A Wilson line realization of quantum groups

Abstract

We study Wilson line operators in three-dimensional Chern–Simons theory on a manifold with boundaries and prove to leading order, through a direct calculation of Feynman integrals, that the merging of parallel Wilson lines reproduces the coproduct on the quantum group $U_\hbar ({\mathfrak {g}})$. We outline a connection of this theory with the moduli spaces of local systems defined by Goncharov and Shen.

1 Introduction

Topological quantum field theories like Chern–Simons theory have long been known to have connections to quantum groups and their representations. The goal of this paper is to give an explicit realization of this connection in the setting of perturbative three-dimensional Chern–Simons theory with boundary conditions, by showing that the tensor product in the category of representations of the quantum group can be realized from the operation of merging two parallel Wilson lines.

In recent papers [4, 5] Costello, Witten and Yamazaki constructed a four-dimensional conformal Chern–Simons theory in which they realized the representation theory of the Yangian. They suggest in section 7.8 of their second paper that a three-dimensional theory with boundaries can be constructed from their four-dimensional theory by restricting to U(1)-invariant fields. The corresponding three-dimensional theory was constructed explicitly by the first author in [1]. Concretely, the set-up is Chern–Simons theory on a manifold $M = \mathbb {R}^2 \times I$ with a semi-simple Lie algebra $\mathfrak {g}$. The relevant set of boundary conditions on the gauge field $A\in \Omega ^1(M,\mathfrak {g})$ comes from defining a Manin triple $({\mathfrak {l}}_-,{\mathfrak {l}}_+,\mathfrak {g})$ and restricting A to take value in subalgebras ${\mathfrak {l}}_+$ (resp. ${\mathfrak {l}}_-$) on the upper (resp. lower) boundary. Since the gauge symmetry of the action is broken by the boundary conditions, this theory permits a set of gauge invariant operators given by open Wilson lines associated to representations of $\mathfrak {g}$ and extending to infinity along $\mathbb {R}^2$.

In the present paper, we consider a product on the set of Wilson lines in the above setting coming from merging two parallel lines. By computing the leading order Feynman amplitude for a gauge boson coupling to the pair of merging lines, we show that this product agrees with the leading order deformation of the tensor product in ${{\,\textrm{Rep}\,}}(U_\hbar (\mathfrak {g}))$. It was argued in [1] that, in the same theory, the leading order contribution to the expectation value of a pair of crossing Wilson lines is given by the classical r-matrix. Together these results suggest that the category of Wilson line operators is equivalent to the category ${{\,\textrm{Rep}\,}}(U_\hbar (\mathfrak {g}))$ as a braided monoidal category.

One motivation for considering this 3d Chern–Simons theory with boundary conditions instead of the 4d conformal version is its close connection to the moduli spaces of local systems on punctured surfaces considered by Goncharov and Shen [9]. In fact, as we will discuss in the final section of this paper, our construction can be seen as a realization through perturbation theory of the “geometric avatar of a TQFT” described in section 5 of the paper of Goncharov and Shen.

2 The Gauge theory

The gauge theory that we will be concerned with is three-dimensional Chern–Simons theory defined by the action:

$$\begin{aligned} S_\text {CS}(A)=\frac{1}{2\pi }\int _{M}\Big <A\wedge dA+\frac{1}{3}A\wedge [A, A]\Big > \end{aligned}$$

(2.1)

where the gauge field (connection) $A\in \Omega ^1(M,\mathfrak {g})$ is a 1-form on M taking values in the Lie algebra $\mathfrak {g}$ of the gauge group and ${\langle {,\,}\rangle }$ is an invariant symmetric bilinear form on $\mathfrak {g}$. In the present paper we will take $M=\mathbb {R}^2\times I$ where $I=[-1,1]$ and we take $\mathfrak {g}$ to be a complex semi-simple Lie algebra.

In order to have a well-defined theory in the presence of boundaries, we must impose boundary conditions on the gauge field. Specifically, when varying the action with respect to the gauge field, $A\rightarrow A+\delta A$ where $\delta A$ is an exact 1-form, we pick up a boundary term:

$$\begin{aligned} \frac{1}{2\pi }\int _{\mathbb {R}^2\times \{-1,1\}} \left<A\wedge \delta A\right>, \end{aligned}$$

(2.2)

and we must impose boundary conditions on A and $\delta A$ ensuring that this term vanishes on each boundary. At the same we want that the restriction of the gauge theory to each boundary component is in itself a well-defined, gauge invariant theory (see [4] section 9.1 for more elaboration on this). By the second requirement, choosing a set of boundary conditions amounts to specifying subalgebras ${\mathfrak {l}}_+,{\mathfrak {l}}_-\subset \mathfrak {g}$ and imposing that A and $\delta A$ take value in ${\mathfrak {l}}_+$ (resp. ${\mathfrak {l}}_-$) at the upper (resp. lower) boundary. It was argued in [4] that a we get a valid set of boundary conditions giving rise to quantum group structures by choosing ${\mathfrak {l}}_+$ and ${\mathfrak {l}}_-$ so that the triple $(\mathfrak {g},{\mathfrak {l}}_+,{\mathfrak {l}}_-)$ is a Manin triple. In other words ${\mathfrak {l}}_+$ and ${\mathfrak {l}}_-$ must be non-intersecting, half-dimensional, isotropic subalgebras of $\mathfrak {g}$ such that $\mathfrak {g}={\mathfrak {l}}_+\oplus {\mathfrak {l}}_-$.

3 Manin triples and quantum groups

3.1 Constructing a Manin triple

Not all semi-simple Lie algebras admit the structure of a Manin triple (for example if the Lie algebra has odd dimension). Following the construction of [4] (section 9.2) we can modify $\mathfrak {g}$ to accommodate for this by adding another copy of the Cartan subalgebra. We give here the construction in full detail.

