Quantum traces for $\mathrm{SL}_{n}(\mathbb{C})$: The case $n=3$

Let $\mathfrak{S}$ be an oriented punctured surface of finite topological type, namely $\mathfrak{S}$ is obtained by removing a finite subset $P$, called the set of punctures, from a compact oriented surface $\overline{\mathfrak{S}}$. In particular, note that $\mathfrak{S}$ may have non-empty boundary, $\partial\mathfrak{S}\neq\varnothing$. We require that there is at least one puncture, that each component of $\partial\overline{\mathfrak{S}}$ is punctured (that is, intersects $P$), and that the Euler characteristic $\chi(\mathfrak{S})$ of the punctured surface $\mathfrak{S}$ satisfies $\chi(\mathfrak{S})<d/2$ where $d$ is the number of components of $\partial\mathfrak{S}$. Note that each component of $\partial\mathfrak{S}$ is an ideal arc. These topological conditions guarantee the existence of an ideal triangulation $\lambda$ of the punctured surface $\mathfrak{S}$, namely a triangulation $\overline{\lambda}$ of the surface $\overline{\mathfrak{S}}$ whose vertex set is equal to the set of punctures $P$. See Figure 4 for some examples of ideal triangulations. The ideal triangulation $\lambda$ consists of $\epsilon=-3\chi(\mathfrak{S})+2d$ edges $E$ and $\tau=-2\chi(\mathfrak{S})+d$ triangles $\mathfrak{T}$.

For simplicity, we always assume that $\lambda$ does not contain any self-folded triangles. Consequently, each triangle $\mathfrak{T}$ of $\lambda$ has three distinct edges. Such an ideal triangulation $\lambda$ always exists (except when $\mathfrak{S}$ is a disk with one internal puncture and one puncture on the boundary). Our results should generalize to allow for self-folded triangles, requiring only minor adjustments (one would need to modify Definition 17–compare [BW11, §2.1], which makes use of the Weyl quantum ordering (§3.1.3)–but the main definition, Definition 35, is un-changed).

The discrete $n$-triangle $\Theta_{n}$ is

as shown in Figure 5. The interior of the $n$-discrete triangle is

Let the punctured surface $\mathfrak{S}$ be equipped with an ideal triangulation $\lambda$, and let $N=\epsilon(n-1)+\tau(n-1)(n-2)/2$; see §2.1.

The associated dotted ideal triangulation consists of $\lambda$ together with $N$ distinct black dots attached to the edges $E$ and triangles $\mathfrak{T}$ of $\lambda$, where there are $n-1$ edge-dots attached to each edge $E$ and $(n-1)(n-2)/2$ triangle-dots attached to each triangle $\mathfrak{T}$ (punctures, that is, triangle vertices, are always drawn as white dots). For each triangle $\mathfrak{T}$ including its three boundary edges $E_{1}$, $E_{2}$, $E_{3}$, these dots are arranged as the vertices of the discrete $n$-triangle $\Theta_{n}$ (minus its three corner vertices) overlaid on top of the ideal triangle $\mathfrak{T}$; see Figures 6 and 7. We talk about boundary-dots or interior-dots depending on whether the dots are on the boundary of interior of the surface.

Given a triangle $\mathfrak{T}$ of $\lambda$, which acquires an orientation from the orientation of $\mathfrak{S}$, and given an edge $E$ of $\mathfrak{T}$, it makes sense to say that an edge-dot on $E$ is to the left or to the right of another edge-dot on $E$ as viewed from the triangle $\mathfrak{T}$; see Figure 6(b).

We always assume that we have chosen an ordering for the $N$ dots lying on the dotted ideal triangulation $\lambda$, so we can talk about the $i$-th dot, $i=1,2,\dots,N$.

Let the punctured surface $\mathfrak{S}$ be equipped with a dotted ideal triangulation $\lambda$.

Let $\mathrm{M}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ denote the algebra of $n\times n$ matrices with coefficients in the commutative classical polynomial algebra $\mathscr{T}_{n}^{1}(\lambda)$ $=$ $\mathbb{C}[X_{1}^{\pm 1/n},X_{2}^{\pm 1/n},\dots,X_{N}^{\pm 1/n}]$; see Definition 9. Let the special linear group $\mathrm{SL}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ be the subset of $\mathrm{M}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ consisting of the matrices with determinant equal to $1$.

