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 represented by its fixed-point restrictions (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

One can also 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 $(b_i)$.

For multiplication matrices in a chosen basis, use matrix_small_quantum_product.

Quantum Arithmetic

GKMtools.quantum_product — Function

quantum_product(G, beta, a, b; show_bar=false, twist_class = nothing)

Calculate the $q^\beta$-coefficient of the equivariant quantum product of the equivariant cohomology classes a and b on G.

Twisting

The experimental optional argument twist_class allows to multiply by any additional EquivariantClass before integrating over $\overline{\mathcal{M}}_{0,3}(X;\beta)$ in the definition of the quantum product.

Progress bar

The optional progress bars for each underlying call to gromov_witten can be activated by setting show_bar=true.

Example

julia> P1 = projective_space(GKMGraph, 1);

julia> beta0 = zero(GKM_second_homology(P1).H2);

julia> p = point_class(P1, 1);

julia> quantum_product(P1, beta0, p, p) == localize(p^2)
true

Let us compute the equivariant quantum products of some point classes on $\mathbb{P}^2$.

julia> P2 = projective_space(GKMGraph, 2);

julia> beta = curve_class(P2, Edge(1, 2));

julia> quantum_product(P2, beta, point_class(P2, 1), point_class(P2, 2))
GKM class with restrictions:
[-t1 + t3, -t2 + t3, 0]

julia> quantum_product(P2, 0*beta, point_class(P2, 1), point_class(P2, 2))
GKM class with restrictions: 
[0, 0, 0]

julia> quantum_product(P2, 2*beta, point_class(P2, 1), point_class(P2, 2))
GKM class with restrictions: 
[0, 0, 0]