Abstract
In the formulation of (2+1)dimensional gravity as a ChernSimons gauge theory, the phase space is the moduli space of flat Poincaré group connections. Using the combinatorial approach developed by Fock and Rosly, we give an explicit description of the phase space and its Poisson structure for the general case of a genus oriented surface with punctures representing particles and a boundary playing the role of spatial infinity. We give a physical interpretation and explain how the degrees of freedom associated with each handle and each particle can be decoupled. The symmetry group of the theory combines an action of the mapping class group with asymptotic Poincaré transformations in a nontrivial fashion. We derive the conserved quantities associated to the latter and show that the mapping class group of the surface acts on the phase space via Poisson isomorphisms.
HWM032
EMPG0302
grqc/0301108
Poisson structure and symmetry
in the ChernSimons formulation of (2+1)dimensional gravity
C. Meusburger^{1}^{1}1 and B. J. Schroers^{2}^{2}2
Department of Mathematics, HeriotWatt University
Edinburgh EH14 4AS, United Kingdom
26 January 2003
1 Introduction
The interest of (2+1)dimensional gravity (two spatial dimensions, one time dimension) is that it can serve as a toy model for regular Einstein gravity in (3+1) dimensions. In the (2+1)dimensional case, the Einstein equations of motion reduce to the requirement that the spacetime be flat outside the regions where matter is located. As a result, Einstein gravity in (2+1) dimensions is much simpler than its (3+1)dimensional analogue. The only degrees of freedom present in (2+1)dimensional gravity are a finite number of global degrees of freedom related to matter and the topology of the spacetime manifold. This gives rise to hope that (2+1)dimensional gravity will allow one to study conceptual questions related to the quantisation of general relativity without being hindered by the technical difficulties present in the (3+1)dimensional case.
As the features of a quantum theory are modelled after their classical counterparts and should reduce to them in the classical limit, an important prerequisite of the quantisation of (2+1) gravity is the investigation of its phase space. This includes among others the parametrisation of the phase space, the determination of its Poisson structure and the identification of symmetries and conserved quantities. Starting with the work of Regge and Nelson [2], these questions have been addressed by a number of physicists, using a variety of methods, see [3] for further background. For us, the recent work by Matschull [4] which is entirely based on the Einstein formulation of (2+1) gravity is going to be a key reference. However, despite these efforts, a complete and explicit description of the phase space and its Poisson structure for arbitrarily many (spinning) particles on a surface with handles is still missing. In view of our opening paragraph it is furthermore desirable that such a description provides a starting point for an equally explicit quantisation of the theory.
It was first noted in [5] and further discussed in [6] that the EinsteinHilbert action in (2+1) gravity can be written as the ChernSimons action of an appropriate gauge group. If one adopts this formulation, a large body of mathematical knowledge becomes available and can be applied to the phase space of (2+1) gravity. The phase space of a ChernSimons theory with gauge group is the space of flat connections modulo gauge transformations, in the following referred to as the moduli space. It inherits a Poisson structure from the canonical Poisson structure on the space of ChernSimons gauge fields. The moduli space of flat connections and its Poisson structure has been investigated extensively in mathematics for the case of compact semisimple Lie groups . Of special interest for this article is the work of Fock and Rosly [7], in which the moduli space arises as a quotient of two finite dimensional spaces and the Poisson structure is given explicitly in terms of a classical matrix [7]. This description of the moduli space was developed further in [8],[9], using the language of PoissonLie groups. Moreover, it is the starting point for a quantisation procedure called combinatorial quantisation, carried out and described in detail in [10], [11] and [12].
The aim of this article is to apply these methods to the ChernSimons formulation of (2+1) gravity in order to give a description of the phase space that allows for a systematic study of its physics and which is amenable to quantisation. One advantage of this approach is its generality. We are able to give an explicit parametrisation of the phase space on a spacetime manifold of topology , where is a genus oriented surface with punctures representing massive particles (possibly with spin) and a connected boundary corresponding to spatial infinity. In this description, the mathematical concepts and parameters introduced in [7] have a natural physical interpretation. We thus obtain a mathematically rigorous framework in which the physics questions can be addressed.
Our article is structured as follows. Sect. 2 contains an introduction to the ChernSimons formulation of (2+1) gravity and its phase space as the moduli space of flat connections. We summarise briefly some mathematical results about the moduli space, in particular the work of Fock and Rosly [7] and Schomerus and Alekseev [12] underlying our description of the phase space.
