aa r X i v : . [ m a t h - ph ] A p r Quasi-Hamiltonian bookkeeping of WZNW defects
C. Klimˇc´ıkInstitut de math´ematiques de Luminy163, Avenue de LuminyF-13288 Marseille, Francee-mail: [email protected]
Abstract
We interpret the chiral WZNW model with general monodromy as an infinite dimen-sional quasi-Hamiltonian dynamical system. This interpretation permits to explain thetotality of complicated cross-terms in the symplectic structures of various WZNW defectssolely in terms of the single concept of the quasi-Hamiltonian fusion. Translated from theWZNW language into that of the moduli space of flat connections on Riemann surfaces,our result gives a compact and transparent characterisation of the symplectic structureof the moduli space of flat connections on a surface with k handles, n boundaries and m Wilson lines. Introduction
The study of WZNW defects has been quite a hot topic since last ten years [3, 4, 8, 10, 11, 12,13, 14, 18, 19, 20, 22, 25, 26, 28]. The idea was to modify the standard WZNW dynamics byconsistent boundary conditions on the world-sheet or by defect lines in the bulk where the groupvalued WZNW field is allowed to jump in a particular way. In the presence of such defects theWZNW classical field equations can still be explicitely solved and the corresponding symplecticstructure on the classical space of solutions can be derived starting from the classical WZNWaction in [16, 27]. The resulting explicit expressions for the symplectic forms turn out to bequite complicated, however.More conceptual understanding of the WZNW symplectic structures in the presence of defectswas proposed in [17], where the language of flat conections on Riemann surfaces was used.This insight was motivated by an older result [7] where the symplectic structure of the bulkWZNW model without defects was identified with that of the moduli space of flat connectionson the annulus. In the paper [17], the phase space of the boundary WZNW model was thenshown to be symplectomorphic to the moduli space of flat connections on the disc with twoWilson lines inserted. The holonomies of the flat connections around the insertion points lie insome conjugacy classes in the group manifold G which are interpreted as ”D-branes”, i.e. assubmanifolds of the target space G on which the open strings end.Following the same philosophy, symplectic structures of several other defects were identifiedwith those of appropriate moduli spaces of flat connections [27]. Thus the jump of the groupvalued WZNW field through a defect line on the world-sheet [13, 18] was shown to lead to themoduli space of flat connections on the annulus with one Wilson line insertion [27]. In this case,the holonomy around the insertion point lies in the same conjugacy class as the jump. Thedictionnary between the WZNW defects and the moduli spaces of flat connections was thenenlarged to yet other types of defects still in [27]. For example, the phase space of the boundaryWZNW model with one bulk defect line turns out to be the moduli space of flat connectionson the disc with three Wilson line insertions. Finally, the last example treated in [27] is that ofpermutation branes [10, 12, 14, 26, 28] which are the boundary conditions for the n -fold directproduct G × G × ... × G WZNW model on a strip world-sheet. It was conjectured in [27] that therelevant moduli space for this situation corresponds to the Riemann surface with n boundariesand two Wilson line insertions.Although the book-keeping of the WZNW defects via the moduli spaces of flat connections isvery elegant, it is more of conceptual importance than of concrete technical utility. In practice,one rather needs to have a description of the relevant symplectic structures on the modulispaces in terms of group-valued holonomies of the flat connections since they correspond to thephysically interpretable WZNW observables. Such description is, however, quite cumbersomealready in the presence of small number of defects, since there arise many cross terms in thesymplectic forms which correspond to ”interactions” of the defects.The goal of the present paper is to propose an alternative conceptual bookkeeping of the WZNWdefects which would be technically more friendly and would use quantities with direct physicalinterpretation. Our main inspiration comes from the approach of Ref. [2], where the symplectic2tructures of the moduli spaces of flat connections on closed surfaces (i.e. without boundaries)were described in terms of the so called quasi-Hamiltonian fusion. Speaking more precisely, themoduli space of flat connections on the compact closed surface with m Wilson line insertionsand k handles was identified in [2] as the following symplectic manifold M mk ≡ ( C − ⊛ C − ⊛ . . . ⊛ C − m ⊛ D ( G ) ⊛ · · · ⊛ D ( G ) | {z } k times ) e . (1.1)Here C − i is the conjugacy class to which belongs the holonomy of the connection around the i th insertion point (the superscript − means the inverse of the standard quasi-Hamiltonianstructure on the conjugacy class), the symbols D ( G ) stand for the so called internally fusedquasi-Hamiltonian double of the structure Lie group G , the operation ⊛ is the fusion of twoquasi-Hamiltonian manifolds and the notation ( M ) e means the symplectic manifold obtainedby the quasi-Hamiltonian reduction of the quasi-Hamiltonian manifold M at the unit level ofthe moment map.The big advantage of the expression (1.1) consists in the fact that not only it gives the explicitcharacterization of the symplectic structures of the moduli spaces in terms of the convenientgroup-like variables but, at the same time, it remains conceptually neat. Indeed, each handle ordefect brings its building block into the expression and all ingredients are glued together usingthe single concept of the quasi-Hamiltonian fusion.In what follows, we shall generalize the formula (1.1), by allowing the presence of the boundarieson the Riemann surface. This change involves the transition from the finite dimensional contextto an infinite-dimensional one, since the moduli spaces of flat connections in the presence ofboundaries are smooth infinite-dimensional symplectic manifolds [6]. Indeed, the flat connec-tions on the closed surfaces correspond roughly to the topological G/G
