f_3    = 1/√(6)e_1+1/√(2)√(3)e^2 i π/3e_2+√(2)/√(3) e^-2 i π/3 e_3+1/√(6)e_4, 
    
      f_4    = 1/√(6)e_1+1/√(2)√(3)e^-2 i π/3e_2+√(2)/√(3)e^2 i π/3e_3+1/√(6)e_4.
 
Then



    Ψ = (
    [            -i/√(2)                  0                  0             i/√(2);             1/√(6)          √(2)/√(3)         1/√(2)√(3)             1/√(6);             1/√(6)  -√(2)/√(3)e^i π/3 1/√(2)√(3)e^2i π/3             1/√(6);             1/√(6)  √(2)/√(3)e^2i π/3 -1/√(2)√(3)e^i π/3             1/√(6);                    ]).

Denote 

    U= Ψ𝒰Ψ^-1 = (
    [           0           0           0           0;           0    3· 2^2/3           0           0;           0           0 3· 2^2/3ϵ^2           0;           0           0           0   3· 2^2/3ϵ;             ]),

and 

    V= ΨμΨ^-1 = (
    [        0  √(3)/2i  √(3)/2i  √(3)/2i; -√(3)/2i        0 -√(3)/6i  √(3)/6i; -√(3)/2i  √(3)/6i        0 -√(3)/6i; -√(3)/2i -√(3)/6i  √(3)/6i        0;          ]).

For a fundamental matrix solution 𝒴(z) to (<ref>), let

    Y(z) = Ψ𝒴(z).

Then Y=Y(z) satisfies the equation

    dYdz=(U+Vz)Y.






Note that there exists unique fundamental matrix solutions Y_L/R to (<ref>), analytic in Π_left/right, respectively, such that 

    Y_L/R(z)∼ G(z) e^zU

as |z| →∞ within the sectors, where G (z) is a formal power series of z^-1 with G(∞)=I (see (<ref>)). Then the corresponding solutions to (<ref>) have the following asymptotic behaviour as |z| →∞
 
    𝒴_L/R(z)    = Ψ^-1Y_L/R(z) 
    
       =(
    [                  i/√(2)e^u_1 z                  1/√(6)e^u_2 z                  1/√(6)e^u_3 z                  1/√(6)e^u_4 z;                              0              1/√(2)√(3)e^u_2 z    -e^-1/3 i π/√(2)√(3)e^u_3 z       -e^i π/3/√(2)√(3)e^u_4 z;                              0               √(2)/√(3)e^u_2 z -i √(2) e^-1/6 i π/√(3)e^u_3 z   -√(2) e^-1/3 i π/√(3)e^u_4 z;                 -i/√(2)e^u_1 z                  1/√(6)e^u_2 z                  1/√(6)e^u_3 z                  1/√(6)e^u_4 z;                                ])   (1+O(1z)).
























  §.§.§ Fundamental matrix solutions 𝒴_L/R and the Stokes matrix

Let S'=(s'_jk) be the Stokes matrix relating Y_L and Y_R in the narrow sector Π_+, i.e. 

    Y_L(z)=Y_R(z) S',     z ∈Π_+.

Then we have
 
    y^L_4k(z)=∑_j=1^4 y^R_4j(z) s'_jk,   for π6< arg z<π3,

Note that y^L/R_4k have the following asymptotic behaviour as |z| →∞

    y^L/R_4k(z)=c_k e^u_k z(1+O(1z)) ,  for  z ∈Π_left/right,

where c_1=-i/√(2) , c_2=1/√(6) , c_3=1/√(6) , c_4=1/√(6). 



By comparing asymptotic behaviours , we can determine the form of S'. For example, by definition, we have 
 
    y^L_44(z)=∑_j=1^4 y^R_4j(z) s'_j4,   for π6< arg z<π3.