Let $Z=X_{i}^{\pm 1}$ be an edge-coordinate in the classical polynomial algebra $\mathscr{T}_{n}^{1}(\lambda)$. For $j=1,2,\dots,n-1$ define the $j$-th elementary edge matrix $\mathbf{S}^{\mathrm{edge}}_{j}(Z)\in\mathrm{SL}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ by

Note the normalizing factor $Z^{-j/n}$ multiplying the matrix on the left (or on the right). Similarly, for any triangle-coordinate $X=X_{i}^{\pm 1}$ in $\mathscr{T}_{n}^{1}(\lambda)$ and for any index $j=1,2,\dots,n-1$ define the $j$-th left elementary triangle matrix $\mathbf{S}^{\mathrm{left}}_{j}(X)\in\mathrm{SL}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ by

and define the $j$-th right elementary triangle matrix $\mathbf{S}^{\mathrm{right}}_{j}(X)\in\mathrm{SL}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ by

Note that $\mathbf{S}^{\mathrm{left}}_{1}(X)$ and $\mathbf{S}^{\mathrm{right}}_{1}(X)$ do not actually involve the variable $X$, so we will denote these matrices by $\mathbf{S}^{\mathrm{left}}_{1}$ and $\mathbf{S}^{\mathrm{right}}_{1}$, respectively.

Let $\mathfrak{T}$ be a dotted ideal triangle, which we think of as sitting inside a larger dotted ideal triangulation $\lambda$; see Figure 6. We assign $n\times n$ matrices with coefficients in the classical polynomial algebra $\mathscr{T}_{n}^{1}(\lambda)=\mathbb{C}[X_{1}^{\pm 1/n},X_{2}^{\pm 1/n},\dots,X_{N}^{\pm 1/n}]$ to various ‘short’ oriented arcs lying on the surface $\mathfrak{S}$.

Recall that we think of the $n$-discrete triangle $\Theta_{n}$ (§2.2) as overlaid on top of the triangle $\mathfrak{T}$, so that there is a one-to-one correspondence between coordinates $X_{i}=X_{abc}$ in $\mathscr{T}_{n}^{1}(\mathfrak{T})\subseteq\mathscr{T}_{n}^{1}(\lambda)$ and vertices $(a,b,c)$ in $\Theta_{n}-\{(n,0,0),(0,n,0),(0,0,n)\}$ (in other words, in the $n$-discrete triangle $\Theta_{n}$ minus its three corner vertices). Note that $X_{abc}$ is a triangle-coordinate if and only if $(a,b,c)$ is an interior point $(a,b,c)\in\mathrm{int}(\Theta_{n})$, otherwise $X_{abc}$ is an edge-coordinate.

We will use the following notational convention. Given an arbitrary family $\mathbf{M}_{i}$ of $n\times n$ matrices, put

First, consider a left-moving arc $\overline{\gamma}$, as shown in Figure 8. We assume $\overline{\gamma}$ has no kinks (Figures 3(a) and 3(b)). Let $\mathbf{X}=(X_{i})$ be a vector consisting of the triangle-coordinates $X_{i}$ (Definition 3). Define the associated left matrix $\mathbf{M}^{\mathrm{left}}(\mathbf{X})$ in $\mathrm{SL}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ by

where the matrix $\mathbf{S}_{j}^{\mathrm{left}}(X_{abc})$ is the $j$-th left elementary triangle matrix; see §2.5. (The dots colored red in the figure correspond to the coordinates appearing in the expression of the matrix associated to the curve.)

Next, consider a right-moving arc $\overline{\gamma}$, as shown in Figure 9. We assume $\overline{\gamma}$ has no kinks. We define the associated right matrix $\mathbf{M}^{\mathrm{right}}(\mathbf{X})$ in $\mathrm{SL}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ by

where the matrix $\mathbf{S}_{j}^{\mathrm{right}}(X_{abc})$ is the $j$-th right elementary triangle matrix; see §2.5.

Next, consider an edge-crossing arc $\overline{\gamma}$, as shown on the left hand or right hand side of Figure 10. Let $Z_{j}$, $j=1,2,\dots,n-1$, be the $j$-th edge-coordinate (Definition 3), measured from right to left as seen by the triangle out of which the arc is moving. Let $\mathbf{Z}=(Z_{j})$ be a vector consisting of the $Z_{j}$’s. Define the associated edge matrix $\mathbf{M}^{\mathrm{edge}}(\mathbf{Z})$ in $\mathrm{SL}_{n}(\mathscr{T}_{n}^{1}(\lambda))$ by

where the matrix $\mathbf{S}_{j}^{\mathrm{edge}}(Z_{j})$ is the $j$-th elementary edge matrix; see §2.5. Note that if the orientation of the edge-crossing arc is reversed, then the edge matrix changes by permuting the coordinates $Z_{j}$ by $Z_{j}\leftrightarrow Z_{n-j}$; see Figure 10.