WZNW model and thesurfaces with boundaries take into account the full field theoretical WZNW dynamics. Inspiteof the infinite-dimensional setting, the result of our generalisation is conceptually as simple asthe expression (1.1). Indeed, we shall argue that the moduli space of flat connections on thesurface with n boundaries, m Wilson lines insertions and k handles reads: M nmk ≡ ( W − ⊛ . . . ⊛ W − | {z } n times ⊛ C − ⊛ C − ⊛ . . . ⊛ C − m ⊛ D ( G ) ⊛ . . . ⊛ D ( G ) | {z } k times ) e , (1.2)where W − is the particular infinite-dimensional quasi-Hamiltonian manifold the pointsof which are quasi-periodic maps with values in G . We shall refer to W − as toquasi-Hamiltonian chiral WZNW model. We shall see, in particular, that the quasi-Hamiltonianlanguage of formula (1.2) is very well suited for bookkeeping of multitude of terms appearingin the explicit description of symplectic forms associated to various WZNW defects.The plan of the paper is as follows: In Section 2, we expose some basic facts about the quasi-Hamiltonian geometry; in particular, we define the quasi-Hamiltonian fusion, quasi-Hamiltonianreduction and explain the contents of the so called equivalence theorem of [2] relating Hamilto-nian loop group LG -spaces to the quasi-Hamiltonian G -spaces. In Section 3, we define the chiralWZNW model as the quasi-Hamiltonian system and explain how it can be obtained from thefull WZNW model via that equivalence theorem just mentioned. Section 4 prepares ingredients3or proving the formula (1.2), namely, it gives an elegant description of the Hamiltonian loopgroup space associated by the equivalence theorem to any quasi-Hamiltonian space. The sec-tion 5 and 6 are respectively devoted to the sides AC and BC of the following triangle diagram(the side AB was largely discussed in [27]):Figure 1: B: WZNW defects C: quasi-Hamiltonian geometryA: Flat connectionsIn particular, in Section 5 we review the definition of the symplectic structures on the modulispace of flat connections and then we prove that those structures are indeed described by theformula (1.2). Finally, in Section 6, we work out the symplectic structures of the bulk, boundaryand defect WZNW models starting from the formula (1.2) and find agreement with the WZNWdefect symplectic structures obtained in [15, 16, 17, 27] by the detailed analysis of the WZNWdynamics. Quasi-Hamiltonian manifold M is acted upon by a simple compact connected Lie group G , itis equipped with an invariant two-form Ω and with a moment map µ : M → G in such a waythat four axioms must hold:1. µ intertwines the G action ⊲ on M with the conjugacy action on G : µ ( g ⊲ x ) = gµ ( x ) g − , g ∈ G, x ∈ M. (2.3)2. The exterior derivative of Ω is given by δ Ω = − µ ∗ ( θ, [ θ, θ ]) . (2.4)3. The infinitesimal action of G ≡
Lie( G ) on M is related to µ and Ω by ι ( ζ M )Ω = 12 µ ∗ ( θ + ¯ θ, ζ ) , ∀ ζ ∈ G . (2.5)4. At each x ∈ M , the kernel of Ω x is given byKer(Ω x ) = { ζ M ( x ) | ζ ∈ Ker(Ad µ ( x ) + Id) } . (2.6)4ere ( ., . ) is the Killing-Cartan form on G , θ and ¯ θ denote, respectively, the left- and right-invariant Maurer-Cartan forms on G and ζ M stands for the vector field on M that correspondsto ζ ∈ G .Three examples of quasi-Hamiltonian manifolds will be important for us: the conjugacy classin G , the so called quasi-Hamiltonian double D ( G ) of the group G and the internally fuseddouble D ( G ). The quasi-Hamiltonian moment map µ for a conjugacy class C ⊂ G is just theembedding C ֒ → G and the quasi-Hamiltonian form α evaluated at f ∈ C is defined by theformula [2] α C f ( v ξ , v η ) = 12 (cid:18) ( η, Ad f ξ ) − ( ξ, Ad f η ) (cid:19) . (2.7)Here v ξ , v η are the vector fields corresponding to the infinitesimal actions of ξ, η ∈ G . Thereis another useful way of representing the quasi-Hamiltonian form α in terms of the followingparametrization of the points on the conjugacy class C : f = ke π i τ k − , (2.8)where τ is in the Weyl alcove and k ∈ G . We have then α C f = 12 ( k − δk, e − π i τ k − δke π i τ ) . (2.9)As a manifold, the double D ( G ) is just the direct product G × G . It is the quasi-Hamiltonian G × G manifold with respect to the G × G action( g , g ) ⊲ ( a, b ) ≡ ( g ag − , g bg − ) , (2.10)moment map µ D = ( µ , µ ) : D ( G ) → G × Gµ ( a, b ) = ab, µ ( a, b ) = a − b − (2.11)and the quasi-Hamiltonian form Ω D defined byΩ D = 12 ( a ∗ θ, b ∗ ¯ θ ) + 12 ( a ∗ ¯ θ, b ∗ θ ) . (2.12)As a manifold, the internally fused double D ( G ) is again the direct product G × G equippedwith the G action g ⊲ ( a, b ) ≡ ( gag − , gbg − ) , (2.13)the moment map µ ( a, b ) ≡ aba − b − (2.14)and the two-form Ω = 12 ( a ∗ θ, b ∗ ¯ θ ) + 12 ( a ∗ ¯ θ, b ∗ θ ) + 12 (( ab ) ∗ θ, ( a − b − ) ∗ ¯ θ ) . (2.15)Let us now list some of the properties of the quasi-Hamiltonian spaces relevant for this paper(see [2] for more details): 5 First of all, a quasi-Hamiltonian manifold M equipped with the same G -action, a form − Ω and a moment map µ − is again quasi-Hamiltonian, it is referred to as the inversequasi-Hamiltonian space and denoted as M − . • Suppose that the unit element e ∈ G is the regular value of the moment map µ . Theaxioms of the quasi-Hamiltonian geometry imply that G ≡
Lie( G ) acts on the unit-levelsubmanifold µ − ( e ) without fixed points and thus µ − ( e ) /G is a symplectic orbifold (notnecessarily manifold because there still may be points in µ − ( e ) with a discrete isotropysubgroup). This orbifold is usually denoted as ( M ) e and it is called the unit-level quasi-Hamiltonian reduction of M . By construction, the pull-back of the symplectic form ω from( M ) e to µ − ( e ) is equal to the restriction of Ω to µ − ( e ), however, we stress that ω is thesymplectic form in the usual sense, whilst Ω is neither closed nor globally non-degeneratein general. • A direct product of two quasi-Hamiltonian manifolds M × M is again a quasi-Hamiltonianmanifold if it is equipped with the diagonal G -action, a moment map being the Lie groupproduct µ µ of the respective moment maps µ for M and µ for M and with a two-formΩ = Ω + Ω + 12 ( µ ∗ θ, µ ∗ ¯ θ ) . (2.16)The quasi-Hamiltonian manifold ( M × M , µ µ , Ω ) is called the fusion product and isdenoted as M ⊛ M . In the case of a multiple fusion M ⊛ M ⊛ . . . ⊛ M n , the mixed termin (2.16) gives rise to a multitude of terms in the resulting reduced symplectic form on( M ⊛ M ⊛ . . . ⊛ M n ) e which look quite awkward without the conceptual quasi-Hamiltonianunderstanding of their origin. It is indeed the purpose of the present paper to go in theopposite direction and to give the quasi-Hamiltonian raison d’ˆetre for the multitude ofcross-terms in the symplectic structures induced by the WZNW defects. • Any G -invariant function H on a quasi-Hamiltonian manifold ( M, µ,
