北京十一选五开奖结果

学霸学习网 这下你爽了

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北京十一选五开奖结果 www.frdg.net hep-th/0104212 April 2001

arXiv:hep-th/0104212v3 29 Jul 2003

Boundary scattering, symmetric spaces and the principal chiral model on the half-line

N.J. MacKay1, B. J. Short2

Department of Mathematics, University of York, York YO10 5DD, U.K.

Abstract We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we ?nd a connection with (type I, Riemannian, globally) symmetric spaces G/H : there is a class of integrable BCs in which the boundary ?eld is restricted to lie in a coset of H ; these BCs are parametrized by G/H × G/H ; there are rational solutions of the BYBE in the de?ning representations of all classical G parametrized by G/H ; and using these we propose boundary S -matrices for the principal chiral model, parametrized by G/H × G/H , which correspond to our boundary conditions.

1 2

[email protected] [email protected]

1

Introduction

The bulk principal chiral model (PCM) – that is, the 1 + 1-dimensional, G × G-invariant nonlinear sigma model with target space a compact Lie group G – is known to have a massive spectrum of particles in multiplets which are (sometimes reducible) representations of G×G. These are irreducible representations, however, of the Yangian algebra of non-local conserved charges, and the multiplets are also distinguished by a set of local, commuting conserved charges with spins equal to the exponents of G modulo its Coxeter number. The corresponding bulk scattering (‘S -’) matrices are constructed from G-invariant (rational) solutions of the Yang-Baxter equation (YBE), whose poles determine the couplings between the multiplets. In this paper we investigate the model on the half-line – that is, with a boundary. Any proposed boundary S -matrices must satisfy the boundary Yang-Baxter equation (BYBE), for which only a limited range of solutions is known ([7, 8, 9, 11, 12, 13] is a selection). For constant solutions of the BYBE (i.e. without dependence on a spectral parameter or rapidity) there is a well-established connection with (quantum) symmetric spaces [37]. We shall ?nd a class of BYBE solutions corresponding to, and parametrized by, the symmetric spaces G/H . These solutions utilize for the bulk S-matrix the rational3 , G-invariant solution of the (usual, bulk) YBE in the de?ning (N -dimensional vector) representation of a classical G, and are themselves rational and N -dimensional, describing the scattering of the bulk vector particle o? the boundary ground state. We make the ansatz that they are constant or linear in rapidity, and thus have at most two channels. The underlying algebraic structures are the twisted Yangians [38], though the relationship remains to be explored. However, we begin by investigating how our solutions might arise as boundary S -matrices, by discussing the principal chiral ?eld on the half-line and boundary conditions which preserve its integrability (see also [31], and [10] for a more general discussion of boundary integrability). We shall ?nd two classes of BCs which are associated with the G/H . In the ?rst, ‘chiral’ class, the ?eld takes its values at the boundary in a coset of H , and the G G space of such cosets is (up to a discrete ambiguity) H ×H . Correspondingly, we use our G G BYBE solutions to construct boundary S -matrices, parametrized by H × H , which preserve the same remnant of the G × G symmetry as the integrable boundary conditions. In the second, ‘non-chiral’ class, for which we do not generally have corresponding boundary S matrices, the boundary ?eld lies in a translate of the Cartan immersion of G/H in G. To

3

before the inclusion of scalar prefactors

1

summarize: a connection between boundary integrability and symmetric spaces emerges naturally in two very di?erent ways: by seeking classically integrable boundary conditions, and by solving the BYBE. The plan of the paper is as follows. In section two, building naturally on the results of [3, 4] for the bulk PCM, we discuss boundary conditions which lead naturally to conservation of local charges. As mentioned, there are two classes of BC, which we call ‘chiral’ and ‘non-chiral’. In section three we ?nd minimal boundary S -matrices, by making ans¨ atze for the BYBE solutions and applying the conditions of crossing-unitarity, hermitian analyticity and R-matrix unitarity, and explain how these are related to symmetric spaces. This section is necessarily rather long and involved, and many of the details appear in appendices. From these, in section four, we construct boundary S -matrices for the PCM, and ?nd that these correspond naturally to the chiral BCs. The key statements of our results for the boundary S -matrices can be found in section 3.4 (for the minimal case, without physical strip poles) and section 4.2 (for the full PCM S -matrices). This paper supersedes the preliminary work of [6].

2

2.1

The principal chiral model on the half-line

The principal chiral model on the full line

We ?rst describe the model on the full line, without boundary. This subsection is largely drawn from [4], and full details may be found there. The principal chiral model may be de?ned by the lagrangian 1 L = Tr ?? g ?1? ? g , 2 (2.1)

where the ?eld g (x? ) takes values in a compact Lie group G. (We could also include an overall, coupling constant, but this may be absorbed into , and will not be important for ?1 our purposes.) It has a global GL × GR symmetry g → gLggR associated with conserved currents ?1 ?1 j (x, t)L j (x, t)R (2.2) ? = ?? g g , ? = ?g ?? g which take values in the Lie algebra g of G: that is, j = j a ta (for j L or j R : henceforth we drop this superscript) where ta are the generators of g, and (with G compact) Tr(ta tb ) = ?δ ab . The equations of motion are ? ? j? (x, t) = 0 , ?? jν ? ?ν j? ? [j? , jν ] = 0 , 2 (2.3)

which may be combined as 1 ?? j+ = ??+ j? = ? [j+ , j? ] 2 (t ± x) (and thus ?± = ?0 ± ?1 ). in light-cone coordinates x± = 1 2 In addition to the usual spatial parity P : x → ?x, the PCM lagrangian has further involutive discrete symmetries. The ?rst, which we call G-parity and which exchanges L ? R, is π : g → g ?1 ? j L ? j R . (2.5) (In the usual QCD e?ective model, ‘parity’ is the combination P π .) Then there is g → α(g ) where α is any involutive automorphism, though only for outer automorphisms may this have a non-trivial e?ect on the invariant tensors and local charges which we shall consider shortly. The canonical Poisson brackets for the model are

a b j0 (x), j0 (y ) a b j0 (x), j1 (y ) a b j1 (x), j1 (y ) c = f abc j0 (x) δ (x?y )

(2.4)

c = f abc j1 (x) δ (x?y ) + δ ab δ ′ (x?y )

(2.6)

= 0

at equal time. These expressions hold for either of the currents j L or j R separately, while the algebra of j L with j R (which we shall not need here) involves only δ ′ (x?y ) terms in the brackets of space- with time-components. This model has two distinct sets of conserved charges, and the two sets commute. The ?rst is the extension of the GL × GR charges to the larger algebra of non-local, Yangian charges [14, 15] Y (gL ) × Y (gR ); we shall not discuss these here. There is also4 an in?nite set of local, commuting charges with spins s equal to the exponents of g modulo its Coxeter number,

∞

q±s =

?∞

a1 a2 an (x)j± (x) . . . j± ka1 a2 ...an j± (x) dx

(2.7)

(where n = s + 1); here, unlike for the Yangian, j L and j R give the same charges (up to a change of sign), of which there is therefore only one set. The primitive invariant tensors k have to be very carefully chosen to ensure the charges commute – for the full story see [4]. Such a set appears to be precisely what is needed for quantum integrability, where it leads to the beautiful structure of masses and interactions described in [16].

4

In this paper we restrict to the classical g, although they also exist for exceptional g [5].

3

2.2

Boundary conditions for the model on the half-line

Varying the bulk action on the half-line ?∞ < x ≤ 0 imposes the additional boundary equation Tr(g ?1 ?1 g.g ?1δg ) = 0 at x = 0 , where the variation is over all δg such that g ?1 δg ∈ g. Clearly the Neumann condition ?1 g |0 = 0 solves this, as does the Dirichlet condition δg |0 = 0, or ?0 g |0 = 0. But we can also impose mixed conditions, in any way such that (g ?1?1 g )a (g ?1?0 g )a = 0 (with the usual summation convention). We begin by considering some simple mixed boundary conditions written in terms of the currents j . A little later we shall generalize these, and write them in terms of the ?elds g . We take as a BC on the currents

a b j+ (0) = Rab j? (0) ,

(2.8)

with each j chosen independently to be either L or R; we refer to the four possibilities as LL, RR, LR and RL. The boundary equation of motion then requires that Rab be an orthogonal matrix. We would also like consistency with the Poisson brackets: if we extend a b the currents’ domains to x > 0 by requiring j+ (x) = Rab j? (?x), then this further requires 2 that R be symmetric, and give an (involutive; α = 1) automorphism α of g via α : ta → Rab tb . Together these imply that R is diagonalizable with eigenvalues ±1, so that we may write g =h⊕k, where h and k are the +1 and ?1 eigenspaces respectively. The h indices then correspond a a a a to Neumann directions j+ = j? ? j1 = 0, the k indices to Dirichlet directions j+ = a a ?j? ? j0 = 0 (all at x = 0). Further, R’s being an automorphism implies that [h, h] ? h , [h, k] ? k , [k, k] ? h ,

precisely the properties required of a symmetric space G/H [17], where H is the subgroup generated by h and invariant under the involution α. For integrability we require a great deal of R. As we have said, there are two in?nite sets of charges. The Yangian charges appear no longer to be conserved on the half-line, even 4

with pure Neumann BCs [19] (naively, at least, it seems that there are remnants only, as we shall see). However, we believe they are not essential for integrability, because precisely half of the local charges remain conserved, with either qs + q?s or qs ? q?s surviving. Our conjecture is that these are enough to guarantee the properties of quantum integrability, such as factorizability of the S -matrix. The ?rst charge is energy, and its conservation on the half-line is just the equation of motion, requiring R to be orthogonal. For the higher charges we take q|s| = qs ± q?s

0

=

?∞

a1 an a1 an ...j+ ± j? ...j? ka1 a2 ...an j+

dx .

Then

0 d a1 an a1 an ...j+ q|s| = ka1 a2 ...an (?? + ?1 ) j+ ± (?+ ? ?1 ) j? ...j? dt ?∞ a1 an a1 an ...j+ ? j? ...j? = ka1 a2 ...an j+ |x=0

dx

=

a1 an ...j? ) |x=0 . kb1 b2 ...bn Ra1 b1 ...Ran bn ? ka1 ...an j?

That this is zero for one choice of sign follows from the result [20] that, for every R and k , kb1 b2 ...bn Ra1 b1 ...Ran bn = ?ka1 ...an , (2.9)

where ? = ±1. (This is obvious, with ? = 1, when α is an inner automorphism, but not at all obvious for outer automorphisms.) So, if we now regard α as acting on the currents, a b α(j± ) = Rab j± , we see that

0

q|s| =

?∞

a1 an a1 an ...j+ + α(j? )...α(j? ) dx ka1 a2 ...an j+

(2.10)

is the charge which remains conserved in the presence of the boundary. For LL and RR conditions its density is the combination which is invariant under the combined action P α of spatial parity P (which exchanges j+ ? j? ) and α, while for LR and RL it is the combination invariant under these together with G-parity, P απ . Further, these charges still commute. In the Poisson bracket of the charges constructed from tensors k (1) and k (2) , a total derivative term which vanished in the bulk now gives an additional contribution proportional to (all at x = 0)

(1) a1 as b1 br a1 as b1 br k kca j+ ...j+ j+ ...j+ ? ?(1) ?(2) j? ...j? j? ...j? 1 ...as cb1 ...br (1) (1) (2)

a1 as b1 br (2) Rd1 a1 ...Rds as Re1 b1 ...Rer br ? ?(1) ?(2) δ d1 a1 ...δ ds as δ e1 b1 ...δ er br j? ...j? = kcd1 ...ds kce j? ...j? 1 ...er a1 br (2) Rd0 a0 Rd1 a1 ...Re0 b0 Re1 b1 ... ? ?(1) ?(2) δ d0 a0 δ d1 a1 ...δ e0 b0 δ e1 b1 ... δ a0 b0 j? ... j? = kd0 d1 ...ds ke 0 e1 ...er

= 0

5

by property (2.9). Finally, it is precisely the q|s| of (2.10) above that still commute with 0 L/R the G × G-generating charges Q = ?∞ j0 dx. 2.2.1 General chiral BCs As we commented when ?rst introducing the BC (2.8), in j+ = α(j? ) we may take each j L L R ?1 as either L or R. The LL and RR conditions are then related: j+ = α(j? ) ? gj+ g = R ?1 α(gj? g ) (all at x = 0). In fact the most general such (we shall call it ‘chiral’) BC is to take ?1 g (0) ∈ kL HkR . (2.11) This is the Dirichlet part of the BC; when we impose the boundary equation-of-motion we supplement it with Neumann conditions within this boundary target space (which we shall henceforth refer to as the D-submanifold) so that the current conditions become

?1 L ?1 L kL j+ kL = α kL j? kL ?1 R ?1 R kR j+ kR = α kR j? kR

(at x = 0).