Let $\mathfrak {h}\subset \mathfrak {g}$ be a Cartan subalgebra and consider the root system $\Phi$ of ${\mathfrak {g}}$ relative to $\mathfrak {h}$ equipped with a polarization $\Phi =\Phi _+\sqcup \Phi _-$. We write ${\mathfrak {n}}_+$ and ${\mathfrak {n}}_-$ for the sum over root spaces $\mathfrak {g}_\alpha$ corresponding to the set of positive roots $\alpha \in \Phi _+$ and negative roots $-\alpha \in \Phi _-$, respectively. Then ${\mathfrak {n}}_-$ and ${\mathfrak {n}}_+$ are isotropic subalgebras and we get a decomposition of $\mathfrak {g}$:

$$\begin{aligned} \mathfrak {g}= {\mathfrak {n}}_-\oplus \mathfrak {h}\oplus {\mathfrak {n}}_+. \end{aligned}$$

Now add to $\mathfrak {g}$ another copy ${\tilde{\mathfrak {h}}}$ of the Cartan subalgebra:

$$\begin{aligned} {\tilde{\mathfrak {g}}}=\mathfrak {g}\oplus {\tilde{\mathfrak {h}}} \end{aligned}$$

with the bracket on $\mathfrak {g}$ trivially extended to ${\tilde{\mathfrak {g}}}$, i.e. $[a,{\tilde{b}}]=0$ for $a\in \mathfrak {g}$ and ${\tilde{b}} \in {\tilde{\mathfrak {h}}}$. We can extend the Killing form on $\mathfrak {g}$ to ${\tilde{\mathfrak {g}}}$ as follows: ${\langle {a,{\tilde{b}}}\rangle }=0$ for all $a\in \mathfrak {g}$, ${\tilde{b}}\in {\tilde{\mathfrak {h}}}$ and ${\langle {{\tilde{a}},{\tilde{b}}}\rangle }={\langle {a,b}\rangle }$ for all $a,b\in \mathfrak {h}$. This gives an invariant symmetric bilinear form on ${\tilde{\mathfrak {g}}}$. Define

$$\begin{aligned} \begin{aligned}&\mathfrak {h}_+=\big \{h+ \textbf{i}{\tilde{h}}\,\big |\,h\in \mathfrak {h}\big \}\subset \mathfrak {h}\oplus {\tilde{\mathfrak {h}}}\\&\mathfrak {h}_-=\big \{h-\textbf{i}{\tilde{h}}\,\big |\,h\in \mathfrak {h}\big \}\subset \mathfrak {h}\oplus {\tilde{\mathfrak {h}}}. \end{aligned} \end{aligned}$$

With this definition ${\langle {,\,}\rangle }$ vanishes on $\mathfrak {h}_+$ and $\mathfrak {h}_-$ and thereby choosing

$$\begin{aligned} {\mathfrak {l}}_-={\mathfrak {n}}_-\oplus {\mathfrak {h}}_- \ \text { and } \ {\mathfrak {l}}_+={\mathfrak {n}}_+\oplus \mathfrak {h}_+. \end{aligned}$$

(3.1)

the triple $({\tilde{\mathfrak {g}}}, {\mathfrak {l}}_+, {\mathfrak {l}}_-)$ is a Manin triple.

Conventions Let r be the rank of $\mathfrak {g}$. We fix a choice of simple roots $\Delta _+ = \{\alpha _i\}_{i=1}^r \subset \Phi _+$ along with a basis $\{H_i\}_{i=1}^r$ of ${\mathfrak {h}}$. Furthermore, we let $X_\alpha$ be a generator of the root space $\mathfrak {g}_\alpha$ normalized so that $[X_\alpha ,X_\beta ] = X_{\alpha +\beta }$ and $[X_{\alpha _i},X_{-\alpha _i}] = H_i$. Using standard notation, we write $E_i, F_i :=X_{\alpha _i}, X_{-\alpha _i}$ for each simple root $\alpha _i\in$ $\Delta _+$ and we write $H_i^+=H_i+\textbf{i}{\tilde{H}}_i$ and $H_i^-=H_i-\textbf{i}{\tilde{H}}_i$. A basis ${\mathcal {B}}_+$ for ${\mathfrak {l}}_+$ can now be given as:

$$\begin{aligned} {\mathcal {B}}_+=\{X_\alpha ,H_i^+\}_{\alpha \in \Phi _+,i=1,\dots , r}. \end{aligned}$$

Let ${\mathcal {B}}_-$ be the basis for ${\mathfrak {l}}_-$ dual to ${\mathcal {B}}_+$ with respect to the Killing form. Then ${\mathcal {B}}={\mathcal {B}}_+\cup {\mathcal {B}}_-$ is a basis for ${\tilde{\mathfrak {g}}}$. We write

$$\begin{aligned} {\mathcal {B}} = \{t_a\}_{a=1,\dots , \dim {\tilde{\mathfrak {g}}}}\, \ \ {\mathcal {B}}_+ = \{t_a\}_{a=1,\dots ,\dim {\tilde{\mathfrak {g}}}/2}. \end{aligned}$$

Finally, we denote by $t^a$ the dual element of $t_a\in {\mathcal {B}}$.

3.2 Quantization

In this subsection we briefly recall the construction of a quantum double via the Drinfel’d double construction. For a detailed exposition we refer the reader to, e.g. [8] section 4.

Let ${\mathfrak {b}}={\mathfrak {n}}_+\oplus {\mathfrak {h}}\subset \mathfrak {g}$ be the Borel subalgebra relative to setup of Sect. 3.1 and let $(a_{ij})$ be the Cartan matrix of $\mathfrak {g}$. The Drinfel’d double ${\mathfrak {D}}_\hbar ({\mathfrak {b}})$ is the algebra over ${\mathbb {C}}[[\hbar ]]$ with generators:

$$\begin{aligned} \{E_i,F_i,H^+_i,H^-_i~|~i=1,\dots ,r\} \end{aligned}$$

and relations

$$\begin{aligned} \begin{aligned}&[H_i^\pm ,E_j]=a_{ij}E_j \\&[H_i^\pm ,F_j]=-a_{ij}F_j \end{aligned} \hspace{1.5cm} \begin{aligned}&[H^\pm _i,H^\pm _j]=[H^\pm _i,H^\mp _j]=0 \\&[E_i,F_j]= \delta _{ij}\frac{e^{\hbar H_i^+/2}-e^{-\hbar H^-_i/2}}{e^{\hbar /2}-e^{-\hbar /2 }} \end{aligned} \end{aligned}$$

(3.2)

along with the quantum Serre relations for $i\ne j$. In the case of ${\mathfrak {s}}{\mathfrak {l}}_n({\mathbb {C}})$ these relations take the form

$$\begin{aligned} \begin{aligned} E_i^2E_{j}-(e^{\hbar /2}+e^{-\hbar /2 })E_iE_{j}E_i + E_iE_{j}^2 = 0\\ F_i^2F_{j}-(e^{\hbar /2}+e^{-\hbar /2 })F_iF_{j}F_i + F_iF_{j}^2 =0. \end{aligned} \end{aligned}$$

