ON INTEGRABLE FIELD THEORIES AS DIHEDRAL AFFINE GAUDIN MODELS BENOÎT VICEDO 7 1 Abstract. We introduce the notion of a classical dihedral affine Gaudin model, 0 2 associatedwithanuntwistedaffineKac-Moodyalgebraegequippedwithanactionof n thedihedralgroupD2T,T ≥1through(anti-)linearautomorphisms. Weshowthat averybroadfamilyofclassicalintegrablefieldtheoriescanberecastasexamplesof a suchclassical dihedralaffineGaudin models. Amongthesearetheprincipalchiral J 7 model on an arbitrary real Lie group G0 and theZT-graded coset σ-modelon any coset of G defined in terms of an order T automorphism of its complexification. 1 0 Most of the multi-parameter integrable deformations of these σ-models recently ] constructedintheliteratureprovidefurtherexamples. Thecommonfeatureshared h byalltheseintegrablefieldtheories,whichmakesitpossibletoreformulatethemas t - classical dihedralaffineGaudinmodels, isthefactthattheyarenon-ultralocal. In p particular,wealsoobtainaffineTodafieldtheoryinitslesser-knownnon-ultralocal e formulation as another example of this construction. h Weproposethattheinterpretationofagivenclassical non-ultralocalintegrable [ field theory as a classical dihedral affine Gaudin model provides a natural setting 1 within which to address its quantisation. At the same time, it may also furnish a v general framework for understanding the massive ODE/IM correspondence since 6 the known examples of integrable field theories for which such a correspondence 5 has been formulated can all beviewed as dihedral affine Gaudin models. 8 4 0 . 1 Contents 0 7 1 1. Motivation and introduction 1 : 2. Real affine Kac-Moody algebras 9 v i 3. Double loop algebras and Takiff algebras 19 X 4. Classical dihedral affine Gaudin models 33 r a 5. Examples of non-ultralocal field theories 67 Appendix A. Notations and some useful lemmas 91 References 93 1. Motivation and introduction The ODE/IM correspondence describes a striking and rather unexpected relation between the theory of linear Ordinary Differential Equations in the complex plane on the one hand, and that of quantum Integrable Models on the other. Concretely, the first instance of such a correspondence was formulated by V. Bazhanov, S. Lukyanov andA.Zamolodchikov forquantum KdVtheory intheseriesofseminalpapers[BLZ1] 1 2 BENOÎTVICEDO – [BLZ5], building on from the original insight of P. Dorey and R. Tateo in [DT1]. These works culminated in the remarkable conjecture of [BLZ5] stating that the joint spectrum of the quantum KdV Hamiltonians on the level L Z subspace of an ≥0 ∈ irreducible module over the Virasoro algebra is in bijection with the set of certain one-dimensional Schrödingeroperators ∂2+V (z)with‘monster’potentials V (z)of − z L L a given form. The justification for this conjecture comes from the central observation that the functional relations and analytic properties characterising the eigenvalues of the Q-operators of quantum KdV theory [BLZ2, BLZ3] on a given joint eigenvector coincide with those satisfied by certain connection coefficients of the associated one- dimensional Schrödinger equation. These ideas were soon extended to other massless integrable field theories associated with higher rank Lie algebras of classical type, see e.g. [DT2, DDT1, BHK, DDMST] and the review [DDT2]. Despite the variety of examples of the ODE/IM correspondence, its mathematical underpinning remained elusive for a number of years. This problem was addressed by B.FeiginandE.Frenkelin[FF2]wheretheyarguedthattheODE/IMcorrespondence forquantumg-KdVtheorycouldbeunderstoodasoriginating fromanaffineanalogue of the geometric Langlands correspondence. To explain this connection we make a brief digression on Gaudin models, which provide a realisation of the global geometric b Langlands correspondence for rational curves over the complex numbers. Let g be a finite-dimensional complex semisimple Lie algebra. The Gaudin model, or g-Gaudin