ScalingStacks

While Lie algebra representations and their tensor products have long played an important role in mathematics, their connection with low-dimensional topology is more recent. This involves quantum groups, which provide a deformation of the classical Lie theory. Reshetikhin–Turaev’s theory give rise to invariants of links and $3$-manifolds [ReTu].

Crane and Frenkel [CrFr] conjectured that there should be a “higher” representation theory where vector spaces are replaced by categories, and this would provide invariants of $4$-manifolds. The notion of higher representations was introduced first for type $A$ [ChRou] and then for general Kac–Moody algebras [Rou1, KhoLau]. In a work in preparation [Rou3], the second author gives a construction of a tensor product for higher representations of Kac–Moody algebras, in an $\infty$-categorical setting. An important feature is that the category underlying a tensor product of higher representations $\mathcal{V}$ and $\mathcal{V}^{\prime}$ of $\mathfrak{g}$ depends on the action of the positive part $\mathfrak{g}^{+}$ of $\mathfrak{g}$ on $\mathcal{V}$ and $\mathcal{V}^{\prime}$, and not just on the categories $\mathcal{V}$ and $\mathcal{V}^{\prime}$ themselves. Evidence for Crane and Frenkel’s program has also been provided by the work of Khovanov [Kho1], Webster [We] and others.

In this article, we consider the case of the super Lie algebra $\mathfrak{gl}(1|1)$. We do not discuss the notion of higher representations of $\mathfrak{gl}(1|1)$ (cf [Rou3]), but we focus on the positive part $\mathfrak{gl}(1|1)^{+}=\mathbb{C}e$, a one-dimensional odd super Lie algebra. The notion of a higher representation of $\mathfrak{gl}(1|1)^{+}$ is due to Khovanov [Kho]: it is the data of a differential category ${\mathcal{V}}$ over $\mathbb{F}_{2}$ together with a differential endofunctor $E$ and an endomorphism $\tau$ of $E^{2}$ with $d(\tau)=1$ satisfying $\tau^{2}=0$ and braid relations. So, a higher representation provides an endofunctor whose square is homotopic to $0$. An equivalent definition is that of an action of the monoidal category ${\mathcal{U}}$ generated by an object $E$ and a map $\tau:E^{2}\to E^{2}$ satisfying the conditions above.

We will allow a more general type of action where $E$ is given by a $({\mathcal{V}},{\mathcal{V}})$-bimodule.

We define a notion of tensor product ${\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}$ of higher representations of $\mathfrak{gl}(1|1)^{+}$, endowing the $2$-category of $2$-representations of ${\mathfrak{gl}}(1|1)^{+}$ on differential categories with a structure of monoidal $2$-category. This does not require working in an $\infty$-categorical setting, as in the Kac–Moody case.

Given $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ two higher representations, we construct a higher representation $\mathcal{V}_{1}{\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}\mathcal{V}_{2}$. A typical object of this category is a pair $(M_{1}\otimes M_{2},\pi)$ where $M_{1}\otimes M_{2}$ is an object of the ordinary tensor product $\mathcal{V}_{1}\otimes\mathcal{V}_{2}$ and $\pi$ is a closed map $M_{1}\otimes E_{2}(M_{2})\to E_{1}(M_{1})\otimes M_{2}$ compatible with $\tau$. More generally, one considers objects obtained from those by taking cones and direct summands. The image by $E$ of the pair above is $(\mathrm{cone}(\pi),\pi^{\prime})$ for some $\pi^{\prime}$.

This construction generalizes immediately to differential categories endowed with two commuting structures of higher representations, but we need a more general construction dealing with two lax-commuting higher representations to handle general gluings of surfaces. We provide three increasingly subtle versions of such a construction. In general, we obtain a differential category without the (full) structure of higher representation.