In Sect. 3 we apply these results to the phase space of (2+1) gravity on an open surface with a single massive particle. We extend the description of the moduli space given by Fock and Rosly to incorporate a boundary representing spatial infinity. This allows us to give an explicit parametrisation of the phase space and its Poisson structure as well as a physical interpretation. Relating the Poisson structure of the one particle phase space to the symplectic structure on the dual of the universal cover of the (2+1)dimensional Poincaré group, we show that it generalises the Poisson structure derived by Matschull and Welling [13] in the metric formulation.
Sect. 4 extends these results to the general case of a genus oriented surface with punctures and a boundary representing spatial infinity. After a discussion of the boundary condition and asymptotically nontrivial gauge transformations, we describe the phase space by means of a graph and derive its Poisson structure. The results are given a physical interpretation and related to the theory of PoissonLie groups. Applying the work of Alekseev and Malkin [8],[9], we introduce a set of normal coordinates that decouple the contributions of different handles and particles.
In Sect. 5, we study the symmetries of the phase space. Following the work of Giulini [14], we identify the symmetry group of (2+1) gravity in the ChernSimons formulation for a spacetime of topology . We determine the quantities that generate asymptotic symmetries via the Poisson bracket and interpret them in physical terms. The action of large diffeomorphisms on the spatial surface is investigated, and we prove that the generators of the (full) mapping class group act on the phase space as Poisson isomorphisms.
Our final section contains comments on the relationship between our results and other approaches to (2+1)dimensional gravity, our conclusions and an outlook. Facts and definitions about the universal cover of the (2+1)dimensional Poincaré group as a PoissonLie group are summarised in the appendix.
2 The ChernSimons formulation of (2+1)dimensional gravity and its phase space as the moduli space of flat connections
2.1 Conventions
In the following we restrict attention to (2+1)dimensional gravity with vanishing cosmological constant in its formulation as a ChernSimons theory on a spacetime of topology , where is a disc with punctures and handles, i.e. an oriented surface of genus with punctures and a connected boundary. The punctures at represent massive particles, the boundary corresponds to spatial infinity.
Throughout the paper we use units in which the speed of light is . Exploiting the fact that in (2+1) gravity Newton’s constant has dimensions of inverse mass, we measure masses in units of . Indices are raised and lowered with the (2+1)dimensional Minkowski metric diag, and Einstein summation convention is employed unless stated otherwise. For the epsilon tensor we choose the convention .
and denote respectively the (2+1)dimensional proper ortochronous Lorentz and Poincaré group and , their universal covers. The Lie algebra of the group is with generators , , , and the commutator
(2.1) 
The generators , , span the Lie algebra of . If we write the elements of as
the group multiplication in is given by
(2.2) 
with denoting the element associated to .
In the following we adopt the conventions of [15] for parametrising the (2+1)dimensional Lorentz group and its covers, in particular the parametrisation via the exponential map
(2.3) 
Note that this map is neither into nor onto, but that we can nevertheless write any element in the form
(2.4) 
Combining the parameters into a threevector , we can characterise elliptic elements of by the condition , parabolic elements by and hyperbolic elements by .
This allows us to express the adjoint of an element as
(2.5)  
where . Elliptic conjugacy classes in the group are characterised by two restrictions on the parameter threevectors ,
(2.6) 
with parameters , .
2.2 (2+1)dimensional gravity as a ChernSimons gauge theory
In Einstein’s original formulation of general relativity, the dynamical variable is a metric on . For the ChernSimons formulation it is essential to adopt Cartan’s point of view, where the theory is formulated in terms of the (nondegenerate) dreibein of oneforms , , and the spin connection oneforms , . The dreibein is related to the metric via
(2.7) 
and the oneforms should be thought of as components of the connection
(2.8) 
The Cartan formulation is equivalent to Einstein’s metric formulation of (2+1) gravity provided that the dreibein is invertible.
The vacuum EinsteinHilbert action in (2+1) dimensions can be written in terms of dreibein and spin connection as
(2.9) 
where denotes the components of the curvature twoform:
(2.10) 
Both the connection and the dreibein are dynamical variables and varied independently. Variation with respect to the spin connection yields the requirement that torsion vanishes:
(2.11) 
Variation with respect to yields the vanishing of the curvature tensor:
(2.12) 
In (2+1) dimensions, this is equivalent to the Einstein equations in the absence of matter.
For the ChernSimons formulation of gravity, dreibein and the spin connection are combined into a Cartan connection [16]. This is a oneform with values in the Lie algebra
(2.13) 
whose curvature
(2.14) 
combines the curvature and the torsion of the spin connection.
