Recent Progress on Conjecture $\mathcal{O}$ and Its Variants: A Survey ^†

Abstract

Conjecture $\mathcal{O}$ (and the Gamma Conjectures), introduced by Galkin, Golyshev, and Iritani stand as pivotal open problems in the quantum cohomology of Fano manifolds, bridging algebraic geometry, mathematical physics, and representation theory. These conjectures aim to decode the structural essence of quantum multiplication by uncovering profound connections between spectral properties of quantum cohomology operators and the underlying geometry of Fano manifolds. Conjecture $\mathcal{O}$ specifically investigates the spectral simplicity and eigenvalue distribution of the operator associated with the first Chern class $c_1$ in quantum cohomology rings, positing that its eigenvalues govern the convergence and asymptotic behavior of quantum products.

1. Introduction

Let X be a N-dimensional Fano manifold, namely a compact complex N-dimensional manifold X with positive first Chern class $c_1\left(X\right)$. Let ${\overline{\mathcal{M}}}_{0,k}(X,\beta )$ denote the moduli stack of k-pointed genus 0 stable maps $(f:C\to X;p_1,p_2,\cdots ,p_k)$ of class $\beta \in H_2(X,\mathbb{Z})$, which has a coarse moduli space ${\overline{M}}_{0,k}(X,\beta )$. Its virtual fundamental class ${\left[{\overline{M}}_{0,k}(X,\beta )\right]}^{virt}$ is of complex degree ${dim}_{\mathbb{C}}X+{\int }_{\beta }{c}_1\left(X\right)+k-3$ in the Chow group $A_{*}\left({\overline{M}}_{0,k}(X,\beta )\right)$. The k-pointed, genus zero Gromov-Witten invariants of degree $\beta$ for ${\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_k\in H^{*}\left(X\right)=H^{*}(X,\mathbb{Q})$ is defined by

$$\begin{array}{c}\hfill {〈{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_k〉}_{0,k,\beta }^X:={\int }_{{\left[{\overline{M}}_{0,k}(X,\beta )\right]}^{virt}}e{v}_1^{*}\left({\gamma }_1\right)\cup e{v}_2^{*}\left({\gamma }_2\right)\cup \cdots \cup e{v}_k^{*}\left({\gamma }_k\right)\end{array}$$

Here $e{v}_i:{\overline{M}}_{0,k}(X,\beta )\to X$$(i=1,2,\cdots k)$ denotes the i-th evaluation map. Set $m:=rank{H}_2(X,\mathbb{Z})$, take any homogeneous basis ${\left\{{\varphi }_i\right\}}_{i=1}^N$ of $H^{*}\left(X\right)$, and let $\left\{{\varphi }^i\right\}$ denote the dual basis of $H^{*}\left(X\right)$ that satisfy $({\varphi }_i,{\varphi }^j)={\int }_X{\varphi }_i\cup {\varphi }^j={\delta }_{i,j}$ with respect to the poincaré pairing. The small quantum cohomology ring $Q{H}^{*}\left(X\right)=(H^{*}\left(X\right){\otimes }_{\mathbb{Q}}\mathbb{Q}[q_1,\cdots ,q_m],{\star }_{\mathbf{q}=(q_1,\cdots ,q_m)})$ is a certain $\mathbf{q}$-deformation of classical cohomology $H^{*}\left(X\right)$. The quantum product of $\alpha ,\gamma \in H^{*}\left(X\right)$ is given by

$$\alpha {\star }_{\mathbf{q}}\gamma :=\sum _{\beta \in H_2(X,\mathbb{Z})}\sum _{i=1}^N{〈\alpha ,\gamma ,{\varphi }_i〉}_{\beta }^X{\varphi }^i{\mathbf{q}}^{\beta }$$

(1)

Here ${\mathbf{q}}^{\beta }={\prod }_{j=1}^m{q}_j^{\beta }$ for $\beta =({\beta }_1,\cdots ,{\beta }_m)$ with a basis of effective curve classes of $H_2(X,\mathbb{Z})$ being fixed a prior. The quantum product is a polynomial product is independent of choices of the basis ${\left\{{\varphi }_i\right\}}_{i=1}^N$.

