ScalingStacks

Let $Z$ be a singular oriented curve. We associate to $Z$ a differential algebra $A(Z)=\bigoplus_{i\geq 0}A_{k}(Z)$.

The algebra $A_{k}(Z)$ has a basis given by “braids”: these are pairs $(I,([\zeta_{i}])_{i\in I})$, where $I$ is a set of $k$ singular points of $Z$ and $[\zeta_{i}]$ is a homotopy class of smooth oriented paths starting at $i$ and ending at a singular point. We require that the end points of $\zeta_{i}$ and $\zeta_{j}$ are distinct if $i\neq j$.

We define $d(I,([\zeta_{i}]))$ to be the sum over intersection points between paths $\zeta_{i}$ of the braid obtained by resolving the intersection point.

The composition $(I^{\prime},[\zeta^{\prime}_{i^{\prime}}])\circ(I,[\zeta_{i}])$ is $0$ or $(I,[\zeta^{\prime}_{\zeta_{i}(1)}\circ\zeta_{i}])$ if a number of conditions are satisfied:

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  $\{\zeta_{i}(1)\}=I^{\prime}$

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  the paths $\zeta^{\prime}_{\zeta_{i}(1)}\circ\zeta_{i}$ are smooth

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  there are no representatives in the homotopy class of the concatenated paths with fewer intersections than $\{\zeta^{\prime}_{\zeta_{i}(1)}\circ\zeta_{i}\}_{i\in I}$.

A singular curve $Z$ with a worst ordinary double points gives rise to a sutured surface $F(Z)$ with an arc decomposition (cf the case of a torus below) and we have $A(F(Z))=A(Z)$: the algebras $A(Z)$ generalize those of [LiOzTh1, Za].

Given a closed embedding of $(0,1]$ (resp. $[-1,0)$) in $Z$ avoiding singular points, we construct a higher representation of $\mathfrak{gl}(1|1)^{+}$ on $A(Z)$. The bimodule $E$ has a basis given by braids where one path starts at $1$ (resp. ends at $-1$).

Given embeddings of $[-1,0)$ and $(0,1]$ in $Z$ as above, we can construct a singular curve $\bar{Z}$ by attaching $[-1,1]$ to $Z$ along $[-1,0)\cup(0,1]$. Our results on algebras associated to the gluing of surfaces is a consequence of the more general result below on singular curves.

A version of this result allowing partially oriented singular curves contains as a special case the construction of nil affine Hecke algebras from nil Hecke algebras.

The reconstruction of the partially wrapped Fukaya categories for the torus depicted in §1.3 corresponding to the following decomposition of the corresponding singular curve: