Seidel elements / Shift operators

This section deals with certain equivariant Gromov–Witten invariants on Seidel spaces. The definition used in this package should be carefully compared to [Iri17, Section 3] and [MO19, Chapter 8] (shift operators). A (non-equivariant) symplectic account can be found in [MS12, Section 11.4] (Seidel representation).

Let $X$ be a (smooth projective) GKM variety with torus action by $T$, and let $\iota\colon \mathbb{C}^\times \rightarrow T$ be a group homomorphism. This gives rise to the $X$-bundle $\pi\colon S_X\rightarrow \mathbb{P}^1$, where $S_X$ is the Seidel space associated to $(X, \iota)$. Recall that $S_X$ is a GKM space with respect to $\widehat{T}:= T\times\mathbb{C}^\times$, where the extra copy of $\mathbb{C}^\times$ acts by rotating the base $\mathbb{P}^1$.

Let $(e_i)$ and $(e^i)$ be dual bases of $H_T^*(X;\mathbb{Q})$ with respect to the $T$-equivariant Poincaré pairing on $X$. Note also that we have a $H_T^*(\text{pt};\mathbb{Q})$-linear pushforward map $(i_\infty)_*\colon H_T^*(X;\mathbb{Q}) \rightarrow H_{\widehat{T}}^*(S_X;\mathbb{Q})$ raising degree by one.

Finally, let $H_2^\text{sec}(S_X;\mathbb{Z})$ be the set of effective section curve classes in $S_X$, i.e., effective curve classes that project to $[\mathbb{P}^1]$ under $\pi\colon S_X\rightarrow\mathbb{P}^1$. The additive group $H_2^\text{sec}(S_X;\mathbb{Z})$ is an $H_2^\text{eff}(X;\mathbb{Z})$-torsor, so after picking some $\beta_0\in H_2^\text{sec}(S_X;\mathbb{Z})$ there is an identification $r\colon H_2^\text{sec}(S_X;\mathbb{Z})\stackrel{\cong}{\longrightarrow} H_2^\text{eff}(X;\mathbb{Z})$ that sends $\beta_0$ to $0$.

The (equivariant) Seidel element associated to $\iota$ is

\[ \mathcal{S}(\iota) := \sum_{\beta\in H_2^\text{sec}(S_X;\mathbb{Z})} \left[GW^{\widehat{T}}_{0,1,\beta}((i_\infty)_*(e_i))\right]_{\hat{t} = 0} e^i q^{r(\beta)} \in QH_T^*(X)\]

where $\hat{t}$ is the equivariant parameter for the extra $\mathbb{C}^\times$ in $\widehat{T}$.