Denote $H^{even}\left(X\right)=H^{even}(X,\mathbb{C})$ the even degree part of cohomology group of X. Set $\mathbf{q}=1$, if X is Fano, then there are finite such curve classes $\beta$ and the above sum is finite, $(H^{even}\left(X\right),{\star }_{\mathbf{q}=1})$ is a finite dimensional algebra.

For Fano manifolds, Galkin–Golyshev–Iritani proposed a conjecture [1,2]. It states that the eigenvalues of the linear operator ${\widehat{c}}_1:=\left(c_1\left(X\right){\star }_{\mathbf{q}=1}\right):H^{even}\left(X\right)\to H^{even}\left(X\right)$, satisfy the following properties.

Conjecture 1 

[Conjecture $\mathcal{O}$]. Suppose that $\rho$ is the Fano index of X, and

$$\begin{array}{c}\hfill {\lambda }_0=sup\left\{\left|\lambda \right|:\lambda \in Spec\left({\widehat{c}}_1\right)\right\}\end{array}$$

is the spectral radius of ${\widehat{c}}_1$. Then:

1. 

${\lambda }_0$ is also an eigenvalue of  ${\widehat{c}}_1$ with multiplicity one;

2. 

If $\lambda$ is an eigenvalue of ${\widehat{c}}_1$ with $\left|\lambda \right|={\lambda }_0$, then $\lambda =\zeta {\lambda }_0$ with ${\zeta }^{\rho }=1$.

Rietsch provided an explicit description of the eigenvalues and eigenvectors of quantum multiplication operators on the quantum cohomology ring of the Grassmannian ([3]), offering all the key ingredients needed for a proof of Conjecture $\mathcal{O}$.

Under mirror symmetry, Fano manifold X is mirror to a holomorphic function $f:Y\to \mathbb{C}$ on a complex manifold Y, and eigenvalues of ${\widehat{c}}_1$ conicides with the critical values $\lambda$ of f. Under homological mirror symmetry, the vanishing cycle associated to ${\lambda }_0$ correspond to the structure sheaf $\mathcal{O}$ of X [1,4].

For toric Fano manifolds, assuming Conjecture $\mathcal{O}$, Galkin, Golyshev and Iritani proved Gamma conjectures I ([1]).

If a basis ${\left\{{\varphi }_i\right\}}_{i=1}^N$ of $H^{*}\left(X\right)$ is choosen, then the linear opeator ${\widehat{c}}_1:=\left(c_1\left(X\right){\star }_{\mathbf{q}=1}\right)$ can be expressed by a certain matrix M:

$$\begin{array}{c}\hfill M=\left(m_{ij}\right)=\left(〈c_1\left(X\right),{\varphi }^j,{\varphi }^i〉\right).\end{array}$$

Conjecture $\mathcal{O}$ can be approached from both A-side and B-side:

• A-side: Conjecture $\mathcal{O}$ can be verified for homogeneous manifolds and some horospherical varieties of Picard rank 1 by computing eigenvalues of ${\widehat{c}}_1$ directly and applying Perron-Frobenius theorem, Quantum Chevalley formula and Quantum Bruhat diagram. It is remarkable that for del Pezzo surface and Fano projective complete intersections, it’s difficult to compute all $m_{ij}$, but it’s relatively easy to get some vanishing or nonnegative results about $m_{ij}$ by topological recursion relation or generalized Perron-Frobenius theorem;
• B-side: Conjecture $\mathcal{O}$ can be verified mainly for toric Fano manifold by mirror symmetry and analysis of critical value of Landau-Ginzburg superpotential.

