Flat $GL(1|1)$-connections and fatgraphs

Recently, the study of super-analogues of Teichmüller spaces and supermoduli spaces achieved much progress. In particular, Penner-type coordinates were discovered in [penzeit], [ipz], [ipz2] for $N=1,N=2$ versions of Teichmüller space and their decorated analogues. In that particular case, these spaces were viewed as a subspace of the character variety $\operatorname{Hom}(\pi_{1}(F),G)/G$, where $\pi_{1}(F)$ is the fundamental group of a hyperbolic Riemann surface $F$ with punctures. For the standard Teichmüller space $T(F)$, $G=PSL(2,\mathbb{R})$, while the super-analogues $ST_{N=1}(F)$, $ST_{N=2}(F)$ cases are related to rank $1$ and rank $2$ supergroups $OSP(1|2)$, $OSP(2|2)$ correspondingly.

Penner coordinates [DTT], [penner] are essential not only in the study of hyperbolic geometry, but they also provide a geometric example of a fundamental algebraic object, known as a cluster algebra. Let us briefly characterize the context. One starts from an ideal triangulation of the Riemann surface, or, equivalently, the dual trivalent fatgraph (aka ribbon graph). Penner coordinates assign a parameter for every edge of triangulation/fatgraph related to a suitably renormalized geodesic length. This provides coordinates for a trivial bundle $\tilde{T}(F)$ over $T(F)$, known as the decorated Teichmüller space. There is a simple transformation to coordinates on Teichmüller space so that these new coordinates are subject to linear constraints. One of the benefits of such coordinates is that the action of mapping class group on $\tilde{T}(F)$ is described in a combinatorial way by embedding into the Ptolemy groupoid. The Ptolemy groupoid is generated by elementary moves on the fatgraphs, called flips. The action of flips on Penner coordinates gives an example of the so-called cluster transformations. The other benefit of Penner coordinates is that they serve as Darboux-type coordinates for the Weil-Petersson 2-form, which makes them useful for the quantization of $T(F)$ [Kashaev97], [Chekhov99].

The supergroups $OSP(1|2)$, $OSP(2|2)$ which give rise to super-Teichmüller spaces both contain $G=SL(2,\mathbb{R})$ as their body subgroup. Penner coordinates have been generalized successfully in both of these cases, leading to the super-analogue of Ptolemy transformation. Currently, there are a lot of attempts to construct a super-analogue of cluster algebras based on these formulas [ovshap],[musikoven], [musikoven2]. The critical ingredient of both constructions were the graph $G$-connections: $\mathbb{Z}_{2}$-graph connections describing the spin structures on $F$ for $ST_{N=1}(F)$, and two such spin structures accompanied by $GL(1)_{+}$-graph connection for $ST_{N=2}(F)$. In particular, part of the decoration for $ST_{N=2}(F)$ was related to the gauge equivalences of $GL(1)_{+}$-graph connections. One of the choices of root systems for $OSP(2|2)$ is such that simple roots are “grey”, namely each of them gives rise to a $GL(1|1)$ subgroup. $GL(1|1)$ is a reductive supergroup of rank $1$, which contains two abelian subgroups as its body. Thus, only the odd coordinates are responsible for non-commutativity.

In this note, we study the first nontrivial case of a character variety related to simple supergroups, namely

which is quite interesting on its own. We will consider this space as the space of $GL(1|1)$-graph connections on the trivalent fatgraph associated with $F$. Then, we will define coordinates on this space through the assignment of specific parameters to edges of the fatgraph. These parameters are related to Gaussian decomposition of $GL(1|1)$. Using these coordinates, we obtain a characterization of the action of the Ptolemy groupoid. Finally, we discuss a Poisson bracket structure on ${\rm M}_{GL(1|1)}$.

We believe that $GL(1|1)$ character variety is essential in the context of the super-analogue of abelianization [hn]. Another important context is the quantum $GL(1|1)$ Chern-Simons theory [Rozansky92], which recently attracted some attention [Aghaei18].

The structure of the paper is as follows. Section 2 reviews some of the notions of super mathematics and necessary facts about the $GL(1|1)$ supergroup. Section 3 discusses fatgraphs, $G$-graph connections, and Ptolemy groupoid actions on trivalent fatgraph $G$-connections. Section 4 is devoted to the construction of a coordinate system on ${\rm M}_{GL(1|1)}$ and its decorated version using its fatgraph $GL(1|1)$-connection realization through Gaussian decomposition of $GL(1|1)$. Section 5 is primarily of a computation nature, where we describe the action of flip transformation using the minimal amount of changing variables in the decorated space. Finally, in Section 6, we discuss the Poisson bracket structure on ${\rm M}_{GL(1|1)}$.

We consider surfaces $F$ with genus $g\geq 0$ and $s\geq 1$ punctures, such that $2g-2+s>0$.

One can reconstruct $F$ from a fatgraph $\tau$ by fattening the edges and verticies along with cyclic ordering of edges at the vertices (see e.g. [DTT] for details). The condition $2g-2+s>0$ is necessary to choose a trivalent fatgraph for a given surface, as the number of vertices of such a graph is equal to $2(2g-2+s)$. There are Whitehead (flip) transformations as in Figure 1, which take the edge $e$ between vertices $u,v$, adjacent to $c,d$ and $a,b$ respectively, shrinks it, and then extends an edge $f$ connecting vertices $u^{\prime},v^{\prime}$ adjacent to $b,c$ and $a,d$ respectively. Flips are known to act transitively on the collection of trivalent fatgraphs of $F$. Altogether they form a Ptolemy groupoid ${\rm Pt}(F)$. Composition of flips are known to give generators for the mapping class group of a surface [DTT].

