Quantum Cohomology

Much of this can be found in [CK99, Chapter 8 and 9.3]. Let $X$ be a GKM space (see Definition). Its (small) equivariant quantum cohomology $QH_T^*(X)$ is given additively by $H_T^*(X;\mathbb{Q})\otimes \widehat{\mathbb{Q}[H_2^\text{eff}(X;\mathbb{Z})]}$, where $\widehat{\mathbb{Q}[H_2^\text{eff}(X;\mathbb{Z})]}$ is the completion of the semigroup ring $H_2^\text{eff}(X;\mathbb{Z})$ of effective curve classes. The element corresponding to $\beta\in H_2^\text{eff}(X;\mathbb{Z})$ is written as $q^\beta$.

The $H_T(\text{pt};\mathbb{Q})$-module $QH_T^*(X)$ is a commutative associative unital $H_T(\text{pt};\mathbb{Q})$-algebra via the (small) equivariant quantum product $\ast$ defined as follows. For every classes $a,b,c\in H_T^*(X;\mathbb{Q})$ we have

\[ \langle a \ast b, c \rangle = \sum_{\beta\in H_2^\text{eff}(X;\mathbb{Z})} GW^T_{X,0,3,\beta}(a,b,c) \cdot q^\beta\]

where:

The equivariant Poincaré pairing is given by $\langle a,b\rangle := \int_X a\cup b\in H_T^*(\text{pt};\mathbb{Q})$, where we use equivariant integration,
We denote by $GW^T_{X,0,3,\beta}(a,b,c)\in H_T^*(\text{pt};\mathbb{Q})$ the equivariant Gromov–Witten invariant for $X$ in class $\beta$ of genus $0$ with $3$ marked points.

Note that setting all the equivariant parameters $t_1,\dots,t_{\dim_\mathbb{C}(T)}$ to zero recovers the standard (small, non-equivariant) quantum product.

Structure Constants

Given any linear basis $e_1,\dots,e_N$ of $H_T^*(X;\mathbb{Q})\otimes \text{Frac}(H_T^*(\text{pt}; \mathbb{Q}))$ over $\text{Frac}(H_T^*(\text{pt}; \mathbb{Q}))\cong \mathbb{Q}(t_1,\dots,t_r)$, the associated structure constants are $(c_{i,j,k})_{i,j,k=1}^N$ where

\[c_{i,j,k}\in \mathbb{Q}(t_1,\dots,t_r)\otimes \widehat{\mathbb{Q}[H_2^\text{eff}(X;\mathbb{Z})]}\]

is the coefficient of $e_k$ in $e_i\ast e_j$. Let us denote the coefficient of $q^\beta$ in $c_{i,j,k}$ by $c_{i,j,k}(\beta)\in\mathbb{Q}(t_1,\dots,t_r)$.

Note

When $(X,T)$ is a compact Hamiltonian or projective GKM space, and $e_1,\dots,e_N$ is chosen to be a linear basis of $H_T^*(X;\mathbb{Q})$ over $H_T^*(\text{point};\mathbb{Q})\cong \mathbb{Q}[t_1,\dots,t_r]$, we have

\[ c_{i,j,k}(\beta)\in\mathbb{Q}[t_1,\dots,t_r]\]

for all $i,j,k,\beta$.

Choice of basis

Since structure constants depend on a choice of basis, let us introduce some common choices and explain structure constants for mixed bases.

The standard basis

In general, when no basis is specified explicitly, all structure constants are computed with respect to the basis $e_1,\dots,e_N$, which we call the standard basis. Namely, let $N$ be the number of vertices of the GKM graph $G$. For each $i\in\{1,\dots,N\}$, let

\[ e_i \in H_T^*(X;\mathbb{Q})\otimes \mathbb{Q}(t_1,\dots,t_r)\]

be class that localizes to $1$ at the $i$-th fixed point and to $0$ at every other fixed point. This uniquely defines a class by the GKM theorem [GKM98].

Note

The $e_i$ are only well-defined as elements of $H_T^*(X;\mathbb{Q})\otimes \mathbb{Q}(t_1,\dots,t_r)$, not of $H_T^*(X;\mathbb{Q})$.
Since the standard basis over $\mathbb{Q}(t_1,\dots,t_r)$, the resulting structure constants $c_{i,j,k}(\beta)$ can fail to be polynomials in $t_1,\dots,t_r$ even when $G$ is the GKM graph of a compact Hamiltonian or projective GKM space $(X,T)$.

In our implementation, $e_i$ is printed as e[i] (see Cohomology).