The constant group elements kL and kR parametrize left- and right-cosets of H in G and may be taken to lie in the Cartan immersion of G/H in G, so that the possible BCs are parametrized by G/H × G/H . (In fact this is true only at the level of the Lie algebras: there is a further discrete ambiguity in the choice of kL, kR . For details of this, and of the Cartan immersion, we refer the reader to appendices 6.1, 5.1.) Our earlier results about conservation and commutation of charges and consistency with the Poisson brackets still apply (generalized here by twisting the currents with an inner automorphism, which does not change the de?nition of the conserved charge q|s| ). Note that when kL = kR = e (where e is the identity element in G), we have g (0) ∈ H , the continuous Dirichlet boundary parameters which determine the D-submanifold are all trivial, and the residual symmetry is H× H . For any kL , kR the case H = G corresponds to the pure Neumann condition, while trivial H , the pure Dirichlet condition, is inadmissible for any non-abelian G. We should point out at this stage that we have not succeeded in ?nding a boundary Lagrangian for any of our mixed BCs. That is, we have no Lagrangian of which the free variation leads to our conditions. The Dirichlet conditions have to be imposed as ‘clamped’ BCs, restricting the boundary variation of g . Let us now examine how much of the G × G symmetry survives. We can see that the ?1 ?1 BC (2.11) is invariant under kL HkL × kR HkR , and we can check that it is precisely the 6

charges generating this subgroup of G × G which are conserved on the half-line. For the 0 a global G-generating charges Qa = ?∞ j0 (x) dx (where subscripts either all L or all R are to be understood), consider d ?1 k Qk dt

0

= k ?1

?∞

?0 j0 dx k

= k ?1 j1 (0)k 1 ?1 k j+ k ? k ?1 j? k x=0 = 2 1 α(k ?1 j? k ) ? k ?1 j? k x=0 , = 2 which is zero on h (only).

2.2.2 General non-chiral BCs If we explore similarly the LR and RL conditions, we ?nd that the condition g (0) ∈ gL G ?1 g H R (2.12)

?1 L (where G/H = {α(g )g ?1|g ∈ G} is the Cartan immersion of G/H in G) leads to gL j0 gL = ?1 R α(gR j0 gR ) (with gL , gR again constant elements of G). If we then apply the boundary equation-of-motion, the condition on the currents becomes ?1 L ?1 R gL j± gL = α gR j? gR

at x = 0.

Unlike our chiral BCs (which are parametrized by G/H ×G/H ), these BCs are parametrized by a single G (again quotiented by a discrete subgroup; see appendix 6.2). In the specialization gL = gR , however, the boundary is parametrized by G/H . Note the inversion of the role of the dimension of H in determining the dimension of the D-submanifold, compared to the chiral case: there (and setting kL = kR = e) we had g (0) ∈ H , whereas here (with gL = gR = e) we have g (0) ∈ G/H . The two extreme non-chiral cases give us nothing new: with trivial H we revert to the free, pure Neumann condition, while at the other extreme of H = G we have the pure Dirichlet condition. As with the chiral case, we can check conservation of the generators of the remnant of the G × G symmetry. This time, because the Cartan immersion of G/H is invariant under Hdiag. (the diagonal subgroup g → hgh?1 ), the surviving global symmetry is Hdiag. conjugated (in G × G) by (gL, gR ), and we may check that d ?1 ?1 gL QL gL + gR QR gR dt = 1 ?1 L L ?1 R R gL (j+ ? j? )g L + g R (j+ ? j? )g R 2 7

x=0

= which is zero precisely on h × h.

1 ?1 R R ?1 R R gR (j+ ? j? )g R ? α (g R (j+ ? j? )g R ) 2

x=0

At this stage it is worth comparing our results with those obtained in the Wess-ZuminoWitten model – that is, with D-branes on group manifolds. There, initial suggestions of a connection with symmetric spaces [21] were supplanted by an understanding that the D-submanifold is actually a ‘twisted’ or ‘twined’ conjugacy class [22, 23], Cα (g0 ) = {α(g )g0g ?1|g ∈ G}. This situation arises because in the WZW model there is only one L R pair of currents, j+ and j? , and so only one, LR, boundary condition. In our case we have ?1 two, LR and RL, conditions, and their interplay further requires that α(g0 ) = g0 . But the space M of such g0 is, for the non-Grassmannian cases, precisely the Cartan immersion √ G/H (which is connected to the identity, so that we can ?nd g0 ), and Cα (g0 ) = {α(g )g0g ?1|g ∈ G} = {α(gg0 2 )(gg0 2 )?1 |g ∈ G} =

?1 ?1

G . H

(2.13)

(For the Grasmannian cases, M is a union of disconnected components, the identityconnected component being G/H – see appendix 5. However, each of the other components is actually a translate of the immersion of G/H ′ for a di?erent H ′ [40], so we obtain no new BCs in this way.) The analogous BC in our case is with gL = gR = e, and the residual symmetry is Hdiag. , necessarily preserved by any BC utilizing a twisted conjugacy class, as in the WZW model. Note that the α = 1, H = G case is purely Dirichlet in our case, whereas in the WZW model g0 is unconstrained and there is still freedom at the boundary. We expect that the non-chiral BCs should remain integrable when a Wess-Zumino term (of arbitrary size) is added, and we plan to explore this in future work. Finally we note the relationship of our work to that on the Gross-Neveu model5 [25]. This model has a single global G = O (N ) invariance, broken by the BC to an H = O (M ) subgroup. Their boundary S -matrix is then diagonal.

2.3

Remarks on quantization

As with the bulk model, we shall assume that our results carry through into the quantum theory: that classically conserved charges remain conserved in the quantum theory; that charges classically in involution do not develop O( 2 ) anomalies in their commutators; and that our BCs therefore lead to quantum conservation of the charges which generate

5

not the generalized chiral GN model, which remains to be investigated

8

the residues of the G × G symmetry. All of this leads to the expectation that boundary scattering factorizes, so that solutions of the BYBE provide boundary S -matrices. The only technique which can give evidence for the continued conservation of the local charges after quantization is Goldschmidt-Witten anomaly counting [26]. This was carried out for the bulk case in [3], where for each classical G at least one non-trivial charge was found to be necessarily conserved in the quantum theory. This was extended to boundary models in [24], and used to prove quantum conservation of the spin-3 charge for G = SO (N ) in [6] for one of our BCs. It is simple to check that for all our BCs and for all classical G, each charge which necessarily survives quantization in the bulk model also survives in the presence of the boundary. We do not give details. Finally, the form of our admissible D-submanifolds is not so surprising when we remember that bulk sigma models on symmetric spaces have particularly nice behaviour after quantization, in that the symmetric spaces preserve their shape under renormalization [41]. We would certainly expect our D-submanifolds to behave similarly nicely.

3

The minimal boundary S -matrices

In this section we construct boundary S -matrices which are minimal — that is, which have no poles on the physical strip. We follow the method used in the bulk case [29], where minimal S -matrices were found by solving the Yang-Baxter equation and applying unitarity, analyticity and crossing symmetry, and the desired pole structure then implemented using the CDD ambiguity. In the boundary case we seek minimal boundary S -matrices by making ans¨ atze to solve the boundary Yang-Baxter equation (BYBE) and applying unitarity, analyticity and the combined crossing-unitarity relation [27]. These minimal solutions will be used to construct PCM boundary S -matrices with the appropriate pole structure in section four. We shall make the ansatz that the boundary S -matrix (the ‘K -matrix’) in the de?ning, N -dimensional, vector representation of a classical group G is in one of the two forms K1 (θ) = ρ(θ)E and K2 ( θ ) = τ (θ ) (I + cθE ) . (1 ? cθ) (3.1)

Here c and E are constants, the latter an N × N matrix; we shall explain the θ-dependent terms below. The crucial point at this stage is the equivalent physical statement that K has at most two ‘channels’: since its matrix structure is at most linear in rapidity θ, it will decompose into one (for K1 ) or two (for K2 ) projectors. This will prove su?cient to yield a set of solutions related to the symmetric spaces in the following 9

Correspondence: for a given G-invariant bulk factorized S -matrix (i.e. a solution of the YBE) in the de?ning representation, the K -matrices of the form (3.1) fall into a set of families in 1 ? 1 correspondence with the set of symmetric spaces G/H , and each family is parametrized by a space of admissible E which is isomorphic to (possibly a ?nite multiple of) the corresponding G/H . Such solutions, we believe, correspond to scattering o? the boundary ground state. We would obtain solutions with many more channels by considering scattering of bulk particles in higher tensor representations or o? higher boundary bound states, or both. These can be obtained by fusion from our solutions6 , and the results of this paper thus lay the foundation for future work in this direction. The calculations of this section will necessarily, because case-by-case and exhaustive, be rather involved. Our strategy is to lead the reader through the implications of the BYBE, unitarity, hermitian analyticity and crossing-unitarity, initially culminating in a precise statement of our solutions in section 3.4. Details of the calculations are relegated to appendices 7 and 8. Then in section 3.5 we explain how our solutions are parametrized by the symmetric spaces; details appear in appendix 5.

3.1

Calculating the boundary S -matrices

Throughout the rest of this paper we shall use the following (somewhat unconventional) notation for the bulk and boundary S -matrices: k

kl Sij (θ ) :

l θ K ij (φ) : j

i φ t

j

i

where θ is the rapidity di?erence between the two in-coming particles which scatter in the ?rst diagram and φ is the rapidity of the in-coming particle re?ecting in the second. We consider the BYBE for two particles in the vector representation

kl np Sij (θ ? φ)(Ijm ? K ln (θ))Smo (θ + φ)(Ioq ? K pr (φ)) =

6

with the exception of the SO(N ) spinorial multiplets

ln pr (Iij ? K kl (φ))Sjm (θ + φ)(Imo ? K np (θ))Soq (θ ? φ )

(3.2)

10

(θ and φ are now the rapidities of the two particles.) We attempt to ?nd solutions of the form (equivalent to (3.1)) K1 (θ) = ρ(θ)E and K2 ( θ ) = τ ( θ ) P ? ? h P+ ciπ (3.3)

where I is the identity matrix and E is a general square matrix of the same size7 . We ?nd that constraints are imposed on the scalar prefactors and the matrix E by the BYBE and unitarity, analyticity and crossing-unitarity. These constraints are in general dependent on the choice of classical group, SU (N ), SO (N ) or Sp(N ), but in all cases involving K2 (θ) we ?nd that E 2 = 1, so that P ± are projectors, as the notation suggests. We note that the constraints imposed on the scalar prefactors will allow us to ?nd them only up to the usual CDD ambiguity. This freedom is then further restricted by taking the K -matrices to be minimal – that is, taking ρ(θ) and τ (θ) to have no poles on the physical strip. We shall also require τ (θ) to have a zero at θ = 1 so that K2 (θ) is ?nite at this c 1 point, since we shall ?nd that c can lie in the physical strip. For the case of SU (N ) the vector representation is not self-conjugate. This allows us to consider the situation where a particle scattering o? the boundary returns as an anti? particle (for an analogous situation see [11]). In this case the K -matrix, K ij (θ), must satisfy a version of the BYBE which for convenience we shall refer to as the ‘conjugated’ BYBE,

n ?p kl ? pr ? Sij (θ ? φ)(Ijm ? K ln (θ))Sm ?q ? ? K (φ)) = o ?(θ + φ)(Io ? ?

where ρ(θ) and τ (θ) are scalar prefactors, h is the dual Coxeter number of the group, c is a parameter and θ + iπx 1 h [x] = P ± = (I ± E ) iπx , 2 θ? h

p ?r ? ln np ? (Iij ? K kl (φ))Sj ?o ? ? K (θ ))So ?q ? (θ ? φ ) . m ? (θ + φ)(Im

(3.4)

We shall ?nd that only K1 (θ) gives a solution of the conjugated BYBE. Analyticity and unitarity become more subtle in this case. Henceforth we shall refer to this case as the ‘conjugating’ case for the SU (N ) model. Even in the ‘non-conjugating’ SU (N ) model, where the representation is preserved under scattering by the boundary, the fact that the vector representation is not self-conjugate leads to subtleties in the calculations. In addition to the standard BYBE of (3.2) there is also the possibility that one or both of the in-coming particles in the BYBE scattering

7

The apparently circuitous involvement of iπ/h ensures that x is an integer, as is usual in the literature.