2. Conjecture $\mathcal{O}$: A-Side

From A-side, Conjecture $\mathcal{O}$ can be proven by the theory of Quantum cohomology and Gromov-Witten invariants. We start with the baby example of ${\mathbb{P}}^1$:

Example: If $X={\mathbb{P}}^1$, then Conjecture $\mathcal{O}$ holds.

The cohomology of ${\mathbb{P}}^1$ is $H^{*}({\mathbb{P}}^1,\mathbb{Z})=\mathbb{Z}\oplus \mathbb{Z}h$, where $h\in H^2({\mathbb{P}}^1,\mathbb{Z})$ hyperplane class. The first Chern class of ${\mathbb{P}}^1$ and Fano index is $c_1\left({\mathbb{P}}^1\right)=2h\Rightarrow r\left({\mathbb{P}}^1\right)=2$ (Fano index). Then the quantum cohomology of ${\mathbb{P}}^1$ and spectral set of ${\widehat{c}}_1$ are as follows:

$$\begin{array}{c}Q{H}^{*}\left({\mathbb{P}}^1\right)=\mathbb{C}[h,q]/(h^2-q)\hfill \\ \Rightarrow \widehat{c_1}{\star |}_{q=1}\left(\begin{array}{c}1\\ h\end{array}\right)=\left(\begin{array}{c}2h\\ 2\end{array}\right)=\left(\begin{array}{cc}0& 2\\ 2& 0\end{array}\right)\left(\begin{array}{c}1\\ h\end{array}\right)\hfill \\ \Rightarrow Spec\left(\widehat{c_1}\right)=\{2,-2\}\hfill \end{array}$$

It is easy to check that 2 is also an eigenvalue of ${\widehat{c}}_1$ with multiplicity one.

For the proof of Conjecture $\mathcal{O}$, the key point is the following Perron-Frobenius theory [5]:

Theorem 1 

(Perron-Frobenius [6]). Let M be an irreducible nonnegative matrix, then

1. 

M has a real positive eigenvalue ${\lambda }_0$ of multiplicity one such that ${\lambda }_0⩾\left|\lambda \right|$ for any eigenvalue λ of M. Moreover, M has a positive eigenvector corresponding to ${\lambda }_0$;

2. 

If $h=h\left(M\right)$ is the index of imprimitivity of M, then the eigenvalues of M of modulus ${\lambda }_0$ are all of multiplicity one, given by the distinct roots of ${\lambda }^h-{\lambda }_0^h=0$. Moreover, the set of eigenvalues of M is invariants under rotation by ${\displaystyle \frac{2\pi }{h}}$.

In particular, for examples of Fano index 1 (i.e., certain del pezzo surface, etc.), the following generalized Perron-Frobenius theorem [7] might be helpful:

Theorem 2 

(Generalized Perron-Frobenius theorem). Let $M=\left(m_{ij}\right)$ be an $n\times n$ real matrix that satisfies the following:

(1) 

${\sum }_{i=1}^n{m}_{ij}>0$ for $j=1,\cdots ,n$;

(2) 

$M^k$ is an irreducible nonnegative matrix for some positive integer k;

Then the spectral radius ${\lambda }_0$ itself is an eigenvalue of M with multiplicity one.

It is also worth mentioning that for nonnegative matrix, the zero pattern of their entries can be used to find out spectral properties of the matrix; one way to encoding this pattern is through the directed graph which can be associated with the quantum Bruhat graph.

Property 1 

(Graph version of Perron-Frobenius theorem [8]). Let M be a nonnegative matrix.

1. 

M is irreducible if and only if the associated directed graph $D\left(M\right)$ is strongly connected;

2. 

If M is irreducible, then the index $h\left(M\right)$ of imprimitivity of M is equal to the index $h\left(D\right(M\left)\right)$ of imprimitivity of the associated directed graph $D\left(M\right)$.

By the result of [5], Conjecture $\mathcal{O}$ holds for general homogeneous manifolds. We have the following theorem:

Theorem 3. 