Given a fatgraph $\tau$ and a Lie (super)group $G$, we define a $G$-graph connection on $\tau$ as follows.

The space of graph connections modulo gauge equivalences can be identified with a more common differential-geometric object. Namely, there is a natural one-to-one correspondence between ${\rm M}_{G}(\tau)$ and the moduli space of flat $G$-connections. This correspondence is constructed as follows: one identifies $M_{A}(e)$, the monodromy of the flat connection $A$ along the oriented edge of the fatgraph $e$, with the group element $g_{e}$. This way, the space $\tilde{\rm M}_{G}(\tau)$ is identified with the space of flat connections modulo gauge transformations which are equal to identity at the vertices. Altogether, compositions of these group elements along the cycles of the fatgraph contain all information about the gauge classes of $A$. However, there is a residual gauge symmetry at the vertices of the fatgraph, which one has to take into account, and that is precisely the equivalence relation for graph connections.

One can formulate this as follows.

For any element $A\in\operatorname{Hom}(\pi_{1}(F),G)/G$, let us denote $A(\tau)$ the corresponding graph connection on a fatgraph $\tau$.

We are interested in understanding the action of ${\rm Pt}(F)$ on graph connections. We require that for any $\Gamma\in{\rm Pt}(F)$ and any cycle $c\in\tau$, if $A$ is the flat connection, the transformed connection $A^{\Gamma}$ is such that $M_{A^{\Gamma}}(\Gamma(c))=M_{A}(c)$. This gives a natural action of ${\rm Pt}(F)$ on $\operatorname{Hom}(\pi_{1}(F),G)/G$.

To write it formally, we look at the elementary flip transformation as in Figure 1, which involves 5 edges.

There are 6 pieces of possible cycles on such a fatgraph, corresponding to a choice of 2 of the 4 boundary edges. The monodromy conservation along these pieces after the flip transformation thus leads to the following equations:

The solution to those equations is unique up to equivalence and given by the following theorem.

One can verify the relations explicitly by the choice of gauge transformation $h_{u^{\prime}}=c^{\prime}c^{-1}$ and $h_{v^{\prime}}=(a^{\prime})^{-1}a$. However, another method is the following trick. One can use gauge transformations to make $g_{e}=1$ and then shrink that edge, since it does not participate in monodromy. This results in a 4-valent vertex. Expanding the edge makes two trivalent vertices and the edge $f$. After that, one makes a gauge transformation to achieve $g_{f}=g_{e}$.

We note that this solution works for any group; however, it is asymmetric. Group elements spread in one direction of the graph but not the other. In the next section we will put the coordinates on ${\rm M}_{G}(\tau)$, represented as graph connections in the case of $G=GL(1|1)$. We will find the formulas for the flip transformations in those coordinates which remove this spreading effect.

We fix a surface $F$ with genus $g\geq 0$ and $s\geq 1$ punctures such that $2g+s-2>0$. We also fix a trivalent fatgraph $\tau\subset F$ with orientation $o$.

Assigning to every edge a group element of $GL(1|1)$, or alternatively assigning an ordered tuple $(a,b;\alpha,\beta)$ in the parametrization (2.5), we obtain a vector in the coordinate system $\tilde{C}(F,o,\tau)$ for the space of $GL(1|1)$-graph connections without factoring by the gauge group equivalences. Thus the chart $\tilde{C}(F,o,\tau)$ gives a diffeomeorphism:

However, we need to take into account that one can change the orientation on edges of the graph, thus leading to an equivalent coordinate system on $\tilde{\rm M}_{G}(\tau)$.

Our goal is now to reformulate the gauge transformations at the vertices of a fatgraph as relations between coordinates in the charts $\tilde{C}(F,o,\tau)$.

It turns out that the edge reversal and the vertex rescalings define an equivalence relation on the chart $\tilde{C}(F,o,\tau)$. The vertex rescalings come from the equivalences on $GL(1|1)$ graph connections provided by the appropriate 1-parameter subgroups. This leads to the following theorem.

It is not hard to see that one can simply constrain the vertex rescalings. To do that let us introduce the following notation: for a given edge with assignment $(a,b;\alpha,\beta)$, which is adjacent to vertex $v$, we denote $a^{v}=a$, $b^{v}=b$, $\alpha^{v}=\alpha$, $\beta^{v}=\beta$, if edge is oriented towards $v$, and $a^{v}=a^{-1}$ $b^{v}=b^{-1}$, $\alpha^{v}=-\alpha$, $\beta^{v}=-\beta$ otherwise. Then, let us give the following definition.

One can see that, for fixed orientation $o$, such constraints pick exactly one element from the equivalence classes provided by vertex rescalings. Moreover, we have the following theorem.

There are benefits of working with the “decorated space” $\tilde{\rm M}_{GL(1|1)}(\tau)$ instead of ${\rm M}_{GL(1|1)}(F)$. As we shall see in the next section, the gauge freedom allows us to write a compact formula for a flip transformation in our coordinates, somewhat in the spirit of Penner coordinates on decorated Teichmüller space.

First we describe the suitable description of the elements of ${M}_{GL(1|1)}(F)$ in terms of coordinates in the extended chart $\tilde{C}(F,o,\tau)$.

Namely, we assign coordinates $(a,b;\alpha,\beta)$, corresponding to a $GL(1|1)$ group element, to an oriented edge as follows: the odd parameters $\alpha,\beta$ are assigned to the terminal and initial points of the edge, while we put the pair of even parameters $(a,b)$ between them, as depicted in Figure 3. This ordering goes along with the Gaussian decomposition from Section 2.

Let us now describe the minimal representation of flip transformation which involves change of the minimal number of parameters. We will fix the odd elements at the ends, as well as eliminate the dependence on fermionic coordinates in the mid-edge after the flip, as depicted in Figure 4.