Observe that the order in which the elementary matrices $\mathbf{S}_{j}$ are multiplied does not matter in the formula for the edge matrix $\mathbf{M}^{\mathrm{edge}}(\mathbf{Z})$, since they are diagonal, but does matter in the formulas for the triangle matrices $\mathbf{M}^{\mathrm{left}}(\mathbf{X})$ and $\mathbf{M}^{\mathrm{right}}(\mathbf{X})$.

Lastly, define the clockwise U-turn matrix $\mathbf{U}$ in $\mathrm{SL}_{n}(\mathbb{C})$ by

We associate to a clockwise U-turn arc (resp. counterclockwise U-turn arc) $\overline{\gamma}$, as shown on the left hand (resp. right hand) side of Figure 11, the U-turn matrix $\mathbf{U}$ (resp. transpose $\mathbf{U}^{\mathrm{T}}$ of the U-turn matrix). Again, we have assumed that $\overline{\gamma}$ has no kinks. Note that $\mathbf{U}^{\mathrm{T}}=-\mathbf{U}$ (resp. $=\mathbf{U}$) when $n$ is even (resp. odd).

Let $\gamma$ be an immersed oriented closed curve in the surface $\mathfrak{S}$ such that $\gamma$ is transverse to the ideal triangulation $\lambda$. We want to calculate the classical trace polynomial $\widetilde{\mathrm{Tr}}_{\gamma}(X_{i}^{1/n})$ in $\mathscr{T}_{n}^{1}(\lambda)$ $=$ $\mathbb{C}[X_{1}^{\pm 1/n},X_{2}^{\pm 1/n},\dots,X_{N}^{\pm 1/n}]$ associated to the immersed curve $\gamma$. We say that the polynomial $\widetilde{\mathrm{Tr}}_{\gamma}(X_{i}^{1/n})$ is obtained from a ‘state sum’, ‘local-to-global’, or ‘transfer matrices’ construction.

More precisely, as we travel along the curve $\gamma$ according to its orientation, assume $\gamma$ crosses edges $E_{j_{k}}$ for $k=1,2,\dots,K$ in that order, and assume $\gamma$ crosses triangles $\mathfrak{T}_{i_{k}}$ for $k=1,2,\dots,K$ in that order. As the curve $\gamma$ crosses the edge $E_{j_{k}}$, moving out of the triangle $\mathfrak{T}_{i_{k-1}}$ into the triangle $\mathfrak{T}_{i_{k}}$, this defines an edge-crossing arc $\overline{\gamma}_{j_{k}}$; see §2.6 and Figure 10. Put $\mathbf{Z}_{j_{k}}=((Z_{j_{k}})_{j^{\prime}})$ and put

the associated edge matrix, where the $(Z_{j_{k}})_{j^{\prime}}\text{'s}$ are the $j^{\prime}=1,\dots,n-1$ edge-coordinates attached to the edge $E_{j_{k}}$ measured from right to left as seen from $\mathfrak{T}_{i_{k-1}}$. As $\gamma$ traverses the triangle $\mathfrak{T}_{i_{k}}$ between two edges $E_{j_{k}}$ and $E_{j_{k+1}}$, it does one of three things:

• •

  the curve $\gamma$ turns left, ending on $E_{j_{k+1}}\neq E_{j_{k}}$, see Figure 8;

• •

  or $\gamma$ turns right, ending on $E_{j_{k+1}}\neq E_{j_{k}}$, see Figure 9;

• •

  or $\gamma$ does a U-turn, thereby returning to the same edge $E_{j_{k+1}}=E_{j_{k}}$, see Figure 11.

We also keep track of the following winding information: for the first and second items above, the number $t_{k}$ of full turns to the right that the curve $\gamma$ makes while traversing the triangle $\mathfrak{T}_{i_{k}}$; and for the third item above, the number $2t_{k}+1$ of half turns to the right that the curve $\gamma$ makes before coming back to the same edge $E_{j_{k}}$. Note $t_{k}\in\mathbb{Z}$. Note that the turning integer $t_{k}$ associated to the curve $\gamma$ as it traverses the triangle $\mathfrak{T}_{i_{k}}$ will only be relevant when $n$ is even. Let the $(X_{i_{k}})_{i^{\prime}}\text{'s}$ be the $i^{\prime}=1,\dots,(n-1)(n-2)/2$ triangle-coordinates attached to the triangle $\mathfrak{T}_{i_{k}}$, and put $\mathbf{X}_{i_{k}}=((X_{i_{k}})_{i^{\prime}})$; see Figure 8. Let $\gamma^{\prime}$ be the curve obtained by ‘pulling tight’ the kinks of $\gamma$. (In particular, $\gamma$ and $\gamma^{\prime}$ are homotopic, but not, in general, regularly homotopic.) Corresponding to the three items above:

