Gauge theory and boundary integrability. Part II. Elliptic and trigonometric cases

We consider the mixed topological-holomorphic Chern-Simons theory introduced by Costello, Yamazaki & Witten on a ℤ2 orbifold. We use this to construct semi- classical solutions of the boundary Yang-Baxter equation in the elliptic and trigonometric cases. A novel feature of the trigonometric case is that the ℤ2 action lifts to the gauge bundle in a z-dependent way. We construct several examples of K -matrices, and check that they agree with cases appearing in the literature.


Introduction
In [1] rational solutions of the boundary Yang-Baxter equation were generated as the vacuum expectation values of Wilson lines in a mixed topological-holomorphic analogue of Chern-Simons theory on a Z 2 -orbifold. This extended the link between gauge theory and quantum integrability developed by Costello, Witten, and Yamazaki in the papers [2][3][4][5] to integrable models with boundary. The mixed topological-holomorphic analogue of Chern-Simons theory underpinning this link was first proposed by Nekrasov in his Ph.D. thesis [6], see also the related papers [7,8]. In this paper we expand this construction to elliptic and trigonometric solutions of the boundary Yang-Baxter equation. We begin by reviewing the CWY approach to 2d quantum integrable lattice models. We shall be brief, and refer the reader to [2,4] for more detailed discussions of the theory -1 -JHEP06(2020)080

CWY theory
CWY theory is defined on a four-manifold M = Σ × C, where Σ is some 2-dimensional real manifold and C is a Riemann surface admitting a closed, holomorphic 1-form ω. The condition that ω be closed, holomorphic, and nowhere vanishing on C is very restrictive, forcing (C, ω) to be biholomorphic to (C, dz), (C * , du/u = dz), or (E τ , dz). These three choices generate rational, trigonometric, and elliptic quasi-classical solutions to the YBE, respectively. In this paper will concentrate on the trigonometric and elliptic cases. We will often use real coordinates r = (x, y) on Σ, and complex coordinates (z,z) on C. We denote these coordinates collectively by w ∈ M .
The action of CWY theory is where is a partial connection on a G-bundle over M . Note that A is only a partial connection since it has no A z component. The theory is topological in Σ and holomorphic in C, i.e. it is invariant under diffeomorphisms of M which are independent of C. We can only study this theory perturbatively, for example because the periods of ω/ are not naturally quantized, The simplest observables in CWY theory are Wilson lines supported on a straight line γ ⊂ Σ and at point z 0 ∈ C. The vacuum expectation value of the following configuration of crossing Wilson lines IR freedom of the theory after gauge fixing together with diffeomorphism invariance in Σ ensures that the R-matrix is local to the point of crossing, so that vevs of configurations in which several Wilson lines cross reduce to the product of R-matrices. The R-matrices automatically obey the Yang-Baxter equation

The boundary Yang-Baxter equation
In this paper we will be concerned with integrable lattice models with boundaries, and with boundary conditions preserving their bulk integrability. Investigations of such boundary conditions date back to work of Skylanin [11] and Olshanski [12,13], and have been extensively studied since. Boundary conditions on a spin chain are encoded in a K-matrix which again depends meromorphically on the spectral parameter z. Here we interpret V and V as the state spaces of an incoming and outgoing excitation reflecting off the boundary, and W as a space of boundary states. The boundary conditions preserve integrability if the K-matrix obeys the boundary Yang-Baxter equation R 12 (z 1 − z 2 )K 13 (z 1 )R 21 (z 1 + z 2 )K 23 (z 2 ) = K 23 (z 2 )R 12 (z 1 + z 2 )K 13 (z 1 )R 21 (z 1 − z 2 ) , (1. 3) which may be viewed pictorially as Note that when a line reflects of the boundary its spectral parameter changes sign. We will be primarily concerned with the case V i ∼ = V i . In [1] we showed that solutions of the boundary Yang-Baxter equation may be obtained by placing CWY theory on the orbifold M = M/Z 2 = (Σ × C)/Z 2 . In defining this -4 -JHEP06(2020)080 orbifold, we assume that the 2-dimensional smooth manifold Σ is the double of some manifold with boundary Σ. This means that Σ can be formed by gluing together two copies of Σ along their boundary. Σ hence admits a natural reflection that swaps the two copies of Σ whilst fixing their common boundary. 2 For convenience we choose our coordinates (x, y) on Σ so that the reflection acts as (x, y) → (−x, y). Then the generator Z of the Z 2 symmetry on M is Z : (x, y, z,z) → (−x, y, −z, −z) .
Note that the fixed points of this action, and hence the singular points on the orbifold, are lines in Σ fixed by Z supported at fixed points of z → −z on C. For C = C there is one such fixed point, for C = C * there are two fixed points, and for C = E τ there are four fixed points.
To lift the action of Z to the adjoint bundle of the gauge theory, in [1] we simultaneously acted of the fibres of this bundle with an automorphism σ ∈ Aut(g). The condition that this defines an action of Z 2 restricts the automorphism to be involutive. In full the action of Z on the gauge field is To do gauge theory on the orbifold we only integrate over field configurations on Σ × C which are fixed by this Z 2 action. We use the action S[A]/2, which is a consistent as long as This holds if σ preserves the g-invariant bilinear used in defining the action. In section 3.4 we will generalise this picture somewhat. The condition A = σ(Z * A) in particular implies that the gauge field along the orbifold fixed lines takes values in a subalgebra p ⊂ g that is fixed by σ. Thus, along such lines we can introduce a new family of line operators that (in the simplest case 3 ) live in representations of this subalgebra. We refer to these operators as 'boundary Wilson lines'. Observables in the theory are configurations of bulk Wilson lines in M , together with boundary Wilson lines inserted along the lines of singular points. This configuration of bulk and boundary Wilson lines must be chosen so that no operators coincide in M .
CWY theory on an orbifold generates K-matrices via the vacuum expectation values of the following configurations of Wilson lines: To avoid a global anomaly we also require that Σ be framed in the sense that Σ admits a nowhere vanishing vector field that is equivariant under this reflection. Such Σ include the half-plane, half-cylinder, and annulus. 3 In the rational case, we showed in [1] that boundary Wilson lines really live in representations of the twisted Yangian B(p, g).