11

process is the conjugate vector particle. Thus we need to introduce a minimal K -matrix, ? (θ), describing this scattering. We take K ? (θ) to have the same one or two channel form K as K (θ), that is ? 1 (θ ) = ρ K ?(θ)F or ? 2 (θ ) = K τ ? (θ ) (I + dθF ) , (1 ? dθ) (3.5)

where we have a new set of scalar prefactors and constants, ρ ?(θ), τ ?(θ), d and F . At this point we introduce the diagrammatic algebra used in the calculations we perform. We represent the matrices I , J , E and F , where J is the symplectic form matrix, in the following way: I: J: E: F: J t: = ?( ) and = = (Thus from the properties of J we have multiplication is achieved by concatenation of the diagrams, for example JIJ t E : = = .) Matrix

Using this diagrammatic algebra we can rewrite the K -matrices of (3.1) as K1 (θ) = ρ(θ) K2 ( θ ) = τ (θ ) (1 ? cθ) + cθ (3.6)

This diagrammatic description is used in the calculations presented in the appendices, and makes them much clearer. We substitute either of these K -matrices into the BYBE, taking the minimal S -matrix to be that of the bulk PCM for a particular G, derived in [29] up to a few minor inconsistencies which we have corrected. This yields constraints (dependent on G and on whether we take K1 or K2 ) that E (and F ) must satisfy in order for Ki to be a solution of the BYBE. In fact the constraints from crossing-unitarity on the Ki are identical to those imposed by the BYBE, therefore we shall delay presentation of the constraints until later. (The reader is referred to appendix 7.1 for details of the BYBE calculations.) Instead we go on to consider unitarity, analyticity and crossing-unitarity.

3.2

Unitarity and Analyticity

3.2.1 The non-conjugating cases For the non-conjugating cases the K -matrices are required to satisfy the conditions of unitarity [27] and hermitian analyticity [28] K (θ)K (?θ) = I 12 K (θ) = (K (?θ? ))? . (3.7)

Substituting K1 we obtain ρ(θ)ρ(?θ) = ρ(θ) = ρ(?θ? )? ( )? ,

These matrix equations are equivalent to ρ(θ)ρ(?θ) = =α freedom in our de?nition of ρ(θ) and )? = 1 α ρ(θ) = βρ(?θ? )? ( )? = β

where α and β are constants with β ∈ U (1). Recalling K1 (θ), we see that we have the constraints imposed on the matrix ( and by unitarity and analyticity become =α where α ∈ U (1).

to set β = 1 and ensure α ∈ U (1), so that the (3.8)

Similarly for K2 we obtain τ (θ)τ (?θ) (1 ? c2 θ2 ) + cθ ? c2 θ 2 = τ (?θ? )? (1 + c? θ) = ? c? θ ( )?

τ (θ ) (1 ? cθ) These are equivalent to

If we consider the expression for K2 (θ), we see that we have the freedom to set c to be purely imaginary and choose γ ∈ U (1). Then the constraints on the matrix ( )? = and =γ where γ ∈ U (1) become

(1 ? c2 θ2 ) τ (θ)τ (?θ) = (1 ? γc2 θ2 ) τ (θ ) τ (?θ? )? = (1 ? cθ) (1 + c? θ)

=γ c = ?c? ( )?

as was the case for K1 . In fact we ?nd that the parameters α and γ cannot be freely is that they are equal chosen in U (1); the only choice consistent with the hermiticity of to 1 (see appendix 8.1). Consequently we ?nd that for both K1 and K2 , unitarity and analyticity impose ( )? = and = . (3.9) The corresponding constraints imposed on ρ(θ) and τ (θ) are ρ(θ) = ρ(?θ? )?

? ?

τ (θ) = τ (?θ )

τ (θ)τ (?θ) = 1 . .)

ρ(θ)ρ(?θ) = 1

(3.10) (3.11)

(We note that similar conditions are imposed on ρ ?(θ), τ ?(θ) and 13

3.2.2 SU (N )-conjugating In the case of the conjugated BYBE (3.4), which we call ‘SU (N )-conjugating’, unitarity and analyticity can no longer be applied as straightforwardly. The reason is that we are no longer dealing with a K -matrix that is an endomorphism of the vector representation space ? . In order to apply our conditions we must introduce V , but rather with K (θ) : V → V ? → V and consider the space V ⊕ V ? on which the endomorphism K ′ (θ ) : V ? (θ ) = K 0 K ′ (θ ) K (θ ) 0 acts.

? (θ) which yields We can then apply analyticity and unitarity to K K (θ) = K ′ (?θ? )? K (θ)K ′ (?θ) = I . (3.12)

The conjugated BYBE that K (θ) and K ′ (θ) must satisfy allows only the K1 (θ) form for both. Thus we have K (θ) = ρ(θ) and K ′ (θ ) = ω (θ ) ≡ F are constant

where ρ(θ) and ω (θ) are scalar prefactors, whilst = E and matrices. Substituting these into our conditions yields ρ(θ) = αω (?θ? )? α =( )? ρ(θ)ω (?θ) = =β 1 β .

(3.13)

There is enough rescaling freedom in splitting K (θ) and K ′ (θ) into scalar prefactors and matrix parts that we can consistently set α = β = 1 and det E = 1 ?? det F = 1 (details in appendix 8.2). This leaves us with E = F ?, ρ(θ) = ω (?θ ) 3.3 Crossing-Unitarity

? ?

EF = I, det F = det E = 1, and ρ(θ)ω (?θ) = 1. (3.14)

3.3.1 The non-conjugating cases The boundary S -matrices must also satisfy crossing-unitarity [27], which in our notation is iπ ? ? iπ lk i? l + θ) . (3.15) K ij ( ? θ) = S? jk (2θ )K ( 2 2 14

Substituting the Ki into the crossing-unitarity equation gives constraints on , c, d and the scalar prefactors. We tabulate these below (details in appendix 7.2). Note that in our diagrammatic algebra T r (E ) is represented by the symbol Group K1 (θ)/K2 (θ) SU (N ) K1 ( θ ) SU (N ) K2 ( θ ) SO (N ) ( K1 ( θ ) ( SO (N ) K2 ( θ ) Sp(N ) ( K1 ( θ ) ( Sp(N ) K2 ( θ ) ( )t )t = = ?( , ), )

t

.

Constraints = 0, (θ ? iπ )(θ + iπ ) h h ρ(iπ ? θ)ρ(iπ + θ) = θ2 1 1 = ih +d , π c iπ iπ (θ ? h )(θ + h )(θ ? iπ + 1 )(θ + iπ ? 1 ) c c τ (iπ ? θ)τ (iπ + θ) = 1 1 (θ ? iπ )(θ + iπ )(θ ? d )(θ + d ) = 0 and either )t = , ), ,

ρ(iπ/2?θ ) ρ(iπ/2+θ )

= σo (2θ)

or = (

2ih , cπ t

)t = ?( (

)t = ),

ρ(iπ/2?θ ) = ?[1]σo (2θ) ρ(iπ/2+θ ) τ (iπ/2?θ ) h = [h ] ciπ ?h τ (iπ/2+θ ) 2 2

σo (2θ)

or

) = ?(

τ (iπ/2?θ ) τ (iπ/2+θ )

= ?[1][ h ] 2

ρ(iπ/2?θ ) ρ(iπ/2+θ )

h ?h ciπ 2

σo (2θ)

= 0 and either = , or

ρ(iπ/2?θ ) = ?[1]σp (2θ) ρ(iπ/2+θ ) τ (iπ/2?θ ) h = [h ] ciπ ?h σp (2θ) τ (iπ/2+θ ) 2 2

= σp (2θ)

or

ih =2 , cπ τ (iπ/2?θ ) = τ (iπ/2+θ )

( )t =? , h h h ?[1][ 2 ] ciπ ? 2 σp (2θ)

The functions σo and σp are scalar prefactors for the bulk S -matrices (see appendix 7). 3.3.2 SU (N )-conjugating The SU (N )-conjugating K -matrix must satisfy the crossing-unitarity equation K ij (

?

iπ ? iπ il ? θ) = Sjk (2θ)K lk ( + θ) . 2 2 15

(3.16)

From this we obtain the constraints (see appendix 7) ρ( iπ ? θ) 2 = iπ ρ( 2 + θ) . Since this we have = the signs in ( σu (2θ) ( )t = )t = ?( ). (3.17)

Similarly, the crossing-unitarity equation for K ′ (θ) yields identical constraints on ω (θ) and )t = ±( ) and ( )t = ±( ) must coincide. From

?[1]σu (2θ) (

ρ( iπ ? θ) ω ( iπ ? θ) 2 2 = , (3.18) iπ iπ ρ( 2 + θ) ω ( 2 + θ) which, together with the other constraints on these scalar prefactors, is enough to show we should take ρ(θ) = ω (θ). Now F and ω (θ) are completely ?xed by E and ρ(θ), and the boundary S -matrix for ? is V ⊕V ? ? (θ) = ρ(θ) 0 E (3.19) K E 0 subject to the following constraints, along with (3.17): E ?E = I ρ(θ) = ρ(?θ ) 3.4 The boundary S -matrices

? ?

and and

det E = 1 ρ(θ)ρ(?θ) = 1 . (3.20)

We have obtained a series of constraints on and on ρ(θ) and τ (θ) which must be satis?ed if the proposed K1 (θ) and K2 (θ) are to be boundary S -matrices. The constraints on the scalar prefactors enable us to determine them exactly for each G, providing we assume some extra minimality conditions, namely that they should be meromorphic functions of θ with no poles on the physical strip Im θ ∈ [0, π ]. Having calculated the scalar prefactors 2 (we do not include details of the calculations, as the reader can simply check the results if required) we obtain the boundary S -matrices below. lies We note that in the case of K2 (θ) there is also the possibility that the pole at θ = 1 c on the physical strip and so the scalar prefactor τ (θ) may be required to have a zero at always lies on the physical strip, and the this point. In fact we shall ?nd that one of ± 1 c expressions given below for K2 (θ), for each case, are valid when 1 lies on the physical strip. c 1 When ? c lies on the physical strip instead, the correct expressions for the minimal K2 ?? (together with d ? ?d

matrices can be obtained by the interchange c ? ?c,

for SU (N )). (Note that this leaves c unchanged.) In section 4 we shall add CDD factors making the PCM boundary S -matrices invariant under this interchange. 16

3.4.1 SU (N ) The minimal boundary S -matrices for SU (N ) are Γ K1 ( θ ) = Γ

θ 2iπ ?θ 2iπ θ 2iπ ?θ 2iπ 1 +2 + 1 +2+ 1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

+ +

1 2 1 2

with

=0

(3.21)

where Γ is the gamma function, and K2 ( θ ) = with = ?1 Γ (1 ? cθ) Γ

1 ih 1 ( +d ). π c cd c +d 1 +2 + 1 +2+ 1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

Γ Γ

θ 2iπ ?θ 2iπ

+ +

1 2iπc 1 2iπc

Γ Γ

?θ 2iπ θ 2iπ

+1 + 2 1 +2+

1 2iπd 1 2iπd

(

+ cθ

) (3.22)

(Note that in the limit c, d → ∞, with 3.4.2 SO (N )

?xed, K2 (θ) → K1 (θ), as we would expect.)