• •

  the curve $\gamma^{\prime}$ turns left, defining a left-moving arc $\overline{\gamma}^{\prime}$ and an associated left matrix $\mathbf{M}^{\prime}_{i_{k}}=\mathbf{M}^{\mathrm{left}}(\mathbf{X}_{i_{k}})$, see §2.6 and Figure 8;

• •

  or the curve $\gamma^{\prime}$ turns right, defining a right-moving arc $\overline{\gamma}^{\prime}$ and an associated right matrix $\mathbf{M}^{\prime}_{i_{k}}=\mathbf{M}^{\mathrm{right}}(\mathbf{X}_{i_{k}})$, see §2.6 and Figure 9;

• •

  or the curve $\gamma^{\prime}$ does a clockwise or counterclockwise U-turn, thereby returning to the same edge $E_{j_{k+1}}=E_{j_{k}}$ and defining a U-turn arc $\overline{\gamma}^{\prime}$, see §2.6 and Figure 11.

In the first two cases, where $\overline{\gamma}^{\prime}$ is either left- or right-moving, put

and in the third case, where $\overline{\gamma}^{\prime}$ is a U-turn, put

where $\mathbf{U}$ is the U-turn matrix defined in §2.6. Note, in the third case, that this is consistent with what was said in §2.6, where, in the case $\overline{\gamma}$ has no kinks, $\overline{\gamma}^{\prime}=\overline{\gamma}$ is associated to $\mathbf{U}$ (resp. $\mathbf{U}^{\mathrm{T}}=(-1)^{n-1}\mathbf{U}$) if $\overline{\gamma}$ travels clockwise hence $t_{k}=0$ (resp. counterclockwise hence $t_{k}=-1$).

In this subsection, we assume (for simplicity) the surface $\mathfrak{S}$ has empty boundary, $\partial\mathfrak{S}=\varnothing$. A complete flag $E\in\mathrm{Flag}(\mathbb{C}^{n})$ is a maximal nested sequence of distinct sub-spaces $\left\{0\right\}=E^{(0)}\subsetneq E^{(1)}\subsetneq E^{(2)}\subsetneq\cdots\subsetneq E^{(n)}=\mathbb{C}^{n}$. The group $\mathrm{PSL}_{n}(\mathbb{C})$ acts on the set of complete flags by matrix multiplication.

In [FG06b], Fock–Goncharov define the moduli space of framed local systems, also called the $\mathscr{X}$-moduli space, whose complex points have been denoted in this paper by $\mathscr{R}_{\mathrm{PSL}_{n}(\mathbb{C})}(\mathfrak{S})_{\mathrm{fr}}$. A framed local system over $\mathbb{C}$ is, roughly speaking, a pair $(\rho,\xi)$ where $\rho:\pi_{1}(\mathfrak{S})\to\mathrm{PSL}_{n}(\mathbb{C})$ is a group homomorphism and $\xi$ assigns to each puncture $p$ a complete flag $\xi(p)$ invariant under the monodromy of any peripheral curve $\gamma_{p}$ around $p$, that is, $\rho(\gamma_{p})(\xi(p))=\xi(p)\in\mathrm{Flag}(\mathbb{C}^{n})$. Framed local systems are considered up to equivalence under the action of $\mathrm{PSL}_{n}(\mathbb{C})$, which in particular acts on representations $\rho$ by conjugation.

Fock–Goncharov associate to each ideal triangulation $\lambda$ of $\mathfrak{S}$ a coordinate chart $U_{\lambda}$ for the moduli space $\mathscr{R}_{\mathrm{PSL}_{n}(\mathbb{C})}(\mathfrak{S})_{\mathrm{fr}}$ parametrized by non-zero complex coordinates $X_{i}$. Here, $N$ is the number of dots associated to the ideal triangulation $\lambda$; see §2.3, 2.4. In particular, choosing $\lambda$ and $N$-many coordinates $X_{i}$ determines a representation $\rho=\rho(X_{i})$ up to conjugation. Since $\rho$ is valued in $\mathrm{PSL}_{n}(\mathbb{C})$, the trace $\mathrm{Tr}(\rho(\gamma))$ for any $\gamma\in\pi_{1}(\mathfrak{S})$ is well-defined only up to multiplication by a $n$-root of unity.