Let $M\left(X\right)$ be a matrix corresponding to the operator ${\widehat{c}}_1$ acting on $H^{even}\left(X\right)$ with respect to Schubert basis $\mathcal{S}$ of X, then

1. 

$M\left(X\right)$ is an irreducible, nonnegative, integral matrix;

2. 

The imprimitivity index $h\left(M\right(X\left)\right)$ is equal to the Fano index ρ of X.

Theorem 4 

(Quantum Chevalley formula). For any $i\in I^P$ and $u\in W^P$, the quantum product of Schubert generators ${\sigma }_{s_i}$ and general Schubert class ${\sigma }_u$ is given by

$${\sigma }_{s_i}\star {\sigma }_u=\sum _{\alpha }{h}_{\alpha }\left({\omega }_{{\alpha }_i}\right){\sigma }_v+\sum _{\alpha }{q}^{d\left(\alpha \right)}{h}_{\alpha }\left({\omega }_{{\alpha }_i}\right){\sigma }_{\omega },$$

(2)

where the first sum is over roots $\alpha \in R^{+}$\$R_P^{+}$ for which $v=u{s}_{\alpha }\in W^P$ satisfies $l\left(v\right)=l\left(u\right)+1$, and the second sum is over roots $\alpha \in R^{+}$\$R_P^{+}$ for which ω is the minimal length representative in $\left[u{s}_{\alpha }\right]$ satisfying $l\left(\omega \right)=l\left(u\right)+1-n_{\alpha }$.

Example 1 

(Lagrangian Grassmanian). The Lagrangian Grassmannian $LG\left(n\right)=SG(n,2n)$ and the first Chern class $c_1\left(LG\left(n\right)\right)=(n+1){\sigma }_1$. When $n=2$, the Schubert basis of $LG\left(2\right)$ is $\left\{{\sigma }_0,{\sigma }_1,{\sigma }_2,{\sigma }_{(2,1)}\right\}$.

$$\left(c_1{\star }_{q=1}\right)\left(\begin{array}{c}{\sigma }_0\\ {\sigma }_1\\ {\sigma }_2\\ {\sigma }_{(2,1)}\end{array}\right)=\left(\begin{array}{cccc}0& 3& 0& 0\\ 0& 0& 6& 0\\ 3& 0& 0& 3\\ 0& 3& 0& 0\end{array}\right)\left(\begin{array}{c}{\sigma }_0\\ {\sigma }_1\\ {\sigma }_2\\ {\sigma }_{(2,1)}\end{array}\right)=M\left(\begin{array}{c}{\sigma }_0\\ {\sigma }_1\\ {\sigma }_2\\ {\sigma }_{(2,1)}\end{array}\right).$$

The characteristic polynomial of the matrix M is

$$\begin{array}{c}\hfill |\lambda I-M|={\lambda }^4-108\lambda =\lambda ({\lambda }^3-108).\end{array}$$

All eigenvalues (i.e., Spectral set) of M can be calculated:

$$\begin{array}{c}\hfill {\lambda }_1=0,{\lambda }_2=3\times 4^{{\displaystyle \frac{1}{3}}},{\lambda }_3=3\times 4^{{\displaystyle \frac{1}{3}}}{e}^{{\displaystyle \frac{2\pi i}{3}}},{\lambda }_4=3\times 4^{{\displaystyle \frac{1}{3}}}{e}^{{\displaystyle \frac{4\pi i}{3}}}\end{array}$$

It is easy to check that ${\lambda }_2=3\times 4^{{\displaystyle \frac{1}{3}}}$ belongs to the spectral set of ${\widehat{c}}_1$ with multiplicity one.

In the following, we will have a look at the case of Fano projective complete intersections [1,9,10] which Conjecture $\mathcal{O}$ also holds. Denote by $X=X_{N,(d_1,\cdots ,d_r)}$ the N-dimensional smooth Fano complete intersection X of degree $(d_1,\cdots ,d_r)$ in ${\mathbb{P}}^{N+r}$, where $N⩾3,d_1,\cdots ,d_r⩾2$, Fano index of $X_{N,(d_1,\cdots ,d_r)}$ is $\rho =N+r+1-(d_1+\cdots +d_r)$.

