Supersymmetric Gauge Theories, Quantization of ${\mathcal M}_\mathrm{flat}$, and Conformal Field Theory

Abstract

We review the relations between ${\mathcal N}=2$-supersymmetric gauge theories, Liouville theory and the quantization of moduli spaces of flat connections on Riemann surfaces.

A citation of the form [V:x] refers to article number x in this volume.

Appendices

Appendix 1: Riemann Surfaces: Basic Definitions and Results

This appendix introduces basic definitions and results concerning Riemann surfaces C that will be used throughout the paper. A Riemann surface C is a two-dimensional topological surface S together with a choice of complex structure on S. We will denote by ${\mathcal M}(S)$ the moduli space of complex structures on a two-dimensional surface S, and by ${\mathcal T}(C)$ the Teichmüller space of deformations of complex structures on the Riemann surface C.

1.1 Complex Analytic Gluing Construction

A convenient family of particular coordinates on the Teichmüller spaces ${\mathcal T}(C)$ is defined by means of the complex-analytic gluing construction of Riemann surfaces C from three punctured spheres [Ma, HV]. Let us briefly review this construction.

Let C be a (possibly disconnected) Riemann surface. Fix a complex number q with $|q|<1$, and pick two points $Q_1$ and $Q_2$ on C together with coordinates $z_i(P)$ in a neighborhood of $Q_i$, $i=1,2$, such that $z_i(Q_i)=0$, and such that the discs $D_i$,

$$ D_i\,:=\,\{\,P_i\in C;\,|z_i(P_i)|<|q|^{-\frac{1}{2}}\,\},\qquad i=1,2\,, $$

do not intersect. One may define the annuli $A_i$,

$$ A_i\,:=\,\{\,P_i\in C;\,|q|^{\frac{1}{2}}<|z_i(P_i)|<|q|^{-\frac{1}{2}}\,\},\qquad i=1,2\,. $$

To glue $A_1$ to $A_2$ let us identify two points $P_1$ and $P_2$ on $A_1$ and $A_2$, respectively, iff the coordinates of these two points satisfy the equation

$$\begin{aligned} z_1(P_1)z_2(P_2)\,=\,q\,. \end{aligned}$$

(6.38)

If C is connected one creates an additional handle, and if $C=C_1\sqcup C_2$ has two connected components one gets a single connected component after performing the gluing operation. In the limiting case where $q=0$ one gets a nodal surface which represents a component of the boundary $\partial {\mathcal M}(S)$ defined by the Deligne-Mumford compactification $\overline{{\mathcal M}}(S)$.

By iterating the gluing operation one may build any Riemann surface C of genus g with n punctures from three-punctured spheres $C_{0,3}$. Embedded into C we naturally get a collection of annuli $A_1,\dots ,A_h$, where

$$\begin{aligned} h\,:=\,3g-3+n\,. \end{aligned}$$

(6.39)

The construction above can be used to define a $3g-3+n$-parametric family of Riemann surfaces, parameterised by a collection $q=(q_1,\dots ,q_h)$ of complex parameters. These parameters can be taken as coordinates for a neighbourhood of a component in the boundary $\partial \overline{{\mathcal M}}(S)$ which are complex-analytic with respect to its natural complex structure [Ma].

Conversely, assume given a Riemann surface C and a cut system, a collection ${\mathcal C}=\{\gamma _1,\dots ,\gamma _h\}$ of homotopy classes of non-intersecting simple closed curves on C. Cutting along all the curves in ${\mathcal C}$ produces a pants decompostion, $C\setminus {\mathcal C}\simeq \bigsqcup _{v}C_{0,3}^v$, where the $C_{0,3}^v$ are three-holed spheres.

Having glued C from three-punctured spheres defines a distinguished cut system, defined by a collection of simple closed curves ${\mathcal C}=\{\gamma _1,\dots ,\gamma _h\}$ such that $\gamma _r$ can be embedded into the annulus $A_r$ for $r=1,\dots ,h$.