The final ingredient needed to establish the ChernSimons formulation is a nondegenerate, invariant bilinear form on the Lie algebra
(2.15) 
Then the ChernSimons action for the connection on is
(2.16) 
A short calculation shows that this is equal to the EinsteinHilbert action (2.9). Moreover, the equation of motion found by varying the action with respect to is
(2.17) 
Using the decomposition (2.14) we thus reproduce the condition of vanishing torsion and the threedimensional Einstein equations, as required. This shows that there is a onetoone correspondence of solutions of Einstein’s equations and flat ChernSimons gauge fields with nondegenerate dreibein.
For a spacetime of the form with punctures corresponding to massive particles, an additional condition has to be given regarding the behaviour of the curvature tensor at the punctures. The inclusion of massive particles with spin into (2+1)dimensional gravity in its ChernSimons formulation has been investigated by several authors. As explained in [17], the curvature tensor develops function singularities at the positions of the particles. For a single particle of mass and spin at rest at the curvature tensor is given by
(2.18) 
As a consequence, the holonomy for an infinitesimal circle surrounding the particle is
(2.19) 
As we will show in Sect. 3, the holonomy for a general loop around the particle is related to via conjugation with a element. It is an element of a fixed elliptic conjugacy class parametrised by the particle’s mass and spin as in (2.6).
2.3 The phase space of ChernSimons theory
In a ChernSimons theory with gauge group on manifold , where is a twodimensional surface with connected boundary, two gauge connections and describe the same physical state if they are related by a gauge transformation
(2.20) 
where is a valued function on the surface . It must satisfy an appropriate falloff condition compatible with the conditions imposed on the gauge connections at the boundary. The phase space of a ChernSimons theory with gauge group on a manifold is the space of all physically distinct solutions of the equations of motion (2.17) subject to conditions of the form (2.18) at the punctures. It is the moduli space of flat connections on modulo gauge transformations (2.20). It inherits a Poisson structure from the canonical symplectic structure on the space of gauge connections [18]. The properties of the moduli space have been investigated extensively in mathematics for the case of compact gauge groups . In particular, it has been shown that it is finite dimensional. A review of the mathematical results and further references are given in the book [19].
The FockRosly description of the moduli space
Fock and Rosly gave a description of the moduli space of flat connections on a closed, oriented surface with punctures by means of a graph embedded into the surface [7], see[20] for a pedagogical account. The underlying idea is similar to lattice gauge theory, but  due to the absence of local degrees of freedom  the physical content of the theory is captured entirely by a sufficiently refined^{3}^{3}3As we use only sufficiently refined graphs in this article, we will not explain this concept further. graph without the need to take a continuum limit. According to Fock and Rosly, the moduli space and its Poisson structure on an oriented, surface with punctures can be uniquely characterised by a ciliated fat graph . This is a set of vertices and oriented edges connecting the vertices together with a linear ordering of the incident edges at each vertex. If such a graph is embedded into the surface , the orientation induces a cyclic ordering of the incident edges at each vertex and makes the graph a fat graph. The surface can be completely reconstructed from a sufficiently refined fat graph. The reconstruction of the surface and the Poisson structure on the moduli space requires a ciliated fat graph. It can be obtained from a fat graph embedded into the surface by adding a cilium at each vertex in order to separate the incident edges of minimum and maximum order. Given a smooth connection on the surface and a ciliated fat graph embedded into it, parallel transport along the edges of the graph assigns an element of the gauge group to each oriented edge and thus induces a graph connection: a map from the set of oriented edges into the direct product of copies of the gauge group . Flat connections on the surface induce flat graph connections, for which the ordered product of the group elements assigned to edges around a face of the graph is trivial if the face does not contain any punctures. Similarly, gauge transformations (2.20) induce transformations of the group elements associated to each edge: The group element assigned to the oriented edge by parallel transport transforms according to
(2.21) 
where denotes the position of the vertex edge points to and the position of the vertex at which it starts. This defines a graph gauge transformation, a map from the set of vertices into the direct product of copies of the gauge group. As explained in [7], for any sufficiently refined graph the moduli space on is isomorphic the quotient of the space of flat graph connections modulo graph gauge transformations .
Theorem 2.1
The moduli space on a closed, oriented surface with punctures is isomorphic to the quotient of the space of graph connections modulo graph gauge transformations for any sufficiently refined fat graph
This characterises the moduli space in terms of two finite dimensional spaces. The essential advantage of the approach by Fock and Rosly is that it allows one to express the Poisson structure on the moduli space by a Poisson structure defined on the space of graph connections rather than the Poisson structure on the (infinite dimensional) space of gauge connections. The construction of this Poisson structure involves the assignment of a classical matrix for the Lie algebra to each vertex of the graph. This is an element which satisfies the classical YangBaxter equation (A.4) and whose symmetric part is equal to the tensor representing a nondegenerate invariant bilinear form on ; details are given in the appendix.