The first Chern class of Fano projective complete intersection is:

$$\begin{array}{c}\hfill c_1\left(X_{N,(d_1,\cdots ,d_r)}\right)=\rho H,\end{array}$$

where H is the restriction of hyperplane class of ${\mathbb{P}}^{N+r}$ to X.

The vector space $H\left(X\right):=H^{even}(X,\mathbb{C})$ can be decomposed into two subspaces:

$$\begin{array}{c}\hfill H\left(X\right)=H_{amb}\left(X\right)\oplus H_{prim}\left(X\right),\end{array}$$

where $H_{amb}\left(X\right)={\oplus }_{i=0}^N\mathbb{C}{H}^i$ is the ambient part, and $H_{prim}\left(X\right)$ is the primitive part with $H_{prim}\left(X\right)=0$ if N is odd.

Quantum products of H in $H_{amb}\left(X\right)$ are given by

$$\begin{array}{cc}\hfill H^{\star (N+1)}& =d_1^{d_1}\cdots d_r^{d_r}{H}^{\star (N+1-\rho )},\mathrm{if}1<\rho ⩽N\hfill \\ \hfill {(H+d_1!\cdots d_r!)}^{\star (N+1)}& =d_1^{d_1}\cdots d_r^{d_r}{(H+d_1!\cdots d_r!)}^{\star N},\mathrm{if}\rho =1.\hfill \end{array}$$

Then the spectrum of $\left(c_1\left(X\right){\star }_{q=1}\right)$ on $H_{amb}\left(X\right)$ is

$$Spec\left(c_1\left(X\right){\star }_{q=1}\right)=\left\{0\right\}\cup {\left\{e^{{\displaystyle \frac{2\pi ki}{\rho }}}{(d_1^{d_1}\cdots d_r^{d_r})}^{{\displaystyle \frac{1}{\rho }}}\rho \right\}}_{k=0}^{\rho -1}\mathrm{if}1<\rho ⩽N$$

(3)

$$Spec\left(c_1\left(X\right){\star }_{q=1}\right)=\left\{-d_1!\cdots d_r!,d_1^{d_1}\cdots d_r^{d_r}-d_1!\cdots d_r!\right\}\mathrm{if}\rho =1$$

(4)

Especially, There is some vanishing result of quantum product [9].

$$c_1\left(X\right){\star }_{q=1}{\xi }_i=0,\rho >1,i=1,\cdots ,N^{\prime }$$

(5)

where $N^{\prime }={dim}_{\mathbb{C}}{H}_{prim}\left(X\right)$, ${\left\{{\xi }_i\right\}}_{i=1}^{N^{\prime }}$ is orthonormal basis of $H_{prim}\left(X\right)$.

In conclusion, there are three subcases of Fano projective complete intersections:

• N is odd: $H_{prim}\left(X\right)=0$. Conjecture $\mathcal{O}$ holds by Equations (3) and (4);
• N is even with $\rho >1$. Conjecture $\mathcal{O}$ holds by vanishing result of Quantum product $c_1\left(X\right){\star }_{q=1}$ Equation (5) and Equation (3);
• N is even with $\rho =1$. The eigenvalues of $c_1\left(X\right){\star }_{q=1}$ acting on $H_{amb}\left(X\right)$ are $-d_1!\cdots d_r!$ and $d_1^{d_1}\cdots d_r^{d_r}-d_1!\cdots d_r!$. On the other hand, the eigenvalue of $c_1\left(X\right){\star }_{q=1}$ acting on $H_{prim}\left(X\right)$ is denoted by $\lambda$. Based on genus-one topological recursion relation and Mirror formula, we have q

  $$\lambda ={\displaystyle \frac{1}{d_1!\cdots d_r!c_N-1-N}}\sum _{p=0}^N[c_{N-p}-{\displaystyle \frac{1}{d_1\cdots d_r}}\left(\begin{array}{c}N+1\\ p+1\end{array}\right)]I_p=-d_1!\cdots d_r!$$