Since e^u_4 z is dominated by e^u_1 z,e^u_2 z,e^u_3 z on the sector 0< arg z<2π/3, i.e. |e^u_4 z|<|e^u_k z|,  k=1,2,3, functions y^R_41, y^R_42, y^R_43 will not appear on the right-hand side. 
Then we have 

    y^L_44(z)=y^R_44(z),   for π6< arg z<π3,

which means s'_14=0, s'_24=0, s'_34=0, s'_44=1. By similar dominance arguments, we have s'_11=1, s'_22=1, s'_33=1, s'_23=0, s'_21=0, s'_31=0 . Then S' must have the form

    (
    [ 1 * * 0; 0 1 0 0; 0 * 1 0;   * * 1;   ]).



To explicitly compute S', we have to construct entries of fundamental matrix solutions 𝒴_L/R=(y^L/R_jk)_1≤ j,k ≤ 4 on each sector Π_left/right that have the same asymptotic behaviour as (<ref>). Note that we only have to construct y^L/R_4k,  1≤ k ≤ 4  due to Proposition <ref>.





From (<ref>) and the existence of y^L_44 and y^R_44, φ_1(z) can be analytically continued to the sector -5π/6< arg z<4π/3. Since the asymptotic behaviours of y^L_44, y^R_44 as |z| →∞ are same, we have 

    -z^3/2/2√(2)π^2φ_1(z)= 1/√(6) e^u_4 z(1+O(1z)),  for  -5π/6< arg z<4π/3.

Note that the continued  φ_1(z) may not have the integral representation (<ref>) in the above region. 
Since φ_1(zϵ^2) is also a solution of (<ref>), we have

    -z^3/2/2√(2)π^2φ_1(zϵ^2) = linear combination of  y^R_4k, 1≤ k ≤ 4,   for  -5π/6< arg z<0.

 Rotating z by ϵ^2 in (<ref>), we get 

    -z^3/2/2√(2)π^2φ_1(zϵ^2) = 1/√(6) e^u_2 z(1+O(1z)),  for  -13π/6< arg z<0.

From (<ref>) and the fact e^u_2 z is dominated by e^u_1 z,e^u_3 z,e^u_4 z on the sector -4π/3< arg z<-2π/3, we have 


    -z^3/2/2√(2)π^2φ_1(zϵ^2) = y^R_42(z),  for    -5π/6< arg z<0.

By the existence of y^R_42 on -5π/6< arg z<π/3, -z^3/2/2√(2)π^2φ_1(zϵ^2) can be analytically continued to -13π/6< arg z<π/3 and still have the asymptotic behaviour as in (<ref>). Then we have the following proposition for φ_1(z). 

   The following asymptotic behaviour holds as |z| →∞ 

    -z^3/2/2√(2)π^2φ_1(z)= 1/√(6) e^u_4 z(1+O(1z)),    -5π/6<  z<5π/3.


Therefore, we conclude that 

    y^R_42(z) = -z^3/2/2√(2)π^2φ_1(zϵ^2),  for  z ∈Π_right.









Now φ_2(z) can be analytically continued to the sector π/2< arg z<5π/6 through the identity (<ref>). From the fact e^u_4 z is dominated by e^u_1 z on the whole sector -π/6< arg z<5π/6 and the uniformity of asymptotic behaviour, we can determine the asymptotic behaviour of φ_2(z) as |z| →∞: 

    -√(2)i z^3/2/π^2φ_2(z)= -i/√(2) e^u_1 z(1+O(1z)),  for  -π/6< arg z<5π/6.

By the identity (<ref>) with the variable z rotated by ϵ and similar arguments as above, we finally arrive at the following proposition for φ_2(z).




   The following asymptotic behaviour holds as |z| →∞ 

    -√(2)i z^3/2/π^2φ_2(z)= -i/√(2) e^u_1 z(1+O(1z)),    -5π/6<  z<5π/6.



By the identity (<ref>), φ_2(z) can be further continued to -3π/2< arg z<5π/3, although it may not have the same asymptotic behaviour as (<ref>)  and the integral representation as (<ref>) beyond the sector -5π/6< arg z<5π/6. Since it is a linear combination of solutions, the continued φ_2(z) is still a solution to (<ref>). Now the identity (<ref>) also holds in a larger sector. 