Ω) defines a ”quasi-Hamiltonian dynamics”, in the sense that there is a unique (evolution) vector field v H satisfying the conditions ι ( v H )Ω = δH, ι ( v H ) µ ∗ θ = 0 . (2.17)Here δ stands for the de Rham differential. The Hamiltonian vector field v H is G -invariantand preserves ω and µ [2]. • Perhaps the most remarkable property of the quasi-Hamiltonian spaces is the equivalencetheorem of Ref. [2]. It states that every quasi-Hamiltonian space determines a standardHamiltonian loop group space with proper moment map and vice versa. In this way manystructural questions which can be asked about infinite-dimensional symplectic manifoldsadmitting the Hamiltonian actions of loop groups can be reformulated and solved in an an-alytically more friendly environment, in particular, if the corresponding quasi-Hamiltonianspace turns out to be finite dimensional. Speaking more precisely, the Hamiltonian LG space N with an equivariant moment map Φ : N → L G ∗ and a symplectic form ω givesrise to the quasi-Hamiltonian structure on the manifold Hol( N ) ≡ N/ Ω G where Ω G isthe group of based loops (i.e. loops taking the value e at the distinguished point σ = 0).6n order to make explicite the quasi-Hamiltonian form and the quasi-Hamiltonian mo-ment map on Hol( N ), we need to introduce some technical tools, namely, the space ofquasi-periodic maps W and a map Hol: L G ∗ → W .The space W consists of smooth maps l : R → G with the property l ( σ + 2 π ) = l ( σ ) M, ∀ σ ∈ R . (2.18)The element M ∈ G does not depend on σ and it is called the monodromy of l ∈ W . Forevery A ∈ L G ∗ there si then a unique element w A ∈ W such that A = w A ( σ ) − ∂ σ w A ( σ ) dσ, w A (0) = e. (2.19)We have thus defined the map Hol: L G ∗ → W Hol( A ) := w A . (2.20)The loop group LG acts on L G ∗ by gauge transformations (the Hamiltonian moment mapΦ is equivariant precisely with respect to this action!): g ⊲ A = gAg − − g ∗ ¯ θ, g ∈ LG. (2.21)The transformation (2.21) then induces the following transformation of the holonomy: w g⊲A ( σ ) = g (0) w A ( σ ) g ( σ ) − . (2.22)In particular, w A (2 π ) is gauge invariant with respect to the transformations from thebased loop group Ω G since in this case g (0) = g (2 π ) = e . It is this gauge invariance whichpermits to define the quasi-Hamiltonian moment map µ : Hol( N ) → G as µ := w Φ (2 π ) . (2.23)The quasi-Hamiltonian form Ω on Hol( N ) is constructed as follows. First of all, considera two-form Υ on L G ∗ defined byΥ = 12 Z π dσ (Hol ∗ σ ¯ θ, ∂ σ Hol ∗ σ ¯ θ ) . (2.24)Note that the definition (2.24) makes sense since, for a fixed value of σ , Hol σ is a mapfrom L G ∗ → G . The vector fields corresponding to the infinitesimal action of the groupΩ G on N turn out to be the degeneracy directions of the following two-form on N : ω + Φ ∗ Υ . (2.25)This two-form is therefore the pull-back of some form Ω on Hol( N ), which is nothing butthe quasi-Hamiltonian form on Hol( N ). 7 Quasi-Hamiltonian equivalent of the WZNW model
The full WZNW model [30] is the standard symplectic dynamical system, the phase space P W Z of which admits two different Hamiltonian actions of the loop group LG . One of thoseactions has the equivariant moment map in the sense of Definition 8.2 of [2] (see also Eq.(3.30) of the present paper). Following the discussion at the end of Section 2, we can associateto the equivariant Hamiltonian LG -manifold P W Z the equivalent quasi-Hamiltonian dynamicalsystem on the space Hol( P W Z ) equipped with the corresponding G -action induced by somequasi-Hamiltonian moment map. It is the goal of this section to show that this equivalentquasi-Hamiltonian system is nothing but the quasi-Hamiltonian version of the chiral WZNWmodel.Ideologically, we shall describe here the WZNW model in the language of the twisted Heisenbergdouble [29, 21]. Thus the phase space P W Z of the WZNW model is the cotangent bundle of theloop group LG parametrized by J L ( σ ) ∈ L G and g ( σ ) ∈ LG , however, the symplectic form isnot the canonical one on the cotangent bundle since it contains the additional term (the twist): ω W Z = − δ Z π dσ ( J L , δgg − ) − Z π dσ ( δgg − , ∂ σ ( δgg − )) . (3.26)Here δ is the de Rham differential on P W Z .There are two Hamiltonian actions of the loop group LG on the phase space P W Z : h ⊲ L ( J L , g ) := ( hJ L h − + ∂ σ hh − , hg ) , h ∈ LG ; h ⊲ R ( J L , g ) := ( J L , gh − ) , h ∈ LG. (3.27)The moment maps of these two actions are J L and J R , respectively, where J R := − g − J L g + g − ∂ σ g. (3.28)Indeed, it is easy to check that it holds ι ( v Lξ ) ω W Z = δ Z π ( J L , ξ ) dσ, ι ( v Rξ ) ω W Z = δ Z π ( J R , ξ ) dσ, (3.29)where ξ ∈ L G and v L,Rξ are the respective vector fields corresponding to the infinitesimal actionsof ξ on P W Z . Note the transformation of the right current J R under the action ⊲ R on P W Z byan element h ∈ LG : J R → hJ R h − − ∂hh − . (3.30)We observe that the moment map J R is equivariant following the conventions of Sections 8.1and 8.2 of [2]. However, the left current J L transforms under the action ⊲ L with the oppositesign of the inhomogeneous term: J L → hJ L h − + ∂hh − . (3.31)8e shall refer to the moment map J L as ’anti-equivariant’. We finish the resuming of theWZNW model by defining its Hamiltonian: H W Z = − Z π ( J L , J L ) dσ − Z π ( J R , J R ) dσ. (3.32)Since the phase space P W Z with the right action ⊲ R of LG is the Hamiltonian LG -space in thesense of the definition 8.2 of [2], we can construct the corresponding quasi-Hamiltonian G -spaceHol( P W Z ) following the recipe described at the end of Section 2. This gives the statement ofthe following important Theorem:
Theorem 1 : The quasi-Hamiltonian space
Hol( P W Z ) is the space of quasi-periodic maps W ,the corresponding quasi-Hamiltonian moment map µ : W → G is the inverse monodromy of theelement l ∈ W µ ( l ) = l (2 π ) − l (0) (3.33) and the quasi-Hamiltonian form Ω on W induced by ω W Z on P W Z reads Ω( l ) := 12 (cid:20)Z π ( l − δl, ∂ σ ( l − δl )) dσ + ( δll − | , δll − | π ) (cid:21) . (3.34) Proof.