Starting with two structures of higher representations given by endofunctors $E_{1}$ and $E_{2}$ on a differential category $\mathcal{W}$ and a map $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ (suitably compatible with $\tau$’s), we define a differential category $\Delta_{\sigma}(\mathcal{W})$ by proceeding as in the tensor product case. It will have a structure of higher representation if $\sigma$ is invertible.

The notion of right higher representation coincides with that of (left) higher representation, but it leads to a different version of the construction above. We start with the same structures as above, but write $F_{1}$ instead of $E_{1}$ and $\lambda$ instead of $\sigma$. We define a differential category $\Delta_{\lambda}(\mathcal{W})$ with typical objects pairs $(M,(\upsilon)_{i\geq 1})$ where $M$ is an object of $\mathcal{W}$ and $\upsilon_{i}:E_{2}^{i}F_{2}^{i}(M)\to M$ is a system of compatible maps with respect to $\lambda$ and $\tau$. In order to define a structure of higher representation, we need $F_{1}$ to have a right adjoint $E_{1}$. Using this adjunction, $\lambda$ gives rise to $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ and, when $\sigma$ is invertible, we obtain a structure of higher representation on $\Delta_{\lambda}(\mathcal{W})$.

Finally, starting with a lax action of $\mathcal{U}\times\mathcal{U}$ on $\mathcal{W}$, we define a differential category $\Delta_{E}(\mathcal{W})$.

Our constructions extend an earlier construction of Douglas-Manolescu [DouMa]. They provided a construction of the category underlying a tensor product.

One of the applications of tensor products in higher representation theory is the construction of complicated categories from simpler ones. This is illustrated below in the reconstruction of partially wrapped Fukaya categories of symmetric powers of surfaces from more basic algebras. We formulate the problem in terms of surfaces with an arc decomposition, and then reformulate it again in terms of certain singular curves. We provide another example, the construction of nil affine Hecke algebras from nil Hecke algebras (in type $A$).

The main examples of higher representations of $\mathfrak{gl}(1|1)^{+}$ that we introduce here are on partially wrapped Fukaya categories of symmetric powers of surfaces. These are $A_{\infty}$-categories and the results of §1.3 will be made more precise in §1.4, where we work with differential categories.

Let $\Sigma$ be a compact oriented surface with a finite collection $M$ of marked points in its boundary, and assume that each component of $F$ contains at least one point of $M$.

In the pictures below, we expand each point of $M$ to an interval in the boundary of $\Sigma$. We draw the complement of these intervals in dotted light orange. Here are two views of a genus-one surface with one boundary component and $M$ consisting of one point.

For any $k\geq 0$, Auroux [Au2, Section 3.1] considers a partially wrapped Fukaya category $\mathcal{F}(\mathrm{Sym}^{k}(\Sigma),M)$ of $\mathrm{Sym}^{k}(\Sigma)$ with set of stops $M\times\mathrm{Sym}^{k-1}(\Sigma)$. We write $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ for the direct sum of these categories over all $k\geq 0$ (they vanish for $k$ large enough).

Given a component $I$ of $\partial\Sigma\setminus M$, we define a higher action of $\mathfrak{gl}(1|1)^{+}$ on $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$.

Consider $(\Sigma_{1},M_{1},I_{1})$ and $(\Sigma_{2},M_{2},I_{2})$ two surfaces with chosen intervals as above. We form a new surface $(\Sigma,M)$ with a chosen interval $I$ by gluing $I_{1}$ and $I_{2}$ to the two legs of an open pair of pants.

Theorem 1.3.1 extends a result of Douglas–Manolescu [DouMa]; their theorem corresponds to the special case of Theorem 1.3.1 in which $\Sigma_{1}$ and $\Sigma_{2}$ have only one boundary circle and one marked point each. They prove an equivalence of categories without the statement on compatibility of higher actions.