Theorem 2.2
(Fock, Rosly)

Let be a sufficiently refined ciliated fat graph. Assign a matrix to each vertex , and let be its components with respect to a basis , dim of . Then the following bivector defines a Poisson structure on the space of graph connections
(2.22) where denotes an edge pointing towards vertex , an edge starting at vertex . and , respectively, are the left and right invariant vector fields associated to edge with respect to the basis of and , refer to the ordering of the incident edges at each vertex. The convention is chosen for each edge starting and ending at the same vertex.

Let be equipped with the fold direct product of the Poisson structure on the group that is defined by means of a matrix as described in the appendix. The elements of the group of graph gauge transformations act as Poisson maps with respect to the direct product Poisson structure on and the Poisson structure on defined by (2.22).

As graph gauge transformations are Poisson maps, the Poisson structure defined by (2.22) induces a Poisson structure on the quotient . It is independent of the graph and isomorphic to the Poisson structure induced by the canonical symplectic structure on the space of gauge connections.
The Poisson bivector (2.22) can be decomposed into a part tangential to the gauge orbits and a part transversal to them:
(2.23)  
The first line in formula (2.23) depends only on the antisymmetric parts
(2.24) 
of the matrices assigned to each vertex and is tangential to the gauge orbits. The second part of the bivector and with it the Poisson structure on the moduli space depends only on the symmetric part common to all matrices, the components
(2.25) 
of the matrix representing the bilinear form on the Lie algebra . In particular, this implies that the Poisson structure on the moduli space is not affected by the choice of the matrix at each vertex.
The description using the fundamental group
Alekseev, Grosse and Schomerus specialised this description of the moduli space to the simplest graph that can be used to characterise a closed surface with punctures: a choice of generators of its fundamental group [10], [11] [12]. For a closed surface of genus with punctures, the fundamental group with respect to a basepoint is generated by curves starting and ending at , two curves , , , around each handle and a loop , , around each puncture (see Fig. 1).
Fig. 1
The generators of the fundamental group on a genus g surface with n punctures
Its generators obey a single relation
(2.26) 
where is the group commutator . Via the holonomy, a graph connection assigns an element of the gauge group to each of the generators. Due to equation (2.18), the holonomies of the loops associated to the punctures must be elements of a fixed conjugacy classes , whereas there is no such constraint for the group elements , associated to the curves around the handles. Taking this into account, the space of graph connections is given as
As the graph defined by the fundamental group has only one vertex, the group of graph gauge transformations is the gauge group , acting on the holonomies by global conjugation, and the moduli space is the quotient
(2.28) 
The Poisson bivector (2.22) defines a Poisson structure on the space of graph connections that induces a Poisson structure equivalent to the canonical Poisson structure on the moduli space.
2.4 The phase space of (2+1) gravity as the moduli space of flat connections
The material of the previous two subsections allows us to formulate in precise, technical terms the questions we have to address when applying the FockRosly description of the moduli space to the ChernSimons formulation of (2+1)dimensional gravity.
The first ingredient required to implement the FockRosly description is a classical matrix for the Lie algebra of the Poincaré group. As we shall see in the next section, the required matrix corresponds to the one given in [21] for the Euclidean case and can easily be adapted to the Lorentzian setting.
Second, we have to choose the gauge group . The ChernSimons action only depend on the Lie algebra , and a priori we are free to consider the Poincaré group or one of its covers as the gauge group. In this paper we take the gauge group to be the universal cover , as this leads to various technical simplifications.
Since essentially all mathematical work on the moduli space of flat connections has been carried out in the context of compact gauge groups one might worry about complications caused by working with the noncompact group . However, the difficulties associated to noncompact gauge groups arise when taking quotients by an action of the gauge group. It turns out that in our application to (2+1) gravity, such quotients never arise. This is due to the fact that we work with a surface with boundary and thus closely related to the next challenge, namely the incorporation of a surface with boundary in the FockRosly formalism. The details of how this is done are the subject of the next section. However, one consequence is that we do not need to take the quotient (2.28).
The final issue to be investigated is that of gauge invariance and symmetry. The ChernSimons formulation of (2+1)dimensional gravity is invariant under ChernSimons gauge transformations as well as under diffeomorphisms. However, large diffeomorphisms (not connected to the identity) and asymptotically nontrivial gauge transformations do not relate physically equivalent configurations. They combine in a rather subtle way to form the symmetry group of the theory.