Denote by g R ∈ W the element w J R defined by (2.19), i.e. J R = g − R ∂ σ g R , g R (0) = e. (3.35)We can also conveniently parametrize the current J L as J L = ∂ σ g L g − L , g L (0) = e (3.36)and the field g ( σ ) as g ( σ ) = g L ( σ ) b ( σ ) g R ( σ ) . (3.37)The relation (3.28) then implies that b ( σ ) in fact does not depend on σ and it is therefore equalto g (0). In what follows, we set l ( σ ) := g L ( σ ) b ( σ ) = g L ( σ ) g (0) (3.38)and express straightforwardly the symplectic form ω W Z in terms of the variables l ( σ ) and g R ( σ ): ω W Z = 12 Z π ( l − δl, ∂ σ ( l − δl )) −
12 ( l − δl, δg R g − R ) (cid:12)(cid:12)(cid:12)(cid:12) π − Z π ( δg R g − R , ∂ σ ( δg R g − R )) . (3.39)Because of the fact that g R (0) = e and g (2 π ) = l (2 π ) g R (2 π ) = l (0) = g (0) , (3.40)9e conclude that the quasi-Hamiltonian form (2.25) becomes ω W Z + J ∗ R Υ = ω W Z + 12 Z π ( δg R g − R , ∂ σ ( δg R g − R )) = 12 (cid:20)Z π ( l − δl, ∂ σ ( l − δl )) dσ +( δll − | , δll − | π ) (cid:21) . (3.41)Now Eqs. (2.23) and (3.40) show that the quasi-Hamiltonian moment map is indeed the inversemonodromy of l ∈ W µ ( l ) = g R (2 π ) = l (2 π ) − l (0) . (3.42)Finally, it remains to identify the quasi-Hamiltonian space Hol( P W Z ) with W . Note thatHol( P W Z ) is the space of cosets P W Z / Ω G , so starting from the parametrization ( J L , g ) of P W Z we see that Hol( P W Z ) can be parametrized by means of g L and g (0) as J L = ∂ σ g L g − L , g = g (0).Following (3.38), Hol( P W Z ) can be parametrized also by l ∈ W since g (0) = l (0). From (3.27),we conclude that the G -action on W is given by l ( σ ) → l ( σ ) h − , l ( σ ) ∈ W, h ∈ G. (3.43)Although from the general theorems of Ref. [2] it follows that the triple ( W, Ω( l ) , µ ( l )) given byEqs. (3.34), (3.42) and (3.43) is the quasi-Hamiltonian G -space, we prefer to provide a directproof of this fact in order to make the present paper more self-contained: Theorem 2 : Define a function on W by the formula H ( l ) = − Z π ( ∂ σ ll − , ∂ σ ll − ) dσ, (3.44) the G -action on W by l ( σ ) → l ( σ ) h − , l ( σ ) ∈ W, h ∈ G, (3.45) the moment map µ : W → G by µ ( l ) = l (2 π ) − l (0) (3.46) and the two-form Ω( l ) on W by Ω( l ) := 12 Z π ( l − δl, ∂ σ ( l − δl )) dσ + 12 ( δll − | , δll − | π ) (3.47) The quadruple ( W, Ω( l ) , µ, H ) is then the quasi-Hamiltonian dynamical system.Proof. We immediately observe from (2.18), (3.45) and (3.46) that µ ( h ⊲ l ) = hµ ( l ) h − , (3.48)which means that the first defining quasi-Hamiltonian property (2.3) is verified.A simple bookkeeping of boundary terms gives the second defining quasi-Hamiltonian property(2.4): δ Ω( l ) = 112 ( l − δl | π , [ l − δl | π , l − δl | π ]) −
112 ( l − δl | , [ l − δl | , l − δl | ]) + 12 δ ( δll − | , δll − | π ) =10 −
112 ( µ − l δµ l , [ µ − l δµ l , µ − l δµ l ]) , (3.49)where we have set µ ( l ) = µ l .Let us now verify the third property (2.5). First of all, let ξ W be a vector field on W inducedby the infinitesimal action of an element ξ ∈ G . We infer easily ι ( ξ W ) l − δl = − ξ, ι ( ξ W )( δll − | ) = − l (0) ξl (0) − , ι ( ξ W )( δll − | π ) = − l (2 π ) ξl (2 π ) − , (3.50)hence we find indeed that ι ( ξ W )Ω = 12 ( ξ, δµ l µ − l + µ − l δµ l ) . (3.51)It remains to verify the last property (2.6). First of all we note that W is a submanifold of thegroup R G consisting of all smooth maps from R to G . Therefore any vector field v at a point l of W can be written as the left transport − L l ∗ ζ of some − ζ ∈ Lie( R G ). From this informationwe find ι ( v )( l − δl ) = − ζ (3.52)therefore ι ( v )Ω = Z π ( l − δl, ∂ σ ζ ) dσ −
12 ( ζ , l − δl ) (cid:12)(cid:12)(cid:12)(cid:12) π −
12 ( l ζ l − , δll − | π ) + 12 ( δll − | , l π ζ π l − π ) . (3.53)If v is to be in the kernel of Ω then obviously ∂ σ ζ = 0 and ι ( v )Ω = 12 ( ζ , δµ l µ − l + µ − l δµ l ) = 12 ( µ l ζ µ − l + ζ , δµ l µ − l ) . (3.54)From the last formula, the wanted property (2.6) readily follows.We conclude the demonstration by noting that the Hamiltonian (3.44) is evidently G -invariant,as it should be. Definition : We shall refer to the quasi-Hamiltonian dynamical system ( W, Ω( l ) , µ ( l ) , H ( l )) asto the quasi-Hamiltonian chiral WZNW model. Remark 1 : Historically, the origin of the concept of the chiral WZNW model lies in the attempt to equipthe left and right movers of the WZNW model with independent dynamics. Recall that every solution of theWZNW model in the configuration space LG can be described as the product of left and right movers [15, 16]: g ( σ, τ ) = l ( σ + τ ) r − ( σ − τ ) , σ ∈ [0 , π [ , τ ∈ R , (3.55)where both left and right movers l and r are the elements of W and can be viewed as almost independentcoordinates on the infinite-dimensional phase space P W Z of the theory. Indeed, l and r are tied only by therequirement that they must have the same monodromies in order to insure the periodicity of the WZNW field g ( σ ). In [15], the symplectic form ω W Z was expressed in terms of the left and right movers as ω W Z = Ω( l ) − Ω( r ) , (3.56) here the two-form Ω( l ) is nothing but our quasi-Hamiltonian friend (3.34). The form of the WZNW symplecticform (3.56) suggests that it may be possible to separate completely the left and right movers by allowing theindependent monodromies for them. However, the trouble in doing that was remarked already in [15]. The pointis that the exterior derivatives of the forms Ω( l ) and Ω( r ) do not vanish separately as it can be seen from (3.49)(in fact, in calculating δω W Z , they cancel with each other precisely when the left and right monodromies arethe same). As the solution to the problem of non-closedness of Ω( l ), it was proposed in [15] to add to Ω( l ) atwo-form ρ ( µ ( l )) depending exclusively on the inverse monodromy µ ( l ) and to define the chiral WZNW model asa theory on the phase space W , with the symplectic form Ω( l ) + ρ ( µ ( l )) and the quadratic current Hamiltonian(3.44). The problem with this definition is the ambiguity of the choice of the two-form ρ ( µ ( l )) as well as thefact that, strictly speaking, such ρ exists only on a dense open subset of the group manifold G . In this section,we did not attempt to define the chiral dynamics in the symplectic way, but we adopted the quasi-Hamiltonianpoint of view. Said in other words, we have defined the chiral WZNW model as the quasi-Hamiltonian dynamicalsystem . For this, we did not need to add any term to the two-form Ω( l ) on W , but we let it as it stands. Ofcourse, all this is just a shift of interpretation but it will turn out soon that our quasi-Hamiltonian version ofthe chiral WZNW has some good structural properties, namely it is useful for the compact description of thesymplectic properties of the WZNW defects. We devote this section to the formulation and proof of a technical Theorem 3, which will be ofbig utility in Section 5. It gives a convenient description of the Hamiltonian LG -space equivalentto a given quasi-Hamiltonian space in the sense of the equivalence formulated in Section 2: Theorem 3 : The Hamiltonian LG -manifold ( N, ω, Φ) with equivariant proper moment mapequivalent to a quasi-Hamiltonian G -manifold ( M, Ω , µ ) is given by N = ( M ⊛ W − ) e , i.e. by thequasi-Hamiltonian fusion of M and W − followed by the unit-level quasi-Hamiltonian reduction.The corresponding LG -action on ( M ⊛ W − ) e is given by ( x, l ( σ )) → ( x, h ( σ ) l ( σ )) , x ∈ M, l ( σ ) ∈ W, h ( σ ) ∈ LG (4.57) and the corresponding L G ∗ -valued moment map Φ is given by Φ( x, l ) = − ∂ σ ll − dσ, x ∈ M, l ∈ W. (4.58) Proof.