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The picture on the left represents the Wilson line configuration on M , the double cover of the orbifold. This picture is often useful for calculation. The second picture is the more familiar depiction of the K-matrix as describing the reflection of an excitation off a boundary. To obtain it, we have used the Z 2 action to write all the observables in the region 4 M L = Σ x≤0 × C. Note, however, that the choice M L is artificial and is not determined canonically by the orbifold structure. In each case, {z * , W * } indicates the presence of a set of boundary Wilson lines, one at each singular point in C, in representations W * of p.
(Thus {z * , W * } indicates one, two or four such boundary lines in the rational, trigonometric and elliptic cases, respectively.) Since we work perturbatively, as for the bulk R-matrix, the K-matrices we generate inevitably admit an expansion in . However, unlike the quasi-classical R-matrices, the leading ( = 0) term will not typically be the identity. This is because, in order to obtain the more familiar (second) picture, we have acted on V with the bundle automorphism σ. In particular, suppose σ is an inner automorphism so that V σ ∼ = V . Then, with a single boundary Wilson line, the usual K-matrix would be viewed as a map with the two copies of V canonically identified. However, in our case the identification is non-trivial: inner automorphisms correspond to conjugation by some τ (for σ to be an involution we require τ 2 ∈ Z(G)). Then V σ identified with τ V V where τ V is τ acting in the V representation. Thus, the vacuum expectation of the configuration above computes K(z) τ −1 V rather than the usual K-matrix itself. In other words, the semi-classical K-matrix generated by CWY theory on an orbifold takes the form where Feyn(z − z * ) is the sum of all Feynman diagrams contributing to the expectation value. As in the bulk, these K-matrices inevitably obey the boundary Yang-Baxter equation (1.3).
The semi-classical contributions to the vacuum expectation of the above diagrams are again easy to evaluate in holomorphic gauge. We represent the configuration of Wilson lines on the orbifold via the picture on the left above. From the method of images, a propagator which respects the Z 2 orbifold action is given by where r(z) is the bulk classical r-matrix and σ 2 denotes our automorphism acting on the second factor. In the presence of n boundary Wilson lines there are 1 + n diagrams which contribute. They are the bulk self-interaction and the n bulk-boundary interactions. 5 4 If we so wish, we can then view the theory as being defined on the manifold with boundary ML, albeit with non-local boundary conditions imposed on A at x = 0. 5 We neglect self-interactions on the bulk line which do not arise as a result of the orbifold structure and simply alter its normalization. Similarly we neglect self-interactions of the boundary lines.

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The bulk self-interaction arises from the contribution to the following diagram of the image part of the propagator In terms of the basis {t a } dim g a=1 of g, the contribution of this diagram is  where the linear map gl : U (g) ⊗ U (g) → U (g) acts on X ⊗ Y by gluing the two factors of the tensor product together in order to give XY . (Note that gl is not an isomorphism of associative algebras.) The remaining diagrams describe bulk-boundary interactions, and there as many of these diagrams as there are boundary Wilson lines. They are all represented by the diagram below.
z, V {z * , W * } Inserting the boundary Wilson lines at singular points z * ensures that, when coupled to by a propagator, both the bulk and mirror part of the propagator contribute identically. This means that we can effectively replace the propagator by 2 δ (2) Σ (r − r ) r(z − z ) so that the contribution of the above diagram is just 2r(z − z * )| V ⊗W * . It is then clear that full semi-classical limit of the K-matrix is given by where σ = conj τ . (This formula also applies for σ outer, but we must work a little harder to interpret it correctly.) We will sometimes write (z)τ V for the semi-classical K-matrix k(z), so that with 0 (z) the order self-interaction term and δ (z) the corresponding bulk-boundary interaction.

Elliptic solutions
In this section we will discuss how to generate elliptic K-matrices using gauge theory. We start by reviewing the method for generating elliptic R-matrices described in [4].