There are two types of minimal boundary S -matrix, corresponding to E symmetric or antisymmetric, for each of the K1 (θ) and K2 (θ) forms in the case G = SO (N ). (The symmetric K2 (θ) type was investigated in [8] and, for M = 1, in [35]. The K2 (θ) type with antisymmetric E has been considered in [36].) The minimal boundary S -matrices of the K1 (θ) form are K1 ( θ ) = with ( )t = , Γ Γ Γ Γ

θ 2iπ ?θ 2iπ

+ +

3 4 3 4

Γ Γ

?θ 2iπ θ 2iπ

+ +

1 2 1 2

Γ Γ

θ 2iπ ?θ 2iπ

+1 + 2 1 +2+

1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

+1 + 4 1 +4+

1 2h 1 2h

(3.23)

= 0, and

θ 2iπ ?θ 2iπ

K1 ( θ ) = with ( )t = ?( K2 ( θ ) =

+ +

3 4 3 4

Γ Γ

?θ 2iπ θ 2iπ

+ +

1 2 1 2

Γ Γ

θ 2iπ ?θ 2iπ

+1 + 2 1 +2+

1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

+3 + 4 3 +4+

1 2h 1 2h

(3.24)

), ? ?1 Γ (1 ? cθ) Γ

= 0 . Those of the K2 (θ) form are

θ 2iπ ?θ 2iπ

+ + ×

3 4 3 4

Γ Γ

θ 2iπ ?θ 2iπ

?θ 2iπ θ 2iπ

Γ Γ

θ 2iπ ?θ 2iπ

+1 + 2 1 +2+

?θ 2iπ θ 2iπ

1 2h 1 2h

Γ Γ

1 2iπc 1 2iπc

?θ 2iπ θ 2iπ

+1 + 4 1 +4+ + cθ

1 2h 1 2h

Γ Γ

+ +

1 2iπc 1 2iπc

Γ Γ

+ +

1 2 1 2

+ +

(

)

(3.25)

with (

)t =

,c

=

2ih , π

and

θ 2iπ ?θ 2iπ

K2 ( θ ) =

?1 Γ (1 ? cθ) Γ

+ +

3 4 3 4

Γ Γ

?θ 2iπ θ 2iπ

Γ Γ 17

θ 2iπ ?θ 2iπ

+1 + 2 1 +2+

1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

+3 + 4 3 +4+

1 2h 1 2h

× with ( )t = ?( ), ?

Γ Γ

θ 2iπ ?θ 2iπ

+ +

1 2iπc 1 2iπc

Γ Γ

?θ 2iπ θ 2iπ

+ +

1 2 1 2

+ +

1 2iπc 1 2iπc

(

+ cθ

)

(3.26)

=0 .

(Note that in the limit c → ∞, with c ?xed, the symmetric (respectively antisymmetric) K2 (θ) tends to the symmetric (respectively antisymmetric) K1 (θ), again as expected.)

3.4.3 Sp(N ) For Sp(N ) there are again two minimal solutions of the form K1 (θ): Γ K1 ( θ ) = Γ with ( )t =

θ 2iπ ?θ 2iπ

+ +

3 4 3 4

Γ Γ

?θ 2iπ θ 2iπ

+ +

1 2 1 2

Γ Γ

θ 2iπ ?θ 2iπ

+1 + 2 +1 + 2

1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

+1 + 4 +1 + 4

1 2h 1 2h

(3.27)

, ?

θ 2iπ ?θ 2iπ

= 0 , and Γ Γ

?θ 2iπ θ 2iπ

Γ K1 ( θ ) = Γ with ( )t K2 (θ) are

+ +

3 4 3 4

+ +

1 2 1 2

Γ Γ

θ 2iπ ?θ 2iπ

+1 + 2 +1 + 2

1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

+3 + 4 +3 + 4

1 2h 1 2h

(3.28)

= ?(

),

θ 2iπ ?θ 2iπ

= 0. The two minimal boundary S -matrices of the form

3 4 3 4 ?θ 2iπ θ 2iπ θ 2iπ ?θ 2iπ

?1 Γ K2 ( θ ) = (1 ? cθ) Γ

+ +

Γ Γ

θ 2iπ ?θ 2iπ

Γ Γ

+1 + 2 +1 + 2

?θ 2iπ θ 2iπ

1 2h 1 2h

Γ Γ

1 2iπc 1 2iπc

?θ 2iπ θ 2iπ

+1 + 4 +1 + 4 + cθ

1 2h 1 2h

Γ × Γ

+ +

1 2iπc 1 2iπc

Γ Γ

+ +

1 2 1 2

+ +

(

)

(3.29)

with

(

)t

=

, ?

= 0 , and

θ 2iπ ?θ 2iπ

?1 Γ K2 ( θ ) = (1 ? cθ) Γ

+ +

3 4 3 4

Γ Γ

θ 2iπ ?θ 2iπ

?θ 2iπ θ 2iπ

Γ Γ

θ 2iπ ?θ 2iπ

+1 + 2 +1 + 2

?θ 2iπ θ 2iπ

1 2h 1 2h

Γ Γ

1 2iπc 1 2iπc

?θ 2iπ θ 2iπ

+3 + 4 +3 + 4 + cθ

1 2h 1 2h

Γ × Γ =

+ +

1 2iπc 1 2iπc

Γ Γ

+ +

1 2 1 2

+ +

(

)

(3.30)

with

(

)t

= ?(

), c

2ih . π

?xed, the K2 (θ) with (As c → ∞, with c the respective property, as expected.) 18

(

)t

= ±(

) tends to the K1 (θ) with

3.4.4 SU (N )-conjugating Here it is only K1 (θ) that provides valid boundary S -matrices. There are two minimal possibilities, with symmetric and antisymmetric K1 ( θ ) = Γ Γ

θ 2iπ ?θ 2iπ

, with ( )t = , and (3.31)

+ +

1 4 1 4

Γ Γ + +

1 4 1 4

?θ 2iπ θ 2iπ

+1 + 4 1 +4+

?θ 2iπ θ 2iπ

1 2h 1 2h 1 2h 1 2h

K1 ( θ ) =

Γ Γ

θ 2iπ ?θ 2iπ

Γ Γ

+3 + 4 3 +4+

with (

)t = ?(

).

(3.32)

We have been unable to make contact between our solutions and a rational limit of the trigonometric solutions in [11].

3.5

Constraints on E : the symmetric-space correspondence

We now turn our attention to the constraints imposed on the matrices E . We recall that E? = E and E 2 = I (3.33)

were to be imposed in all cases, except that of SU (N )-conjugating, due to unitarity and analyticity. For the case of SU (N )-conjugating (3.33) is replaced by E?E = I and det E = 1 . (3.34)

The further constraints particular to the di?erent groups were Group SU (N ) SO (N ) Sp(2n) SU (N )conjugating K1 ( θ ) Tr(E ) = 0 Tr(E ) = 0, E t = ±E Tr(E ) = 0, JE t J = ±E E t = ±E K2 ( θ ) 1 (1 + d ) Tr(E ) = ih π c 2ih Tr(E ) = cπ , or Tr(E ) = 0, Et = E E t = ?E ih , Tr(E ) = 0, or Tr(E ) = 2 cπ t t JE J = E JE J = ?E no solution

3.5.1 SU (N ) We begin by considering SU (N ), where in addition to the constraints imposed by unitarity and analyticity we have a single extra constraint on the trace of E . From the ?rst two 19

constraints we can express E as the conjugate of a diagonal matrix X by an SU (N ) matrix, E = Q? XQ where X is of the form X= with Q ∈ SU (N ) IM 0 0 ?IN ?M ,

IM is the M × M identity matrix and 0 ≤ M ≤ N (see appendix 5.2.1). By the cyclicity of trace, if we impose the trace condition for K1 then N must be even and equal to 2M . If we impose the condition for K2 we obtain a condition on c and d, and so ?nd K1 ( θ ) E = Q? 1 IN/2 0 0 ?IN/2 Q1 E = Q? 2 IM 0 0 ?IN ?M K2 ( θ ) Q2

where

1 c

+

1 d

=

π (2M ?N ) ih

with Qi ∈ SU (N ). We can see that the case K1 corresponds to the limit of K2 in which we take M = N , that is the c, d → ∞, as we would expect. 2 Thus we have parametrized the possibilities for E with a matrix Q ∈ SU (N ) and an integer M . Once M is ?xed, the suitable E form a space isomorphic to the symmetric space SU (N ) S (U (M ) × U (N ? M ))

where the correspondence is between an element E = Q? XQ and the left coset of H = S (U (M ) × U (N ? M )) by Q. In the same way the possible E for K1 (θ) form a space isomorphic to SU (N ) (only for N even). S (U (N/2) × U (N/2)) 3.5.2 SO (N ) We now consider SO (N ), where in addition to the constraints associated with SU (N ) we also have E t = ±E . We consider ?rst the case E t = E : E ? = E , E t = E ? E ? = E. So E is a symmetric real matrix, and we can diagonalize it by conjugating with a matrix R ∈ SO (N ). Since E squares to the identity the diagonal matrix must be of the form X= IM 0 0 ?IN ?M .

Then imposing the constraints on the trace of E is the same as for SU (N ) and we have 20

K1 ( θ )

t E = R1

K2 ( θ ) R1

t E = R2

IN/2 0 0 ?IN/2

IM 0 0 ?IN ?M

R2

where c =

2ih π (2M ?N )

with Ri ∈ SO (N ). In a similar way to the SU (N ) case, once M is ?xed, the space of matrices E is isomorphic to SO (N ) , S (O (M ) × O (N ? M )) with the E for K1 (θ) isomorphic to SO (N ) . S (O (N/2) × O (N/2)) Thus, in the same way as for SU (N ), we have an isomorphism between the space of allowed E and the symmetric spaces (see appendix 5.2.2 for more details). The remaining case to consider is that of antisymmetric E . We ?nd in this case (see appendix 5.2.6) that the matrices E form a space isomorphic to two copies of the symmetric space SO (N ) . U (N/2) 3.5.3 Sp(N ) In addition to the SU (N ) constraints, for Sp(N ) we also have JE t J = ±E . We consider ?rst the case JE t J = ?E , with E2 = I and JE t J = ?E ? JE t JE = ?I ? E t JE = J

so that, since we also know E ∈ U (N ), we must have E ∈ Sp(N ). After appealing to an argument involving quarternionic matrices (appendix 5.2.3) we ?nd that the space of allowed E for K2 is isomorphic to Sp(N ) . Sp(M ) × Sp(N ? M ) In the case of K1 (θ) we again require M =

N 2

and the E -space is isomorphic to

Sp(N ) . Sp(N/2) × Sp(N/2)

21

For K1 (θ) with JE t J = +E , E is conjugate over C to IN/2 0 0 ?IN/2 as T r (E ) = 0. We ?nd (see appendix 5.2.7) that the allowed E form a space isomorphic to Sp(N ) . U (N/2) 3.5.4 SU (N )-conjugating The last case to consider is that of SU (N )-conjugating. We recall that in this case the constraints due to unitarity and analyticity were slightly modi?ed, to E?E = I and det E = 1 .

Crossing-unitarity imposed E t = ±E . Taking the symmetric case ?rst, the allowed E form a set {E |E ? E = I, E t = E, det E = 1} which turns out (see appendix 5.2.4) to be isomorphic to SU (N ) . SO (N ) Lastly, we turn to the antisymmetric case {E |E ? E = I, E t = ?E, det E = 1} . This we ?nd (appendix 5.2.5) is isomorphic to {1, ω 2 } × SU (N ) Sp(N ) where any ω s.t. ω N = ?1 is chosen.

4

The PCM boundary S -matrices

In this section we construct the boundary S -matrices for the principal chiral model. We recall [29] that the bulk model S -matrix has G × G symmetry and is constructed as SP CM (θ) = X11 (θ) SL (θ) ? SR (θ) 22 (4.1)

where X11 (θ) is the CDD factor for the PCM and SL,R (θ) are left and right copies of the minimal S -matrix possessing G-symmetry. Following this prescription, we shall use the minimal K -matrices from the previous section to construct boundary S -matrices for the PCM on a half-line. We then go on to explore their symmetries and make connection with the classical results of section two.