model to emphasise the dependence on g, is a quantum integrable spin- chain with long-range interactions [G]. If we let N Z denote the number of sites ≥1 ∈ then the algebra of observables of the model is the N-fold tensor product U(g)⊗N of the universal enveloping algebra U(g) of g. The quadratic Gaudin Hamiltonians are elements of U(g)⊗N given by N Ia(i)I(j) a H := (1.1) i z z i j j=1 − X j6=i where the z , i = 1,...,N are arbitrary distinct complex numbers, Ia and I are i a { } { } dual bases of g with respect to a fixed non-degenerate invariant bilinear form , on h· ·i g, and x(i) is the element of U(g)⊗N with x g in theith tensorfactor and 1’s in every ∈ other factor. The quantum integrability of the model is characterised by theexistence of a large commutative subalgebra Zz(g) ⊂U(g)⊗N with z := {zi}Ni=1∪{∞}, known as the Gaudin algebra, containing in particular the quadratic Gaudin Hamiltonians. LetM ,i = 1,...,N beg-modules. Oneisinterestedinfindingthejointspectrumof i Zz(g) on the spin-chain Ni=1Mi. Note that a joint eigenvalue of the Gaudin algebra defines a homomorphism Zz(g) C sending each element of Zz(g) to its eigenvalue. N → The joint spectrum can therefore be described as a subset of the maximal spectrum of the commutative algebra Zz(g), i.e. the set of all homomorphisms Zz(g) C. It → was shown by E. Frenkel in [F2, Theorem 2.7(1)] that the maximal spectrum of the Gaudin algebra Zz(g) isisomorphic toacertain subquotientof thespace ofLg-valued connections onP1,knownasLg-opers, withregularsingularities inthesetz,whereLg denotes the Langlands dual of the Lie algebra g. In other words, each joint eigenvalue ON INTEGRABLE FIELD THEORIES AS DIHEDRAL AFFINE GAUDIN MODELS 3 oftheGaudinalgebraZz(g)onthegivenspin-chain Ni=1Mi willbedescribedbysuch an Lg-oper. In fact, when all the g-modules M are finite-dimensional irreducibles V i N λi of highest weights λ h∗, [F2, Conjecture 1] states that for each integral dominant i weight λ∞ h∗, the j∈oint spectrum of Zz(g) on the subspace of weight λ∞ singular ∈ vectors in N V is in bijection with the subspace of such Lg-opers with residue at i=1 λi the points z and infinity given by the shifted Weyl orbits of the weights λ and λ i i ∞ N respectively, and with trivial monodromy representation. The description of the maximal spectrum of the Gaudin algebra Zz(g) in terms of Lg-opers also generalises to the case of Gaudin models with irregular singularities; see [FFT,FFRy]. AnotherpossiblegeneralisationofGaudinmodelsisgivenbycyclotomic Gaudinmodels,introducedin[ViY1,ViY2]andmorerecently [ViY3]forthecasewith irregular singularities. A similar description of the corresponding cyclotomic Gaudin algebra of[ViY1]was recently conjectured in[LV]interms ofcyclotomic Lg-opers, i.e. Lg-opers equivariant under an action of the cyclic group. In fact, these descriptions of the various Gaudin algebras in terms of global Lg-opers on P1 follow (conjecturally in thecyclotomiccase)fromthe‘local’versionprovedbyB.FeiginandE.Frenkelintheir seminal paper [FF1] (see also [F3, F4]) which states that the space of singular vectors in the vacuum Verma module Vcrit(g) at the critical level over the untwisted affine 0 Kac-Moody algebra g, which naturally forms a commutative algebra, is isomorphic to the algebra of functions on the space of Lg-opers on the formal disc. b TheapparentsimilaritybetweenthedescriptionofthejointspectrumoftheGaudin algebra on any given spin-chain in terms of certain Lg-opers and the statement of the ODE/IM correspondence for quantum KdV theory is more than just a coincidence. Indeed, as argued in [FF2], quantum g-KdV theory can be regarded as a generalised Gaudinmodelassociatedwiththeuntwisted affineKac-Moodyalgebrag,org-Gaudin model for short, with a regular singularity at the origin and an irregular singularity of b themildestpossibleformatinfinity. Unfortunately,muchlessisknowatpresentabout b b Gaudin models