  (6)

  where $c\left(X\right)={\sum }_{p=0}^N{c}_p{H}^p$, $I_a={〈{\tau }_{a-1}\left(H^{N-a}\right)〉}_{0,1}^X$

  $$\begin{array}{c}\hfill {\displaystyle \frac{{(1+x)}^{N+r+1}}{{\prod }_{i=1}^r(1+d_{i}x)}}=\sum _{p=0}^{\infty }{c}_p{x}^p,(d_1\cdots d_r)(d_1!\cdots d_r!)[{\displaystyle \frac{{\prod }_{i=1}^r{\prod }_{m=1}^{d_i}(1+\frac{d_i}{m}x)}{{(1+x)}^{N+r+1}}}-1]=\sum _{a=0}^{\infty }{I}_a{x}^a\end{array}$$

Recently, Conjecture $\mathcal{O}$ from A-side has been verified for the following cases (with key points for proof) listed below in a similar fashion:

• Flag manifold $G/P$ [5]: quantum Chevalley formula, Frobenius-Perron theorem;
• ${\mathbb{P}}^n,Gr(k,n)$ [3]: quantum Pieri rule;
• $LG\left(n\right),OG\left(n\right)$ [11,12]: quantum Giambelli formula;
• Fano projective complete intersections [9]: topological recursion relation (TRR);
• Blow-ups of 4-dim quadrics $B{l}_{{\mathbb{P}}^0}{Q}^4,B{l}_{{\mathbb{P}}^2}{Q}^4$ [13]: blow-up formulae, Frobenius-Perron theorem;
• Del Pezzo surfaces [7]: Vanishing results, Topological recursive relation and generalized Perron-Frobenius theorem;
• Odd-symplectic Grassmannian $IG(k,2n+1)$ [14]: quantum Chevalley formula and quantum Bruhat graph;
• Some horospherical varieties of Picard rank 1 [15]: quantum Chevalley formula and quantum Bruhat graph;
• Cayley Grassmannian [16];
• Bott-Samelson variety $Z({\alpha }_1,{\alpha }_2,{\alpha }_1)$ (in Type $A_2$) [17]: Chevalley matrices.

3. Conjecture $\mathcal{O}$ for Toric Fano Manifolds: B-Side

A toric manifold X of complex dimension N is a complex manifold containing an complex torus as open dense subset, such that the action of the torus on itself extends to the whole manifold. Geometric information of $X:=X_{\Sigma }$ can be determined by the combinatorics of its associated fan $\Sigma$ in ${\mathbb{R}}^N$. Here we always assume $X_{\Sigma }$ to be smooth and Fano, namely $\Sigma =\Sigma (\Delta )$ is the normal fan of a convex polytope $\Delta$ with nice properties. Thus $X_{\Sigma }$ is also denoted as $X_{\Delta }$ by abuse of notations.

Let $f=f_{\Sigma }=f_{\Delta }$ denote the Landau-Ginzburg superpotential mirror to the toric Fano manifold $X_{\Sigma }=X_{\Delta }$. It is a Laurent polynomial, which can be read off immediately from $\Sigma$. To be precise, if we denote by $b_1,\cdots ,b_m\in {\mathbb{Z}}^N$ the primitive generators of rays in $\Sigma$, then the Picard number of $X_{\Sigma }$ is equal to $m-N$, and f is given by

$$\begin{array}{c}\hfill f:{\left({\mathbb{C}}^{\times }\right)}^N\to \mathbb{C};\mathbf{x}\mapsto f\left(\mathbf{x}\right)={\mathbf{x}}^{b_1}+\cdots +{\mathbf{x}}^{b_m},\end{array}$$

where $\mathbf{x}:=(x_1,\cdots ,x_N)$ and ${\mathbf{x}}^{b_i}=x_1^{b_{i1}}\cdots x_N^{b_{iN}}$ for $b_i=(b_{i1},\cdots ,b_{iN})$. Mirror symmetry between quantum cohomology of $X_{\Sigma }$ and oscillatory integrals of f has been proved by [18,19]. As a remarkable property, there exists an isomorphism $\Phi$ of $\mathbb{C}$-algebras between quantum cohomology ring and Jacobian ring [1,2,4]:

$$\begin{array}{c}\hfill \Phi :Q{H}^{*}{\left(X\right)|}_{q=1}\underrightarrow{\cong }Jac\left(f\right)=\mathbb{C}[x_1^{\pm },\cdots ,x_N^{\pm }]/(x_1{\partial }_{x_1}f,\cdots ,x_N{\partial }_{x_N}f).\end{array}$$

Theorem 5. 

With notations as above, we have $\Phi \left(c_1\left(X\right)\right)=\left[f\right]$. Specifically, the eigenvalues of $\widehat{c_1}$, including their multiplicities, coincide with the critical values of f.

Here we notice that eigenvalues of the linear operator on $Jac\left(f\right)$ induced by the function multiplication by f are precisely the critical values of f. Thus, the analysis of eigenvalues of $\widehat{c_1}$ naturally turn into analysis of critical values of LG superpotential f.

Moreover, the restriction ${f|}_{{\left({\mathbb{R}}_{>0}\right)}^N}$ is a real function on ${\left({\mathbb{R}}_{>0}\right)}^N$ that admits a global minimum at a unique point $x_{con}in{\left({\mathbb{R}}_{>0}\right)}^N$. Such ${\mathbf{x}}_{con}$ is called the conifold point of f. Let $T:=f\left({\mathbf{x}}_{con}\right)$.

Definition 1. 

We say that a Fano toric manifold X with Fano index ρ satisfies the Property $\mathcal{O}$, if its mirror Laurent polynomial $f=f_{\Sigma }$ satisfies the following three conditions:

1. 

every critical value u of f satisfies $\left|u\right|⩽T$;

2. 

the conifold point ${\mathbf{x}}_{con}$ is the unique critical point contained in $f^{-1}\left(T\right)$;

3. 

for any critical value u of f with $\left|u\right|=T$, $u=\zeta T_{con}$ with ${\zeta }^{\rho }=1$.

In conclusion, Conjecture $\mathcal{O}$ from B-side holds for the following cases (with key points):

• $Gr(k,n)$ [1]: Hori-Vafa mirror superpotential and conifold points;
• Toric del Pezzo surfaces [7]: Hori-Vafa mirror superpotential;
• $B{l}_{{\mathbb{P}}^r}{\mathbb{P}}^n$ [20]: Hori-Vafa mirror superpotential;
• ${\mathbb{P}}_{{\mathbb{P}}^n}(\mathcal{O}\oplus \mathcal{O}\left(n\right)),n$ even or n = 1 [21]: Hori-Vafa mirror superpotential;
• ${\mathbb{P}}_{{\mathbb{P}}^n}(\mathcal{O}\oplus \mathcal{O}(n-1)),n$ odd or n = 2 [21]: Hori-Vafa mirror superpotential.

4. Summary and Future Perspective

This survey paper presents different facets (both A-side and B-side) of Conjecture $\mathcal{O}$ and lists many known examples of Fano manifolds which Conjecture $\mathcal{O}$ holds. On the A-side, the main procedure is to compute all the eigenvalues of the matrix of linear operator ${\widehat{c}}_1$ and get the necessary conditions for (generalized) Perron-Frobenius theorem; On the B-side, the main procedure is to analyse critical values (especially, conifold points) of Landau-Ginzburg superpotential by optimization method. Recently, it is expected that Conjecture $\mathcal{O}$ will be modified and updated by changing $\mathbf{q}$ in the extended kahler moduli space.