3 The phase space for a single massive particle
3.1 The open spacetime containing a single particle
We begin our investigation of the phase space of (2+1)dimensional gravity with the case of an open universe of topology , containing a single massive particle. In the metric formalism, a spacetime with a particle of mass and spin that is at rest at the origin, is described by the conical metric [22]
(3.1) 
where the mass of the particle is restricted to the interval . Deficit angle and time shift of the cone are given by
(3.2) 
A dreibein and spin connection leading to this metric via (2.7) are
(3.3)  
With the conventions given in Sect. 2, they can be combined into a ChernSimons gauge field
(3.4) 
As the spatial surface has a boundary representing spatial infinity, we must impose an appropriate boundary condition on the gauge field and restrict to gauge transformations which are compatible with this condition. We derive the boundary condition in the ChernSimons formulation from a corresponding boundary condition in the metric formalism. In a reference frame where the massive particle is at rest at the origin, the particle’s centre of mass frame) the boundary condition in the metric formalism is the (trivial) condition that the metric be conical of the form (3.1). The gauge transformations compatible with this condition must reduce to spatial rotations and translations in the time direction outside an open region containing the particle.
In the ChernSimons formulation of (2+1)dimensional gravity, the metric (3.1) corresponds to a gauge field (3.4) with , given by (3.3). The admissible gauge transformations are asymptotically constant ChernSimons gauge transformations. Spatial rotations are implemented by ChernSimons gauge transformations (2.20) of , where is constant and takes the value
with constant outside an open region containing the particle. For time translations we take
with constant outside an open region containing the particle. These asymptotically nontrivial transformations differ in their physical interpretation from regular gauge transformations which vanish outside a region containing the particle. They are not related to gauge degrees of freedom but physically meaningful transformations acting on the phase space. We return to this question in the next section when we consider a universe containing an arbitrary number of particles.
Note, however, that in the ChernSimons formulation it is not necessary to limit oneself to the particle’s centre of mass frame. In order to study the dynamics of a single particle we should admit general Poincaré transformations with respect to the centre of mass frame. Then the admissible transformations are ChernSimons transformations of the form (2.20) with outside an open region containing the particle. The corresponding boundary condition on the gauge field is the requirement that the gauge field be obtained from (3.4) by an asymptotically constant gauge transformation . i.e. that it be of the form .
3.2 Phase space and Poisson structure
The simplest graph describing the open spacetime containing a single consists of a single vertex at the boundary and a loop around the particle. The loop can be built up from an edge connecting vertex and particle and an (infinitesimal) circle around the particle as pictured in Fig. 2.
\epsfboxsp.eps
Fig. 2
The holonomy of a loop around a puncture
Using expression (3.3), we calculate that the holonomy around the infinitesimal circle is the element given by (2.19). If we write for the group element obtained by parallel transport along the edge connecting vertex and particle, the holonomy of the loop starting and ending at is
(3.5) 
Defining
(3.6) 
and
(3.7) 
we can also write the holonomy as
(3.8) 
where
(3.9) 
The fact that the holonomy is an element of a fixed conjugacy class determined by mass and spin of the particles results in constraints on the parameters and
(3.10) 
Graph gauge transformations arise from asymptotically nontrivial gauge transformations (2.20) and act on the graph by conjugating the holonomy. The moduli space on the surface would be obtained by dividing this transformations out. However, as asymptotically nontrivial transformations are physically meaningful, we do not want to do this. The extended phase space is then simply the group parametrised by the parameter threevectors and . The physical phase space is obtained from it by imposing the constraints (3.10), which selects the conjugacy class given by (3.5).
The Poisson structure on the phase space is defined by the FockRosly bivector (2.22). For the graph introduced above it is given by
(3.11) 
As we do not divide by graph gauge transformations, the choice of the matrix is no longer arbitrary and different matrices lead to different Poisson structures on the phase space. The correct choice of the matrix has to be determined from physical considerations: the components of the vectors , should have the Poisson brackets expected for momentum and angular momentum threevector of a free relativistic particle and act as infinitesimal generators of asymptotic Poincaré transformations. As it turns out, this requires the matrix
(3.12) 
The right and left invariant vector fields on are given by
(3.13)  
where all functions are evaluated at parametrised in terms of and as in (3.8)^{4}^{4}4Note that the analogous relation between the Euclidean group element and and is stated incorrectly at one point in [21]. The expression in (3.13) is defined properly by
(3.14) 
where denotes the projector onto the direction of the momentum . Note that there is still a coordinate singularity in the limit , where the coordinates (3.6) no longer provide a good parametrisation of the holonomy .
