Peng Zhou

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===== The rank-one setup =====

We begin with \(\mathfrak g=\mathfrak{sl}_2\) and fix \(d\geq 0\). On the algebraic side, we consider the \(\hbar\)-deformed nilHecke algebra \(NH_d^\hbar\) over \(\mathbb Z[\hbar]\). It is generated by dots \(x_1,\ldots,x_d\) and elementary crossings \(s_1,\ldots,s_{d-1}\), subject to
\[
\begin{gathered}
x_a x_b=x_b x_a,\qquad s_a^2=0,\qquad s_a s_b=s_b s_a\quad\text{if }|a-b|>1,\cr
s_a s_{a+1}s_a=s_{a+1}s_a s_{a+1},\qquad s_a x_b=x_b s_a\quad\text{if }b\neq a,a+1,\cr
x_a s_a-s_a x_{a+1}=\hbar,\qquad s_a x_a-x_{a+1}s_a=\hbar.
\end{gathered}
\]
At \(\hbar=1\) this is the usual nilHecke algebra. We refer to Khovanov--Lauda for its standard properties and its role in the categorification of \(U_q^+(\mathfrak{sl}_2)\).

Recall that
\[
\mathbb I=T^*[-1,1]=\{z\in\mathbb C\mid -1\leq \operatorname{Re}(z)\leq 1\}.
\]
By rescaling, we may replace \(\mathbb I\) by \(\mathbb C\), remembering the two boundary components of \(\mathbb I\) as stops at \(z=+\infty\) and \(z=-\infty\).

The degree-\(d\) open \(\mathfrak{sl}_2\) Zastava space is
\[
Z_d=\{(P,Q)\in\mathbb C[z]\times\mathbb C[z]\mid P\text{ is monic of degree }d,\ \deg Q<d,\ \gcd(P,Q)=1\}.
\]
Equivalently, a point of \(Z_d\) determines the based rational function \(Q(z)/P(z)\). It carries the divisor map
\[
\pi:Z_d\longrightarrow \operatorname{Sym}^d(\mathbb C),\qquad (P,Q)\longmapsto \operatorname{div}(P).
\]

Let \(\Delta\subset\operatorname{Sym}^d(\mathbb C)\) denote the big diagonal, and set
\[
\operatorname{Sym}^d(\mathbb C)^\circ=\operatorname{Sym}^d(\mathbb C)\setminus\Delta,
\qquad
Z_d^\circ=\pi^{-1}\bigl(\operatorname{Sym}^d(\mathbb C)^\circ\bigr).
\]
We write
\[
(\mathbb C^d)^\circ:=\{(y_1,\ldots,y_d)\in\mathbb C^d\mid y_a\neq y_b\text{ for }a\neq b\}
\]
for the ordered cover of \(\operatorname{Sym}^d(\mathbb C)^\circ\), and define
\[
\widetilde Z_d^\circ:=Z_d^\circ\times_{\operatorname{Sym}^d(\mathbb C)^\circ}(\mathbb C^d)^\circ.
\]
On \(\widetilde Z_d^\circ\) we may write \(P(z)=\prod_{a=1}^d(z-y_a)\) and define
\[
u_a:=\operatorname{Res}_{z=y_a}\frac{Q(z)}{P(z)}\,dz=\frac{Q(y_a)}{P'(y_a)}.
\]
Since \(P\) and \(Q\) are relatively prime, \(u_a\neq 0\) for every \(a\), and these coordinates give a trivialization
\[
\widetilde Z_d^\circ\cong(\mathbb C^d)^\circ\times(\mathbb C^*)^d.
\]

Write \(Q(z)=c_{d-1}z^{d-1}+\cdots+c_0\). The Whittaker superpotential is the globally defined function \(W_d:Z_d\to\mathbb C\) given by \(W_d(P,Q)=c_{d-1}\), equivalently the total residue of the rational one-form \(Q(z)P(z)^{-1}dz\) at its finite poles. On \(\widetilde Z_d^\circ\), partial fractions give
\[
\frac{Q(z)}{P(z)}=\sum_{a=1}^d\frac{u_a}{z-y_a},
\qquad
W_d=\sum_{a=1}^d u_a.
\]

There is a single standard generating object in degree \(d\), which we denote by \(T_d\). Over \(\widetilde Z_d^\circ\), it is obtained from \(d\) disjoint standard arcs in the \(y\)-plane and the standard Lagrangian in each \(\mathbb C_u^*\)-factor; we specify these representatives and their wrapping in the next subsubsection. We define Floer theory over \(\mathbb Z[\hbar]\), where a holomorphic disk meeting the divisor \(\pi^{-1}(\Delta)\) with intersection number \(m\) is weighted by \(\hbar^m\). We write
\[
\mathcal A_d^{\mathrm{geo}}:=HW^*(T_d,T_d;\mathbb Z[\hbar]).
\]
The goal of the rank-one calculation is to construct an isomorphism
\[
NH_d^\hbar\cong\mathcal A_d^{\mathrm{geo}}.
\]