An important deformation of the complex structure of C is the Dehn-twist: It corresponds to rotating one end of an annulus $A_r$ by $2\pi$ before regluing, and can be described by a change of the local coordinates used in the gluing construction. The coordinate $q_r$ can not distinguish complex structures related by a Dehn twist in $A_r$. It is often useful to replace the coordinates $q_r$ by logarithmic coordinates $\tau _r$ such that $q_r=e^{2\pi i \tau _r}$. This corresponds to replacing the gluing identification (6.38) by its logarithm. In order to define the logarithms of the coordinates $z_i$ used in (6.38), one needs to introduce branch cuts on the three-punctured spheres, an example being depicted in Fig. 4.

Fig. 4

A sphere with three punctures, and a choice of branch cuts for the definition of the logarithms of local coordinates around the punctures.

By imposing the requirement that the branch cuts chosen on each three-punctured sphere glue to a connected three-valent graph $\Gamma$ on C, one gets an unambiguous definition of the coordinates $\tau _r$. We see that the logarithmic versions of the gluing construction that define the coordinates $\tau _r$ are parameterized by the pair of data $\sigma =({\mathcal C}_\sigma ,\Gamma _\sigma )$, where ${\mathcal C}_\sigma$ is the cut system defined by the gluing construction, and $\Gamma _\sigma$ is the three-valent graph specifying the choices of branch cuts. In order to have a handy terminology we will call the pair of data $\sigma =({\mathcal C}_\sigma ,\Gamma _\sigma )$ a pants decomposition, and the three-valent graph $\Gamma _\sigma$ will be called the Moore-Seiberg graph, or MS-graph associated to a pants decomposition $\sigma$. The construction outlined above gives a set of coordinates for the neighbourhood ${\mathcal U}_\sigma$ of the boundary component of ${\mathcal T}(C)$ corresponding to $\sigma$.

The gluing construction depends on the choices of coordinates around the punctures $Q_i$. There exists an ample supply of choices for the coordinates $z_i$ such that the union of the neighbourhoods ${\mathcal U}_{\sigma }$ produces a cover of ${\mathcal M}(C)$ [HV]. For a fixed choice of these coordinates one produces families of Riemann surfaces fibred over the multi-discs ${\mathcal U}_{\sigma }$ with coordinates q. Changing the coordinates $z_i$ around $Q_i$ produces a family of Riemann surfaces which is locally biholomorphic to the initial one [RS].

1.2 The Moore-Seiberg Groupoid

Let us note [MS, BK] that any two different pants decompositions $\sigma _2$, $\sigma _1$ can be connected by a sequence of elementary moves localized in subsurfaces of $C_{g,n}$ of type $C_{0,3}$, $C_{0,4}$ and $C_{1,1}$. The elementary moves are called the B, F, Z and S-moves, respectively. Graphical representations for the elementary moves F, S and B are given in Figs. 1, 2 and 3, respectively. The Z-move is just the change of distinguished boundary component in a three-punctured sphere.

One may formalize the resulting structure by introducing a two-dimensional CW complex ${\mathcal M}(C)$ with set of vertices ${\mathcal M}_{\mathfrak 0}(C)$ given by the pants decompositions $\sigma$, and a set of edges ${\mathcal M}_{\mathfrak 1}(C)$ associated to the elementary moves. The Moore-Seiberg groupoid is defined to be the path groupoid of ${\mathcal M}(C)$. It can be described in terms of generators and relations, the generators being associated with the edges of ${\mathcal M}(C)$, and the relations associated with the faces of ${\mathcal M}(C)$. The classification of the relations was first presented in [MS], and rigorous mathematical proofs have been presented in [FG1, BK]. The relations are all represented by sequences of elementary moves localized in subsurfaces $C_{g,n}$ with genus $g=0$ and $n=3,4,5$ punctures, as well as $g=1$, $n=1,2$. Graphical representations of the relations can be found in [MS, FG1, BK].