4.1

The CDD factors

Introducing the CDD factor, X11 (θ), into the bulk S -matrix for the PCM requires that we introduce an extra factor, Y11 (θ) (or Y1? 1 (θ ) in the case of SU (N ) conjugating), into the boundary S -matrix in order to satisfy crossing-unitarity. We construct the boundary S -matrix for the PCM as KP CM (θ) = Y11 (θ) KL (θ) ? KR (θ) (4.2)

where KL,R (θ) are left and right copies of the same type of minimal K -matrix, chosen from among the possibilities classi?ed in section three. In order that KP CM (θ) satisfy the crossing-unitarity equation with SP CM (θ) we require Y11 (iπ ? θ)Y11 (iπ + θ) = 1 Y11 ( iπ ? θ) 2 = X11 (2θ) Y11 ( iπ + θ) 2

iπ Y1? 1 ( 2 ? θ) = X11 (2θ) iπ Y1? 1 ( 2 + θ)

SU (N ), SO (N ) and Sp(N ), SU (N ) ? conjugating. (4.3)

The CDD factors for the bulk PCM S -matrices are X11 (θ) = (2)θ = X1? 1 (iπ ? θ ) X11 (θ) = (2)θ (h ? 2)θ SU (N ), SO (N ) and Sp(N ),

iπx ) 2h iπx ) 2h

(4.4)

We ?nd [30] the following candidates for the Y functions: Y11 (θ) = ?(1 ? h)θ h +2 Y11 (θ) = ? 2 h Y1? +2 1 (θ ) = 2 SU (N )

θ

sinh ( θ + 2 where (x)θ = θ sinh ( 2 ?

.

θ

h +1 2 h +1 2

θ

(1 ? h)θ

SO (N ) and Sp(N ) SU (N ) ? conjugating,

(4.5)

θ

23

and note that none of these factors have poles on the physical strip. We still have freedom to multiply by an arbitrary boundary CDD factor. That is, we can replace any of the above Y (θ) factors by g (θ)Y (θ), where g (θ) is a CDD factor. This allows us to introduce simple poles into the PCM boundary S -matrices. In the case where we construct a PCM S -matrix using left and right copies of K2 (θ), we wish to introduce the CDD factor8 g (θ ) = g (θ ) = h ciπ h ciπ h? h diπ h h? ciπ SU (N ),

θ

θ

SO (N ) and Sp(N )

θ

(4.6)

θ

which gives a simple pole at θ = 1 , corresponding to the formation of a boundary bound c state.

4.2

The boundary S -matrices

We now list the full PCM boundary S -matrices for the various G. We make use of the relation π . Γ(z )Γ(1 ? z ) = sin (πz ) We will also require the scalar factors Γ η (θ ) = Γ ? (θ ) = λ (θ ) = 1 4 π 2 c2 Γ Γ

θ 2iπ ?θ 2iπ

+ +

1 2 1 2

+ +

1 2h 1 2h

Γ Γ

?θ 2iπ θ 2iπ

+ +

1 2 1 2

,

Γ ν (θ ) = Γ

θ 2iπ ?θ 2iπ

+1 + 2 1 +2+

1 2iπc 1 2iπc

1 2h 1 2h

Γ Γ

θ 2iπ ?θ 2iπ

?θ 2iπ θ 2iπ

,

?θ 2iπ θ 1 +2 2iπ

Γ

1 4 π 2 c2 Γ

1 ?θ +M + 2iπc Γ 2iπ N M 1 θ θ + N + 2iπc Γ 2iπ 2iπ

1 1 ?θ 1 1 +2 + 2iπc Γ 2 +2 ? 2iπc Γ 2θ + iπ iπ 1 θ 1 1 ?θ + 2iπc Γ 2iπ + 2 ? 2iπc Γ 2iπ + 1 + ?θ 2iπ

Γ Γ

1 2iπc 1 2iπc

1 +1? M ? 2iπc Γ 2θ + N iπ M 1 ?θ + 1 ? N ? 2iπc Γ 2iπ + 1 + θ 2iπ ?θ 2iπ

1 ? 2iπc +1? θ 2iπ ?θ 2iπ

1 2iπc

,

Γ Γ

and

Γ ?n,m (θ) = Γ

+ +

n 4 n 4

Γ Γ

?θ 2iπ θ 2iπ

+ +

m 4 m 4

+ +

1 2h 1 2h

1 ? 2iπc +1?

1 2iπc

.

The PCM S -matrices can then be written as follows.

1 We are assuming, as in section 3.4, that M ≤ N 2 so that c is on the physical strip. As stated in section 3.4, the resulting PCM boundary S -matrix will possess a c ? ?c (and d ? ?d for G = SU (N )) symmetry and so will be correct for M ≥ N 2 also. 8

24

4.2.1 SU (N ) We have found two types of boundary S -matrix for SU (N ), KP CM (θ) = ?(1 ? h)θ ?(θ) ν (θ)(I + cθEL ) ? ν (θ)(I + cθER ) (whose c → ∞ limit: KP CM (θ) = ?(1 ? h)θ η (θ)EL ? η (θ)ER is a valid PCM boundary scattering matrix) where9 EL/R ∈ and KP CM (θ) = ?(1 ? h)θ λ(θ) ν (θ)(I + cθEL ) ? ν (θ)(I + cθER ) where EL/R ∈ 4.2.2 SO (N ) For SO (N ) three types have been found, KP CM (θ) = ? h +2 2 h +1 2 (1?h)θ ?(θ) ν (θ)?3,3 (θ)(I +cθEL )?ν (θ)?3,3 (θ)(I +cθER ) (4.10) SU (N ) . S (U (N ? M ) × U (M )) (4.9) SU (N ) , S (U (N/2) × U (N/2)) (4.8) (4.7)

θ

θ

(whose c → ∞ limit: KP CM (θ) = ? h +2 2

θ

h +1 2

θ

(1 ? h)θ η (θ)?3,3 (θ)EL ? η (θ)?3,3 (θ)ER

(4.11)

is a valid PCM boundary scattering matrix) where10 EL/R ∈ SO (N ) × {+1, ?1} , U (N/2)

KP CM (θ) = ?

9

h +2 2

θ

h +1 2

θ

(1 ? h)θ η (θ)?3,1 (θ)EL ? η (θ)?3,1 (θ)ER

(4.12)

We are using the symmetric space notation G/H here to denote the relevant translated Cartan immersion. Details are given in appendix 5. 10 The factor {+1, ?1} indicates that the space containing EL/R is a twofold copy of the symmetric space – no group structure is implied. See appendix 5.2.6 for details.

25

where EL/R ∈ and KP CM (θ) = ? where EL/R ∈ 4.2.3 Sp(N ) h +2 2 h +1 2

SO (N ) , S (O (N/2) × O (N/2))

θ

θ

(1?h)θ ?(θ) ν (θ)?3,1 (θ)(I +cθEL )?ν (θ)?3,1 (θ)(I +cθER ) (4.13) and c = 2ih in ?(θ). π (2M ? N )

SO (N ) S (O (N ? M ) × O (M ))

Three types of KP CM (θ) have also been found for Sp(N ), KP CM (θ) = ? h +2 2 h +1 2 (1?h)θ ?(θ) ν (θ)?3,1 (θ)(I +cθEL )?ν (θ)?3,1 (θ)(I +cθER ) (4.14)

θ

θ

(whose c → ∞ limit: KP CM (θ) = ? h +2 2

θ

h +1 2

θ

(1 ? h)θ η (θ)?3,1 (θ)EL ? η (θ)?3,1 (θ)ER

(4.15)

is a valid PCM boundary scattering matrix) where EL/R ∈ Sp(N ) , U (N/2)

KP CM (θ) = ? where

h +2 2

θ

h +1 2

θ

(1 ? h)θ η (θ)?3,3 (θ)EL ? η (θ)?3,3 (θ)ER

(4.16)

EL/R ∈ and KP CM (θ) = ? where EL/R ∈ h +2 2 h +1 2

Sp(N ) , Sp(N/2) × Sp(N/2))

θ

θ

(1?h)θ ?(θ) ν (θ)?3,3 (θ)(I +cθEL )?ν (θ)?3,3 (θ)(I +cθER ) (4.17) and c = 26 2ih in ?(θ). π (2M ? N )

Sp(N ) Sp(N ? M ) × Sp(M ))

4.2.4 SU (N )-conjugating Lastly, we have found two types of representation-conjugating boundary S -matrix for SU (N ) h h KP CM (θ) = +2 +1 ?1,1 (θ)EL ? ?1,1 (θ)ER (4.18) 2 2 θ θ where EL/R ∈ and KP CM (θ) = where EL/R ∈ {1, ω 2} × 4.3 h +2 2 SU (N ) , SO (N ) ?1,3 (θ)EL ? ?1,3 (θ)ER (ω N = ?1) . (4.19)

θ

h +1 2

θ

SU (N ) Sp(N )

Symmetries of the PCM boundary S -matrices

We now consider the symmetries possessed by the PCM boundary S -matrices. Before looking at the surviving group symmetries at the boundary, we ?rst point out a symmetry possessed by those S -matrices constructed from the K2 (θ)-type minimal solution. 4.3.1 M ? N ? M symmetry This was ?rst noted, for SU (N ) diagonal boundary scattering, in [33]. The Grassmannian symmetric spaces SU (N ) S (U (M ) × U (N ? M )) SO (N ) S (O (M ) × O (N ? M )) Sp(N ) Sp(M ) × Sp(N ? M )

are all invariant under M ? N ? M . Consequently, we might expect that the PCM boundary S -matrices constructed using matrices EL/R lying in translated Cartan constructions of these symmetric spaces would also respect this symmetry. This is exactly what we ?nd for the KP CM (θ) matrices (4.9), (4.13) and (4.17). To see this invariance, we consider how the exchange M ? N ? M a?ects the degrees of freedom in the K -matrices. The matrices EL/R are constructed as

?1 EL/R = UL/R XUL/R

where UL/R ∈ SU (N ), SO (N ) or Sp(N ) and 27

(4.20)

X=

ˇL/R = UL/R XU ˇ ?1 . Taking traces we see that c ? ?c Thus under M ? N ? M, EL/R → E L/R (and d ? ?d) under the exchange. Now note that the scalar factor ?(θ) is invariant under ˇ P CM (θ) c ? ?c and λ(θ) is invariant under c ? ?c, M ? N ? M . So KP CM (θ) → K where ˇL/R ) . (I + cθEL/R ) → (I ? cθE Now ˇL/R = UL/R (?X ˇ )U ?1 = UL/R OXO ?1U ?1 ?E L/R L/R where O= 0 IN ?M ±IM 0 .

0 IM 0 ?IN ?M

,

so under M ? N ? M

ˇ = X→X

0 IN ?M 0 ?IM

.

We choose the sign ± to ensure that det O = 1 and (noting that when N and M are even O ∈ Sp(N )) we see that UL/R ∈ G = SU (N ), SO (N ) or Sp(N ) =? (UL/R O ) ∈ G. Thus, if we denote by KP CM (θ; UL , UR ) the matrix constructed using

?1 EL/R = UL/R XUL/R

(4.21)

ˇ P CM (θ; UL , UR ) the image of this under the exchange M ? N ? M , we have and by K ˇ P CM (θ; UL , UR ) = KP CM (θ; UL O, UR O ) . K (4.22)

So we see that the KP CM (θ) matrices do respect this invariance of the symmetric spaces, in the sense that the action of the exchange M ? N ? M on the K -matrices is simply a translation in the parameter space. A further consequence of this emerges if we consider the pole structure of these K iN matrices. Restricting to the choice of parameter c = d = π(22 in the case G = SU (N ), M ?N ) so that it is analogous to the G = SO (N ), Sp(N ) cases, there is exactly one simple pole or θ = ? 1 (since M = N ). If we interpret the simple on the physical strip at either θ = 1 c c 2 pole as the formation of a boundary bound state at this rapidity, then the bound state is + + ? ? in a representation projected onto by either PL ? PR or PL ? PR , respectively, where 1 ± PL/R = (I ± EL/R ) = UL/R 2 We note 1 (I + X ) = 2 IM 0 0 0 and 28 1 (I ? X ) = 2 0 0 0 IN ?M . 1 ?1 (I ± X ) UL/R 2 (4.23)

lies on the physical strip as M ? N , and so the boundary bound state We ?nd that ± 1 c 2 representation is always the smaller of the two projection spaces. We plan to investigate further the spectrum of boundary bound states in future work, but for the moment we return to consider the surviving remnant of group symmetry at the boundary and make connections with the classical boundary conditions of section two.

4.3.2 Boundary group symmetry in the non-conjugating cases We recall [29] that the principal chiral model in the bulk possesses a global G × G symmetry, respected by the bulk S -matrices. In section two we saw that the introduction of a boundary in the classical PCM generally breaks the G × G symmetry, so that only a remnant survives, the nature of which is dictated by the boundary condition. In particular we saw in section 2.2.1 that the boundary condition (2.11),

?1 g (0) ∈ kL HkR

where kL/R parametrize left/right cosets of H ∈ G,

?1 ?1 preserves kL HkL × kR HkR .