For the general case see, e.g. [8] section 4.2. The quantized universal enveloping algebra of $\mathfrak {g}$ is constructed from the double as

$$\begin{aligned} U_\hbar (\mathfrak {g}):={\mathfrak {D}}_\hbar ({\mathfrak {b}})/{\langle {H_i^+-H_i^-}\rangle }. \end{aligned}$$

It holds that ${\mathfrak {D}}_\hbar ({\mathfrak {b}})$ has the structure of a quasi-triangular Hopf algebra with co-product:

$$\begin{aligned}&\Delta E_i=1\otimes E_i +E_i\otimes 1+\frac{\hbar }{4}(E_i\otimes H^+_i-H^+_i\otimes E_i)+{\mathcal {O}}(\hbar ^2) \nonumber \\&\Delta F_i=1\otimes F_i +F_i\otimes 1 +\frac{\hbar }{4}(F_i\otimes H^-_i-H^-_i\otimes F_i) + {\mathcal {O}}(\hbar ^2)\nonumber \\&\Delta H^\pm _i= 1\otimes H^\pm _i +H^\pm _i\otimes 1. \end{aligned}$$

(3.3)

Notice that this realizes the usual co-product on the universal enveloping algebra as the limit $\hbar \rightarrow 0$ and we have that ${\mathfrak {D}}_\hbar ({\mathfrak {b}}) \cong {\mathfrak {D}}({\mathfrak {b}})[[\hbar ]]$ as ${\mathbb {C}}[[\hbar ]]$ modules, where ${\mathfrak {D}}({\mathfrak {b}})=U({\tilde{\mathfrak {g}}})$.

Remark 1

Often in the theory of quantum groups one defines

$$\begin{aligned} K_i^\pm = q^{H_i^\pm }, \quad q=\exp {(\hbar /2)} \end{aligned}$$

for which the (non-perturbative) co-product takes the form

$$\begin{aligned} \Delta E_i=(K_i^+)^{-1/2}\otimes E_i +E_i\otimes (K_i^+)^{1/2}. \end{aligned}$$

Since we are realizing the co-product in the setting of perturbation theory, it will be more convenient to use Eqs. (3.2) and (3.3) as our convention.

Lemma 1

The leading order correction to the co-product on a general basis element $t_a\in {\mathcal {B}}_+$ is given by

$$\begin{aligned} \Delta _{(1)}t_a = \frac{1}{2}\sum _{b,c=1}^{n/2}\big ({f_a}^{bc}~t_{b,V}\otimes t_{c,V'}\big ). \end{aligned}$$

(3.4)

Recall the definition of the structure constant:

$$\begin{aligned}{}[t_a,t_b]=\sum _{c=1}^{n}{f_{ab}}^ct_c \, \ \ f_{abc}={\langle {[t_a,t_b],t_c}\rangle }. \end{aligned}$$

(3.5)

Proof

One checks that this formula agrees with the co-product in Eq. (3.3) on the algebra generators $E_i,H^+_i$ and that it commutes with the bracket. $\square$

The R -matrix Another part of the quasi-triangular Hopf algebra structure is an R-matrix element $R \in {\mathfrak {D}}_\hbar ({\mathfrak {b}}) \otimes {\mathfrak {D}}_\hbar ({\mathfrak {b}})$ given by

$$\begin{aligned} R=1 + \sum \hbar ^k r^{(k)}, \end{aligned}$$

(3.6)

where each $r^{(k)}$ is an element of ${\mathfrak {D}}({\mathfrak {b}})\otimes {\mathfrak {D}}({\mathfrak {b}})$. The element $r:=r^{(1)}$ is known as the classical R-matrix and is given by

$$\begin{aligned} r= \sum _{a=1}^{n/2} t_a \otimes t^a . \end{aligned}$$

(3.7)

The category of representations of ${\mathfrak {D}}_\hbar ({\mathfrak {b}})$ is a braided monoidal category with monoidal product coming from the co-product in Eq. (3.3) and braiding coming the R-matrix in Eq. (3.7).

4 Perturbation theory

4.1 The propagator

The remainder of this paper studies the expectation value of operators in the theory in the setting of perturbation theory. An essential ingredient for this is constructing a propagator, which can be thought of as the probability amplitude for a gauge boson traveling between two points on the manifold. The propagator in the present setting is a Lie algebra valued two-form ${P}\in \Omega ^2\big ((M\times M){\setminus }{{\,\textrm{diag}\,}},\,{\tilde{\mathfrak {g}}}\otimes {\tilde{\mathfrak {g}}}\big )$ such that P is a Green’s function for the differential operator d, that is

$$\begin{aligned} dP(x,y)=\delta ^{(3)}(x,y)\,{\mathscr {C}}({\tilde{\mathfrak {g}}}), \end{aligned}$$

(4.1)

where $\delta ^{(3)}(x,y)$ is the three-dimensional Dirac delta distribution localized at $x=y$ and ${\mathscr {C}}({\tilde{\mathfrak {g}}})\in {\tilde{\mathfrak {g}}}\otimes {\tilde{\mathfrak {g}}}$ is the Casimir element of ${\tilde{\mathfrak {g}}}$.

Moreover we impose boundary conditions on the propagator coming from the boundary conditions on the gauge field in Eq. (3.1): Write $\partial _+M$ for the upper boundary $\partial _+M=\mathbb {R}^2\times \{1\}$ and $\partial _-M$ for the lower boundary $\partial _-M=\mathbb {R}^2\times \{-1\}$. We require that

• (i) the restrictions $P\big |_{\partial _+M\times M}$ and $P\big |_{M\times \partial _-M}$ take values in ${\mathfrak {l}}_+\otimes {\mathfrak {l}}_-$,

• (ii) the restrictions $P\big |_{\partial _-M\times M}$ and $P\big |_{M\times \partial _+M}$ take values in ${\mathfrak {l}}_-\otimes {\mathfrak {l}}_+$.

A two-form satisfying Eq. (4.1) along with the above boundary constraints can be constructed as follows: Let $\omega =f{{\,\textrm{vol}\,}}_{S^2}\in \Omega ^2(S^2)$ where ${{\,\textrm{vol}\,}}_{S^2}$ is the unit volume form on $S^2$ given in terms of the coordinates on $\mathbb {R}^3$ by

$$\begin{aligned} {{\,\textrm{vol}\,}}_{S^2}=x\,dy\wedge dz+y\,dz\wedge dx+z\, dx\wedge dy, \end{aligned}$$

and $f\in C^\infty (S^2)$ satisfying the following properties:

• (i) f is only supported in a small neighborhood of “the north pole” $x_{np}=(0,0,1)$

• (ii) f is symmetric under rotations around the axis through $x_{np}=(0,0,1)$ and $x_{sp}=(0,0,-1)$

• (iii) $\int _{S^2}f{{\,\textrm{vol}\,}}_{S^2}=1$.