associated with general Kac-Moody algebras; see however [MV, F1]. In particular, there is currently no known analogue of the Feigin-Frenkel isomorphism for describing the space of singular vectors in the suitably completed vacuum Verma module over the double affine, or toroidal, Lie algebra g. It is not even clear what the critical level should be in this setting. Nevertheless, the notion of an affine oper, or g-oper, on P1 can certainly be defined [F1] and so it ibbs tempting to speculate that the description of the spectrum of the g-Gaudin Hamiltonians in terms of Lg-opers persists when g is replaced by an affine Kac-Moody algebra. b In this spirit, the explicit form of the Lg-opers which ought to describe the joint spectrumofthequantumg-KdVHamiltoniansoncertainirreduciblemodulesoverthe W-algebraassociatedwithgwasconjecturedin[FF2],byusingasafinite-dimensional b analogy a certain description of the finite W-algebra for a regular nilpotent element b in terms of Lg-opers. Remarkably, when g = sl so that also Lg = sl , these sl -opers 2 2 2 were shown to coincide exactly, after a simple change of coordinate on P1, with the one-dimensional Schrödinger operators wbrittebn down in [BLZ5b]. Tbhis resultbnot only confirms the idea that the ODE/IM correspondence can be thought of as a particular instance of the geometric Langlands correspondence but alsoprovides strong evidence 4 BENOÎTVICEDO in support of the general claim that the joint spectrum of the higher Hamiltonians of an affine Gaudin model can be described in terms of affine opers for the Langlands dual affine Kac-Moody algebra. Another approach to testing the proposed link between the joint spectrum of the quantum g-KdV Hamiltonians and Lg-opers of the prescribed form is to follow the same strategy originally used to establish the ODE/IM correspondence for quantum KdV theory. Specifically, one should compare the functional relations and analytic b b properties of the joint eigenvalues of the Q-operators of quantum g-KdV theory on joint eigenvectors in the irreducibles over the W-algebra associated with g, with those satisfied by the connection coefficients of the associated Lg-opers. This programme b was initiated in [Su] and was further developed very recently in [MRV1, MRV2] where some remarkable functional relations, referred toas the QQb-system, were obtained for certain generalised spectral determinants of the ODE associated with the Lg-opers of [FF2] corresponding to highest weight states in represeentations of the W-algebra. Even more recently in [FH], the very same QQ-system was shown to arise as reblations in the Grothendieck ring K (O) of the category O of representations of the Borel 0 subalgebra of the quantum affine algebra Uqe(g) for an untwisted affine Kac-Moody algebra g. Analogous relations were alsoconjectured toholdwhen g isatwisted affine Kac-Moody algebra. Since non-local quantum g-KdV Hamiltonians can be associated b withelementsofK (O)bytheconstructionof[BLZ2,BLZ3,BHK],thejointspectrum b 0 b of these Hamiltonians also satisfy the QQ-systebm, thereby providing further evidence in favour of the ODE/IM correspondence for quantum g-KdV theory. e The recent developments towards formulating and ultimately proving the ODE/IM b correspondence for quantum g-KdV theory, which we briefly recalled above, can be summarised in the following commutative diagram b quantum [FF2] g-Gaudin [FF2] Lg-opers g-KdV theory model b b [FH] [MRVb1, MRV2] QQ-system (1.2) The top line of this diagram, correspondieng to the work [FF2], consisted of two steps. The first was to reinterpret quantum g-KdV theory as a particular affine g-Gaudin model. Thesecond,whichwerepresentbyadashedarrowtoemphasiseitsconjectural status,wastomakeuseoftheexistingdescriptionofthespectrumofg-Gaudinmodels b b intermsofLg-opersasananalogy. Thebigopenproblemhereistoestablishtheaffine counterpart of the latter statement to put the second step on a firm mathematical footing. Indeed, this would promote the top line in the above diagram to a proof of the ODE/IM correspondence for quantum g-KdV theory. While the top line is still partly conjectural, the bottom part of the diagram provides a solid link between both sides of the ‘KdV-oper’ correspondence of [FbF2] through the common QQ-system. Untilrelativelyrecently,thestudyoftheODE/IMcorrespondencehadbeenlimited e to describing integrable structures in conformal field theories only. This left open the ON INTEGRABLE FIELD THEORIES AS DIHEDRAL AFFINE GAUDIN MODELS 5 important question of whether similar ideas could be used to describe the spectrum of massive quantum integrable field theories aswell. The firstexampleof such amassive ODE/IM correspondence was put forward by S. Lukyanov and A. Zamolodchikov for quantumsine-Gordonandsinh-Gordontheoriesintheirpioneeringpaper[LZ]. Specif- ically, they showed that thefunctional relations andanalytic properties characterising the vacuum eigenvalues of the Q-operators of quantum sine/sinh-Gordon theory were the same as those satisfied by certain connection coefficients of the auxiliary linear problem of the classical modified sinh-Gordon equation for a suitably chosen classical solution. Subsequently, various higher rank generalisations of this massive ODE/IM correspondence for quantum affine g-Toda field theories were also conjectured, when g is of type A for rank 3 in [DFNT] and for general rank n in [AD], and more recently forageneraluntwisted affineKac-Moodyalgebra gin[IL1,IL2]aswellasexamples of e twisted type in [IS]. Another important quantum integrable field theory for which a e massive ODE/IM correspondence has been conjectured in [Lu], and further studied in e [BL,BKL],istheFateev model[Fa]. Itcanbeviewedasatwo-parameter deformation of the SU principal chiral model and as such it is equivalent [HRT] to the so called 2 SU bi-Yang-Baxter σ-model [K2]. Here as well the correspondence is a conjectured 2 link between the spectrum of the Fateev model on the one hand, and solutions of the classical modified sinh-Gordon equation on the other. A noteworthy feature of the massive ODE/IM correspondence for quantum affine g-Toda field theory in the non-simply-laced case is the appearance of the Langlands dual Lg of the affine Kac-Moody algebra g on the ODE side. This strongly suggests that the geometric Langlands correspondence may also underly the massive ODE/IM e correspondence. Oneof theaims ofthe presentpaperis tomakethefirststeptowards e e generalising theabovepicture in(1.2)forquantum g-KdV theory tomassivequantum integrablefieldtheories. Infact,asinthecaseofg-(m)KdVtheory,onetypicallystarts fromadescriptionoftheclassical integrablefieldtheory. Therefore,inafirstinstance, b one is faced with the initial problem of how to quantise the given classical integrable b field theory. We will argue that both problems are in fact closely related. The most effective approach for quantising a given classical integrable field theory and establishing its quantum integrability is the quantum inverse scattering method [FT1,KS],whosemathematicalunderpinninggaverisetothetheoryofquantumaffine algebras. In particular, it can be used to obtain functional equations such as Baxter’s TQ-relation and the QQ-system, all of which follow from corresponding relations in the Grothendieck ring of category O. Unfortunately, the quantum inverse scattering methodiswell knowntoeapplyonly undertherestrictiveassumption thattheclassical integrable field theory one startswith is ultralocal. Sincethe mainfocus ofthe present paper is to address the problem of quantising classical integrable field theories which violate this condition, we begin by briefly recalling why this condition is necessary in the standard quantum inverse scattering method. The starting point of the classical inverse scattering method, as crystalised