Elliptic solutions of the Yang-Baxter equation
The gauge invariant data determining a classical solution of CWY theory is a stable holomorphic G-bundle over C. For the vacuum to be isolated, the tangent space to the moduli space of such bundles at the vacuum must be trivial. When C = E τ , the topological class of such a bundle is determined by an element of π 1 (G). The stability condition amounts to requiring that the Lie algebra of the automorphism group of the holomorphic G-bundle be trivial, or equivalently that the automorphism group be discrete. This constrains the gauge group to be G = PSL n (C) ∼ = π 0 (Aut(sl n C)) for n ≥ 2, and the topological class of the PSL n (C)-bundle to be a generator ζ ∈ Z n ∼ = π 1 (PSL n (C)).
To construct the bundle explicitly we introduce the matrices A, B ∈ SL n C obeying where ε = exp(2πi/n) is an n th root of unity. 6 Up to conjugation and scalar multiplication, we can without loss of generality take the components of A and B to be Viewed as elements of PSL n (C), A and B commute, and hence can be used to define a flat G-bundle over E τ with monodromies A ζ and B around the two cycles. Forgetting the ∂ z + A z component of the covariant derivative, we get a holomorphic G-bundle over Σ whose automorphism group is Z 2 n . This is the vacuum we will expand around. As a holomorphic bundle over E τ the adjoint bundle of the gauge theory is given by The L i,j are holomorphic line bundles over the elliptic curve with vanishing Chern class and corresponding to the points (i + jτ )/n in the Jacobian variety. The set {t i,j } (i,j)∈In form a basis of sl n C. This basis obeys and so diagonalizes the conjugacy action of A ζ and B. Now we've fixed the vacuum we can construct the propagator and determine the nextto-leading order contribution to the quasi-classical R-matrix. Recall that in holomorphic gauge the relevant component of the propagator is P xy (w, w ) = r(z − z ) δ (2) (r − r ) 6 What we're really doing here is choosing a conjugacy class in the set of pairs of commuting elements in PSLn(C). This is precisely the data required to define a flat PSLn(C) bundle over an elliptic curve. where r(z) is the elliptic classical r-matrix taking the form Here the w i,j (z) are given by The w i,j (z) obey the quasi-periodicity conditions and so are meromorphic sections of L i,j . They are the unique such sections with a simple pole with residue 1 at the origin, as befits the propagator.

Elliptic solutions of the boundary Yang-Baxter equation
Now consider CWY theory on the orbifold M = ( In lifting this Z 2 action to the adjoint bundle we have the freedom to simultaneously act with the involutive bundle morphism σ : Ad P → Ad P preserving the vacuum gauge field A = 0. It is straightforward to determine that σ acts on the fibres as a constant involutive automorphism of sl n C which inverts the monodromies around the two cycles This then fixes σ in terms of two parameters ξ, η ∈ Z n to be A derivation of this is included in appendix C. Now let's consider the behaviour of the gauge field at orbifold singularities. These are the four lines The quotient E τ /{z ∼ −z} has four conical singularities corresponding to the four fixed points, and embeds in 3-dimensional space as a 'pillowcase'. Pulling back the gauge field to these lines we learn that 7 7 Note that the ambiguity in the choice of map z * → −z * can be absorbed into a different choice of σ.
This shows that the subalgebra of sl n C in which A y takes values depends on which of the four singular lines we pull back to. Since we have the effect of σ * is to shift the non-trivial monodromies as The positive eigenspace of this automorphism, which we shall denote by p * depends on n, ξ * , and η * . We then have the following possibilities for p * : • If n is odd then dim p * = (n 2 − 1)/2 and p * ∼ = sl (n+1)/2 C ⊕ sl (n−1)/2 C ⊕ C.
We can see that for n odd the subalgebras on the singular lines are all isomorphic, but for n even the subalgebra on one of the four lines is not isomorphic to the other three. Note that the σ * are all inner automorphisms, and in particular their positive eigenspaces always contain an abelian summand. By inserting Wilson lines in 1-dimensional representations of these abelian summands along the associated singular lines we will be able to generate continuous families of K-matrices. Let's do this explicitly now.

Construction of elliptic solutions
Given the bulk propagator in holomorphic gauge, or equivalently the classical elliptic rmatrix, we can directly apply equation (1.4) to determine the next-to-leading order contribution to the K-matrix in . The term arising from self-interactions of the bulk Wilson line on the orbifold is determined by the representation of the bulk Wilson line, and is given by Here, and in the sequel, the primed sum indicates that we remove the term proportional to the identity. If we choose the bulk Wilson line to be the fundamental representation 8 of sl n C then we find that t 2 i,j = ε −ζ −1 ij t 2i,2j , so the self-interaction contribution becomes where we have used the action of σ on t i,j given above. 8 To avoid an anomaly on the bulk line, we must choose it to be in a representation of g which lifts to a representation of the associated elliptic quantum group.

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To determine the contribution of bulk-boundary interactions we first need to identify the generators of the abelian summand in each of the h * . We will consider the cases where n is odd/even separately.
The simplest case to deal with is n odd, as in this case the four algebras p * are isomorphic. In terms of n, ξ * , and η * the generator of C in p * is where by 2 −1 we mean the inverse of 2 modulo n, which exists since n is odd. Our representation of p * is given explicitly by Q * →q * ∈ C. The contribution of the bulkboundary interactions is then To combine this with the self-interaction contribution, it is helpful to first make the change of variables in the sum in equation (2.3). Subsequently applying the identity allows us to rewrite the self-interaction term as Thus the full O( ) contribution to K(z)τ −1 is We delay a discussion of this solution until after we've dealt with the case where n is even.
The case n = 2m (n even) is less straightforward as the four Lie subalgebras p * are no longer all isomorphic. The abelian generators in p * are easily expressed using the formula where again the primed sum indicates that we remove the term proportional to the identity. This appears only if ξ * ≡ η * ≡ 0 (mod 2). Choosing 1-dimensional representations Q * →q * for the four boundary Wilson lines, bulk-boundary interactions generate a contribution where the p * are given by Combining this with the self-interaction term shows that the full next-to-leading order contribution to K(z)τ −1 is in the even case, where now plays the role of the 'charges' of the boundary lines. Now that we've found the semi-classical contributions (z) for both even and odd n, let's discuss them. The formulas we have obtained agree with the semi-classical limits of the four-parameter elliptic K-matrices appearing in [14], as we demonstrate in appendix A. The four continuous parameters of these solutions can be interpreted as the effective charges of each of the four boundary Wilson lines. (We will make a similar observation in the case of trigonometric K-matrices.) From the point of view of gauge theory, a surprising feature of the K-matrices appearing in [14] is that they are independent of the parameter playing the role of in the bulk R-matrix. From the gauge theory perspective, this arises as follows. The next-to-leading order bulk Wilson line self-interaction term is inevitably proportional to and contains no free parameters. However, we can cancel this contribution with an appropriate choice of the boundary chargesq * . Having achieved this cancellation, the remaining (z) is proportional to the boundary charges, which may now be rescaled to absorb . This ensures that the next-to-leading order contribution to the K-matrix can always be rescaled so as to remove . We anticipate that in the full perturbative expansion of the K-matrix, may similarly be absorbed by the same shifts and rescalings of the boundary charges. 9 In this sense the K-matrix would be 'independent' of .
It would be disappointing if the described independence was a general property of elliptic K-matrices that gauge theory obscures. Fortunately however, it appears to be special to the particular choice of (fundamental) representation we made for the bulk Wilson line. For generic representations we find that self-interactions of the bulk Wilson lines cannot be compensated by any choice of boundary parameters. For example, we claim that if we choose the bulk Wilson line to be in the adjoint representation, 10 then the K-matrix does depend on . 9 The normalization of the K-matrix is fixed by the condition that the 'Sklyanin determinant' be 1.
This constraint depends on the R-matrix, and so generically we would expect the normalization to be dependent. 10 For this example to be valid, we need the adjoint representation of slnC to lift to a representation of the elliptic quantum group. We believe that [15] demonstrates that the adjoint of slnC does indeed lift.