Furthermore, let $\phi :M\times M{\setminus }{{\,\textrm{diag}\,}}\rightarrow S^2$ be the map

$$\begin{aligned} \phi (x,y)=\frac{y-x}{|y-x|} \end{aligned}$$

(4.2)

and let R be the orientation reversing map, $R:S^2\rightarrow S^2$, $R(x)=-x$. We now define the propagator as the pull back

$$\begin{aligned} P=\phi ^*\big (\omega ~r^+-R^*\omega ~r^-\big ), \end{aligned}$$

(4.3)

where $r^+\in {\mathfrak {l}}_+\otimes {\mathfrak {l}}_-$ and $r^-\in {\mathfrak {l}}_-\otimes {\mathfrak {l}}_+$ are uniquely determined by the constraint in Eq. (4.1). To see this, notice first that since P is the pull back of a top-dimensional form on $S^2$ it holds that $dP(x,x')$ vanishes for all $x,x'$ with $x\ne x'$. Now fix $x'=0$ and consider the integral of dP(x, 0) when x is in the unit ball around 0. by Stokes’ theorem we have

$$\begin{aligned} \int _{x\in B}dP(x,0)=\int _{S^2}P(x,0)=\int _{S^2}\big (\omega (x)\, r^+-R^*\omega (x)\, r^-\big )=r^++r^-. \end{aligned}$$

This fixes $r^+$ and $r^-$, namely

$$\begin{aligned} \begin{aligned} r^+=r \, \ \ r^-= T\circ r. \end{aligned} \end{aligned}$$

(4.4)

where r is the classical R-matrix given in Eq. (3.7) and T is the map that swaps the tensor factors.

4.2 Feynman diagrams

In perturbation theory, the expectation value of an observable is computed as an expansion in the parameter $\hbar$ in terms of a set of weighted graphs (Feynman diagrams). The weight of a given graph is determined from a set of Feynman rules derived from the Chern–Simons action in Eq. (2.1). By a Feynman diagram in the present setting we mean the following:

Definition 1

A Feynman diagram is a directed trivalent graph with leaves (external half-edges) and with the half-edges decorated by elements of ${\mathcal {B}}$ such that: A half-edge labeled by $t_a\in {\mathcal {B}}_+$ is connected by an edge to a half-edge labeled by $t^a\in {\mathcal {B}}_-$ with the edge orientation going from ${\mathcal {B}}_-$ to ${\mathcal {B}}_+$.

Feynman rules The Feynman rules outlined below associates to any Feynman diagram $\Gamma$ a differential form on the space of embeddings of the vertices of $\Gamma$ into $\mathbb {R}^2\times I$. The weight of a Feynman diagram is computed as the integral of the associated differential form over the space of embeddings.

• (1) An edge going from a vertex at $p\in \mathbb {R}^2\times I$ to a vertex at $q\in \mathbb {R}^2\times I$ contributes a two-form $\hbar \,\phi ^*\omega (p,q)$ coming from the propagator (see Remark 2 below).

• (2) An internal vertex with incident edges labeled by basis elements $t_a,t_b,t_c\in {\mathcal {B}}$ contributes a factor structure constant $\frac{1}{\hbar }f_{abc}$.

Recall that the structure constant is given by $f_{abc}={\langle {[t_a,t_b],t_c}\rangle }$.

• (3) An external half-edge labeled by $t_a\in {\mathcal {B}}$ and connected a vertex at $p\in \mathbb {R}^2\times I$ contributes a gauge field $A^a(p)$.

Remark 2

Write $P=\sum _{ab}P^{ab}t_a\otimes t_b$. We note that, in a free Chern–Simons theory (with no boundary conditions), one would consider Feynman diagrams with unoriented edges and with half-edges labeled by general elements of ${\mathcal {B}}$. To an edge with half-edges labeled by $t_a$ and $t_b$, the Feynman rules would associate the component $P^{ab}(x,y)$. However, as seen from Eq. (4.3), the boundary conditions in the present theory split the propagator into two parts corresponding to the two edge orientations, and we can therefore choose as a convention to define Feynman diagrams with oriented and sum over all edge orientations.

4.3 Wilson lines in perturbation theory

A common set of gauge invariant observables to study in Chern–Simons theory is the so-called Wilson loops. Given a closed loop $\gamma \subset M$ and a representation V of ${\mathfrak {g}}$ the associated Wilson loop is defined as the trace of the holonomy of the gauge field around $\gamma$:

$$\begin{aligned} W_V(\gamma )&={{\,\textrm{Tr}\,}}_V\bigg ({\mathscr {P}}\exp \int _\gamma A\bigg )\nonumber \\&:={{\,\textrm{Tr}\,}}(1_V) + \int _\gamma dx^i A_i^a(x)\,{{\,\textrm{Tr}\,}}(t_{a,V}) \nonumber \\&\quad + \int _\gamma dx^i\int ^x dx'^j A^a_i(x) A^b_j(x')\,{{\,\textrm{Tr}\,}}(t_{a,V}\,t_{b,V})+\cdots \end{aligned}$$

where ${\mathscr {P}}$ means the path ordering of the exponential and we use the notation $t_{a,V}$ to denote the basis element $t_a$ acting in the representation V. In this paper, we consider instead a set of operators called Wilson lines coming from omitting the trace and replacing the closed loop $\gamma$ with an open line L extending to infinity along $\mathbb {R}^2$. In the setting of perturbation theory, we think of a Wilson line L(V) simply as a pair (L, V), and we allow a gauge field $A^a$ to couple to L(V) by inserting a basis element $t_{a,V}$ at the corresponding point on L. In other words, we expand the definition of Feynman diagrams to include graphs with univalent vertices along L, with the additional Feynman rule that a vertex on L with incident half-edge labeled by $t_a$ contributes an element $t_{a,V}$ (Fig. 1).