More generally, given $(\Sigma,M)$ with two disjoint chosen intervals $I_{1},I_{2}$ in $\partial\Sigma\setminus M$, we form a new surface $(\overline{\Sigma},\overline{M})$ by gluing the two legs of an open pair of pants to $I_{1}$ and $I_{2}$. Theorem 1.3.1 generalizes to say that $\mathcal{F}(\mathrm{Sym}^{*}\overline{\Sigma},\overline{M})$ is equivalent to $\Delta\mathcal{F}(\mathrm{Sym}^{*}\Sigma,M)$ with its diagonal action.

This makes it possible to recover $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ for any $(\Sigma,M)$ from the case of a disk with two points in the boundary. As an illustration, consider the genus-one surface $(\Sigma,M)$ shown above; the partially wrapped Fukaya category $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ is described by the “torus algebra”, a standard example in bordered Heegaard Floer homology. We can cut $(\Sigma,M)$ along three arcs as shown; we are left with two rectangles. Gluing the cuts back together, the torus algebra can then be recovered from one application of the tensor product followed by two applications of the more general $\Delta$ construction.

In the case of the first symmetric power $\operatorname{Sym}\nolimits^{1}(\Sigma)=\Sigma$, a general construction of Fukaya categories by a gluing procedure is given by Haiden, Katzarkov and Kontsevich in [HaiKaKon].

A general theory of partially wrapped Fukaya categories and how they glue is provided by Ganatra, Pardon and Shende in [GaPaShe], but this doesn’t apply directly to our case.

Heegaard Floer homology, defined by Ozsváth–Szabó [OsSz1, OsSz2, OsSz3], is a set of invariants for 3- and 4-dimensional manifolds. For a 3-manifold $Y$, the Heegaard Floer invariant of $Y$ (an abelian group) is defined by choosing a Heegaard decomposition of $Y$ as two handlebodies glued along a genus-$g$ surface $\mathcal{H}$, then computing a Lagrangian intersection Floer homology group between two Lagrangian submanifolds in $\mathrm{Sym}^{g}(\mathcal{H})$ induced by the two handlebodies.

In bordered Heegaard Floer homology [LiOzTh1, Za], there are also extended Heegaard Floer invariants for 2d surfaces and 3d cobordisms. Let $F$ be the data of a surface $(\Sigma,M)$ with a set of points $M\subset\partial\Sigma$ as above, equipped with a choice of arc decomposition. To such a surface, bordered Heegaard Floer associates a differential algebra $A(F)$. Auroux [Au2] has shown that the algebra $A(F)$ is the endomorphism algebra of a generating object of $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ determined by the arc decomposition.

Our constructions are based on the combinatorics of $A(F)$ and we do not work directly with $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$. Given a component of $\partial\Sigma\setminus M$, we define a differential bimodule $E$ over $A(F)$, and a bimodule endomorphism $\tau$ of $E\otimes_{A(F)}E$, that yield a higher representation of $\mathfrak{gl}(1|1)^{+}$.

Theorem 1.3.1 follows now from the following result.

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:

• •

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

• •

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

• •

  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:

We expect that our constructions (in particular Theorem 1.3.1) will be part of a $4$-dimensional TQFT.

This article is the first step towards a higher representation-theoretic reconstruction of Heegaard Floer theory, which would be a fulfilment of the program of Crane and Frenkel for ${\mathfrak{gl}}(1|1)$. This article focuses on dimensions $1$ and $2$, where homotopic phenomena can be avoided.

It is natural to ask if our constructions extend to dimension $0$. Work in progress of the first author and Reeshad Arian [ArMa] shows this is possible at the decategorified level.

We are pursuing two directions for the extension to dimension $3$.

In a work in preparation, the first author extends the tensor product functoriality to the $A_{\infty}$-setting. This is a key step to construct morphisms of $2$-representations from Heegaard Floer diagrams by cutting them into pieces.

Work in preparation of the second author [Rou4] provides a construction of invariants of links in $S^{3}$. Appropriate $t$-structures are used to handle the homotopic phenomena.