1.3 Uniformization

The classical uniformization theorem ensures existence and uniqueness of a hyperbolic metric, a metric of constant negative curvature, on a Riemann surface C. In a local chart with complex analytic coordinates y one may represent this metric in the form $ds^2=e^{2\varphi }dyd\bar{y}$, with $\varphi$ being a solution to the Liouville equation $\partial \bar{\partial }\varphi =\mu e^{2\varphi }dy d\bar{y}$.

The solutions to the Liouville equation may be parameterized by a function t(y) related to $\varphi$ as

$$\begin{aligned} t:=\,-(\partial _y\varphi )^2+\partial _y^2\varphi \,. \end{aligned}$$

(6.40)

t(y) is holomorphic as a consequence of the Liouville equation. The solution to the Liouville equation can be reconstructed from t(y) by first finding the solutions to

$$\begin{aligned} (\partial _y^2+t(y))\chi =0\,. \end{aligned}$$

(6.41)

Picking two linearly independent solutions $\chi _\pm$ of (6.41) with $\chi _+'\chi _--\chi _-'\chi _+=1$ allows us to represent $e^{2\varphi }$ as $e^{2\varphi }=-(\chi _+\bar{\chi }_--\chi _-\bar{\chi }_+)^{-2}$. The hyperbolic metric $ds^2=e^{2\varphi }dyd\bar{y}$ may then be written in terms of the quotient $A(y):=\chi _+/\chi _-$ as

$$\begin{aligned} ds^2\,=\,e^{2\varphi }dyd\bar{y}\,=\,\frac{\partial A\bar{\partial }{\bar{A}}}{(\mathrm{Im}(A))^2}\,. \end{aligned}$$

(6.42)

It follows that A(y) represents a conformal mapping from C to a domain $\Omega$ in the upper half plane ${\mathbb U}$ with its standard constant curvature metric. The monodromies of the solution $\chi$ are represented on A(y) by Moebius transformations. These Moebius transformations describe the identifications of the boundaries of the simply-connected domain $\Omega$ in ${\mathbb U}$ which represents the image of C under A. C is therefore conformal to ${\mathbb U}/\Gamma$, where the Fuchsian group $\Gamma$ is the monodromy group of the differential operator $\partial _y^2+t(y)$.

Appendix 2: Moduli Spaces of Flat Connections

In this appendix we shall review some of the basic definitions and results concerning the moduli spaces ${\mathcal M}_\mathrm{flat}(C)$.

1.1 Moduli of Flat Connections and Character Variety

We will consider flat $\mathrm{PSL}(2,{\mathbb C})$-connections $\nabla =d-A$ on Riemann surfaces C. Let ${\mathcal M}_\mathrm{flat}(C)$ be the moduli space of all such connections modulo gauge transformations.

Given a flat $\mathrm{PSL}(2,{\mathbb C})$-connection $\nabla =d-A$, one may define its holonomy $\rho (\gamma )$ along a closed loop $\gamma$ as $\rho (\gamma )={\mathcal P}\exp (\int _\gamma A)$. The assignment $\gamma \mapsto \rho (\gamma )$ defines a representation of $\pi _1(C)$ in $\mathrm{PSL}(2,{\mathbb C})$. As any flat connection is locally gauge-equivalent to the trivial connection, one may characterize gauge-equivalence classes of flat connections by the corresponding representations $\rho :\pi _1(C)\rightarrow \mathrm{PSL}(2,{\mathbb C})$. This allows us to identify the moduli space ${\mathcal M}_\mathrm{flat}(C)$ of flat $\mathrm{PSL}(2,{\mathbb C})$-connections on C with the so-called character variety

$$\begin{aligned} {\mathcal M}_\mathrm{char}(C):=\mathrm{Hom}(\pi _1(C),\mathrm{PSL}(2,{\mathbb C}))/\mathrm{PSL}(2,{\mathbb C})\,. \end{aligned}$$

(6.43)

The moduli space ${\mathcal M}_\mathrm{flat}(C)$ has a natural real slice, the moduli space ${\mathcal M}^{{\mathbb R}}_\mathrm{flat}(C)$ of flat $\mathrm{PSL}(2,{\mathbb R})$-connections.

1.2 The Teichmüller Component

There is a well-known relation between the Teichmüller space ${\mathcal T}(C)$ and a connected component of the moduli space ${\mathcal M}_\mathrm{flat}^{{\mathbb R}}(C)$ of flat $\mathrm{PSL}(2,{\mathbb R})$-connections on C. This component is called the Teichmüller component and will be denoted as ${\mathcal M}_\mathrm{flat}^{{\mathbb R},0}(C)$. The relation between ${\mathcal T}(C)$ and ${\mathcal M}_\mathrm{flat}^0(C)$ may be described as follows. To a hyperbolic metric $ds^2=e^{2\varphi }dyd\bar{y}$ let us associate the connection $\nabla =\nabla '+\nabla ''$,

$$\begin{aligned} \nabla ''=\bar{\partial },\qquad \nabla '\,=\,\partial +M(y)dy, \qquad M(y)\,=\,\bigg (\,\begin{matrix} 0 &{} -t \\ 1 &{} 0 \end{matrix}\,\bigg )\,, \end{aligned}$$

(6.44)

with t constructed from $\varphi (y,\bar{y})$ as in (6.40). This connection is flat since $\partial _y\bar{\partial }_{\bar{y}}\varphi =\mu e^{2\varphi }$ implies $\bar{\partial }t=0$. The Fuchsian group $\Gamma$ characterizing the uniformization of C is nothing but the holonomy $\rho$ of the connection $\nabla$ defined in (6.44).

The Fuchsian groups $\Gamma$ fill out the connected component ${\mathcal M}_\mathrm{char}^{{\mathbb R},0}(C)\simeq {\mathcal T}(C)$ in ${\mathcal M}^{{\mathbb R}}_\mathrm{flat}(C)$ called the Teichmüller component.

1.3 Fock–Goncharov Coordinates

Let $\tau$ be a triangulation of the surface C such that all vertices coincide with marked points on C. An edge e of $\tau$ separates two triangles defining a quadrilateral $Q_e$ with corners being the marked points $P_1,\ldots ,P_4$. For a given local system $({\mathcal E},\nabla )$, let us choose four sections $s_i$, $i=1,2,3,4$ that obey the condition $\nabla s_i = 0,$ and are eigenvectors of the monodromy around $P_i$. Out of the sections $s_i$ form [FG1, GMN2]

$$\begin{aligned} {\mathcal X}_e^{\tau } := -\frac{(s_1\wedge s_2)(s_3\wedge s_4)}{(s_2\wedge s_3)(s_4\wedge s_1)}, \end{aligned}$$

(6.45)

where all sections are evaluated at a common point $P\in Q_e$. It is not hard to see that ${\mathcal X}_e^{\tau }$ does not depend on the choice of P.

There exists a simple description of the relations between the coordinates associated to different triangulations. If triangulation $\tau _e$ is obtained from $\tau$ by changing only the diagonal in the quadrangle containing e, we have

$$\begin{aligned} {\mathcal X}_{e'}^{\tau _e}\,=\,\left\{ \begin{aligned}&{\mathcal X}_{e'}^{\tau }\, \big (1+ ({\mathcal X}_e^{\tau })^{-\mathrm{sgn}(n_{e'e})}\big )^{-n_{e'e}} \;\;&\mathrm{if}\;\;e'\ne e\,,\\&({\mathcal X}_e^{\tau })^{-1}\;\;&\mathrm{if}\;\;{e'= e}\,. \end{aligned}\right. \end{aligned}$$

(6.46)

This reflects part of the structure of a cluster algebra that ${\mathcal M}_\mathrm{flat}^{}(C)$ has.

1.4 Trace Functions

The trace functions

$$\begin{aligned} L_\gamma :=\nu _\gamma \mathrm{tr}(\rho (\gamma ))\,, \end{aligned}$$

(6.47)

represent useful coordinate functions for ${\mathcal M}^{{\mathbb C}}_\mathrm{flat}(C)$. The signs $\nu _\gamma$ will be chosen such that the restriction to $L_{\gamma }$ to the Teichmüller component ${\mathcal M}_\mathrm{char}^{{\mathbb R},0}(C)$ satisfies $L_{\gamma }=2\cosh (l_\gamma /2)>2$, where $l_\gamma$ is the length of the hyperbolic geodesic on ${\mathbb U}/\Gamma$ isotopic to $\gamma$.

The coordinate functions $L_\gamma$ generate the commutative algebra ${\mathcal A}(C)\simeq \mathrm{Fun}^\mathrm{alg}({\mathcal M}_\mathrm{flat}(C))$ of functions on ${\mathcal M}_\mathrm{flat}(C)$. The well-known relation $\mathrm{tr}(g)\mathrm{tr}(h)=\mathrm{tr}(gh)+\mathrm{tr}(gh^{-1})$ valid for any pair of SL(2)-matrices g, h implies that the geodesic length functions satisfy the so-called skein relations,

$$\begin{aligned} L_{\gamma _1} L_{\gamma _2}\,=\,L_{S(\gamma _1,\gamma _2)}\,, \end{aligned}$$

(6.48)

where $S(\gamma _1,\gamma _2)$ is the loop obtained from $\gamma _1$, $\gamma _2$ by means of the smoothing operation, defined as follows. The application of S to a single intersection point of $\gamma _1$, $\gamma _2$ is depicted in Fig. 5. The general result is obtained by applying this rule at each intersection point, and summing the results.

Fig. 5

The symmetric smoothing operation

1.5 Topological Classification of Closed Loops

With the help of pants decompositions one may conveniently classify all non-selfintersecting closed loops on C up to homotopy. To a loop $\gamma$ let us associate the collection of integers $(r_e,s_e)$ associated to all edges e of $\Gamma _{\sigma }$ which are defined as follows. Recall that there is a unique curve $\gamma _e\in {\mathcal C}_\sigma$ that intersects a given edge e on $\Gamma _\sigma$ exactly once, and which does not intersect any other edge. The integer $r_e$ is defined as the number of intersections between $\gamma$ and the curve $\gamma _e$. Having chosen an orientation for the edge $e_r$ we will define $s_e$ to be the intersection index between e and $\gamma$.

Dehn’s theorem (see [DMO] for a nice discussion) ensures that the curve $\gamma$ is up to homotopy uniquely classified by the collection of integers (r, s), subject to the restrictions

$$\begin{aligned} \begin{aligned} \mathrm{(i)} \quad&r_e\ge 0\,,\\ \mathrm{(ii)} \quad&\mathrm{if}\;\;r_e=0\;\Rightarrow \;s_e\ge 0\,,\\ \mathrm{(iii)} \quad&r_{e_{\mathfrak 1}}+r_{e_{\mathfrak 2}}+r_{e_{\mathfrak 3}}\in 2{\mathbb Z}\;\,\mathrm{whenever}\;\, \gamma _{e_{\mathfrak 1}},\gamma _{e_{\mathfrak 2}},\gamma _{e_{\mathfrak 3}}\;\,\text {bound the same trinion}. \end{aligned} \end{aligned}$$

(6.49)

We will use the notation $\gamma _{(r,s)}$ for the geodesic which has parameters $(r,s):e\mapsto (r_e,s_e)$.

1.6 Generators and Relations

The pants decompositions allow us to describe ${\mathcal A}(C)$ in terms of generators and relations. As set of generators for ${\mathcal A}(C)$ one may take the functions $L_{(r,s)}\equiv L_{\gamma _{(r,s)}}$. The skein relations imply various relations among the $L_{(r,s)}$. It is not hard to see that these relations allow one to express arbitrary $L_{(r,s)}$ in terms of a finite subset of the set of $L_{(r,s)}$.

Let us temporarily restrict attention to surfaces with genus zero and $n=4$ boundaries. The Moore-Seiberg graph $\Gamma _{\sigma }$ will then have only one internal edge, allowing us to drop the index e labelling the edges. Let us introduce the geodesics $\gamma _s=\gamma _{(0,1)}$, $\gamma _t=\gamma _{(2,0)}$ and $\gamma _u=\gamma _{(2,1)}$. The geodesics $\gamma _s$ and $\gamma _t$ are depicted as red curves on the left and right half of Fig. 1. We will denote $L_{k}\equiv L_{\gamma _k}$, where $k\in \{s,t,u\}$. The trace functions $L_s$, $L_t$ and $L_u$ generate ${\mathcal A}(C)$.

These coordinates are not independent, though. Further relations follow from the relations in $\pi _1(C)$. It can be shown (see e.g. [Go09] for a review) that the coordinate functions $L_s$, $L_t$ and $L_u$ satisfy an algebraic relation of the form

$$\begin{aligned} P(L_s,L_t,L_u)\,=\,0\,. \end{aligned}$$

(6.50a)

The polynomial P in (6.50) is explicitly given as⁶

$$\begin{aligned} P(L_s, L_t, L_u) :=&-L_s L_t L_u + L_s^2 + L_t^2 + L_u^2 \nonumber \\&+L_s (L_3L_4 + L_1L_2) + L_t (L_2L_3 + L_1L_4) + L_u (L_1L_3 + L_2L_4) \nonumber \\&-4 + L_1^2+L_2^2+L_3^2+L_4^2+L_1L_2L_3L_4\,. \end{aligned}$$

(6.50b)

In the expressions above we have denoted $L_{i}:=L_{\gamma _i}$, where $\gamma _i$, $i=1,2,3,4$ represent the boundary components of $C_{0,4}$, labelled according to the convention defined in Fig. 1.

1.7 Trace Functions in Terms of Fock-Goncharov Coordinates

Assume given a path $\varpi _{\gamma }$ on the fat graph homotopic to a simple closed curve $\gamma$ on $C_{g,n}$. Let the edges be labelled $e_i$, $i=1,\ldots ,r$ according to the order in which they appear on $\varpi _{\gamma }$, and define $\sigma _i$ to be 1 if the path turns left at the vertex that connects edges $e_i$ and $e_{i+1}$, and to be equal to $-1$ otherwise. Consider the following matrix,

$$\begin{aligned} \mathrm{X}_{\gamma } = \mathrm{V}^{\sigma _r}\mathrm{E}(z_{e_r})\cdots \mathrm{V}^{\sigma _1} \mathrm{E}(z_{e_1}), \end{aligned}$$

(6.51)

where $z_e=\log X_e$, and the matrices $\mathrm{E}(z)$ and $\mathrm{V}$ are defined respectively by

$$\begin{aligned} \mathrm{E}(z) = \bigg (\begin{array}{cc} 0 &{} +e^{+\frac{z}{2}}\\ -e^{-\frac{z}{2}} &{} 0 \end{array}\bigg ),\quad \mathrm{V} = \bigg (\begin{array}{cc} 1 &{} 1 \\ -1 &{} 0 \end{array}\bigg ). \end{aligned}$$

(6.52)

Taking the trace of $\mathrm{X}_{\gamma }$ one gets the hyperbolic length of the closed geodesic isotopic to $\gamma$ via [F97]

$$\begin{aligned} L_{\gamma } \equiv 2\cosh (l_{\gamma }/2) = |\mathrm{tr}(\mathrm{X}_{\gamma })|. \end{aligned}$$

(6.53)

We may observe that the classical expression for $L_{\gamma }\equiv 2\cosh \frac{1}{2}l_{\gamma }$ as given by formula 6.53 is a linear combination of monomials in the variables $u_e^{\pm 1}\equiv e^{\pm \frac{z_e}{2}}$ of the very particular form (5.4).