We start by checking, that the formula (4.57) consistently defines the LG -action onthe quasi-Hamiltonian quotient ( M ⊛ W − ) e . First of all, the monodromy of the configuration h ( σ ) l ( σ ) is the same as that of l ( σ ) for every h ( σ ) ∈ LG therefore the action (4.57) survivesthe unit-level reduction constraint µµ − l = e . On the top of that, the action (4.57) obviouslycommutes with the quasi-Hamiltonian G -action (3.45) on W , it descends therefore to the G -quotient.In what follows, we find more convenient to describe the space ( M ⊛ W − ) e differently. Forthat, consider the quasi-Hamiltonian G × G action on M × W − , that is the G -action on M and the action (3.45) on W − . Now the diagonal subaction, the quotient with respect to which12e consider, permits a global slice given by the requirement l (0) = e . We shall denote by ˜ l the elements of W for which this requirement is respected, i.e. ˜ l (0) = 0 and we parametrize( M ⊛ W − ) e as ( M ⊛ W − ) e = { ( x, ˜ l ) ∈ M × W, ˜ l (0) = e, µ ( x )˜ l (2 π ) = e } . (4.59)We infer from (3.34) that, in the parametrization (4.59), the symplectic form ω on ( M ⊛ W − ) e obtained form the quasi-Hamiltonian reduction reads ω = Ω − Z π (˜ l − δ ˜ l, ∂ σ (˜ l − δ ˜ l )) dσ. (4.60)In order to verify that (4.58) gives the moment map of the LG -action (4.57), we have to char-acterize this action in the parametrization (4.59). We distinguish two cases: the action of thebased loops from Ω G and the action of the constant loops from G . We find( x, ˜ l ) → ( x, h ˜ l ) , h ∈ Ω G ; (4.61)( x, ˜ l ) → ( h ⊲ x, h ˜ lh − ) , h ∈ G. (4.62)Here h ⊲ x stands for the G -action on M .Denote by v ξ the vector field corresponding to the infinitesimal action (4.61) of an element ξ ∈ Lie(Ω G ). Then we find easily ι ( v ξ ) ω = − Z π (˜ l − ξ ˜ l, ∂ σ (˜ l − δ ˜ l )) dσ + 12 Z π (˜ l − δ ˜ l, ∂ σ (˜ l − ξ ˜ l )) dσ == − Z π ( ξ, ˜ l∂ σ (˜ l − δ ˜ l )˜ l − ) dσ = − δ Z π ( ξ, ∂ σ ˜ l ˜ l − ) dσ (4.63)We note that all boundary terms in the computation (4.63) vanished because of ξ (0) = 0.Denote by v ξ the vector field corresponding to the infinitesimal action (4.62) of an element ξ ∈ Lie( G ). Then we find easily ι ( v ξ ) ω = ι ( v ξ )Ω − Z π (˜ l − ξ ˜ l − ξ, ∂ σ (˜ l − δ ˜ l )) + 12 Z π (˜ l − δ ˜ l, ∂ σ (˜ l − ξ ˜ l − ξ )) == 12 ( ξ, δµµ − + µ − δµ ) + 12 ( ξ, ˜ l − δ ˜ l ) | π + 12 (˜ l − δ ˜ l, ˜ l − ξ ˜ l ) | π − Z π ( ξ, ˜ l∂ σ (˜ l − δ ˜ l )˜ l − ) == 12 ( ξ, δµµ − + µ − δµ ) + 12 ( ξ, δ ˜ l (2 π )˜ l (2 π ) − + ˜ l (2 π ) − δ ˜ l (2 π )) − δ Z π ( ξ, ∂ σ ˜ l ˜ l − ) dσ. (4.64)Following (4.59), the first two terms on the r.h.s. of (4.64) vanish because of the constraint µ ( x )˜ l (2 π ) = e . Combining this fact with (4.63), we conclude that Φ = − ∂ σ ˜ l ˜ l − dσ ∈ L G ∗ isindeed the moment map of the LG -action (4.57) on the symplectic manifold ( M ⊛ W − ) e .It remains to prove that the Hamiltonian LG -space (( M ⊛ W − ) e , ω, Φ) is equivalent to the quasi-Hamiltonian G -space ( M, Ω , µ ) in the sense of the equivalence discussed at the end of Section 2.13or that we shall determine the equivalent system ( M ′ , Ω ′ , µ ′ ) to (( M ⊛ W − ) e , ω, Φ) and thenshow that ( M ′ , Ω ′ , µ ′ ) and ( M, Ω , µ ) are isomorphic as the quasi-Hamiltonian spaces. Let usfirst prove that the quotient M ′ ≡ ( M ⊛ W − ) e / Ω G indeed coincides with M as manifold. Forthat, we use the following well-known parametrization of the space W of quasi-periodic mapsused in [15, 16]: l ( σ ) ≡ h ( σ ) e i τσ g − , (4.65)where h ( σ ) ∈ LG , g ∈ G and τ is the element of the Weyl alcove. It follows from (4.65), inparticular, that the elements ˜ l ( σ ) can be parametrized as l ( σ ) ≡ k ( σ ) g e i τσ g − , (4.66)where k ( σ ) is in the based loop group Ω G . The quotient M ′ = ( M ⊛ W − ) e / Ω G can be thereforeidentified with the set of elements ( x, g e i τσ g − ) ∈ M × W such that g e i2 πτ g − = µ ( x ) − andthis set, in turn, can be directly identified with M .Following Eq. (2.25), the quasi-Hamiltonian form Ω ′ on M which corresponds to the Hamilto-nian LG -space (( M ⊛ W − ) e , ω, Φ) is given by the formulaΩ ′ = ω − Z π ( δg R g − R , ∂ σ ( δg R g − R )) = Ω − Z π (˜ l − δ ˜ l, ∂ σ (˜ l − δ ˜ l )) dσ − Z π ( δg R g − R , ∂ σ ( δg R g − R )) , (4.67)where g R ∈ W , g R (0) = e is defined by g − R ∂ σ g R dσ = Φ = − ∂ σ ˜ l ˜ l − dσ. (4.68)We thus see that g R = ˜ l − and Ω ′ = Ω . (4.69)The fact that the induced G -action on M ′ is clearly given by the restriction of the action(4.62) on the first term, i.e. by the G -action on M . Finally, the moment map µ ′ is given by g R (2 π ) = ˜ l (2 π ) − = µ ( x ) which finishes the proof.Repeating step by step the proof of Theorem 3, we obtain also Corollary : The manifold ( M ⊛ W ) e is the Hamiltonian LG -space with the LG -action given by(4.57) and the anti-equivariant moment map given by Φ − = ∂ σ ll − dσ. (4.70) Remark 2 : Theorem 3 can be easily generalized to the case where the manifold M is the quasi-Hamiltonian G × G -space and we transform to the loop group language only one copy of G . Wefind then that the resulting fusion/reduction ( M ⊛ W − ) e does not give a symplectic manifoldbut it yields the quasi-Hamiltonian G -space with respect to the copy of G which we ”did nottouch”. For example, if M is the quasi-Hamiltonian double D ( G ) then a one-line computation14hows that ( D ( G ) ⊛ W − ) e = W − , or, said in other words, D ( G ) acts as identity with respectto the partial fusion.To a given Hamiltonian LG -space ( N, ω ) with the equivariant moment map Φ, one can canon-ically associate its ”loop-reversal” Hamiltonian LG -space ( N, ω ) with the anti-equivariant mo-ment map Φ − (the term anti-equivariant was defined by means of Eq. (3.31)). To do that, wedefine the loop-reversal map I : S → S as I ( σ ) = 2 π − σ, σ ∈ [0 , π [ . (4.71)Let now act the loop group LG on N as h ⊲ I y := ( I ∗ h ) ⊲ y, h ∈ LG, y ∈ N, (4.72)where ⊲ stands for the original LG -action with the equivariant moment map Φ and ⊲ I standsfor the new action defined in terms of the original one and of the pull-back I ∗ of the map I . Itis easy to see that the new action ⊲ I has the anti-equivariant moment map Φ − = − I ∗ Φ. Indeed,we have ι ( I ∗ ξ ) ω = δ Z (Φ , I ∗ ξ ) = δ Z ( − I ∗ Φ , ξ ) . (4.73)We have now the following proposition Theorem 4 : The anti-equivariant Hamiltonian LG -space ( M ⊛ W ) e is isomorphic to the loopreversal of the equivariant space ( M ⊛ W − ) e .Proof. The quasi-Hamiltonian space ( W, Ω( l ) , µ ( l )) corresponding to the chiral WZNW modelhas an interesting property that its quasi-Hamiltonian inverse ( W, − Ω( l ) , µ ( l ) − ) is isomorphicto the original space ( W, Ω( l ) , µ ( l )). This isomorphism I ∗ : W → W is simply the extension ofthe pull-back of the loop reversal map and, with a slight abuse of notation, we have denoted itagain by I ∗ : ( I ∗ l )( σ ) := l (2 π − σ ) . (4.74)To see that this is isomorphism, we just check that the G -action (3.45) commutes with I ∗ andit holds I ∗ Ω = − Ω and µ ( I ( l )) = µ ( l ) − .The existence of the isomorphism I ∗ obviously implies that the reduction/fusion ( M ⊛ W ) e isisomorphic to ( M ⊛ W − ) e as symplectic manifold, but not necessarily as LG -space. Indeed, thesymplectic form Ω + Ω( l ) on ( M ⊛ W ) e can be rewritten in the coordinates ˆ l ≡ I ∗ l on W asΩ − Ω(ˆ l ) and in the same coordinates the action (4.57) of the loop group becomes the action( x, ˆ l ( σ )) → ( x, h (2 π − σ )ˆ l ( σ )) , x ∈ M, ˆ l ( σ ) ∈ W, h ( σ ) ∈ LG. (4.75)Thus we see that the change of coordinates l → ˆ l gives the loop reversed action ⊲ I of LG on( M ⊛ W − ) e . 15 Flat connections and the proof of formula (1.2)
In this section, we wish to deal with the side AC of the triangle on Fig. 1 and to prove theformula (1.2).Let G be a compact simple connected and simply connected Lie group, G its Lie algebra and Σbe a Riemann surface with boundaries ∂ Σ. Denote by G (Σ) the group of smooth maps from Σto G . The group G (Σ) naturally acts on the space of connections on the trivial bundle Σ × G which we denote as Ω (Σ , G ): A g = gAg − − g ∗ (¯ θ ) , A ∈ Ω (Σ , G ) , g ∈ G (Σ) . (5.76)The space Ω (Σ , G ) is symplectic; its symplectic form ω is defined by ω = Z Σ ( δA ∧ , δA ) , (5.77)where δ stands for the de Rham differential on the infinite-dimensional manifold Ω (Σ , G ) and( ., . ) is the Killing-Cartan form on G . It turns out [1] that the action (5.76) is symplectic withthe moment map Ψ given by h Ψ( A ) , ξ i ≡ Z Σ ( dA + A , ξ ) + Z ∂ Σ ( A, ξ ) , (5.78)where ξ ∈ Lie( G (Σ)) ≡ Ω (Σ , G ) and d is the de Rham differential on the surface Σ. Note thatwe take the orientation on ∂ Σ opposite to the induced orientation on Σ as in [2, 24].The object of central interest for us is obtained by a partial symplectic reduction of the fullconnection space Ω (Σ , G ) by the subgroup G ∂ (Σ) of G (Σ) consisting of map sending the bound-aries to the unit element e of G . The moment map of this action is given just by the first termin (5.78) with ξ ∈ Lie( G ∂ (Σ)) ⊂ Ω (Σ , G ) . Setting the moment map to the zero value (flatconnections!) and factoring the corresponding 0-level set by the partial gauge group G ∂ (Σ) weobtain the principal actor of our game: M (Σ) ≡ Ω (Σ , G ) //G ∂ (Σ) (5.79)It was proved in [6], that in the case of non-empty boundary the moduli space M (Σ) is a smoothsymplectic manifold. Needless to say, for Σ being the annulus, M (Σ) is the phase space of thestandard WZNW model.Denote by G ( ∂ Σ) the factor group G (Σ) /G ∂ (Σ). Obviously, G ( ∂ Σ) can be identified with thegroup of smooth maps from ∂ Σ to G and it acts on the moduli space M (Σ) in the Hamiltonianway. The equivariant moment map of this residual action is given by the second term onthe r.h.s. of (5.78) where A is the restriction of the representant of the class [ A ] ∈ M (Σ) toΩ ( ∂ Σ , G ).If the boundary ∂ Σ has r +1 connected components then to each one corresponds the equivariantHamiltonian action of a copy of the loop group LG on M (Σ). The explicit description of the16anifold M (Σ) with k -handles was given in [23] as M (Σ) = (cid:26) ( a, c, ζ ) ∈ G k × G r × ( L G ∗ ) r +1 (cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 [ a i − , a i ] = Y i =1 Hol( ζ i ) (cid:27) . (5.80)where c = e and [ ., . ] stands for the group commutator. In this description, the action of h = ( h , . . . , h r ) ∈ ( LG ) r +1 is given by h ⊲ a i = Ad h (0) a i , h ⊲ c j = h (0) c j h j (0) − , h ⊲ ζ j = Ad h j ζ j − d h j h − j . (5.81)The equivariant moment map is the projection to the ( L G ∗ ) r +1 -factor.The expression for the symplectic form on M (Σ) in the parametrization (5.80) is complicatedand it was not given in [23]. We shall find now an alternative description of the space M (Σ) inwhich the structure of the symplectic form becomes transparent and it is given in terms of thequasi-Hamiltonian fusion. Theorem 5 : Let Σ be a Riemann surface with k handles and r + 1 boundaries. Then M (Σ) = (cid:18) W − ⊛ W − ⊛ . . . ⊛ W − | {z } r +1 times ⊛ D ( G ) ⊛ D ( G ) ⊛ . . . ⊛ D ( G ) | {z } k times (cid:19) e , (5.82) where D ( G ) is the internally fused quasi-Hamiltonian double of G .Proof. We shall start with the quasi-Hamiltonian G × G × ... × G | {z } r +1 times -equivalent of the Hamiltonian LG × LG × ... × LG | {z } r +1 times -space M (Σ) as obtained in [2]:Hol( M (Σ)) = D ( G ) ⊛ D ( G ) ⊛ .. ⊛ D ( G ) | {z } r times ⊛ D ( G ) ⊛ D ( G ) ⊛ . . . ⊛ D ( G ) | {z } k times . (5.83)Here D ( G ) is the standard quasi-Hamiltonian double. Using Theorem 4 and Remark 2, thequasi-Hamiltonian representation (5.82) of M (Σ) follows directly. Corollary : The moduli space of flat connections on the surface with n boundaries, m Wilsonlines insertions and k handles reads: M nmk (Σ) ≡ ( W − ⊛ . . . ⊛ W − | {z } n times ⊛ C − ⊛ C − ⊛ . . . ⊛ C − m ⊛ D ( G ) ⊛ . . . ⊛ D ( G ) | {z } k times ) e . (5.84) Proof.