Turning our attention to the PCM boundary S -matrices, we ?nd that KP CM (θ; kL , kR ) is invariant under exactly this symmetry. That is,

?1 ?1 KP CM (θ; kL , kR ), kLHkL × kR HkR = 0.

(4.24)

We begin with the Grassmannian cases, where it is enough to show that

?1 ?1 = 0, (I + cθEL ) ? (I + cθER ), kLhL kL × kR hR kR

where (for subscripts L and R) h are arbitrary elements of H , E = kXk ?1 and k ∈ G = SU (N ), SO (N ) or Sp(N ). But this is immediate: Xh = hX , since H is constructed to be precisely those elements in G which commute with X . For the case G/H = SO (2n)/U (n) (respectively G/H = Sp(2n)/U (n)) we note (appendix 5.2.6, resp. 5.2.7) that E = ikJk ?1 where k ∈ G, and that H = U (n) is constructed as those elements in G satisfying Jh = hJ , giving the required result.

4.3.3 Boundary group symmetry in the SU (N )-conjugating case The cases of SU (N )/SO (N ) and SU (N )/Sp(N ) are a little more subtle. Performing similar calculations to the above (and again leaving the L/R su?x implicit) we ?nd, on 29

constructing E as in appendices 5.2.4, 5.2.5, that E khk ?1 = (khk ?1 )? E , which implies

?1 ?1 ?1 ?1 ? KP CM (θ; kL, kR ) (kLHL kL × k R HR k R ) = (kLHL kL × k R HR k R ) KP CM (θ; kL , kR ) . (4.25) ?L ? V ?R . It Such a result is not surprising, since in this case KP CM (θ) : VL ? VR → V is straightforward to obtain a symmetry relation in which, as earlier in section 3.2.2, we ?L ) ? (VR ⊕ V ?R ). We consider a boundary S -matrix which is an endomorphism of (VL ⊕ V do not give details.

4.4

Concluding summary

?1 When G/H is a symmetric space, the classical boundary condition g (0) ∈ kL HkR preserves the local PCM conserved charges necessary for integrability. Thus, as stated in section 2.2.1, the possible BCs are parametrized by a moduli space G/H × G/H 11 . We have also found boundary S -matrices which are parametrized by G/H × G/H . Further, we ?nd that the global symmetry which survives in the presence of this BC is precisely that which commutes with the boundary S -matrix KP CM (θ; kL, kR )12 . So we ?nish with this Claim: The principal chiral model on G is classically integrable with boundary condition ?1 g (0) ∈ kL HkR , where kL/R ∈ G and G/H is a symmetric space; and it remains integrable at the quantum level, where its boundary S -matrix is KP CM (θ; kL , kR ).

Additional comment for v3 Recently a paper [34] has appeared which deals with osp(m|n) spin chain models. Its results are related to ours for the G = SO (N ) and G = Sp(N ) cases. Acknowledgments We should like to thank Tony Sudbery and Ian McIntosh for discussions of symmetric spaces. Our thanks also go to G? erard Watts for pointing out an error in our original discussion of crossing unitarity. NJM would like to thank Patrick Dorey and Ed Corrigan for helpful discussions, and Bernard Piette, Paul Fendley and Jonathan Evans for email exchanges. BJS would like to thank Gustav Delius, Brett Gibson and Mark Kambites for discussions. Finally NJM thanks the Centre de Recherches Math? ematiques, U de Montr? eal, where this work was begun during the ‘Quantum Integrability 2000’ program, for hospitality and ?nancial support, and BJS thanks the UK EPSRC for a D.Phil. studentship.

11 12

up to a discrete ambiguity, as further explained in appendix 6.1 with the subtlety noted above in the case of SU (N )-conjugated

30

Finally, we should like to thank G? erard Watts for pointing out an error (in our use of the crossing-unitarity relation) in earlier versions which has led to the changes (namely the slightly greater freedom in some scalar prefactors) in v3.

5

Appendix: Symmetric spaces and the Cartan immersion

Under the action of an involutive automorphism α (which may or may not be inner), a Lie algebra splits into eigenspaces g = h ⊕ k of eigenvalue +1 (h) and ?1 (k), with [h, h] ? h , [h, k] ? k , [k, k] ? h .

The subgroup H generated by h is compact, and we have taken G to be compact (type I) rather than maximally non-compact (type III). For the classical groups these are the groups G = SU (N ), SO (N ) and Sp(N ) (where the argument of Sp is understood always to be even) themselves, along with those described in the table below. The dimension is dimG?dimH , and the automorphism is given by its action on U in the de?ning representation, where X is the diagonal matrix with M +1s and N ? M ?1s and J is the symplectic form matrix, ?1 which is block-diagonal with N/2 blocks 0 and satis?es J 2 = ?IN . 1 0 symmetric space SU (N )/S (U (N ? M ) × U (M )) SO (N )/SO (N ? M ) × SO (M ) Sp(N )/Sp(N ? M ) × Sp(M ) SU (N )/SO (N ) SU (N )/Sp(N ) SO (2n)/U (n) Sp(2n)/U (n) dimension 2 M (N ? M ) M (N ? M ) M (N ? M )

N (N +1) 2 N (N ?1) 2

automorphism U → XUX U → XUX U → XUX U → U? U → ?JU ? J U → ?JUJ U → ?JUJ

?1 ?1

n(n ? 1) n(n + 1)

(We refer to the ?rst three as the ‘Grassmannian’ cases.)

5.1

The Cartan immersion

The Cartan immersion constructs G/H as a subspace of G (due to Cartan, and described brie?y in [17] or more fully in [18] (vol.II, sect.10, prop.4). Lifting α in the natural way 31

from the algebra to the group (so that α(h) = h for all h ∈ H ), under we have G/H gH → α(g )g ?1 = { α (g )g ? 1 | g ∈ G } .

(This statement, of course, depends crucially on the fact that we have chosen H so that it consists of all elements of G invariant under α; for the more general case see [39, 40].) We then have α(k ) = k ?1 for all k ∈

G H

?→ G. De?ning (5.1)

M = { k ∈ G | α (k ) = k ? 1 } ,

it turns out [39, 40] that G/H is in 1-1 correspondence with M0 , the identity-connected component of M. In the non-Grassmannian cases M = M0 , but in the Grassmannian cases M is a union of disconnected components, each of which is the Cartan immersion of a di?erent G/H ′ [40]. In order to make connections with subsection 3.5 we consider translations of the Cartanimmersed G/H . In the Grassmannian cases we translate by left-multiplying by the diagonal matrix X . (In the unitary and orthogonal cases if det X = ?1 then the resulting construction is no longer a subset of SU (N ) or SO (N ), but lies in the determinant ?1 part of U (N ) or O (N ), respectively.) In the case of SU (N )/SO (N ) we do not need to translate the Cartan construction. For SU (N )/Sp(N ) we translate by J (which has determinant 1). In the remaining cases of SO (2n)/U (n) and Sp(2n)/U (n) we translate by iJ , which has determinant (?1)n and so will be a translation into the determinant ?1 part of O (2n) if and only if n is odd. The full set of translated Cartan immersions is then SU (N ) S (U (M ) × U (N ? M )) SO (N ) S (O (M ) × O (N ? M )) Sp(N ) Sp(M ) × Sp(N ? M ) SU (N ) SO (N ) SU (N ) Sp(N ) SO (2n) U (n) Sp(2n) U (n) ? = {UXU ? |U ∈ SU (N )} ? = {UXU t |U ∈ SO (N )} ? = {UXU ?1 |U ∈ Sp(N )} ? = {U ? U ? |U ∈ SU (N )} ? = {U ? JU ? |U ∈ SU (N )} ? = {iUJU t |U ∈ SO (2n)} ? = {iUJU ?1 |U ∈ Sp(2n)} ,

and we treat each of these in turn in appendices 5.2.1 - 5.2.7. 32

5.2

The Boundary S -matrix Constraints

The aim of this appendix is to show in every case that the above constructions of the symmetric spaces can be described in terms of constraints (those from subsection 3.5) on a single complex N × N matrix E ∈ Gl(N, C). 5.2.1 {UXU ? |U ∈ SU (N )} = {E |E ? = E, E 2 = I, T r (E ) = 2M ? N } {UXU ? |U ∈ SU (N )} ? {E |E ? = E, E 2 = I, T r (E ) = 2M ? N } is obvious.

Now if E ? = E then ? U ∈ SU (N ) s.t. E = UDU ? , where D is diagonal. E 2 = I ? D 2 = I so D has diagonal entries ± 1 Thus, after possible reordering of the diagonal entries (which we absorb into U ) D= IM 0 ? 0 ?IN ?M ?

The constraint on T r (E ) implies D = X and so we have E = UXU ? , as required. 5.2.2 {UXU t |U ∈ SO (N )} = {E |E ? = E, E 2 = I, E t = E, T r (E ) = 2M ? N } {UXU t |U ∈ SO (N )} ? {E |E ? = E, E 2 = I, E t = E, T r (E ) = 2M ? N } is obvious.

Now E ? = E and E t = E imply that E is a real symmetric matrix, therefore ? U ∈ SO (N ) s.t. E = UDU t , where D is diagonal As in the case above the conditions E 2 = I and T r (E ) = 2M ? N require that D = X , so we have E = UXU t , as required. 5.2.3 {UXU ?1 |U ∈ Sp(N )} = {E |E ? = E, E 2 = I, JE t J = ?E, T r (E ) = 2M ? N } {UXU ?1 |U ∈ Sp(N )} ? {E |E ? = E, E 2 = I, JE t J = ?E, T r (E ) = 2M ? N } is obvious.

In order to show the converse we consider the following correspondence13 . Consider the representation of H by the 2 × 2 complex matrices

13

Thanks to Ian McIntosh for providing this suggestion.

a = a0 e + a1 i + a2 j + a3 k → A = a0 E + a1 I + a2 J + a3 K 33

where E= 1 0 0 1 I= i 0 0 ?i J= 0 ?1 1 0 K= 0 ?i ?i 0

If we de?ne complex conjugation on H in the standard way (note that this corresponds to hermitian conjugation of the matrices), and consider quaternions of unit length

2 2 2 aa? = (a2 0 + a1 + a2 + a3 )e = e

these correspond to elements of SU (2) = Sp(2) under the map. (Recall our de?nition of Sp(2n), as being the subset of U (2n) satisfying the condition AJAt = J . In the n = 1 case this is simply the constraint det A = 1, and so SU (2) = Sp(2).) This correspondence can be ? a11 a12 ? a21 a22 ? A=? . . . ? . . . an1 an2 generalized to Sp(2n) in the ? ? A11 A12 . . . a1n ? ? . . . a2n ? ? A21 A22 ?→? . . . .. . . ? . . . ? . . . . . ann An 1 An 2 following way. Consider ? . . . A1n . . . A2n ? ? ?=A . .. . . . ? . . . Ann

where the quaternion, aij → Aij , the 2 × 2 block, in the way described above. We de?ne ? ? J 0 ... 0 ? 0 J 0 ? 0 ?1 ? ? J =? . ; where J = ? . . . 1 0 ? . ? . 0 0 J

then the conditions AA? = I (where I is the quaternionic identity matrix) and A ∈ Sp(2n) are in exact correspondence. We now appeal to the fact that the (quaternionic) unitary matrix A can be diagonalized Q? AQ = D where Q, D ∈ U (n, H) and D is diagonal

Under the isomorphism we have established between U (n, H) and Sp(2n) this statement corresponds to Q? AQ = D where Q, D ∈ Sp(2n) and D is of the form ? ? 0 D=? ? ? D1 0 . D2 . . .. .. . . 0 0 Dn ? ? ? ? ?

with Di ∈ SU (2) .

34

We can diagonalize each SU (2) block Di by conjugating by some Pi ∈ SU (2). If we form the matrix P by placing these SU (2) blocks, in order, down the diagonal then we have P ? DP = X where X is diagonal .

Since SU (2) = Sp(2) we ?nd that P ∈ Sp(2n). Thus, for A ∈ Sp(2n), we have obtained the conjugation Q? P ? AP Q = X where Q, P, X ∈ Sp(2n) and X is diagonal .