Inserting the vector fields (3.13) and the matrix into equations (3.11), we obtain the Poisson structure in terms of the parameters and , ,
(3.15) 
These equations together with the constraints (3.10) suggest an interpretation of the parameters as the components of the particle’s threemomentum, as its energy and the spatial components as its momentum. Similarly, the vector , in the following referred to as angular momentum threevector, contains the particle’s angular momentum and the spatial components , associated to the Lorentz boosts. It can be seen immediately that the constraints (3.10) are Casimir functions of this Poisson structure. Their values parametrise its symplectic leaves, the conjugacy classes in .
3.3 Physical Interpretation
The components of the graph used to construct the Poisson structure can be given a physical interpretation as follows. The vertex represents an observer. The element assigned to the edge connecting it with the particle gives the Poincaré transformation relating the observer’s reference frame to centre of mass frame of the particle. The vector determines the translation in time and the position, a spatial rotation and/or Lorentz boosts.
The nonstandard relation (3.9) between the angular momentum threevector , the momentum and the position threevector was first discussed in detail for the case of spinless particles in [13] , where it was derived entirely in terms of the metric formulation of (2+1) gravity. In that paper the authors also pointed out that it leads to nonstandard commutation relations of the position coordinates. In order to understand this noncommutativity from our point of view, recall that the usual (nongravitational) formula for the angular momentum threevector of a free relativistic particle is obtained using the coadjoint action (instead of the conjugation action above)
(3.16) 
leading to the familiar formula for the angular momentum threevector
(3.17) 
Note that the formula (3.9) approaches (3.17) in the limit . More generally the relationship between and is given by the Lorentz transformation
(3.18) 
Since this transformation, which is the inverse of (3.14), plays an important role in the rest of the paper, we express it more concretely as a power series valid for any value of the momentum threevector
(3.19) 
In the case at hand, where is a timelike vector of length , this yields
(3.20) 
where we again used the projector defined after (3.14). Using the adjoint or vector representation of the , i.e. , one checks that
(3.21) 
Note that in terms of the transformed position vector
(3.22) 
the expression for (3.9) takes on the “nongravitational” form
(3.23) 
This formula goes some way towards explaining the nonstandard commutation relations of the position coordinates . They differ from those of the (in terms of which has the familiar form) because and are related by the dependent transformation .
3.4 The Relation to the Dual of the Poisson Lie group
For the case of vanishing spin , our description of the oneparticle phase space agrees with the description derived by Matschull and Welling [13], [4] in a completely independent way and generalises their results to the case of arbitrary spin. In order to see this, we have to describe the Poisson structure in terms of a symplectic form rather than a Poisson bivector. It follows from the general results in [8] that the Poisson bivector (3.11) gives the Poisson structure of the dual PoissonLie group . In the appendix we show that, as a group, the dual is the direct product . Elements can be factorised uniquely in the form (A.21) , where and can naturally be thought of as elements of . The symplectic leaves are precisely the conjugacy classes of . The symplectic form on those symplectic leaves can be written in terms of the parametrisation (3.5) as [8]
(3.24) 
Explicitly, we find that is the rightinvariant oneform on
(3.25) 
with values in the Lie algebra , and
(3.26) 
Using
and  (3.27) 
one computes
(3.28) 
which can be written as the exterior derivative of the symplectic potential
(3.29) 
This agrees with the symplectic potential derived in [13] in the spinless case, provided we identify their with our , and generalises this expression to the case of non vanishing spin. Finally, note that the symplectic potential can be written very compactly as
(3.30) 
4 The phase space of N massive particles on a genus g surface with boundary
4.1 Boundary conditions and invariance transformations
After describing the phase space of an open universe with a single particle, we extend our description to a general spacetime manifold with handles and particles of masses and spins . As we did for the single particle universe, we must first impose an appropriate boundary condition on the gauge fields at spatial infinity and require the corresponding asymptotic behaviour of the gauge transformations. Again, our choice of the boundary condition is modelled after the boundary condition in the metric formalism. In a reference frame where the centre of mass of the universe is at rest at the origin, the physically sensible boundary condition at spatial infinity is the requirement that the metric be asymptotically conical of the form (3.1) [4], i. e. that the universe asymptotically appears like a single particle. Because of the invariance of the conical metric (3.1) under rotations and time translations, the centre of mass condition does not uniquely determine a reference frame. However, we can imagine using a distant particle to fix the orientation and time origin of a distinguished centre of mass frame. This is analogous to the use of distant fixed stars for selecting a reference frame in our (3+1) dimensional universe.