The fixed point basis

The fixed point basis is given by $f_1,\dots,f_N$, where $N$ is the number of vertices of $G$ and

\[ f_i = \left( \prod_{\epsilon\in E(G)_i} \alpha(\epsilon) \right) e_i \in H_T^*(X;\mathbb{Q}).\]

Mathematically, we have $\prod_{\epsilon\in E(G)_i} \alpha(\epsilon) = e_T(T_{p_i}X)$ when $G$ is realized by the GKM space $X$ and $p_i\in X^T$ corresponds to the $i$-th vertex of $G$.

Note

Sometimes, we use the notation $f_i = PD(p_i)$ as $f_i$ is the equivariant Poincaré dual of $p_i\in X^T$.
The collection $(f_i)_{i=1}^N$ is not a $\mathbb{Q}[t_1,\dots,t_r]$-linear basis for $H_T^*(X;\mathbb{Q})$, but it is a $\mathbb{Q}(t_1,\dots,t_r)$-linear basis for $H_T^*(X;\mathbb{Q})\otimes \mathbb{Q}(t_1,\dots,t_r)$.
The element $f_i$ can be obtained computationally as point_class(i, G) (see point_class).

Mixing the bases

In some functions (such as QH_structure_constants) we use two different bases to define $c_{i,j,k}(\beta)$. Namely, given two bases $(a_i)$ and $(b_i)$, we let $c_{i,j,k}(\beta)$ be the coefficient of $b_k q^\beta$ in $a_i \ast a_j$. In this case, we say that $(c_{i,j,k}(\beta))$ are the structure constants with respect to the input basis $(a_i)$ and output basis $(b_i)$.

GKMtools.QH_structure_constants — Function

QH_structure_constants(G::AbstractGKM_graph; refresh::Bool=false)

Return the structure constants of the equivariant quantum cohomology $QH_T^*(X)$ where $X$ is the GKM space realizing the GKM graph. The structure constants are returned with respect to the fixed point basis as input basis and the standard basis as output basis (see Mixing the bases). This has the advantage that all entries are polynomials in the equivariant parameters when G is the GKM graph of a compact Hamiltonian or projective GKM space.

Warning

This requires is_strictly_nef(G)==true, as this guarantees that there are at most finitely many curve classes $\beta$ with non-zero coefficients for $q^\beta$.
If is_strictly_nef(G)==false, use the method of QH_structure_constants below that specifies a specific $\beta$.

Note

As this computation might be expensive, the result is stored in G for later use. If the requested structure constants have been computed before, they will not be computed afresh unless the optional argument refresh is set to true.

Output format:

The output type is a Julia dictionary of type Dict{CurveClass_type, Array{Any, 3}}. For each curve class $\beta$, the corresponding value is a 3-dimensional array of type Array{Any, 3}. If ans denotes this dictionary, then ans[beta][i, j, k] is the the $q^\beta$-coefficient of $PD(v_i) \ast PD(v_j)$ localized at $v_k$, where $v_i, v_j, v_k$ represent the fixed points with indices $i,j,k$, respectively, and $PD$ represents the equivariant Poincaré dual (cf. fixed point basis).

Optional arguments:

refresh::Bool: false by default. If true, then this will overwrite any previously calculated $QH_T$ structure constants of G.

Example

In the following example, we calculate the structure constants of $\mathbb{P}^1$. The fixed point basis $f_1,f_2$ (see above) is given by f1 = (t1 - t2)*e[1] and f2 = (-t1 + t2)*e[2]. Let us check that

\[f_1 \ast f_1 = (t_1 - t_2) f_1 + q = (t_1 - t_2)^2 e_1 + q(e_1 + e_2)\]

noting that $e_1+e_2=1$. Similarly, we will check that

\[f_1 \ast f_2 = q = q(e_1 + e_2)\]

and

\[f_2 \ast f_2 = (-t_1 + t_2) f_2 + q = (-t_1 + t_2)^2 e_2 + q(e_1 + e_2).\]

The following code verifies these identities.

julia> P1 = projective_space(GKM_graph, 1);

julia> S = QH_structure_constants(P1; show_progress=false);

julia> curve_classes = collect(keys(S));

julia> S[curve_classes[1]] # curve class 0
2×2×2 Array{Any, 3}:
[:, :, 1] =
 t1^2 - 2*t1*t2 + t2^2  0
 0                      0

[:, :, 2] =
 0  0
 0  t1^2 - 2*t1*t2 + t2^2

julia> S[curve_classes[2]] # line class
2×2×2 Array{Any, 3}:
[:, :, 1] =
 1  1
 1  1

[:, :, 2] =
 1  1
 1  1