    The identity (<ref>) holds for -5π/6<  z<5π/3.



Now we can determine some entries of fundamental matrix solutions besides y^R_42: 

    y^L_44(z)   =-z^3/2/2√(2)π^2φ_1(z),  for  z ∈Π_left,  
     
     y^R_44(z)   =-z^3/2/2√(2)π^2φ_1(z),  for  z ∈Π_right,
     
     y^L_42(z)   =z^3/2/2√(2)π^2φ_1(zϵ^-1),  for  z ∈Π_left,
     
     y^R_43(z)   =z^3/2/2√(2)π^2φ_1(zϵ),  for  z ∈Π_right,
     
     y^L_41(z)   =√(2)i z^3/2/π^2φ_2(zϵ^-1),  for  z ∈Π_left,
     
     y^R_41(z)   =-√(2)i z^3/2/π^2φ_2(z),   for  z ∈Π_right.





 The only remained entry is y^L_43. For y^L_43, let some coefficients be undetermined: 

    -z^3/2/2√(2)π^2φ_1(zϵ^-2)=y^L_43(z)+γ_2”y^L_42(z),  for π/2< arg z<4π/3,

and 

    z^3/2/2√(2)π^2φ_1(zϵ)=y^L_43(z)+γ_1”y^L_41(z)+γ_4”y^L_44(z),  for π/6< arg z<π.

Subtract (<ref>) and (<ref>), we get

    -z^3/2/2√(2)π^2φ_1(zϵ^-2)-z^3/2/2√(2)π^2φ_1(zϵ)=γ_2”y^L_42(z)-γ_1”y^L_41(z)-γ_4”y^L_44(z) ,  for π/2< arg z<π.

We then replace all φ_1 by φ_2 in (<ref>) by using (<ref>) and apply (<ref>) to φ_2. By comparing cofficients, we obtain  

    y^L_43(z)=-z^3/2/2√(2)π^2φ_1(zϵ^-2)+5y^L_42(z),  for π/6< arg z<2π/3,

and 

    y^L_43(z)=z^3/2/2√(2)π^2φ_1(zϵ)-4y^L_41(z)+5y^L_44(z),  for π/2< arg z<4π/3.






Note that we have finished the construction of 𝒴_L/R. Comparing 𝒴_L and 𝒴_R, we get the Stokes matrix S' 

    S' = (
    [   1   4   4   0;   0   1   0   0;   0   5   1   0;  -4  -5 -11   1;     ]) .

We then triangularize S' by a permutation P of canonical coordinates: 

    S = PS'P^-1= [   1  -4 -11  -5;   0   1   4   4;   0   0   1   5;   0   0   0   1;     ],

where 

    P = (
    [ 0 0 0 1; 1 0 0 0; 0 0 1 0; 0 1 0 0;   ]).





  §.§.§ Central connection matrix


Let C^'=(c_jk^')_1≤ j,k ≤ 4 be the central connection matrix relating 𝒴_top(z) and 𝒴_R(z), i.e. 

    𝒴_R(z) =𝒴_top(z)C^' ,


To compute C^', we have to find the expansion of Y^R at z=0 and compare the coefficents to determine c_jk^'. For example, 

    y_44^R(z)   =-z^3/2/2√(2)π^2φ_1(z)=-z^3/2/2√(2)π^2∫_Λ_1Γ (s)^4/Γ(s+1/2)2^-2 s e^π  i s z^-3 s ds 
       = -z^3/2/2√(2)π^22π i∑_n=0^∞res_s=-n(Γ (s)^4/Γ(s+1/2)2^-2 s e^π  i s z^-3 s)
        
       =z^3/2(9 i /2 √(2)π ^3/2(log z)^3+i (162 γ -54 i π ) /12 √(2)π ^3/2(log z)^2 +i (162 γ ^2-108 i γπ -15 π ^2) /12 √(2)π ^3/2log z.
        
       .       +i (-12 ζ (3)+54 γ ^3-54 i γ ^2 π -15 γπ ^2+i π ^3)/12 √(2)π ^3/2)+z^9/2(9 i/√(2)π ^3/2(log z)^3.
        