Suppose that r + 1 = n + m . First of all, we convert into W − via Theorem 4 andRemark 2 only n − D ( G ) in the fusion product (5.83). Then we use the fact provedin [2], that the inclusion of the Wilson line with holonomy in a conjugacy class C i amounts tothe reduction of D ( G ) at C i and it is equal to C − i . On the remaining m factors D ( G ) we thusperform the reduction at a tuple of the conjugacy classes C = ( C , ..., C m ) to obtain the desiredformula (5.84). 17 Symplectic geometry of defects
So far we have been dealing with the side AC of the triangle on Figure 1 and we have provedthe quasi-Hamiltonian formula (1.2) expressing the symplectic structure of the moduli spaceof flat connections on the surface with n boundaries, m Wilson lines insertions and k handles.We shall now turn to the side BC of the triangle and perform the explicit evaluation of thesymplectic structures of several particular WZNW defects starting from the formula (1.2) andapplying successively the formula (2.16). In all cases, we shall find the perfect agreement withthe results obtained before from the detailed analysis of the WZNW dynamics [15, 16, 17, 27].This fact confirms that the concept of the quasi-Hamiltonian fusion is the unique structuralingredient explaining the multitude of terms in the defect symplectic forms. Before doing theactual calculations, we should comment on two things: 1) We note that the fusion productintroduced in Section 2 is commutative only on the isomorphism classes of quasi-Hamiltonianspaces. Although the isomorphism between M ⊛ M and M ⊛ M is described explicitely in[2], in practice it turns out to be more convenient to reshuffle the order of the fused manifoldsto ensure a direct comparison of the symplectic forms issued from the formula (1.2) with thesymplectic forms of the corresponding WZNW defects as obtained previously in [15, 16, 17, 27].2) In physical literature the loop group actions on symplectic manifolds related to the WZNWdynamics are often considered with the anti-equivariant moment map [15, 16, 17, 27]. Wealready know from Section 4, that this is a mere convention since the loop reversal map changesthe anti-equivariant moment map into the equivariant. In order to match the same convention,sometimes we perform the transition from equivariant to anti-equivariant at the level of theformula (1.2). As it was proved in Theorem 4, this amounts simply to replacing W − by W .1. Bulk WZNW model with no defects.
This is an important warm up case to start with. It is well-known that the phase spaceof the standard WZNW model is the moduli space of flat connections on the surface withtwo boundaries [7]. Conventionally, the action of the loop group corresponding to one ofthe boundaries is taken to be anti-equivariant and the other one equivariant. Followingour main formula (1.2), this phase space should therefore coincide with the symplecticmanifold ( W ⊛ W − ) e . Let us see that this is indeed true. Following the fusion formula(2.16) we obtain Ω W ⊛ W − = Ω( l ) − Ω( r ) −
12 ( µ − l δµ l , µ − r δµ r ) . (6.85)Note the opposite sign of Ω( r ) and the inverse of the moment map µ r related to the factthat the right sector correspond to the inverse quasi-Hamiltonian space W − in the senseexplained in Section 2. Now the quasi-Hamiltonian reduction of W ⊛ W − at the unit levelof the fused moment map µ l µ − r = e makes the last term on the r.h.s. of (6.85) disappearand we are left with Ω W ⊛ W − (cid:12)(cid:12)(cid:12)(cid:12) µ l = µ r = Ω( l ) − Ω( r ) , (6.86)where the configurations l, r ∈ W have the same monodromies. But this coincides withthe expression of the standard WZNW symplectic form ω W Z in terms of the left and rightmovers as given by (3.56) (cf. also [15]). 18.
Bulk WZNW model with one defect.
The defect in the bulk WZNW model means that the WZNW configuration field g ( σ )is allowed to jump at some point σ of the loop; we choose σ = 0. As shows theanalysis of [13], the preservation of the full LG × LG symmetry requires that the jumpof the configuration field must lie in some conjugacy class C ⊂ G . Following our generalphilosophy of relying on the formula (1.2), the WZNW symplectic form in the presence ofthe defect should be given by the unit-level reduction of the fusion product W ⊛ W − ⊛ C .Following the formulae (1.2), (2.7), (2.16) and (3.34), we find that the quasi-Hamiltonianform on W ⊛ W − ⊛ C restricted to the unit value of the product moment map µ l µ − r µ reads: Ω W ⊛ W − ⊛ C (cid:12)(cid:12)(cid:12)(cid:12) µ = µ r µ − l = Ω( l ) − Ω( r ) + α C µ r µ − l + 12 ( µ − l δµ l , δµ − r µ r ) . (6.87)Here we recall that α C µ r µ − l is the quasi-Hamiltonian form (2.7),(2.9). Our expression (6.87)coincides with Eq. (106) of [27], where the symplectic structure of the WZNW model withone defect was described ( µ l,r ≡ γ − L,R , α C = − ω in the notation of [27]).3. Bulk WZNW model with two defects.
Again from the formulae (1.2), 2.7), (2.16) and (3.34), we find the quasi-Hamiltonian formon the fusion product W ⊛ W − ⊛ C ⊛ C restricted to the unit value of the product momentmap µ l µ − r µ µ :Ω W ⊛ W − ⊛ C ⊛ C (cid:12)(cid:12)(cid:12)(cid:12) µ r µ − l = µ µ = Ω( l ) − Ω( r )+ α C µ + α C µ + 12 ( µ − l δµ l , δµ − r µ r )+ 12 ( µ − δµ , δµ µ − ) . (6.88)This expression is equivalent to Eq. (121) of [27], where the symplectic structure of theWZNW model with two defects was derived. To see this, some more work is needed. Firstof all, we have to identify the notations here and in [27]: µ l,r ≡ γ − L,R , µ ≡ ˜ d β and µ ≡ d α .Then we have to reexpress Ω( l ) in terms of the parametrization (4.65) of l ∈ W : l ( σ ) ≡ h ( σ ) e i τσ g − , (6.89)where h ( σ ) is strictly periodic (therefore it is an element of LG ), τ is in the Weyl alcoveof G and g is in G . With this parametrization, we obtainΩ( l ) = 12 Z π (cid:20) ( h − δh, ∂ σ ( h − δh )) − δ ( τ, h − δh ) (cid:21) dσ ++ 2 π i( δτ, g − δg ) + 12 ( g − δg , e π i τ g − δg e − π i τ ) . (6.90)Inserting (6.90) into (6.88), we obtain the formula which coincides with Eq. (121) of [27].4. Boundary WZNW model with open string ending on the conjugacy classes C and C . It was found in [17], that the symplectic structure of the boundary WZNW model withopen string ending on two conjugacy classes C and C is the same as that of the moduli19pace of flat connections on the disc with two Wilson lines insertions with the holonomiesin C and C . Following our quasi-Hamiltonian dictionnary, we shall evaluate the unit-levelreduction of the fusion product C ⊛ C − ⊛ W . Thus, assembling the quasi-Hamiltonianform on the conjugacy classes (2.7), the expression of the quasi-Hamiltonian form issuedfrom the fusion (2.16) and the chiral WZNW form (3.34), we find the following formulafor the quasi-Hamiltonian form on the fusion product C ⊛ C − ⊛ W restricted to the unitvalue of the product moment map µ µ − µ l :Ω C ⊛ C − ⊛ W (cid:12)(cid:12)(cid:12)(cid:12) µ − l µ = µ = Ω( l ) + 12 ( µ δµ − , δµ l µ − l ) − α C µ + α C µ . (6.91)This expression coincides with Eq. (53) of [17], where the symplectic structure of theboundary WZNW model was first determined (to see it, one must identify µ − l ≡ γ , µ ≡ h and µ ≡ γh ).5. Boundary WZNW model with one defect.