Note that Ellis, Pevtkova and Vértesi in [ElPeVe] construct homological invariants of tangles in the setting of bordered Heegaard Floer theory in which ${\mathfrak{gl}}(1|1)$-categorifications appear. Note also that [DouLiMa] considered $3$-manifolds with codimension-$2$ corners in the setting of [DouMa].

We expect that our algebraic constructions will provide a blueprint for the construction of higher categorical structures and $3$ and $4$-dimensional invariants associated to (ordinary) simple Lie algebras.

We gather in §2 a number of basic definitions and facts involving differential categories and bimodules. Most differential vector spaces we encounter come with bases, and we formalize this aspect in the notion of “differential pointed sets” and corresponding differential pointed categories.

We consider Hecke algebras in §3. We study in §3.1 the differential algebra structure on nil Hecke algebras of Coxeter groups over a field of characteristic $2$ and we describe adjunctions for induction and restriction functors, in the case of finite Coxeter groups. An important fact is that those Hecke algebras are the graded algebras associated with the filtration of the group algebra with respect to the length function. The remainder of §3 is devoted to the case of symmetric groups and their affine versions. We introduce in §3.2.6 positive submonoids of the affine symmetric groups and we provide a description by generators and relations of their nil Hecke algebras.

Section §4 is devoted to the development of the $2$-representation theory of ${\mathfrak{gl}}(1|1)^{+}$. We introduce the monoidal category ${\mathcal{U}}$. Our main construction is that of a tensor product operation on $2$-representations, and more generally, of a diagonal action given two (lax) commuting $2$-representation structures. We also consider a more complicated “dual” construction in §4.4. In §5, we recast our functorial constructions into bimodule constructions. We formulate our constructions in the differential ungraded setting.

In §6, we construct bimodules and $2$-representations associated with nil Hecke algebras. In §6.1, we describe explicitly the structures of $2$-representation coming from the left and the right action of the monoidal category ${\mathcal{U}}$ on itself and we show that the diagonal category arising from these commuting left and right actions corresponds to Hecke algebras of positive affine symmetric groups. It is a remarkable fact that those can be recovered from the Hecke algebras of the ordinary symmetric groups. We introduce in §6.2 a categorical version of affine symmetric groups and their Hecke algebras.

We develop in §7 an extension of Lipshitz-Ozsváth-Thurston [LiOzTh1] and Zarev’s [Za] theory of strand algebras associated with matched circles and intervals. Instead of considering curves with matchings, we consider the corresponding quotient spaces, where the matched points are identified. We start in §7.1 with $1$-dimensional spaces, which we define as complements of a finite set of points in a $1$-dimensional finite CW-complex. In §7.2, we define our objects of interest, the singular curves. They are $1$-dimensional spaces together with an additional structure at singular points, and a partially defined orientation. They arise as quotients of smooth curves, or, equivalently, as curves in ${\mathbf{R}}^{n}$ with transverse intersections of branches. This leads to a notion of admissible paths, those paths that lift to a smooth model for the curve (§7.3). We introduce in §7.4 the differential categories of strands associated to a curve. They are defined as graded categories associated with a filtered category, in a way similar to the constructions of §3.1. We show in §7.4.3 that strand categories on unoriented $S^{1}$ correspond to the categories built from nil Hecke algebras of affine symmetric groups.

The final section §8 shows that the strand category of a glued curve is obtained as a tensor (or more general diagonal) construction from the strand category of the original curve. This provides some sort of $1$-dimensional field theory, which is really part of a $2$-dimensional field theory for surfaces with extra structure. This gives a categorical mechanism by which strand categories can be computed by cutting the curve into basic building blocks. We start in §8.1 by constructing a structure of $2$-representation associated with an unoriented “end” of a curve. We describe in §8.2 how the strand categories behave under the gluing of two ends of a curve. This requires to solve a combinatorial generators and relation problem generalizing Proposition 3.2.9. When the gluing operation does not create an $S^{1}$, we show in §8.3 that the resulting $2$-representation is the one obtained from the diagonal action.

We thank Ciprian Manolescu for several useful conversations.