Fig. 1

Feynman rule for the coupling of a gauge field $A^a$ to the Wilson line L(V)

5 Quantum groups and Wilson lines

5.1 Merging parallel Wilson lines

The study of the remainder of the paper will be the product of two parallel Wilson line operators in the limit when the lines come close together. We fix a set of coordinates (x, y, z) on $\mathbb {R}^2\times I$ with (x, y) coordinates in $\mathbb {R}^2$ and z the coordinate along I. Let L(V) be a Wilson line supported at $x=z=0$ and $L_\varepsilon (V')$ a Wilson line supported at $x=\varepsilon$, $z=0$. We write $L(V)L_\varepsilon (V')$ to mean the disjoint union of the lines L and $L_\varepsilon$ such that a gauge field $A^a$ couples to the line L by inserting an element $t_{a,V}\otimes 1_{V'}$ and to the line $L'$ by inserting an element $1_V\otimes t_{a,V'}$. In general, the coupling of an external gauge field to the two Wilson lines is given by a perturbative expansion in $\hbar$ using the Feynman rules in Sect. 4.2:

$$\begin{aligned} {\mathcal {A}}_a\big (L(V)L_\varepsilon (V')\big ) =\sum _{k=0}^\infty \hbar ^k{\mathcal {A}}_a^{(k)}\big (L(V)L_\varepsilon (V')\big ) \in {{\,\textrm{End}\,}}(V\otimes V') \end{aligned}$$

(5.1)

where each element ${\mathcal {A}}_a^{(k)}\big (L(V)L_\varepsilon (V')\big ) \in {{\,\textrm{End}\,}}(V\otimes V')$ is computed as the weighted sum over Feynman diagrams with a single external half-edge (leaf) labeled by $t_a$ and with the number of internal edges minus the number of internal vertices equal to k. In the limit $\varepsilon \rightarrow 0$ one would expect Eq. (5.1) to reproduce the expression for an external gauge field coupling to a single Wilson line at L. It is however not immediately clear what representation should be associated to the merged Wilson line.

Fig. 2

The classical level Feynman diagrams for an external gauge field coupling to the two Wilson lines

At the classical level the gauge field simply couples to each line individually as shown in Fig. 2, and the corresponding Feynman amplitude is given by

$$\begin{aligned} {\mathcal {A}}^{(0)}_a\big (L(V) L_\varepsilon (V')\big )=\int _{q\in L}A^{a}(q)\,t_{a,V}\otimes 1_{V'}+\int _{q'\in L_\varepsilon }A^a(q')\,1_{V}\otimes t_{a,V'}. \end{aligned}$$

Taking the limit $\varepsilon \rightarrow 0$ on the right-hand side in the above we get

$$\begin{aligned} \int _{q\in L}A^a(x)\,\big (t_{a,V}\otimes 1_{V'}+1_V\otimes t_{a,V'}\big ), \end{aligned}$$

which is the expression for a gauge field coupling to a single Wilson line at L in the tensor product representation $V\otimes V'$. Hence, at the classical level we have

$$\begin{aligned} \begin{aligned} \lim _{\varepsilon \rightarrow 0}{\mathcal {A}}^{(0)}_{a}\big (L(V) L_\varepsilon (V')\big )={\mathcal {A}}^{(0)}_a\big (L(V\otimes V')\big ). \end{aligned} \end{aligned}$$

(5.2)

The object of the remainder of this paper is to carry out the computation of the leading order contribution ${\mathcal {A}}_{a}^{(1)}\big (L(V)L_\varepsilon (V')\big )$ in the limit $\varepsilon \rightarrow 0$. As we shall see, this gives a correction to the tensor product $V\otimes V'$ in Eq. (5.2) which agrees with the leading order quantum deformation of the tensor product in ${\mathfrak {D}}_\hbar ({\mathfrak {b}})$ given in Eq. (3.3). This is expressed in the following theorem:

Theorem 1

It holds that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}{\mathcal {A}}_a^{(1)}\big (L(V) L_\varepsilon (V')\big )={\mathcal {A}}_a^{(1)}\big (L(V\otimes _\hbar V')\big ), \end{aligned}$$

where $V\otimes _\hbar V'$ is the tensor product in ${{\,\textrm{Rep}\,}}{\mathfrak {D}}_\hbar ({\mathfrak {b}})$ defined via the co-product in Eq. (3.3).

Notice that Lemma 1 in Sect. 3.2 defines the relevant co-product on a general basis element $t_a\in {\mathcal {B}}_+$ and it follows that:

$$\begin{aligned} {\mathcal {A}}_a^{(1)}\big (L(V\otimes _\hbar V')\big )=\frac{1}{2}\sum _{b,c=1}^{\dim {\tilde{\mathfrak {g}}}/2}\big ({f_a}^{bc}~t_{b,V}\otimes t_{c,V'}\big ) \int _{q\in L}A^a(q). \end{aligned}$$

(5.3)

We conjecture Theorem 1 to hold at all orders in perturbation theory. However, explicitly computing the contributing Feynman integrals at higher orders appears to be too difficult a task, and a proof would therefore require different techniques.

5.2 The configuration space of vertices

The differential form associated to a Feynman diagram $\Gamma$ is defined on the configuration space of vertices of $\Gamma$. We here give a definition of the relevant configuration space in the presence of Wilson lines $L,L_\varepsilon$, and we refer the reader to [3] for a more general definition. As we are interested in studying the limit when $\varepsilon$ goes to 0 it will be convenient to think of $\varepsilon$ as a parameter in the configuration space.

Definition 2

For $n_1, n_2, m\in {\mathbb {Z}}_{\ge 0}$ define ${{\,\textrm{Conf}\,}}_{n_1,n_2,m}$ to be the space of points

$$\begin{aligned} \{\varepsilon ,q_1,\dots q_{n_1},q'_1,\dots , q'_{n_2},p_{1},\dots , p_m\}, \end{aligned}$$

where $\varepsilon \in [0,\infty )$ and

$$\begin{aligned}&q_1,\dots q_{n_1}\in L \ \text { with } \ q_i\ne q_j\,, \\&q'_1,\dots , q'_{n_2}\in L_\varepsilon \ \ \text { with } \ q'_i\ne q'_j\,, \\&p_1,\dots , p_m\in (\mathbb {R}^2\times I){\setminus }\{q_1,\dots ,q_{n_1},q'_1,\dots , q'_{n_2}\} \ \text { with } \ p_i\ne p_j. \end{aligned}$$

Furthermore, consider the projection onto the first factor

$$\begin{aligned} {{\,\textrm{Conf}\,}}_{n_1,n_2,m}\rightarrow (0,\infty ). \end{aligned}$$

We denote by ${{\,\textrm{Conf}\,}}_{n_1,n_2,m}^\varepsilon$ the fiber of this projection over a fixed $\varepsilon \in (0,\infty )$.

5.3 Contributing diagrams at leading order

Fig. 3

Contributing diagrams

Fig. 4

Vanishing diagrams

Lemma 2

The only Feynman diagrams contributing to ${\mathcal {A}}^{(1)}_a\big (L(V)L_\varepsilon (V')\big )$ are the ones shown in Fig. 3.