       .  +i (-216+324 γ -108 i π ) /12 √(2)π ^3/2(log z)^2+i (144-432 γ +324 γ ^2+144 i π -216 i γπ -30 π ^2) /12 √(2)π ^3/2log z .
        
       .  +i (-24 ζ (3)+144 γ -216 γ ^2+108 γ ^3-48 i π +144 i γπ -108 i γ ^2 π +20 π ^2-30 γπ ^2+2 i π ^3)/12 √(2)π ^3/2)
        
                                                                        +O(z^15/2).



 From (<ref>),  

    y_44^R   (z) = z^3/2(9 c^'_1,4 (log z)^3+9 c^'_2,4 (log z)^2+3 c^'_3,4log z+c^'_4,4)+z^9/2(18 c^'_1,4 (log z)^3.
       
       . +(18 c^'_2,4-36 c^'_1,4) (log z)^2 +(24 c^'_1,4-24 c^'_2,4+6 c^'_3,4) log z+8 c^'_2,4-4 c^'_3,4+2 c^'_4,4)+O(z^15/2).

Compare the coefficents we get 

    c^'_1,4   = i/2 √(2)π ^3/2,
    
      c^'_2,4   = π +3 i γ/2 √(2)π ^3/2,
    
      c^'_3,4   = 54 i γ ^2+36 γπ -5 i π ^2/12 √(2)π ^3/2,
    
      c^'_4,4   = -12 i ζ (3)-54 i γ ^3-54 γ ^2 π +15 i γπ ^2+π ^3/12 √(2)π ^3/2.

In a similar way, we compare y_41^R, y_42^R, y_43^R with right hand side of (<ref>) and we can obtain C^'. After the permutation P of canonical coordinates, we get



    C = C^'P^-1   =  (
    [                                                 i/2 √(2)π ^3/2                                                   i/√(2)π ^3/2;                                          π +3 i γ/2 √(2)π ^3/2                                               3 i γ/√(2)π ^3/2;                        54 i γ ^2+36 γπ -5 i π ^2/12 √(2)π ^3/2                                i (54 γ ^2+7 π ^2)/6 √(2)π ^3/2; -12 i ζ (3)-54 i γ ^3-54 γ ^2 π +15 i γπ ^2+π ^3/12 √(2)π ^3/2                      i (-4 ζ (3)+18 γ ^3+7 γπ ^2)/2 √(2)π ^3/2;                                                                ].
    
       .    [                                                 -i/2 √(2)π ^3/2                                                  i/2 √(2)π ^3/2;                                           π -3 i γ/2 √(2)π ^3/2                                       -3 (π -i γ )/2 √(2)π ^3/2;                        -54 i γ ^2+36 γπ +5 i π ^2/12 √(2)π ^3/2                     i (54 γ ^2+108 i γπ -53 π ^2)/12 √(2)π ^3/2;    12 i ζ (3)+(π -3 i γ ) (18 γ ^2+12 i γπ -π ^2)/12 √(2)π ^3/2 -4 i ζ (3)+18 i γ ^3-54 γ ^2 π -53 i γπ ^2+17 π ^3/4 √(2)π ^3/2;                                                                 ]).










 §.§ Proof of the refined Dubrovin conjecture <ref> for LG(2,4)





Let 𝒰 be the tautological bundle on LG(2,4), and 𝒰^*:= (𝒰,𝒪). The collection 

    ( 𝒪, 𝒪(1), Σ^(2,1)𝒰^*, 𝒪(2) )

is a full exceptional collection <cit.> (cf. <cit.>), where Σ is the Schur functor. Since the semisimplicity of the small quantum cohomology of LG(2,4) at q=1 is known <cit.>, we have proved part 1 of Conjecture <ref> for LG(2,4).



Now twist (<ref>) by ∧^2 𝒰^*: 

    ( 𝒪⊗∧^2 𝒰^*, 𝒪(1)⊗∧^2 𝒰^*, Σ^(2,1)𝒰^*⊗∧^2 𝒰^*, 𝒪(2)⊗∧^2 𝒰^* ).