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The orbifold singularities z * should have significance beyond simply providing four free parameters for the K-matrix. To see this, recall that the category of representations of the (elliptic) quantum group has the property that , so the tensor product of representations becomes non-commutative. Nonetheless for generic values of the spectral parameters, the bulk R-matrix provides an non-trivial intertwiner that fails when z 1 = z 2 where the R-matrix becomes singular. The R-matrix thus provides us with a braided tensor category and, according to the general results of [16], any such category is equivalent to a category of representations of a Hopf algebra. Thus knowing the collection of R-matrices is really equivalent to knowing the elliptic quantum group itself. For integrable systems in the presence of a boundary, the charges of the bulk quantum group will be broken to a (left) coideal subalgebra. In the rational case, this is the twisted Yangian B(p, g) ⊂ Y(g), but the appropriate algebraic structure in the elliptic case appears not to have been identified. The orbifold perspective strongly suggests that representations W of this coideal subalgebra should have the property that the K-matrix provides an isomorphism between the tensor product of a representation V (z) of the elliptic quantum group with W , and the tensor product where V (z) is replaced by the representation V σ (−z) obtained by action of the orbifold Z 2 on V (z). Again, this isomorphism will fail when z coincides with any of the four singular lines z * .

Trigonometric
In this section we will discuss how to generate trigonometric K-matrices using gauge theory. Thus, in this section, we choose C to be the cylinder C = R×S 1 = C/{z ∼ z +2πi}. Again, we start by reviewing the method for generating trigonometric R-matrices described in [4].

Trigonometric solutions of the Yang-Baxter equation
To generate trigonometric solutions of the YBE from gauge theory, one must first find an isolated vacuum around which to perturb. This vacuum depends on a choice of boundary conditions as Re z → ±∞ in C. To retain the full topological invariance of the theory in Σ, these C boundary conditions are independent of Σ. The naïve boundary condition A → 0 as Re z → ±∞ turns out to be too stringent, and instead Costello et al. [4] require that the gauge field takes values in certain subspaces g ± ⊂ g as x → ±∞ respectively. For the boundary conditions to be consistent with gauge transformations, g ± must in fact be Lie subalgebras of g and infinitesimal gauge transformations are also restricted to lie in these subalgebras as z → ±∞. Furthermore, g ± should be isotropic with respect to the g-invariant bilinear used to define the action, so as to eliminate a possible boundary term when varying S[A]. To impose the least restrictive condition possible, [4] choose g ± each to be middle dimensional. Finally, to ensure the classical equations of motion to not admit -13 -JHEP06(2020)080 deformations so that we are perturbing around an isolated minimum, one requires that g + ∩ g − = ∅. Together these conditions imply that (g, g − , g + ) form a Manin triple. The nondegenerate invariant bilinear on g provides an identification g − ∼ = g * + . We can use this identification to extend a basis {t a } dim g + a=1 of g + to a basis of all of g by writing {t a } dim g + a=1 for the dual basis viewed as a basis of g − . Using this basis it's straightforward to write down the inverse of the invariant bilinear on g where c ±,∓ ∈ g ± ⊗ g ∓ . Given such a Manin triple, CWY theory generates as the classical r-matrix [4,9]. A generic simple Lie algebra cannot be given the structure of a Manin triple, since if g = g + ⊕ g − as a vector space and dim g + = dim g − , then g itself must certainly be even dimensional. However, it is always possible to construct a Manin triple starting with a given simple Lie algebra g 0 by forming the direct sum of Lie algebras where h is a second copy of the Cartan subalgebra of g 0 . The Lie algebra g is not simple since it has a non-trivial centre h. The g-invariant bilinear used in defining the action of CWY theory is where , g 0 denotes a symmetric invariant bilinear on g 0 proportional to the Killing form, and , h denotes its restriction to the Cartan. To determine one of the Lagrangian subalgebras, say g + , one picks a decomposition g 0 = n − ⊕ h ⊕ n + , where h is a choice of Cartan and n ± are the subalgebras of positive and negative root spaces for a given base. Then This definition makes a choice of identification between the Cartan h of g 0 and the centre h of g. The other Lagrangian subalgebra g − is similarly defined as where M allows for a different identification between h and h in g − . The linear map M must be orthogonal to ensure that g − is isotropic, and +1 must not be an eigenvalue of M if g + and g − are to be disjoint.
To write the r-matrix in the basis adapted to this Manin triple, we let {e µ , f µ } µ∈Φ + ∪ {h µ } µ∈∆ be the standard Chevalley basis of g 0 with respect to our choice of base ∆. Here Φ + denotes the set of positive roots for this base. By adjoining { h µ } µ∈∆ we can extend this -14 -JHEP06(2020)080 to a basis of g. We write κ for the restriction of our g 0 -invariant bilinear to h, and use it to raise and lower indices in both copies of the Cartan. Finally we define the antisymmetric bivector A by is the orthogonal transformation introduced above. Then the classical r-matrix becomes in this basis. This agrees with a family of trigonometric classical r-matrices appearing in [9]