Proof

Recall that the diagrams contributing to ${\mathcal {A}}_a^{(1)}\big (L(V)L_\varepsilon (V)\big )$ has the number of internal edges minus the number of internal vertices equal to one. This gives precisely the diagrams shown in Figs. 3 and 4. Consider first the diagram on the left-hand side of Fig. 4. Since the lines are in the same plane parallel to the boundary, the form vanishes due to the propagator only being nonzero in a small neighborhood of the north pole. Consider now diagram on the right-hand side of Fig. 4. The associated configuration space is ${{\,\textrm{Conf}\,}}_{2,0,1}$. Let $G<{{\,\textrm{Homeo}\,}}(\mathbb {R}^3)$ be the subgroup of scalings and translations along L and consider the quotient map:

$$\begin{aligned} {{\,\textrm{Conf}\,}}_{2,0,1}\rightarrow {{\,\textrm{Conf}\,}}_{2,0,1}/G \end{aligned}$$

(5.4)

The subgroup G is two-dimensional and hence the space ${{\,\textrm{Conf}\,}}_{2,0,1}/G$ has dimension $5-2=3$. On the other hand, let $P_1\wedge P_2\in \Omega ^4({{\,\textrm{Conf}\,}}_{2,0,1})$ be the product of propagators associated to the internal edges, that is,

$$\begin{aligned} P_1\wedge P_2\,(q_1,q_2,p):=\phi ^*\omega (q_1,p)\wedge \phi ^*\omega (q_2,p). \end{aligned}$$

By definition the propagator is invariant under scalings and translations along the L and hence the form $P_1\wedge P_2$ factors through the quotient map in Eq. (5.4). By dimensional counting, this implies that $P_1\wedge P_2$ vanishes. $\square$

Consider therefore the diagrams in Fig. 3. We can assume that $t_a \in {\mathcal {B}}_+$ since the computation for $t_a\in {\mathcal {B}}_-$ is entirely analogous. In this case, the only contribution to the expectation value comes from the diagram on the right-hand side of Fig. 3. In fact, the internal vertex of diagram on the left-hand side of Fig. 3 has all three incident half edges labeled by elements $t_a,t_b,t_c\in {\mathcal {B}}_+$. By the Feynman rules in Sect. 4.2 this vertex is assigned a structure constant $f_{abc}={\langle {[t_a,t_b],t_c}\rangle }$ which is zero since the Killing form vanishes on ${\mathfrak {l}}_+$. The Feynman amplitude coming from the diagram on the right-hand side of Fig. 3 takes the form

$$\begin{aligned} \begin{aligned} {\mathcal {A}}_a^{(1)}\big (L(V) L_\varepsilon (V')\big )&=\sum _{b,c=1}^{\dim {\tilde{\mathfrak {g}}}/2}\big ({f_a}^{bc}~t_{b,V}\otimes t_{c,V'}\big )~{\mathcal {I}}_\varepsilon , \end{aligned} \end{aligned}$$

(5.5)

where

$$\begin{aligned} {\mathcal {I}}_\varepsilon :=\int _{{{\,\textrm{Conf}\,}}_{1,1,1}^\varepsilon }A^a(p)\wedge \phi ^*\omega (p,q)\wedge \phi ^*\omega (p,q'). \end{aligned}$$

(5.6)

5.4 A configuration space compactification

Since the propagator is only defined away from the diagonal, it is not clear what will happen to the integral ${\mathcal {I}}_\varepsilon$ in Eq. (5.6) when $p\rightarrow q, q'$. In order compute $\lim _{\varepsilon \rightarrow 0}{\mathcal {I}}_\varepsilon$, we must therefore define a partial compactification $\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}$ of ${{{\,\textrm{Conf}\,}}}_{1,1,1}$ in the direction $\varepsilon \rightarrow 0$ such that the integrand extends smoothly to the corresponding boundary $\partial _\varepsilon \,\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}$. Then we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} {\mathcal {I}}_\varepsilon = \int _{\partial _\varepsilon \overline{{{\,\textrm{Conf}\,}}}_{1,1,1}}A^a(p)\wedge \phi ^*\omega (p,q)\wedge \phi ^*\omega (p,q'). \end{aligned}$$

(5.7)

To this aim, we use the so-called Fulton-MacPherson configuration space compactification. This compactification was originally due to Fulton and MacPherson [7] and applied to Chern–Simons perturbation theory by Axelrod and Singer [2] and (in the presence of Wilson loops) Bott and Taubes [3]. In this compactification $\partial _\varepsilon \,\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}$ can be divided into the disjoint union of strata

$$\begin{aligned} \partial _\varepsilon \, \overline{{{\,\textrm{Conf}\,}}}_{1,1,1}=\bigcup _{i=1}^3\partial _i\,\overline{{{\,\textrm{Conf}\,}}}_{1,1,1} \end{aligned}$$

corresponding to the following cases:

• (a) The internal vertex p remains far from the lines compared to $\varepsilon$ as $\varepsilon \rightarrow 0$.

• (b) The internal vertex p moves close to the lines and at least one vertex q or $q'$ remains far from $\varepsilon \rightarrow 0$.

• (c) All three vertices move close to each other as $\varepsilon \rightarrow 0$.

Lemma 3

We get no contribution to Eq. (5.7) coming from the boundary stratum $\partial _1\,\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}$ corresponding to case (a) in the above.

Proof

When p is far from the lines the integrand in Eq. (5.6) extends smoothly to the boundary coming from allowing $\varepsilon \rightarrow 0$, and the corresponding boundary stratum takes the form

$$\begin{aligned} \partial _1\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}=\big \{(q,q',p)\in L\times L\times (\mathbb {R}^2\times I{\setminus } L)~\big |~p\ne q,q'\big \}. \end{aligned}$$

The contribution to Eq. (5.7) is given by

$$\begin{aligned} \int _{p\in (\mathbb {R}^2\times I){\setminus } L}A(p)\wedge \bigg (\int _{q\in L}\phi ^*\omega (q,p)\bigg )\wedge \bigg (\int _{q'\in L}\phi ^*\omega (q',p)\bigg ), \end{aligned}$$

(5.8)

which is zero since the last two factors are identical one forms. $\square$

Lemma 4

We get no contribution to Eq. (5.7) coming from the boundary stratum $\partial _2\,\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}$ corresponding to case (b) in the above.

