Subsection 7.2.5. (0PH8) · ScalingStacks

7.2.5. Sutured surfaces and topological field theories

We define a sutured surface to be a quadruple $(F,\Lambda,S^{+},S^{-})$ where $F$ is a compact oriented surface, $\Lambda$ is a finite subset of $\partial{F}$ , and $S^{+}$ and $S^{-}$ are unions of components of $\partial{F}-\Lambda$ such that $\Lambda=\overline{S^{+}}\cap\overline{S^{-}}$ and $\partial{F}-\Lambda=S^{+}\cup S^{-}$ (this is [Za, Definition 1.2] without the topological restrictions). Note that $(F,\Lambda,S^{+},S^{-})$ is determined by the data of $(F,S^{+})$ : we have $\Lambda=\overline{S}^{+}-S^{+}$ and $S^{-}=\partial F-\overline{S}^{+}$ . A sutured surface is representable by a chord diagram (as we define it) if and only if each component of $F$ (not $\partial{F}$ ) intersects $S^{+}$ and $S^{-}$ nontrivially.

Let $({\mathcal{Z}},{\mathbf{a}})$ be a chord diagram. We define a sutured surface $(F,\Lambda,S^{+},S^{-})$ :

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the oriented surface $F$ is obtained from ${\mathcal{Z}}\times[0,1]$ by adding $1$ -handles at $\{(z,0),(z^{\prime},0)\}$ for all pairs $\{z,z^{\prime}\}$ in ${\mathbf{a}}$
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$S^{+}=\bigl({\mathcal{Z}}\times\{1\}\bigr)\cup\bigl(\partial{\mathcal{Z}}\times(\frac{1}{2},1]\bigr)$

When ${\mathcal{Z}}$ is a union of intervals, this is Zarev’s construction [Za, §2.1].

Let $Z$ be a singular curve giving rise to $({\mathcal{Z}},{\mathbf{a}})$ . The oriented surface $F$ can be identified with $Z\times[0,1]$ and paths in $Z$ give rise to paths in $F$ .

The sutured surface $F$ also comes with an arc decomposition: for each $\{z,z^{\prime}\}$ in ${\mathbf{a}}$ , we have an arc $\omega_{\{z,z^{\prime}\}}$ with set of end points $\{(z,1),(z^{\prime},1)\}$ in $S^{+}$ corresponding to the $1$ -handle added at $\{(z,0),(z^{\prime},0)\}$ .

0P9L

Example 7.2.13. In the table below, the first row depicts some chord diagrams. The second and third rows show the corresponding sutured surfaces with the $S^{+}$ part of the boundary in green and with the arcs $\omega_{z}$ in red; the second row applies the above construction directly, and the third row gives an alternate perspective. The fourth row shows the sutured surfaces as open-closed cobordisms (with empty source and with target colored in green); this interpretation is discussed in §7.2.5.

Under the strands algebra construction of §8.1, the first and second columns give rise to simple 2-representations of $\mathcal{U}$ , categorifying the vector representation and its dual.

Tensor powers of the algebra of the first column give algebras very similar to the one considered by Tian [Ti]; in fact, Tian’s algebras were an important early clue in the development of the present work. Tensor powers of the algebra of the second column are studied from the Heegaard Floer perspective by the first-named author in [Man].

The algebra of the third column is the $n=3$ case of a family of algebras considered in [ManMarWi, LePo]. For general $n$ , these are isomorphic to the algebras ${\mathcal{B}}(n)=\oplus_{k=0}^{n}{\mathcal{B}}(n,k)$ used by Ozsváth and Szabó in their theory of bordered knot Floer homology [OsSz4, OsSz5, OsSz6] (their notation is slightly different). The middle summand of the algebra of the fourth column is the undeformed version of a curved $A_{\infty}$ -algebra used by Lipshitz-Ozsváth-Thurston [LiOzTh2, LiOzTh3] to define bordered $HF^{-}$ for $3$ -manifolds with torus boundary. The middle summand of the algebra of the fifth column is the well-known “torus algebra” from bordered Floer homology. The fifth and sixth columns together illustrate our perspective on cornered Floer homology; following Zarev’s ideas, we view the cornered Floer gluing theorem as recovering the algebra of two matched intervals glued end-to-end, rather than as the invariants of two matched intervals with distinguished endpoints being glued to form a pointed matched circle.

The first, fifth, and sixth columns give algebras that are among Zarev’s strands algebras $\mathcal{A}(\mathcal{Z})$ , although the first diagram is degenerate (equivalently, its sutured surface has closed circles in $S^{-}$ ). The second, third, and fourth columns do not satisfy the restrictions that Zarev imposes. As far as we are aware, our strands categories below give the first detailed description of strands algebras associated to general chord diagrams with circles as well as intervals; less formal descriptions have appeared previously, cf. [Au2, Proposition 11]. As indicated by Lipshitz-Ozsváth-Thurston’s work [LiOzTh2, LiOzTh3], curved $A_{\infty}$ -deformations of the algebras appear necessary in the general setting when defining modules and bimodules for 3-manifolds with boundary, although in special cases like Ozsváth-Szabó’s bordered knot Floer homology (third column) this complication should be avoidable.

A sutured surface can be viewed as a morphism in the 2d open-closed cobordism category with empty source; if $(F,\Lambda,S^{+},S^{-})$ is a sutured surface, the corresponding open-closed cobordism has target given by $S^{+}$ and non-gluing boundary given by $S^{-}$ . See the bottom row of the figure in Example 7.2.13; the targets of these open-closed cobordisms are shown in green and the non-gluing boundary is shown in black.

Let us consider how the end-to-end gluings of chord diagrams covered by our results in §8 can be viewed in terms of open-closed cobordisms. When gluing two distinct intervals of a chord diagram end-to-end, the corresponding sutured surface gets glued as in the top-left picture below: the two intervals marked in blue are glued together to form the top-middle picture. However, we can also consider the top-middle picture as arising from the top-right picture; in this latter case the gluing is an instance of composition (with an open pair of pants) in the open-closed cobordism category. Similarly, when self-gluing the two endpoints of an interval of a chord diagram, the sutured surface gets glued as in the bottom-left picture below, producing the bottom-middle picture; we can also think of the bottom-middle picture as arising from the bottom-right picture, which is another instance of composition in the open-closed category.

One could try to view our constructions as giving part of the structure of an open-closed 2d TQFT valued in a category whose objects are dg 2-categories and whose morphisms are certain dg 2-functors. In particular, this hypothetical open-closed TQFT would assign a dg 2-category of 2-representations of ${\mathcal{U}}$ to an interval. To an open-closed cobordism with empty source, the open-closed TQFT would assign an object of the dg 2-category of the target, encoding the data of a lax multi-2-action of ${\mathcal{U}}$ for the interval components of the target. Our approach doesn’t quite realize that. We associate 2-representations of ${\mathcal{U}}$ to chord diagrams or singular curves rather than directly to surfaces.

One can also consider the extent to which such a theory would extend to a point. Things are considerably simpler for the decategorified version of the theory, where one sees many relationships with other work on 3d TQFTs; this will be addressed in more detail in a follow-up paper [ArMa].

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2