As explained in the single particle discussion in Sect. 3, we do not need to restrict attention to centre of mass frames in the ChernSimons formulation. Rather we impose the boundary condition that the gauge field at spatial infinity be related to a connection of the form (3.4) by a gauge transformation given asymptotically by a constant element of the Poincaré group, which physically represents Poincaré transformations of the observer relative to the distinguished centre of mass frame. In a universe with several particles and possibly handles, the parameters and in (3.3) stand for the total mass and spin of the universe. Note that we do not impose a fixed value for and at this stage. However, we will find in Sect. 5.2 that and remain constant during the time evolution of the universe. Our boundary condition can be rephrased by saying that the holonomy around the boundary representing infinity is in an elliptic conjugacy class. The equations of motion guarantee that the class remains fixed during the time evolution. Our strategy in the following discussion of the phase space will be to work in a centre of mass frame and then generalise our results to a general frame.
The starting point for our definition of the phase space of ChernSimons theory is the space of solutions of the equations of motion that satisfy the boundary condition. In a centre of mass frame this is the space of flat connections on which are of the form (3.4) for some value of and in an open neighbourhood of the boundary. The physical phase space is obtained as a quotient of this space by identifying solutions which represent the same physical state. This means that we have to determine for each invariance of the theory, i. e. for each bijection of the space to itself, if it has an interpretation as a gauge transformation to be divided out of the phase space or gives rise to a physically meaningful symmetry transformation between different states.
The first type of invariance transformation in ChernSimons theory are gauge transformations (2.20) which are compatible with the boundary conditions at spatial infinity. In a centre of mass frame they are given by the group of valued functions on which are a constant rotation or timetranslation in a neighbourhood of the boundary and act on the gauge field according to (2.20). It follows from the fact that is contractible (and in particular, simply connected) that all of these transformations are small, i. e. connected to the identity transformation and obtained by exponentiating infinitesimal transformations. Asymptotically trivial transformations which are the identity in a neighbourhood of the boundary form a subgroup . They are generated by a gauge constraint and transform different descriptions of the same physical state into each other. As redundant transformations without physical meaning, they have to be divided out of the phase space, and field configurations related by them have to be identified. The situation is different for asymptotically nontrivial ChernSimons gauge transformations. They are not generated directly by a gauge constraint but would require an additional boundary term in the action [23], [14]. They do not correspond to gauge degrees of freedom but give rise to symmetries acting on the phase space and have a physical interpretation of Poincaré transformations with respect to the distinguished centre of mass frame.
The second type of invariance present in ChernSimons theory arises from the fact that it is a topological theory. It is invariant under the group of all orientation preserving diffeomorphisms compatible with the boundary conditions at spatial infinity. This is the group of orientation preserving diffeomorphisms that reduce to a global spatial rotation in an open neighbourhood of the boundary. Its elements act on the space via pull back with the inverse
(4.1) 
Among the elements of we further distinguish the subgroup of asymptotically trivial diffeomorphisms, i. e. diffeomorphisms of which keep the boundary fixed pointwise. Diffeomorphisms in which are connected to the identity are called small diffeomorphisms and form a normal subgroup . In a ChernSimons theory, diffeomorphisms are not a priori related to gauge transformations and there is no reason to consider them as redundant transformations between field configurations describing the same physical state. However, Witten showed in [6] that infinitesimal diffeomorphisms can onshell be written as infinitesimal ChernSimons gauge transformations as follows. The infinitesimal transformation of a flat connection generated by a vector field is given by the Lie derivative
(4.2) 
Using the formula and the flatness of , this can be written as an infinitesimal gauge transformation
(4.3) 
with generator
(4.4) 
where denotes the contraction with . By exponentiating, we see that any two field configurations and related by a small, asymptotically trivial diffeomorphism are also related by an asymptotically trivial ChernSimons gauge transformation and therefore have to be identified. Small diffeomorphisms that reduce to global rotations at the boundary correspond to asymptotically nontrivial ChernSimons gauge transformations and therefore represent physical transformations with respect to the centre of mass frame. However, the correspondence between diffeomorphisms and ChernSimons gauge transformations does not hold for large diffeomorphisms, i. e. diffeomorphisms that are not infinitesimally generated. Those diffeomorphisms are not related to gauge degrees of freedom but are physically meaningful transformations acting on the phase space.
We conclude that the physical phase space of ChernSimons theory is obtained from the space of flat gauge connections that satisfy the boundary conditions by identifying connections , if and only if they are related by an asymptotically trivial ChernSimons gauge transformation. This implies division by small, asymptotically trivial diffeomorphisms, whereas large diffeomorphisms and asymptotically nontrivial ChernSimons transformations give rise to physical symmetries. We will return to this question and give a more mathematical treatment in Sect. 5.1, where we determine the symmetry group of the theory.