Proof

This follows from the property that $\omega$ is only nonzero in a small neighborhood of the north pole. In fact, assume that p is approaching some point $q\in L$ and that $q'$ remains far from p as $\varepsilon \rightarrow 0$. Because the Wilson lines are in the same plane parallel to the boundary it holds that, given any $\eta >0$ there is a $\delta >0$ such that, if we define $U\subset {{\,\textrm{Conf}\,}}_{1,1,1}$ to be the neighborhood where $|p-q'|>\eta$ and $|p-q|<\delta$ then $\phi ^*\omega (p,q')=0$ for all $(p,q,q')\in U$. The situation is illustrated in Fig. 5$\square$

Fig. 5

Neighborhood of ${{\,\textrm{Conf}\,}}_{1,1,1}$ where p is close to q and far from $q'$

By Lemmas 3 and 4, the only contribution to Eq. (5.7) comes from the boundary stratum $\partial _3{{\,\textrm{Conf}\,}}_{1,1,1}$ corresponding to all three vertices coming together as $\varepsilon \rightarrow 0$. To define the corresponding boundary stratum we need the following definition:

Fig. 6

The space ${\mathcal {H}}$

Definition 3

Let $S_r$ be the “right” half of the unit circle with center (0, 0), i.e.

$$\begin{aligned} S_r = \big \{(x,y,0)\in \mathbb {R}^2\times I ~ \big |~x^2+y^2=1 \,, \ x >0 \big \} \end{aligned}$$

We define

$$\begin{aligned} {\mathcal {H}}=\big \{(u,v)\in S_r\times \mathbb {R}^3{\setminus } \{0\}~\big |~v\ne u\big \}. \end{aligned}$$

The space ${\mathcal {H}}$ is illustrated in Fig. 6.

Lemma 5

We can define a partial compactification of ${{\,\textrm{Conf}\,}}_{1,1,1}$ in the direction where all three vertices come together, such that the compactified space is a manifold with boundary and corresponding boundary stratum $\partial _3\,\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}$ is given by

$$\begin{aligned} \partial _3\, \overline{{{\,\textrm{Conf}\,}}}_{1,1,1}=L\times {\mathcal {H}}. \end{aligned}$$

Proof

For some small $\eta >0$, define $U\subset {{\,\textrm{Conf}\,}}_{1,1,1}$ by

$$\begin{aligned} U=\big \{(\varepsilon ,q,q',p)\in {{\,\textrm{Conf}\,}}_{1,1,1}~\big |~ |p-q|<\eta \text { and } |q'-q|<\eta \big \}. \end{aligned}$$

Furthermore, define $V\subset (0,\eta )\times L\times {\mathcal {H}}$ by

$$\begin{aligned} V= \big \{(t,q_0,(u,v))\in (0,\eta )\times L\times {\mathcal {H}}~\big |~|v|t<\eta \big \}, \end{aligned}$$

(5.9)

There exists a diffeomorphism $\varphi :V\rightarrow U$ defined by $(t,q_0,(v,u))\mapsto (\varepsilon ,q,q',p)$, where

$$\begin{aligned} \begin{aligned} \varepsilon =u_y \, \ \ q=q_0 \, \ \ q'=q_0+t u\, \ \ p = q_0+t v, \end{aligned} \end{aligned}$$

(5.10)

with $u_y$ denoting the y-coordinate of u. This implies that

$$\begin{aligned} \overline{{{\,\textrm{Conf}\,}}}_{1,1,1}:={{\,\textrm{Conf}\,}}_{1,1,1}\cup _V {\overline{V}}, \end{aligned}$$

where

$$\begin{aligned} {\overline{V}}= \big \{(t,q_0,(u,v))\in [0,\eta ]\times L\times {\mathcal {H}}~\big |~|v|t<\eta \big \}, \end{aligned}$$

is a manifold with boundary. Letting all three vertices come together in ${{\,\textrm{Conf}\,}}_{1,1,1}$ corresponds to letting $t\rightarrow 0$ in $\overline{V}$ and the lemma follows. $\square$

5.5 Proof of Theorem 1

We are now equipped to prove Theorem 1. Notice first that with the change of coordinates given in Eq. (5.10), we have

$$\begin{aligned} A(p)=A(q+tv)\, \ \ \phi ^*\omega (p,q)= \phi ^*\omega (0,v)\, \ \ \phi ^*\omega (p,q')= \phi ^*\omega (v,u). \end{aligned}$$

All of the above forms extends continuously to the boundary $\partial _3\,\overline{{{\,\textrm{Conf}\,}}}_{1,1,1}$ corresponding to the limit $t\rightarrow 0$. Hence, by Eq. (5.7) and Lemmas 3, 4 and 5 in the previous subsection, it holds that

$$\begin{aligned} \begin{aligned} \lim _{\varepsilon \rightarrow 0}{\mathcal {I}}(\varepsilon ) =\int _{q\in L}A^a(q)\int _{(u,v)\in {\mathcal {H}}}\phi ^*\omega (v,0)\wedge \phi ^*\omega (v,u). \end{aligned} \end{aligned}$$

(5.11)

By Eqs. (5.3) and (5.5), proving Theorem 1 now amounts to showing that the second integral in Eq. (5.11) contributes a factor of 1/2. This is the goal of the present subsection.

Fig. 7

The space $\mathcal {C}$

Proof of Theorem 1

Let $\mathcal {C}\supset {\mathcal {H}}$ be the space obtained from allowing u to be in the full circle $S\subset \mathbb {R}^2\times \{0\}$ (Fig. 7). That is, we define

$$\begin{aligned} \mathcal {C}= \big \{(u,v)\in S\times \mathbb {R}^3\}. \end{aligned}$$

Due to the rotation symmetry of $\omega$ (see Sect. 4.1), it holds that

$$\begin{aligned} \int _{(u,v)\in {\mathcal {H}}} \phi ^*\omega (v,0)\wedge \phi ^*\omega (v,u)=\frac{1}{2}\int _{(u,v)\in \mathcal {C}} \phi ^*\omega (v,0)\wedge \phi ^*\omega (v,u). \end{aligned}$$

(5.12)

In fact, we are going to modify the space of integration even further: Recall from Sect. 4.1 that $\omega$ is only supported in a small neighborhood of the north pole. Hence, the only contribution to the integral in the right-hand side of Eq. (5.12) comes from when $v\in \mathbb {R}^3_-:=\mathbb {R}^2\times (-\infty ,0)$. Defining $\mathcal {C}_-\subset \mathcal {C}$ by

$$\begin{aligned} \mathcal {C}_- =\{(u,v)\in S\times \mathbb {R}^3_-\}, \end{aligned}$$

we can therefore replace $\mathcal {C}$ with $\mathcal {C}_-$ in Eq. (5.12):

$$\begin{aligned} \int _{(u,v)\in {\mathcal {H}}} \phi ^*\omega (v,0)\wedge \phi ^*\omega (v,u)=\frac{1}{2}\int _{(u,v)\in \mathcal {C}_-} \phi ^*\omega (v,0)\wedge \phi ^*\omega (v,u).\nonumber \\ \end{aligned}$$