Taking U = P Q, we have shown the following to be true: For all A ∈ Sp(2n) ? U ∈ Sp(2n) s.t. U ?1 AU = X where X is diagonal. Now, recalling the set {E |E ? = E, E 2 = I, JE t J = ?E, T r (E ) = 2M ? N }, we have E2 = I and JE t J = ?E ? EJE t = J .

Thus E ∈ Sp(N ) and we apply our result above to give E = UXU ?1 for some U ∈ Sp(N ). We ?nd that X 2 = I and T r (X ) = 2M ? N , so we have X= IM 0 0 ?IN ?M .

(Note: any X satisfying the above is conjugate to this X via some Sp(2n) matrix, which we absorb into U .) Thus we have E = UXU ?1 with the required X . 5.2.4 {U ? U ? |U ∈ SU (N )} = {E |E ? E = I, E t = E, det E = 1} {U ? U ? |U ∈ SU (N )} ? {E |E ? E = I, E t = E, det E = 1} is obvious.

Now if E ? E = I then ? Q ∈ SU (N ) s.t. E = QDQ? , where D is diagonal. If we impose the condition E t = E , we have Q? DQt = QDQ? ? DQt Q = Qt QD . Since D is diagonal, we can ?nd a diagonal matrix C s.t. C 2 = D and with the property CQt Q = Qt QC . Thus, we have E = QC 2 Q? = Q? Qt QC 2 Q? = Q? CQt QCQ? = U ?U ? setting 35 Q? CQt = U ? .

We see that det U = det C ? = ±1, and so we have U ∈ {V ∈ U (N )| det V = ±1}. Next, we show that the two sets {U ? U ? |UU ? = I, det U = 1} and {U ? U ? |UU ? = I, det U = ?1} are in fact the same. This is trivially true for N = 1, so we assume N ≥ 2 and suppose we have an element E belonging to the ?rst set, that is E = U ? U ? for U ∈ SU (N ). We consider U ′ = UT where ? ? 0 1 0 ? T =? 1 0 (note: T ∈ O (N ), det T = ?1), 0 IN ?2 U′ U′

? ?

and we see that

= U ? T T tU ? = U ?U ? = E.

We note that U ′ ? U ′ ? is a member of the second set, right multiplying by T in this way is a self-inverse operation, and thus we have shown the two sets to be equal. Thus, we have established the equality {U ? U ? |U ∈ SU (N )} = {E |E ? E = I, E t = E, det E = 1} , as required for section 3.5.4. 5.2.5 {U ? JU ? |U ∈ U (N ), det U = ±1} = {E |E ? E = I, E t = ?E, det E = 1} {U ? JU ? |U ∈ U (N ), det U = ±1} ? {E |E ? E = I, E t = ?E, det E = 1} is obvious.

Now if E ? E = I then ? Q ∈ SU (N ) s.t. E = QDQ? , where D is diagonal. If we impose the condition E t = ?E , we have Q? DQt = ?QDQ? ? DQt Q = ?Qt QD . We denote the diagonal entries of D by Di , so that D has entries Di δij . If we denote the entries of Qt Q by Pij then the condition above becomes Di δij Pjk = ?Pij Dj δjk ? Di Pik = ?Dk Pik .

36

We see that Pik = 0 ? Di = ?Dk . By a suitable rearrangement of the diagonal entries of D (which we absorb by rede?ning Q) we take D to be of the form ? ? d1 In1 0 ... ... 0 . ? ? . . ?d1 Im1 0 ? 0 ? where ni ≥ mi ≥ 0, ? . ? . . ? ? . . . ni ≥ 1, D=? . . . 0 d2 In2 ? ? . ? d .. .. j = ±di if j = i. ? . ? . . . 0 0 ... ... 0 ?dk Imk Then Qt Q (which is symmetric) ? 0 P1 0 t ? P1 0 0 ? ? ? 0 0 0 ? t t QQ=? ? 0 0 P2 ? . . . ? . . ? . ? 0 0 ... 0 0 must have the form ? 0 ... 0 0 0 ... 0 0 ? ? . . ? . . . . ? P2 ? . . . . 0 . . ? ? ? . .. . . 0 ? . ? . . . . . . 0 Pk ? t . . . . . . 0 Pk 0

where Pi is of size ni rows by mi columns.

We recall that Qt Q ∈ SU (N ), so that the ni rows of Pi are orthonormal with respect to the inner product on Cmi . Consequently, mi ≥ ni and so we must have ni = mi , with Pi ∈ SU (ni ), for all i. ? E where We now decompose D into two diagonal matrices D = D ? ? ? iIn1 0 ?id1 I2n1 0 ... 0 ? . ? ? ? 0 ?iIn1 . . 0 ? id I ? ? ? 2 2 n .. 2 ? =? D ?, E = ? . . .. ? ? ? . . 0 . ? iInk 0 0 ... 0 ?idk I2nk 0 ?iInk ? t Q = Qt QD ? DQ E Qt Q = ?Qt QE . ?

These matrices satisfy

? ? ? ?. ? ?

? = 1. We now choose Recall det E = 1 ? det D = 1, so since det E = 1 we also have det D 2 t t ? and CQ Q = Q QC whose determinant will be ±1. a diagonal matrix C such that C = D (Note that all possible C will have the same determinant.) Now we consider E = QDQ? = Q? Qt QC E CQ? = QC 2 E Q? C E = E C as C and E are diagonal 37

= Q? CQt QE CQ? .

We set R = QE Q? , then QE = RQ so that E = Q? CQt RQCQ? . Now we consider the properties of R R? = QE ? Q? = ?QE Q? = ?R Rt = Q? E Qt = Q? E Qt QQ? = ?Q? Qt QE Q? = ?R R2 = QE Q? QE Q? = ?I as E 2 = ?I . ? R? = R

Further, det R = 1 so we have R ∈ SO (N ) and R2 = ?I . Thus, appealing to the argument contained in the next section 5.2.6, we can ?nd a matrix O ∈ O (N ) such that R = OJO t. Substituting this into our expression for E we have E = Q? CQt OJO tQCQ? . Setting U ? = Q? CQt O → U ? = O t QCQ? we have E = U ? JU ? where U ∈ U (N ), det U = ±1 .

We have shown {U ? JU ? |U ∈ U (N ), det U = ±1} = {E |E ? E = I, E t = ?E, det E = 1}, as required. However, unlike the last subsection 5.2.4, here the det U = ±1 subsets are di?erent, for consider: suppose U ? JU ? = V ? JV ? where U, V ∈ U (N ), det U = 1, det V = ?1 ? det (V t U ? ) = ?1 : then U ? JU ? = V ? JV ? ? V t U ? JU ? V = J ? V t U ? ∈ Sp(N ) ? det (V t U ? ) = 1

The subsets of U (N ) such that det = ±1 are isomorphic via multiplication by ω where ω = ?1. Thus

N

{U ? JU ? |U ∈ U (N ), det U = ?1} = ω 2 {U ? JU ? |U ∈ SU (N )} So we have {E |E ? E = I, E t = ?E, det E = 1} = {1, ω 2} × {U ? JU ? |U ∈ SU (N )}, as required by subsection 3.5.4. 5.2.6 {iUJU t |U ∈ O (2n)} = {E |E ? = E, E 2 = I, E t = ?E, } {iUJU t |U ∈ O (2n)} ? {E |E ? = E, E 2 = I, E t = ?E, } is obvious.

Now, E ? = E, E t = ?E ? E ? = ?E , so we consider F = ?iE . Then F is a real matrix satisfying F tF = I and F 2 = ?I . 38

Since F is orthogonal there exists a matrix R ∈ SO (2n) such that Rt F R has the form ? O1 0 . O2 . . .. .. . . 0 0 On ? ? 0 ? ? ? ? ? ? ?

where Oi ∈ O (2) .

2 Since F 2 = ?I each Oi = ?I2 , the only solutions for which Oi ∈ O (2) are

Oi = ±? = ±

0 ?1 1 0

.

We note that it is possible to conjugate ?? by the matrix 0 1 1 0 ∈ O (2)

to obtain ?. Thus we can ?nd a matrix R′ ∈ O (2)?n ? O (2n) such that R′t Rt F RR′ = J . If we set U = RR′ then we have E = iUJU t for U ∈ O (2n) as required.

We note that {iUJU t |U ∈ O (2n)} = {iUJU t |U ∈ SO (2n)}, since iUJU t = iV JV t ? V t UJU t V = J ? det U = det V . Both sets {iUJU t |U ∈ SO (2n)} and {iUJU t |U ∈ O (2n), det U = ?1} are constructions of the symmetric space SO (2n) U (n) and so {E |E ? = E, E 2 = I, E t = ?E } is isomorphic to two copies of the symmetric space, as stated in section 3.5.2. 5.2.7 {iUJU ?1 |U ∈ Sp(2n)} = {E |E ? = E, E 2 = I, JE t J = E } {iUJU ?1 |U ∈ Sp(2n)} ? {E |E ? = E, E 2 = I, JE t J = E } is obvious. ? V t U ∈ Sp(2n) for U, V ∈ O (N )

39

To establish the converse, we consider the conditions on E E 2 = I and JE t J = E ? EJE t = ?J If we let F = ?iE then the conditions on F are F ? F = I, F 2 = ?I and F JF t = J .

E ? = E and E 2 = I ? E ? E = I .

Thus F ∈ Sp(2n), so from 5.2.3 we can ?nd a V ∈ Sp(2n) such that F = V DV ?1 , where D is a diagonal matrix. Since F 2 = ?I we must also have D 2 = ?I , so the entries of D must be ±i. As D ∈ Sp(2n) we cannot choose the signs of these diagonal entries completely arbitrarily and we ?nd we are restricted to ? ±I 0 . ±I . . .. .. . . 0 ?

? ? 0 D=? ?

? ? ?, 0 ? ±I

recalling I =

i 0 0 ?i

and with each ± freely chosen. However, we have K IK? = ?I and so we can further conjugate in Sp(2n) (which we absorb into V ) to ensure that D has all n ± signs set to +. Now we notice that J ∈ Sp(2n) and so, as above, there exists some W ∈ Sp(2n) such that D = W JW ?1. If we set U = V W then we have obtained F = UJU ?1 where U ∈ Sp(2n). Recalling that E = iF , we see that we have obtained E = iUJU ?1 as required.

6

Appendix: discrete ambiguities in boundary parameters

?1 First, recall the G × G invariance of the Lagrangian under g → gL ggR . Of course this G×G should really be Z(G) (where Z(G) is the centre of G, which is ?nite), since for z ∈ Z(G) we have g = zgz ?1 . The physics literature for the bulk principal chiral model generally is not concerned with this. In the same way we do not explore in the text the ambiguities in our boundary parameters, but we wish here at least to state them precisely, and to prove that they are ?nite in number.

40

6.1

Chiral BCs

?1 The ambiguity here arises because there may be non-trivial g1 , g2 such that g1 Hg2 = H. This requires ?1 ?1 α(g1 )hα(g2 ) = g1 hg2 ?h ∈ H ,

and thence

?1 ?1 g1 α(g1 )h = hg2 α (g 2 )

?h ∈ H. G ∩ C (H ) , H

So (by setting h = e) we see that

?1 ?1 g1 α (g 1 ) = g 2 α (g 2 ) = k

where

k∈

where C (H ) < G is the centralizer of H . But, for a symmetric space, H is a maximal Lie G subgroup (there is no Lie subgroup of greater dimension which contains H ), so H ∩ C (H ) is ?nite, and so the solutions g1 ∈ Hx, g2 ∈ Hy have x = y (since the Cartan immersion is 1-1) and are also ?nite in number.

6.2

Non-chiral BCs

G . H

G ?1 g2 = Here the potential ambiguity is that there may be non-trivial g1 , g2 such that g1 H G In contrast to the chiral case, we can push g1 through H : ?1 ?1 g 1 { α (g )g ? 1 | g ∈ G } g 2 = { α (α (g 1 )g )g ? 1 | g ∈ G } g 2

?1 = {α(α(g1)g )(α(g1)g )?1|g ∈ G}α(g1)g2 G ?1 α (g 1 )g 2 . = H

G G So the boundary is parametrized by G, up to g0 such that H g0 = H . This requires ?1 ?1 ?1 ?1 ?1 α(kg0 ) = g0 k for all k ∈ G/H ?→ G, so k α(g0 ) = g0 k . This must hold for k = e, so g0 ∈ M of (5.1), and commutes with every element of G/H ?→ G.