A final general comment we should make concerns the dreibein in the decomposition of a connection according to (2.13). In Einstein’s metric formulation of gravity the dreibein is assumed to be invertible, but there is no such requirement in ChernSimons theory. In fact, as pointed out and discussed in [24], gauge orbits under gauge transformations may pass through a connection with a degenerate dreibein. In [24] it is argued that this leads to the identification of spacetimes in the ChernSimons formulation of (2+1) gravity which would be regarded as physically distinct in the metric formulation. We will address this issue in the final section of this paper, after we have given a detailed description of the phase space of ChernSimons theory.
4.2 The description of the space time by a graph
The description of the phase space in the formalism of Fock and Rosly depends on a graph. As in the case of a single massive particle, the graphs used in the construction of the Poisson structure on the moduli space have a intuitive physical interpretation. For a (sufficiently refined) graph that is embedded into the surface in such a way that the punctures representing massive particles lie on different faces of the graph and at least one vertex is mapped to the boundary at spatial infinity, the vertices of the graph can be thought of as observers located at different spacetime points. As the spacetime manifold is locally flat, the reference frames of these observers should be inertial frames related by Poincaré transformations. These Poincaré transformations are given by the graph connection. The element assigned to an (oriented) edge can be interpreted as the Poincaré transformation relating the reference frames of the observers at its ends: the translation vector describes the shift in position and time coordinate, the Lorentz transformation the relative orientation of their coordinate axes and their relative velocity. The observers located at different points in the spacetime can determine topology and matter content of their universe by exchanging information about these Poincaré transformations relating their reference frames. Due to the flatness of the graph connection, observers situated at vertices surrounding a face without particles will find that the (ordered) product of the Poincaré transformations along the boundary of the face is equal to the identity. If the face contains a massive particle, the same procedure yields an element of the conjugacy class determined by mass and spin of the particle, thus allowing a measurement of these quantities.
The interpretation of graph connections as Poincaré transformations between reference frames of observers situated at the vertices is complemented by the physical interpretation of graph gauge transformations (2.21). A graph gauge transformation involves an individual transformation at each vertex. As the Poincaré transformations relating adjacent observers have the right transformation property (2.21), the graph gauge transformations can be interpreted as changes of the reference frame for each observer. In this interpretation it is obvious that all transformations not affecting the vertices at the boundary correspond to gauge degrees of freedom. They do not alter the physical state of the universe but only its description in terms of reference frames of different observers. The situation is different, however, for a vertex situated at the boundary. With the interpretation of such a vertex as an observer, the graph gauge transformations affecting the vertex are seen to (uniquely) characterise the asymptotic symmetries discussed above. They correspond to rotations, boosts and translations of the observer with respect to the distinguished centre of mass frame of the universe.
4.3 The phase space of massive particles and handles
Taking into account the special role of the boundary, the most efficient FockRosly graph for a surface is a set of curves representing the generators of the fundamental group with basepoint on the boundary as pictured in Fig. 3.
\epsfboxpi.eps
Fig. 3
The generators of the fundamental group of the surface
The fundamental group is generated by the equivalence classes of the curves around each particle, curves around each handle and a curve around the boundary at spatial infinity, subject to a relation similar to (2.26). By solving the relation, the fundamental group can be presented as the free group generated by the curves around the particles and handles
(4.5) 
The holonomy along these curves assigns elements , , , , of the gauge group to each of the generators , , . Asymptotically nontrivial ChernSimons gauge transformations (2.20) induce graph gauge transformations that act on these group elements via simultaneous conjugation. As explained in the previous subsection, these are physical symmetries and not divided out of the physical phase space.
The same is true for the group of asymptotically trivial, large diffeomorphisms. Mathematically, this is called the mapping class group and defined as
(4.6) 
In our description of the spacetime by means of an embedded graph, asymptotically trivial, large diffeomorphisms map the graph to a different, topologically inequivalent graph. If we include the large diffeomorphisms as symmetries acting on the phase space, our phase space is not simply given as the quotient of the space of graph connections modulo (asymptotically trivial) graph gauge transformations corresponding to a fixed graph. Rather, we consider topologically distinct graphs and include an additional label specifying the graph. In the case where the graph is given by a set of generators of the fundamental group, this can be made more explicit.
The starting point for our definition of the phase space is one fixed graph which consists of the generators of the fundamental group shown in Fig. 3. With respect to this graph, the phase space is simply the product
(4.7) 
where