(5.13)

The integral on the right-hand side of Eq. (5.13) can be computed using purely geometric arguments. Let $S_+^2$ be the upper half of the unit sphere and consider the map $\Phi :\mathcal {C}_-\rightarrow S^2_+\times S^2_+{\setminus } {{\,\textrm{diag}\,}}$, given by

$$\begin{aligned} \Phi (u,v)=(\phi (v,0),\phi (u,v))=\left( -\frac{v}{|v|},\frac{u-v}{|u-v|}\right) . \end{aligned}$$

Lemma 6

The map $\Phi$ is a diffeomorphism.

Proof

An inverse map $\Phi ^{-1}$ is constructed as follows: Let $(a,b)\in S^2_+\times S^2_+{\setminus } {{\,\textrm{diag}\,}}$. For any $u\in S$ write $r^u_a$ for the ray going out from u and pointing along the vector $-a$ and write $r_b$ for the ray going out from 0 and pointing along the vector $-b$. Because $a\ne b$, as we move u around the circle we encounter exactly one point $u_{ab}$ for which the rays $r^u_a$ and $r_{b}$ intersect. Denoting the corresponding point of intersection by $v_{ab}$ we obtain an inverse map by defining $\Phi ^{-1}(a,b)=(u_{ab},v_{ab})$. $\square$

From Lemma 6 and the property that $\omega$ integrates to one on $S^2$ it now follows that

$$\begin{aligned} \begin{aligned} \int _{\mathcal {C}_-}\phi ^*\omega (v,0)\wedge \phi ^*\omega (v,u)=\int \limits _{S^2_+\times S^2_+}\omega (a)\wedge \omega (b)=1. \end{aligned} \end{aligned}$$

Notice that we can include the diagonal ${{\,\textrm{diag}\,}}\subset S^2_+\times S^2_+$ in the integral because $\omega$ extends continuously to the diagonal which is a subspace of co-dimension one. Inserting this back into Eq. (5.11) we get

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}{\mathcal {I}}_\varepsilon =\frac{1}{2}\int _{q\in L}A(q). \end{aligned}$$

By Eqs. (5.3) and (5.5) this completes the proof of Theorem 1. $\square$

6 Outlook to moduli spaces of local systems

To a Lie group G and a surface S with punctures, boundaries, and marked points on the boundaries, Goncharov and Shen [9] construct a moduli space $Loc_{S,G}$ which parameterizes G local systems on S along with some extra data at the punctures, boundaries, and marked points of S. These spaces are closely related to the ${\mathcal {X}}$ moduli spaces of “framed” local systems on S originally constructed by Fock and Goncharov [6], with a slight modification to allow for cutting and gluing of surfaces. In particular, one associates to each marked point on the boundary the conjugacy class of the Borel subgroup $B\subset G$.

Fig. 8

The gluing map on surfaces

Fig. 9

The cutting map

There is a quantization of the ring of regular functions on $Loc_{S,G}$ which is denoted by ${\mathcal {O}}_\hbar (Loc_{S,G})$. Goncharov and Shen construct a natural gluing operation on these quantized spaces coming from gluing surfaces along the boundary segments between two marked points; when S is obtained by gluing $S'$ and $S''$ along boundary segments between marked points one obtains a map

$$\begin{aligned} {\mathcal {O}}_\hbar (Loc_{S,G}) \xrightarrow {Glue} {\mathcal {O}}_\hbar (Loc_{S',G})\otimes {\mathcal {O}}_\hbar (Loc_{S'',G}). \end{aligned}$$

To see how this construction relates that of the present paper, consider a disk with one puncture and two marked points on its boundary. There is a map

$$\begin{aligned} \kappa : {\mathfrak {D}}_\hbar ({\mathfrak {b}}) \rightarrow {\mathcal {O}}_\hbar (Loc_{\odot ,G}) \end{aligned}$$

which is given explicitly on generators, see [10] for a very nice exposition on this in the $\mathfrak {sl}_n$ case and see [11] for the general ADE case. The coproduct is given in ${\mathcal {O}}_\hbar (Loc_{\odot ,G})$ as follows: Take two copies of the punctured disk and glue them along boundary segments between marked points to obtain a twice punctured disk with two marked points on its boundary (see Fig. 8). We denote the twice punctured disk by T. By bringing the two punctures close together and cutting out a small circle around the two punctures one obtains a new once punctured disk (see Fig. 9). This construction gives a map

$$\begin{aligned} {\mathcal {O}}_\hbar (Loc_{\odot ,G}) \xrightarrow {Cut} {\mathcal {O}}_\hbar (Loc_{T,G}) \xrightarrow {Glue} {\mathcal {O}}_\hbar (Loc_{\odot ,G}) \otimes {\mathcal {O}}_\hbar (Loc_{\odot ,G}). \end{aligned}$$

(6.1)

which agrees with the coproduct in ${\mathfrak {D}}_\hbar ({\mathfrak {b}})$. Similarly, the braiding on ${\mathfrak {D}}_\hbar ({\mathfrak {b}})$ is given on ${\mathcal {O}}_\hbar (Loc_{\odot ,G})$ as the map twisting the two punctures around each other:

$$\begin{aligned} {\mathcal {O}}_\hbar (Loc_{T,G}) \xrightarrow {Braid} {\mathcal {O}}_\hbar (Loc_{T,G}) \end{aligned}$$

(6.2)

In other words there are commutative diagrams:

Thus the analogy of this in 3-dimension Chern–Simons theory should now be apparent from the construction in this paper: The punctures in the moduli spaces of Goncharov and Shen translate into Wilson line operators in our theory extending to infinity in the time direction. Cutting a disk around each Wilson line tangent to the boundaries gives a punctured disk two marked points on its boundary. The opposite Borel subgroups assigned to the marked points at the top and bottom can then be thought of as coming from the boundary conditions in the theory and the operation of merging two Wilson lines corresponds to the operation of gluing punctured disks together. We expect that much of the formalism described by Goncharov and Shen (the modular functor conjectures of sections 2.5 and 5 of [9]) can be realized within Chern–Simons theory by exploring this connection further.