Such g0 form a group, which must be ?nite: for suppose not, that its algebra is generated by k0 ? k. Then [k0 , k] = 0. Also [[h, k0 ], k] ? [[k, k0 ], h] + [[h, k], k0 ], both of which are empty, so [h, k0 ] ? k0 . Thus [k0 , g] ? k0 , and k0 is an ideal, and is therefore trivial. Note the specialization (mentioned in the text) when g1 = g2 : the boundary is then ?1 parametrized by α(g1)g1 and thus by G/H (again quotiented, here by Z(G/H ), those elements of G/H which commute with all of G/H ). It is straightforward to propose a compatible PCM boundary S -matrix, though we do not do so here. 41

7

Appendix: boundary Yang Baxter and crossing-unitarity calculations

In this section we include a representative selection of the BYBE and crossing-unitarity calculations required to obtain the various constraints on the boundary S -matrices that we have considered in this paper. We hope they will be su?ciently illustrative that the interested reader can perform any calculations not presented here for themselves. First we list the minimal bulk S -matrices derived in [29] SU (N ) : σu (θ) hθ ) (1 ? 2 iπ hθ ? 2iπ hθ ? 2iπ ? hθ 2iπ + +

hθ 2iπ

σo (θ) SO (N ) : hθ ) (1 ? 2 iπ Sp(N ) : σp (θ) hθ ) (1 ? 2 iπ

? (h 2

hθ ) 2iπ

hθ 2iπ hθ ) 2iπ 14

? (h 2

where h is the dual Coxeter number and the scalar prefactors σ are Γ σu (θ) = ? Γ Γ σo (θ) = ? Γ Γ σp (θ) = ? Γ

θ 2iπ ?θ 2iπ θ 2iπ ?θ 2iπ θ 2iπ ?θ 2iπ

+ +

1 h 1 h 1 2 1 2 1 2 1 2

Γ Γ

+ + + +

1 h 1 h 1 h 1 h

Γ Γ Γ Γ

θ 2iπ ?θ 2iπ θ 2iπ ?θ 2iπ

+ + + +

?θ 2iπ θ 2iπ ?θ Γ 2 iπ θ Γ 2iπ ?θ Γ 2 iπ θ Γ 2iπ

Γ Γ Γ Γ

?θ 2iπ θ 2iπ ?θ 2iπ θ 2iπ

+1 + 2 +1 + 2 +1 + 2 +1 + 2

1 h 1 h 1 h 1 h

where Γ is the gamma function, and ? SU (N ) ? N N ? 2 SO (N ) h= ? N + 2 Sp(N ) . σ ?u (θ) = σu (iπ ? θ) .

The ? sign in the prefactor for SU (N ) re?ects the fact that this sign is an arbitrary choice. However, from consideration of the boundary bootstrap, details of which are outside the scope of this paper, the indications are that a ? sign is required for the model with non-conjugating boundary conditions, whilst a + sign is the preferred choice for representation conjugating BCs.

14

Note that

42

7.1

BYBE calculations

Recall the boundary Yang Baxter equation

kl np Sij (θ ? φ) Ijm ? K ln (θ) Smo (θ + φ) (Ioq ? K pr (φ)) =

ln pr Iij ? K kl (φ) Sjm (θ + φ) (Imo ? K np (θ)) Soq (θ ? φ ) .

For clarity of the calculations we introduce the notation u= hθ , 2iπ v= hφ , 2iπ u0 = h . 4

In any BYBE calculation the scalar prefactors cancel, and we consider here only the matrix part of the equation.

7.1.1 The SU (N ) case with K1 boundary S -matrix Substituting into the BYBE with the bulk S -matrix for the SU (N ) PCM and K1 gives ? (u ? v ) ? (u + v ) ? (u + v ) ? (u ? v ) =

Expanding out and cancelling where possible, we are left with = Thus, for the equation to be satis?ed we require the condition :=α for some constant α.

7.1.2 The Sp(N ) case with K2 boundary S -matrix Substituting into the BYBE with the bulk S -matrix for the Sp(N ) PCM and K2 gives ? (u ? v ) + t(u ? v ) +c ?v ? (u + v ) where t(u) = c ?=

2iπc . h u 2u0 ?u

+c ?u = +c ?u

? (u + v ) +c ?v × ? (u ? v )

+ t(u + v )

×

+ t(u + v )

+ t(u ? v )

and c ? is related to the original S -matrix constant c by the relation

Expanding out, cancelling where possible (noting that the terms involving 43

and cancel after some simple algebra) and rearranging (some less trivial algebra!) we are left with ?c ?uvt(u ? v )t(u + v )(2 + c ? +2? cuvt(u ? v )t(u + v ) c ?2 uv (u ? v ) ? +c ?2 uvt(u ? v ) ? ? = 0. ?c ?2 uv (u + v )t(u ? v ) ?c ?2 uv (u ? v )t(u + v ) In order for this to hold we are forced to have =α =β for some constant α i.e. ( )t = β ) ? ? ?

(Note: then β 2 = 1.) We ?nd that the equation is then satis?ed provided 2β ? 2 ? c ? Since β = ±1 we must have :=α ( )t = ?? c ? = 0 or ( and )t = ?( ) ?? c ? = ?4 . = 0.

7.1.3 The SU (N )-conjugating case Recall the conjugated BYBE

n ?p kl ? pr ? Sij (θ ? φ)(Ijm ? K ln (θ))Sm ?q ? ? K (φ)) = o ?(θ + φ)(Io ? ?

p ?r ? ln np ? (Iij ? K kl (φ))Sj ?o ? ? K (θ ))So ?q ? (θ ? φ ) . m ? (θ + φ)(Im

The K2 boundary S -matrix does not satisfy the above equation, but K1 does, under some constraints. Substituting in and using our simplifying notation we get ? (u ? v ) ? (2u0 ? u ? v ) ? (u ? v ) =

? (2u0 ? u ? v ) Expanding out and cancelling all possible terms leaves = 44

So to satisfy the conjugated BYBE we must have (

)t = ±

.

? ?V ? case considered There are two other BYBEs to consider in addition to the V ? V → V ? ?V ? → V ? V BYBE will be similar to the above, so that if we denote by so far. The V ? → V boundary S -matrix then we must have ( )t = ± . the matrix part of the V ? →V ? ? V , where the BYBE is The last case to consider is V ? V

kl ln np pr ? S? ij (θ ? φ) Ijm ? K (θ ) Smo (θ + φ) (Ioq ? K (φ)) = pr ? ln ? n ?p kl I? ?o ? ? K (θ )) So ?q (θ ? φ) . i? j ? K (φ) S? jm ? (θ + φ) (Im ? ? ? ?

Substituting into this, again with simpli?ed notation, we get ? (2u0 ? u + v ) ? (u + v ) ? (u + v ) ? (2u0 ? u + v ) = .

Expanding out and cancelling all the terms we can we have ? (2u0 ? u + v ) ? (2u0 ? u + v ) This equation is satis?ed provided = =α and = =β . ? (u + v ) ? (u + v ) = .

and ( )t = ± , we have β = ±α and so have only one independent Since ( )t = ± parameter. The conditions imposed for the conjugated SU (N ) case are thus ( )t = ± and ( )t = ± and = =α .

7.2

Crossing-unitarity calculations

Recall the crossing-unitarity equation iπ ? ? iπ lk i? l ? θ) = S? + θ) . jk (2θ )K ( 2 2 i? l We note that it is S? (2θ) that is required here, which is the crossed S -matrix. We obtain jk it by taking the standard S -matrix substituting iπ ? 2θ for 2θ and turning the matrix diagrams through 90o. (We note that the process of crossing doesn’t alter the S -matrix for the SO (N ) and Sp(N ) cases, but we go through the process anyway to illustrate the SU (N ) cases.) We again make use of some simplifying notation. K ij ( 45

7.2.1 The SO (N ) case with K2 boundary S -matrix Substituting into the crossing-unitarity equation we have τ ( iπ ? θ) 2 iπ (1 ? c( 2 ? θ)) +c ?(u0 ? u) σo (iπ ? 2θ)τ ( iπ + θ) 2 = × (1 ? 2uo + 2u)(1 ? c( iπ + θ)) 2 u0 ? u ? 2(u0 ? u) + +c ?(u0 + u) u ? ? ? ? ? ? ? ? ? ? (u 0 + u ) ? u0 ? u ? ?2(u0 ? u) + u ?2? c(u0 ? u)(u0 + u) c ?(u0 ? u)(u0 + u) + u N +c ? )t = ± ? ? ? ? ? ? ? ?

?

τ ( iπ 2 τ ( iπ 2

? θ) + θ)

+c ?(u0 ? u)

σo (iπ ? 2θ)× (1 ? c( iπ ? θ)) 2 = (1 ? 2uo + 2u)× + θ)) (1 ? c( iπ 2

In order for this to be satis?ed it is necessary to impose ( coe?cients of the terms we ?nd 1 ?2 u ? θ) τ ( iπ σo (iπ ? 2θ)(u0 + u) 2 = iπ (1 ? 2u0 + 2u) τ ( 2 + θ)

. Considering

? θ)) (1 ? c( iπ 2 iπ (1 ? c( 2 + θ))

For the coe?cients of the terms to be consistent with this we require a constraint on which depends on the choice of ±, altogether we have the matrix constraints ( )t = ?? c ? = ?4 or ( )t = ?( ) ?? c ? = 0.

From the crossing symmetry of the S -matrix we can simplify the constraint on the scalar prefactor, obtaining τ ( iπ ? θ) (u0 + u)(1 ? 2u)(1 ? c( iπ ? θ)) 2 2 = σo (2θ) iπ iπ τ ( 2 + θ) (u0 ? u)(1 ? 2u)(1 ? c( 2 + θ)) which can be written as ? θ) τ ( iπ 2 = iπ τ ( 2 + θ)

h 2 h ciπ h 2

?[1]

?

h 2 h ciπ

σo (2θ) ?

h 2

(

)t = )t = ? .

σo (2θ) (

7.2.2 The SU (N )-conjugating case The K -matrix for SU (N )-conjugating must satisfy a conjugated version of the crossingunitarity equation, ? iπ ? iπ il (2θ)K lk ( + θ) . K ij ( ? θ) = Sjk 2 2 46

For this case it is not the crossed S -matrix we require, but the standard S -matrix. Substituting in, we have ρ( iπ σu (2θ) ? θ) = 2 (1 ? 2u) ? 2u ρ( iπ + θ) . 2

which can be written as

On expanding this, we see that ( )t = ± is required. Then the condition on the scalar prefactor becomes ρ( iπ ? θ) (1 ? 2u) 2 = σu (2θ) , iπ (1 ? 2u) ρ( 2 + θ) ? θ) ρ( iπ 2 = iπ ρ( 2 + θ) σu (2θ) ( )t = )t = ? .

?[1]σu (2θ) (

8

Appendix: unitarity and hermitian analyticity calculations

In this section we prove the two results, concerning complex parameters that could consistently be set to 1, stated in section 3.2.

8.1

The non-conjugating case

We start from (3.8), E? = E and E 2 = αI where α ∈ U (1).

Since E is hermitian we can diagonalize it as D = QEQ? where Q ∈ SU (N ). Then we have D ? = QE ? Q? = QEQ? = D ? D ? = D. Further, D 2 = QEQ? QEQ? = QE 2 Q? = αI. Now D ? = D ? α ∈ R+ so α = 1, as stated in 3.2.1.

47

8.2

The conjugating case

We start from (3.13) αE = F ? EF = βI and and ρ(θ) = αω (?θ? )? 1 ρ(θ)ω (?θ) = . β

We can use the rescaling freedom in K (θ) = ρ(θ)E and K ′ (θ) = ω (θ)F ,

1 E → λE, ρ(θ) → λ ρ(θ),

1 F → κF, ω (θ) → κ ω (θ ),

to set both α and β to 1. Once this has been done some rescaling freedom still remains: the phase shift λ = eiψ , κ = e?iψ leaves the constraints unchanged. Thus, we can also insist that det E = 1 ?? det F = 1 as stated in 3.2.2.

48

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