Denote the objects of (<ref>) by E_k, 1≤ k ≤ 4. The following proposition can be easily verified.   


    The twisted collection (<ref>) is a full exceptional collection. 
 


The Euler matrix S̃ = (χ(E_j,E_k))_1≤ j,k ≤ n can be computed:   

    S̃= (
    [  1  5 16 14;  0  1  4  5;  0  0  1  4;  0  0  0  1;    ]).

The Gamma class Γ̂^- of LG(2,4) is: 

    Γ̂^-=1+3 γσ_1+1/6(54 γ ^2+π ^2) σ_2+1/2(-4 ζ (3)+18 γ ^3+γπ ^2)σ_2,1.

The graded Chern characters are

    Ch(𝒪)   =1,
     
    Ch(𝒪(1))   =1+2 i πσ_1-4 π ^2 σ_2-83  i π ^3 σ_2,1,
     
    Ch(Σ^(2,1)𝒰^*)   =2+6 i πσ_1-16 π ^2 σ_2-32 i π ^3 σ_2,1,
     
    Ch(𝒪(2))   =1+4 i πσ_1-16 π ^2 σ_2-643  i π ^3 σ_2,1, 
     
    Ch(∧^2 𝒰^*)   =1+2 i πσ_1-4 π ^2 σ_2-83  i π ^3 σ_2,1.

Then we can compute the matrix  

    C_Γ   = (
    [                                                 i/2 √(2)π ^3/2                                                 i/2 √(2)π ^3/2;                                          π +3 i γ/2 √(2)π ^3/2                                         -π -3 i γ/2 √(2)π ^3/2;                        54 i γ ^2+36 γπ -5 i π ^2/12 √(2)π ^3/2                      i (54 γ ^2+36 i γπ -5 π ^2)/12 √(2)π ^3/2; -12 i ζ (3)-54 i γ ^3-54 γ ^2 π +15 i γπ ^2+π ^3/12 √(2)π ^3/2 -12 i ζ (3)+54 i γ ^3-54 γ ^2 π -15 i γπ ^2+π ^3/12 √(2)π ^3/2;                                                                ].
    
             .
    [                                                     i/√(2)π ^3/2                                                   i/2 √(2)π ^3/2;                                           -2 π +3 i γ/√(2)π ^3/2                                        -3 (π -i γ )/2 √(2)π ^3/2;                        i (54 γ ^2+72 i γπ -17 π ^2)/6 √(2)π ^3/2                      i (54 γ ^2+108 i γπ -53 π ^2)/12 √(2)π ^3/2; 2 (π ^3-6 i ζ (3))+54 i γ ^3-108 γ ^2 π -51 i γπ ^2/6 √(2)π ^3/2  -4 i ζ (3)+18 i γ ^3-54 γ ^2 π -53 i γπ ^2+17 π ^3/4 √(2)π ^3/2;                                                                  ]),

whose columns are given by coordinates of

    i/(2π)^3/2Γ̂^-∪  e^-iπ c_1∪Ch(E_k),   1≤ k ≤ 4,

with respect to the basis σ_0,  σ_1,  σ_2,  σ_2,1. 












    Conjecture <ref> holds for the Lagrangian Grassmanian LG(2,4).


    Note that we have proved part 1 of Conjecture <ref> for LG(2,4). Now consider the action of braid group ℬ_4, which is generated by 3 elementary braids β_12, β_23, β_34. The following sequence transforms S into S̃^-1 and C into C_Γ:

    
  * The action by β^-1_23, 
    
  * Change of signs by diag(1,-1,-1,1).
    
    Therefore, we have proved part 2 and 3 of Conjecture <ref> for LG(2,4).  

    
Acknowledgements.
This work was done during my master thesis project. I would like to thank Di Yang for his advice, helpful discussions and encouragements. The work was supported by CAS No. YSBR-032, by National Key R and D Program of China 2020YFA0713100, and by NSFC No. 12371254.