Trigonometric solutions of the boundary Yang-Baxter equation
We now consider the theory on the orbifold M = (Σ × R × S 1 )/Z 2 . As usual we take the This map swaps Re z = +∞ and Re z = −∞. Above, we saw that in the trigonometric case, different boundary conditions are imposed on A in these two limits, with A lying in g ± as Re z → ±∞, respectively. To handle this situation we must generalise our orbifold construction so that the lift of the Z 2 action to the adjoint bundle also exchanges the different boundary conditions. The simplest way to achieve this is to require that the involutive automorphism σ : g → g exchanges the subalgebras g − and g + , i.e. we require σ(g + ) = g − . In terms of the basis {t a } dim g + a=1 of g + and the associated dual basis {t a } dim g + a=1 of g − , the +1 eigenspace of σ is spanned by {u a } dim g + a=1 where u a = 1 2 (t a + σ(t a )) .
It is then straightforward to write down the semi-classical contribution to the K-matrix using equation (1.4). It is given by

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where the three factors in the tensor product are representations of the bulk and two boundary Wilson lines respectively, and where Note that for σ to be a symmetry of the theory it is essential that σ preserves our chosen invariant bilinear on g, which we have used in the above. Since g is not semisimple this is not an immediate consequence of the fact that σ is an automorphism. The classification of involutive automorphisms swapping g + and g − for a Manin triple (g, g + , g − ) follows from standard Lie theory and is given in appendix D. The outcome is that such σ exist only for certain choices of M , and are completely determined by a choice of involutive automorphism of the simple Lie algebra g 0 . We denote this involution by χ. The data we use in defining χ is an involutive automorphism γ of the Dynkin diagram of g 0 , together with an element λ ∈ h which is invariant under then natural action of γ on the Cartan. Explicitly is the Chevalley involution, and where Γ extends the action of γ to all of g 0 by Γ : (e µ , f µ , h µ ) → (e γ(µ) , f γ(µ) , h γ(µ) ) for µ ∈ ∆ .
We get an associated solution of the boundary Yang-Baxter equation whenever where by χ here we mean its restriction to the Cartan. The action of σ on h is given by M • χ. In particular, if γ is the identity, (3.2) implies that M itself is involutive. In this case, since M has no +1 eigenvalue, M is fixed to be M = −1.
We now construct explicit examples of such automorphisms and their associated Kmatrices for some low dimensional g.

Examples of trigonometric solutions
We begin by considering g = sl 2 C ⊕ C with basis {e, f, h, h}. The Dynkin diagram of sl 2 C admits only the trivial diagram automorphism γ = id, so χ = exp(ad µh ) • ω for some µ ∈ C. This involutive automorphism can be realised explicitly as conjugation by

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where Λ = e 2µ . As γ is trivial we must have M = −1. Indeed, this is the only orthogonal transformation of a 1-dimensional space with no +1 eigenvalue. Thus σ acts on h trivially. The positive eigenspace of σ is the ableian Lie algebra We may insert boundary Wilson lines at u = e z = ±1 in representations of this algebra.
To define a K-matrix we need to make appropriate choices of representations for the bulk and boundary Wilson lines. We will take the bulk Wilson line to be in the tensor product of the fundamental of sl 2 C with the charge s representation of h ∼ = C. Bulk R-matrices associated to Wilson lines with non-vanishing charges for h correspond to a generalization of the 6-vertex model with non-vanishing horizontal and vertical fields [4]. Such representations were referred to as inadmissible in [5].
We will choose the boundary Wilson lines at u = ±1 to be in 1-dimensional representations of the abelian Lie algebra (3.4) given by (Λ −1/2 e + Λ 1/2 f )/ √ 2 → q ± and h/ √ 2 → r ± . The next-to-leading contribution to the K-matrix is then f .
The first three terms here are proportional to the identity, and so can be absorbed into the overall normalization of the K-matrix. The terms that cannot be removed in this way are The full K-matrix can then be expressed as where the dressing function and where (q, ξ, r) ∈ C are defined by q + = q sinh(ξ/2), q − = q cosh(ξ/2) and r = r + + r − . Hence -17 -

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To compare this with K-matrices appearing in the literature, it is convenient to first symmetrize the R-matrix, conjugating it by 0 e z/4 e −z/4 0 ∈ P SL 2 C as mentioned in [4]. After performing the corresponding transformation of the K-matrix, we arrive at This matches the expression appearing in [17]. As a second example, let's also consider g = sl 3 C ⊕ C 2 . For the sake of simplicity we restrict our attention to involutive automorphisms for which χ is inner. Since the Chevalley involution of sl 3 C is in fact an outer automorphism, we must choose γ to be the non-trivial outer automorphism of sl 3 C. This simply swaps the two simple roots, i.e. γ = (1 2). Then χ = ad λ • Γ • ω as above. In this case the most general γ-invariant choice of λ is µ(h 1 + h 2 ), for some µ ∈ C. Acting on the Chevalley generators where Λ = e µ . The action of χ on the remaining generators is determined by the fact it is involutive. Explicitly, χ can be realised as conjugation by the matrix Since γ is no longer trivial there exist non-trivial M ∈ O( h) for which In this example, this constraint fixes M to be a non-trivial complex rotation. This includes the possibility that M = −1. For the sake of simplicity we will restrict our attention to this standard case. Under this assumption, σ acts trivially on h, and its positive eigenspace is the direct sum p = p 0 ⊕ h, where Here γ = α (1) + α (2) for α (1) and α (2) the simple roots of sl 3 C, so e γ = [e 1 , e 2 ] and f γ = [f 2 , f 1 ]. p 0 is a subalgebra of sl 3 C isomorphic to sl 2 C ⊕ C.
is the generator of the abelian summand of this subalgebra.

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We are now in a position to compute the leading order contribution to the corresponding K-matrix. We will take the bulk Wilson line to be in the fundamental representation of sl 3 C with no charge under the h factor, or equivalently in the language of [5] we will take the bulk Wilson line to be admissible. We will take the boundary Wilson lines at u = ±1 to be in 1-dimensional representations of p 0 of charge q ± respectively, and to have no charge under h. (These assumptions are made to simplify the resulting K-matrix.) Under these assumptions we can write down the next-to-leading order contribution to the K-matrix explicitly. By repeating similar calculations as for g 0 = sl 2 C we arrive at the expression This form is highly suggestive. It is clear that, with an appropriate choice of the parameters q + and q − , we can eliminate the contribution from the self-interaction of the bulk Wilson line. In particular, if we introduce the parameters q 1 = 1 + √ 6(q + + q − ) and q 2 = √ 6(q + − q − ), the above k-matrix simplifies to This cancellation of the bulk self-interactions with boundary charges was observed in the elliptic case also, and seems to be a common feature of K-matrices associated with the A n−1 ∼ = sl n C family of classical Lie algebras, at least in the fundamental representation. We emphasise again that this is not a generic feature of all K-matrices, indeed there exist even rational K-matrices which depend explicitly on . From the above formula we arrive the final expression for the K-matrix where the overall scale factor is This coincides with one of the three families solutions constructed for g = sl 3 C in [18]. A direct comparison is performed in appendix B.

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It is conspicuous that we are missing the remaining two families of trigonometric K-matrices associated to g 0 = sl 3 C obtained in [18]. Studying these solutions carefully one finds that the analogue of the classical limit for these K-matrices is not z-independent. This motivates us to consider a generalization of the construction presented so far.

Generalization to z-dependent automorphisms
To accommodate K-matrices with z-dependent classical limits, we choose to lift the action of the map Z : (x, y, z,z) → (−x, y, −z, −z) to the adjoint bundle of the gauge theory not by simultaneously acting on its fibres with a constant involutive automorphism, but rather using a bundle morphism σ Z : Ad P → Ad P . We choose this bundle morphism such that the following diagram commutes where σ Z sends the fibre over (x, y, z,z) to fibre over (−x, y, −z, −z) whilst also acting with an involutive automorphism σ(z) of g. If the adjoint bundle had non-trivial topology then this automorphism σ(z) would not exist globally, but in the trigonometric case the adjoint bundle is trivial. The bundle morphism is then completely determined by the holomorphic map σ(z) from C * to the automorphism group of g. To preserve the vacuum A = 0, this automorphism must be independent of Σ, but in contrast to the elliptic case considered earlier, in the trigonometric case σ may vary holomorphically with z. (We show in appendix C that nothing new arises if we allow this generalization in the elliptic case.) For σ Z to be involutive σ(z) must obey σ(z)σ(−z) = 1. Note that for S[A] = S[σ Z (Z * A)], we require that σ Z preserves the g-invariant bilinear on the fibres. We still require our automorphism to exchange the boundary conditions at ±∞. We also require that our automorphism map non-singular field configurations to non-singular field configurations.
To compute the next-to-leading order contribution to the K-matrix for a z-dependent automorphism σ(z) we proceed exactly as before. In holomorphic gauge, the propagator is the only difference being that σ is now z-dependent. We find that the classical k-matrix is given by with the two terms again coming from evaluating the Feynman diagrams describing selfinteraction of the bulk Wilson line, and interactions between the bulk and boundary Wilson lines.