1.8 Fenchel-Nielsen Coordinates for ${\mathcal M}_\mathrm{flat}^{{\mathbb R},0}(C)$

One may express $L_s$, $L_t$ and $L_u$ in terms of the Fenchel-Nielsen coordinates l and k [Ok, Go09]. Explicit expressions are for $C_{0,4}$,

$$\begin{aligned}&L_s\,=\,2\cosh (l/2)\,, \end{aligned}$$

(6.54a)

$$\begin{aligned}&L_t\big ((L_s)^2-4\big )\,=\, 2(L_2L_3+L_1L_4)+L_s(L_1L_3+L_2L_4) \\&+2\cosh (k) \sqrt{c_{12}(L_s)c_{34}(L_s)}\,, \nonumber \end{aligned}$$

(6.54b)

$$\begin{aligned}&L_u\big ((L_s)^2-4\big )\,=\, L_s(L_2L_3+L_1L_4)+2(L_1L_3+L_2L_4) \\&+2\cosh ((2k-l)/2) \sqrt{c_{12}(L_s)c_{34}(L_s)}\,,\nonumber \end{aligned}$$

(6.54c)

where $L_i=2\cosh \frac{l_i}{2}$, and $c_{ij}(L_s)$ is defined as

$$\begin{aligned} c_{ij}(L_s)&\,=\,L_s^2+L_i^2+L_j^2+L_sL_iL_j-4\ . \end{aligned}$$

(6.55)

These expressions ensure that the algebraic relations $P_e(L_s,L_t,L_u)=0$ are satisfied. By complexifying (l, k) one gets (local) coordinates for ${\mathcal M}_\mathrm{flat}^{{\mathbb C}}(C)$ [NRS].

1.9 Poisson Structure

There is also a natural Poisson bracket on ${\mathcal A}(C)$ [Go86], defined such that

$$\begin{aligned} \{\,L_{\gamma _1},\, L_{\gamma _2}\,\}\,=\,L_{A(\gamma _1,\gamma _2)}\,, \end{aligned}$$

(6.56)

where $A(\gamma _1,\gamma _2)$ is the loop obtained from $\gamma _1$, $\gamma _2$ by means of the anti-symmetric smoothing operation, defined as above, but replacing the rule depicted in Fig. 5 by the one depicted in Fig. 6. This Poisson structure coincides with the Poisson structure coming from the natural symplectic structure on ${\mathcal M}_\mathrm{flat}(C)$ which was introduced by Atiyah and Bott.

Fig. 6

The anti-symmetric smoothing operation

The resulting expression for the Poisson bracket $\{\,L_s,L_t\,\}$ can be written elegantly in the form

$$\begin{aligned} \{\,L_s,L_t\,\}\,=\,\frac{\partial }{\partial L_u} P(L_s,L_t,L_u)\,. \end{aligned}$$

(6.57)

It is remarkable that the same polynomial appears both in (6.50) and in (6.57), which indicates that the symplectic structure on ${\mathcal M}_\mathrm{flat}$ is compatible with its structure as algebraic variety.

The Fenchel-Nielsen coordinates are known to be Darboux-coordinates for ${\mathcal M}_\mathrm{flat}(C)$, having the Poisson bracket

$$\begin{aligned} \{\,l,\,k\,\}\,=\,2 \,. \end{aligned}$$

(6.58)

The Poisson structure is also rather simple in terms of the Fock-Goncharov coordinates,

$$\begin{aligned} \{{\mathcal X}_e^{\tau }, {\mathcal X}_{e'}^{\tau }\} = n_{e,e'}\, {\mathcal X}_{e'}^{\tau }\,{{\mathcal X}_e^{\tau }}, \end{aligned}$$

(6.59)

where $n_{e,e'}$ is the number of faces e and $e'$ have in common, counted with a sign.