From the formulae (1.2), (2.7), (2.16) and (3.34), we find the quasi-Hamiltonian form onthe fusion product C − ⊛ C ⊛ W ⊛ C restricted to the unit value of the product momentmap µ − µ µ l µ :Ω C − ⊛ C ⊛ W ⊛ C (cid:12)(cid:12)(cid:12)(cid:12) µ = µ µ l µ = Ω( l ) + 12 ( µ δµ − , δµ µ − ) + 12 ( µ − l δµ l , δµ µ − ) − α C µ + α C µ + α C µ . (6.92)In order that this expression coincide with Eq. (139) of [27], where the symplectic structureof the boundary WZNW model with one defect was computed, we must insert (6.90) into(6.92) and identify µ ≡ h , µ ≡ d α , µ ≡ h π and µ l ≡ γ − . References [1] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans.Roy. Soc. London Ser. A, (1982) 523-615[2] A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. DifferentialGeom. (1998) 445-495, [arXiv: dg-ga/9707021][3] A. Alekseev and V. Schomerus, D-branes in the WZW model, Phys. Rev. D (1999)061901, [arXiv:hep-th/9812193][4] C. Bachas and M. Gaberdiel, Loop operators and the Kondo problem, JHEP (2004)065, [arXiv:hep-th/0411067][5] M. F. Chu, P. Goddard, I. Halliday, D. I. Olive and A. Schwimmer, Quantization of theWess-Zumino-Witten model on a circle, Phys. Lett. B (1991) 71[6] S. Donaldson, Boundary value problems for Yang-Mills fields, J. Geom. Phys. (1992)89-122 207] S. Elitzur, G. W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quan-tization of the Chern-Simons-Witten theory, Nucl. Phys. B (1989) 108[8] S. Elitzur and G. Sarkissian, D-branes on a gauged WZW model, Nucl. Phys. B (2002)166, [arXiv:hep-th/0108142][9] F. Falceto and K. Gaw¸edzki, Lattice Wess-Zumino-Witten model and quantum groups, J.Geom. Phys. (1993) 251, [arXiv:hep-th/9209076][10] J. M. Figueroa-O’Farrill and S. Stanciu, D-branes in AdS(3) x S(3) x S(3) x S(1), JHEP (2000) 005, [arXiv:hep-th/0001199][11] J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. I: Partitionfunctions, Nucl. Phys. B (2002) 353, [arXiv:hep-th/0204148][12] J. Fuchs, I. Runkel and C. Schweigert, Boundaries, defects and Frobenius algebras, Fortsch.Phys. (2003) 850 [Annales Henri Poincare (2003) S175], [arXiv:hep-th/0302200][13] J. Fuchs, C. Schweigert and K. Waldorf, Bi-branes: Target space geometry for world sheettopological defects, J. Geom. Phys. (2008) 576, [arXiv:hep-th/0703145][14] M. R. Gaberdiel and S. Schafer-Nameki, D-branes in an asymmetric orbifold, Nucl. Phys.B (2003) 177, [arXiv:hep-th/0210137][15] K. Gaw¸edzki, Classical origin of quantum group symmetries in Wess-Zumino-Witten con-formal field theory, Commun. Math. Phys. (1991) 201, [ arXiv:hep-th/0103118][16] K. Gaw¸edzki, Boundary WZW, G/H, G/G and CS theories, Annales Henri Poincar´e (2002) 847-881, [arXiv:hep-th/0108044][17] K. Gaw¸edzki, I. Todorov and P. Tran-Ngoc-Bich, Canonical quantization of theboundary Wess-Zumino-Witten model, Commun. Math. Phys. (2004) 217,[arXiv:hep-th/0101170][18] A. Gorsky and N. Nekrasov, Relativistic Calogero-Moser model as gauged WZW theory,Nucl. Phys. B (1995) 582-608, [arXiv:hep-th/9401017][19] K. Graham and G. M. T. Watts, Defect lines and boundary flows, JHEP (2004) 019,[arXiv:hep-th/0306167][20] M. Kato and T. Okada, D-branes on group manifolds, Nucl. Phys. B499 (1997) 583-595,[arXiv: hep-th/9612148][21] C. Klimˇc´ık, Quasitriangular WZW model, Reviews Math.Phys. (2004) 679-808,[arXiv:hep-th/0103118][22] C. Klimˇc´ık and P. ˇSevera, Open strings and D-branes in WZNW models, Nucl. Phys. B (1997) 653, [arXiv:hep-th/9609112] 2123] E. Meinrenken and C. Woodward, Moduli spaces of flat connections on 2-manifolds, cobor-dism, and Witten’s volume formulas , Advances in Geometry Progress in Mathematics (1999) 271-295, [arXiv:dg-ga/9707018][24] E. Meinrenken and C. Woodward, Cobordism for Hamiltonian loop group actions and flatconnections on the punctured two-sphere, Mathematische Zeitschrift (1999) 133-168,[arXiv:dg-ga/9707019][25] V. B. Petkova and J. B. Zuber, Generalised twisted partition functions, Phys. Lett. B (2001) 157, [arXiv:hep-th/0011021][26] A. Recknagel, Permutation branes, JHEP (2003) 041, [arXiv:hep-th/0208119][27] G. Sarkissian, Canonical quantization of the WZW model with defects and Chern-Simonstheory, Int. J. Mod. Phys. A (2010) 1367, [arXiv:0907.3395][28] G. Sarkissian and M. Zamaklar, Symmetry breaking, permutation D-branes on groupmanifolds: Boundary states and geometric description, Nucl. Phys. B (2004) 66,[arXiv:hep-th/0312215][29] M. A. Semenov-Tian-Shansky, Poisson-Lie groups, the quantum duality principle and thetwisted quantum double, Teoret. Mat. Fiz. (1992) 1292-1307, [arXiv:hep-th/9304042][30] E. Witten, Nonabelian bosonization in two dimensions, Commun. Math. Phys.92