Examples of K-matrices associated to z-dependent automorphisms
In appendix E we consider how z-dependent bundle morphisms can be constructed from automorphisms of the loop algebra Lg. The simplest example of a z-dependent automorphism arises for g = sl 2 C ⊕ C, and acts on the sl 2 C summand as conjugation by This automorphism corresponds to a non-trivial transposition of the Dynkin diagram of the affine algebra sl 2 C. It extends to act on h ∼ = C as h → − h. The Lie subalgebras of g that survive at the fixed points u = ±1 are g ∼ = sl 2 C and span C {h} ∼ = C respectively. 11 To generate solutions of the boundary Yang-Baxter equation without boundary degrees of freedom we must insert boundary Wilson lines in 1-dimensional representations of these algebras. There exist no such representations for sl 2 C, so we can only insert a boundary Wilson line at u = −1 in the representation h/ √ 2 → q ∈ C. We will take the reflecting bulk Wilson line to be in the fundamental representation of sl 2 C and to be uncharged under h.
After some algebra, one arrives at the K-matrix 12 where F (z; ) is some irrelevant overall scaling. To symmetrise the R-matrix we conjugate it by 0 e z/4 e −z/4 0 , under which K(z) transforms to This solution belongs to the same family of trigonometric K-matrices appearing in [17] as those constructed using constant σ. Note that this solution has vanishing off diagonal terms. When σ was assumed to be constant the off diagonal terms were necessarily nonvanishing, and so we have in fact generated a family of solutions which we were previously missing.
Finally we apply this construction to sl 3 C ⊕ C 2 , which will allow us to generate less trivial examples. Consider the z-dependent automorphism acting on sl 3 C as conjugation by Replacing z by z − iπ, or equivalently swapping the sign of u, in τ (z) swaps the subalgebras at ±1, and will generate K-matrix essentially equivalent to the one constructed here. 12 For ease of comparison to the K-matrices in [17] we have included in the argument of the hyperbolic functions here. Nonetheless we stress that this K-matrix is valid only to next-to-leading order in .

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This corresponds to the non-trivial transposition of the Dynkin diagram of the affine algebra sl 3 C that swaps the 0 th and 1 st simple roots. (There is a similar automorphism swapping the 0 th and 2 nd roots, which we will comment on later.) Under the simplifying assumption that M = −1, this automorphism extends to the whole of g by acting on h as The subalgebras that survive at the points u = ±1 are both isomorphic to the direct sum of span C { h 2 } ⊂ h with sl 2 C ⊕ C ⊂ g 0 . The abelian summand of sl 2 C ⊕ C here is generated by To construct a K-matrix, we will take the reflecting bulk Wilson line to be in the fundamental representation and uncharged under h. We similarly take our boundary Wilson lines at u = ±1 to be uncharged under h, and as usual will assume them to be in 1-dimensional representations of sl 2 C ⊕ C ⊂ g 0 labelled by the charges q ± respectively. A somewhat tedious calculation shows that the leading order contribution to the K-matrix then takes the form If we introduce the parameters q 1 and q 2 by q 1 = 1 + √ 6(q + + q − ) and q 2 = √ 6(q + − q − ), the above k-matrix simplifies to Note the by now familiar fact that the self-interactions of the bulk R-matrix can be eliminated with a suitable choice of boundary parameters. From the above formula we arrive the final expression for the K-matrix (3.6) where F (z; ) is an irrelevant overall scale factor. This coincides with one of the two remaining families of solutions constructed for g = sl 3 C in [18]. Choosing allows us to generate the final family of solutions.

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Given this generalization to include z-dependent automorphisms, it is natural to ask whether we can now generate all trigonometric K-matrices? Unfortunately there exist notable omissions. The inclusion of z-dependent automorphisms allows us to associate families of K-matrices to involutive automorphisms of loop algebras exchanging the two standard Borel subalgebras. In [23], however, K-matrices were associated to a more general class of involutive automorphisms. These were required to be of the 'second kind', defined by the condition that the intersection of either of the standard Borel subalgebras with its image under the automorphism was finite-dimensional.
It is natural to speculate whether the construction presented here could be generalized to include such automorphisms. One possible approach would be to relax the condition that σ swap the Lagrangian subalgebras at u = 0 and ∞. If we do this, however, we can no longer interpret the theory as being defined on an orbifold, and the method of images cannot be used to construct the propagator.
Finally in [19] and [20] solutions of the boundary Yang-Baxter equation are obtained in the elliptic and trigonometric cases which we have been unable to interpret as arising perturbatively from some classical limit. It would be interesting to fit these solutions into this story.

Comments on uniqueness
In this section we make a few brief comments regarding the uniqueness of K-matrices generated using gauge theory. The argument presented here is equally applicable in both the trigonometric and elliptic cases. It is analogous to that appearing in [5] for bulk R-matrices.
We start by assuming we have a solution to the boundary Yang-Baxter equation with boundary degrees of freedom, K (z), for some solution to the Yang-Baxter equation in the bulk, R (z). We will take these solutions to be quasi-classical, so that they have asymptotic form If W is 1-dimensional, and V is the fundamental representation of sl n C, a constraint on the normalization of the K-matrix, known as the Sklyanin determinant condition, ensures that δK(z)(τ −1 V (z) ⊗ 1 W ) is traceless. This was demonstrated for the rational case in [1], and the result carries over to the elliptic and trigonometric cases straightforwardly. For more general representations V the line of argument is similar. One uses invariant tensors in symmetric and antisymmetric powers of V to define analogues of the Sklyanin determinant, and the resulting constraints can be used to show that δK(z)(τ −1 V (z)⊗1 W ) lies in a subspace of End(V ⊗ W ) ∼ = End V isomorphic as a Lie algebra to g.
If the representation W is not 1-dimensional, then we believe that further constraints arise at the boundary, although we leave the details of this for future work. We now restrict attention to 1-dimensional W .

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The boundary Yang-Baxter equation then ensures that δK(z)(τ −1 V (z)⊗1 W ), which can now be naturally viewed as an element of g, is tangent to the space of classical k-matrices at k(z). If the classical k-matrices generated using gauge theory are 'locally' of the most general form possible, i.e. the tangent space of classical k-matrices generated using gauge theory saturates the whole tangent space at k(z), then we can generate all quasi-classical K-matrices of the form by allowing the parameters defining the K-matrix to take values in C[[ ]] as opposed to C. Examples of the parameters which we permit to take values in include the charges on boundary Wilson lines. Essentially this means that if CWY theory generates all classical k-matrices in a neighbourhood of k(z) then it generates the most general quasi-classical K-matrix with semi-classical limit k(z).
Unfortunately classical k-matrices have not been studied in the same way as classical r-matrices, so we are unable to guarantee that the k-matrices we generate are 'locally' of the most general form.
A Asymptotic behaviour of elliptic K-matrices in the → 0 limit In this appendix we take the semi-classical limit of the elliptic K-matrix appearing in [14], and verify that the next-to-leading order contribution matches the result arrived at using gauge theory. In this appendix we will always take ζ = 1.
Expressed in terms of the basis in terms of the θ-functions θ a b (z|τ ) = m∈Z exp(πi(m + a) 2 τ + 2πi(m + a)(z + b)). It is easy to see that the next-to-leading order contribution to the R-matrix in the semi-classical limit is the classical r-matrix from equation (2.1), with ζ = 1. Now we turn our attention to the elliptic K-matrices appearing in [14]. They depend on the 4 free complex parameters λ * , indexed by the pair r * , s * ∈ {0, 1}, together with a further free complex parameter Λ. This gives 5 complex degrees of freedom, one of which describes the overall scale of the K-matrix. We can eliminate this redundancy by requiring r * ,s * ∈{0,1} λ * (−) nr * s * = 1 .
Whilst this family of K-matrices is independent of the parameter appearing in the Rmatrix, the K-matrices generated using gauge theory are formal power series in . The resolution of this apparent inconsistency is that the parameter Λ is of order in the perturbative expansion, and so should be viewed as an element of C[[ ]]. To get the correct semi-classical limit it is enough to assume that Λ = Q for Q ∈ C. 13 We have expressed the R-matrix in terms of the natural parameters appearing in the gauge theory description, and have picked a normalization in which its classical limit is the identity.

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With respect to the standard basis {e β α } α,β ∈ Zn of End(C n ), we have where K α β (z; r * , s * ) = . and we have to chosen to normalize the K-matrix conveniently so that its classical limit is Conjugating by τ gives the following automorphism Viewed as an automorphism of sl n C via the embedding sl n C ⊂ End(C n ), this coincides with one of the automorphisms discussed in subsection 2.2. In particular it corresponds to ξ = η = 0. Given τ we can form which is somewhat easier to work with. Decomposing L(z) as To make sense of the expression on the right hand side we determine its quasi-periodicities and pole structure. For the quasi-periodicities we find L i,j (z + 1; r * , s * ) = ε i L i,j (z; r * , s * ) , L i,j (z + τ ; r * , s * ) = ε j e 2Q /n L i,j (z; r * , s * ) .
so it is sufficient to determine the locations and residues of poles in the fundamental domain.
In the fundamental domain, L i,j has a simple pole at z = 0, 1/2, τ /2 and (1+τ )/2. However, the residue at this pole differs qualitatively for n even and odd, so we treat the two cases separately.
In this form it is straightforward to take the semi-classical limit. We find that for i, j ≡ 0 (n) Up to the addition of a term at order which is proportional to the identity, we find that Here q * = −Qq * , and w i,j (z − z * ) is the unique meromorphic function with a single simple pole at the origin obeying Res z=0 w i,j (z) = 1 , w i,j (z + 1) = ε i w i,j (z) and w i,j (z + τ ) = ε j w i,j (z) .
Here q * = −Qq * e πia * b * /n , so that the q * are unconstrained complex parameters. This exactly matches the result derived in section 2.3.
B Asymptotic behaviour of trigonometric K-matrices in the → 0 limit In this appendix we directly take the semi-classical limit of the 3 × 3 trigonometric K-matrices appearing in [18]. We then verify that in these limits they coincide with the results obtained in sections 3.3 and 3.5. Up to rescaling and z, and an overall z-dependent factor, the bulk R-matrix appearing in [18] is Here {e j i } 3 i,j=1 are the elementary 3 × 3 matrices. The next-to-leading order term in the small limit of the above coincides with the classical r-matrix appearing in equation (3.1) in the fundamental representation, modulo the addition of a term proportional to the identity.
Comparing this K-matrix with equation (3.5), we make the identifications γ = Λ, = Λ −1 , α = q 2 /4, and β = q 1 /4. The above constraint then determines the δ order by order in . We learn that Through a rescaling of the K-matrix and a redefinition of its parameters we can take ±1 to be +1 in the above expression for δ. Making these identifications the agreement of the classical limit of this K-matrix with that appearing in equation (3.5) is manifest. The second of the three classes takes the form where the parameters α, β, γ, δ, ∈ C obey the constraint γ = (δ − α)(δ + α) .
Comparing this K-matrix with equation (3.6), we make the identifications γ = Λ, = Λ −1 , α = q 1 /4, and β = q 2 /4. The above constraint then determines the δ order by order in . We learn that where as above we can without loss of generality take ±1 to be +1. The agreement of the classical limit of this K-matrix with that appearing in equation (3.6) is now manifest. The final class of solutions can be generated in a similar manner to the second.

C Determination of allowed automorphisms in the elliptic case
In this appendix we will show that, in the elliptic case, the only allowed bundle morphisms σ Z : Ad P → Ad P we can use to construct the orbifold gauge theory are those that were considered in section 2.2. In particular, unlike the trigonometric case, holomorphy and compactness ensure that there are no non-constant morphisms. We first describe Ad P more explicitly. We begin by noticing that M = Σ × E τ is the quotient of the covering space M = Σ × C by (p, z) ∼ (p, z − a − bτ ) for a, b ∈ Z. We can pull back Ad P to M whereupon, since C is contractible (and we assume Ad P