by A. Reiman and M. Semenov-Tian-Shansky in [RS, S1], is to identify the phase space of the given classical integrable field theory with a coadjoint orbit in the smooth dual Gˆ∗ of a hyperplane Gˆ in a certain central extension of the double loop algebra G. 1 1 6 BENOÎTVICEDO The latter consists of smooth maps from the circle S1 to the loop algebra g((z)), or possibly its twist by some finite-order automorphism of g. The smooth dual is defined relative to a certain bilinear form on G given in terms of a model dependent rational function ϕ(z), called the twist function, as ((X,Y )) := dθres X(θ,z),Y (θ,z) ϕ(z)dz (1.3) ϕ z ZS1 (cid:10) (cid:11) forX,Y G. ThePoissonbracketonGˆ∗istheKostant-KirillovR-bracketassociated ∈ 1 with some solution R EndG of the modified classical Yang-Baxter equation. Given any pair of differentiab∈le functionals f and g on Gˆ∗, their Poisson bracket at a generic 1 point (L,1) Gˆ∗, where L G, may be written as ∈ 1 ∈ {f,g} (L,1) = dfL,(adL ◦R+R∗◦adL −(R+R∗)∂θ)·dgL ϕ (1.4) where dfL(cid:0) G de(cid:1)note(cid:0)(cid:0)s the Fréchet derivative of f at (L,1) and R∗ is(cid:1)(cid:1)the adjoint ∈ operator of R with respect to (1.3). In this language the theory is said to be ultralocal if R∗ = R, otherwise it is non-ultralocal. − It is well known that the integrals of motion of the classical integrable field theory can be obtained from spectral invariants of the monodromy ML of the differential operator∂ +L,whichisvaluedintheloopgroupG((z)). Givenanysmoothfunctional θ φonG((z))whichiscentral,theFréchetderivativeofthefunctionalφM : L φ(ML) 7→ on Gˆ∗ defines an element of G, and hence its Poisson bracket (1.4) with any other 1 smoothfunctionalf onGˆ∗ iswelldefined. Inparticular, theinvolutionpropertyofthe 1 integrals of motion is established by showing that for any pair of central functionals φ and ψ on G((z)) we have φM,ψM = 0 [S2]. If a functional φ on G((z)) is not central, { } however, then the Fréchet derivative dφM will in general exhibit a jump discontinuity atthebasepointofML. Inthecaseofanultralocaltheorywherethe ∂θ-termin(1.4) isabsent,thebracketnaturallyextendstosuchfunctionalsf andg withdiscontinuous Fréchetderivatives. Onecanthenevaluate φM,ψM forarbitrarysmoothfunctionals { } φ and ψ [S2], yielding the celebrated Sklyanin bracket on G((z)). The quantisation of the latter then serves as a starting point for the quantum inverse scattering method. By contrast, in the non-ultralocal case the bracket φM,ψM is clearly ill-defined for { } arbitrary smooth functionals φ and ψ. This issue has precluded the direct application of the quantum inverse scattering method to a wide range of important integrable field theories due to their non-ultralocal nature. Although generalisations of the quantum inverse scattering method capable of also accommodating non-ultralocal systems do exist, see for instance [FM, HK, SS], these remain applicable only toavery restricted class ofnon-ultralocal systems. Faced with thislimitation, thecommon strategyforquantising agiven non-ultralocal systemisto attempt to ‘ultralocalise’ it by different means. These include modifying the classical field theory itself by altering its twist function, see e.g. [FR] (and also [SS, DMV1]), finding a suitable gauge transformation which will bring it to an ultralocal form, see e.g. [BLZ1,RT],orpossiblybyfindingadualdescriptionofthetheorywhichwouldbe ultralocal. Yet such attempts at ‘curing’ a classical integrable field theory of its non- ultralocality ultimately work only in a limited number of cases. Let us mention also ON INTEGRABLE FIELD THEORIES AS DIHEDRAL AFFINE GAUDIN MODELS 7 some alternative approaches to dealing with the problem of non-ultralocality which have been put forward recently in [MW] for the Alday-Arutyunov-Frolov model and very recently in [Sch] for the λ-deformation of the AdS S5 superstring. 5 × In fact, classical g-(m)KdV theory is one of those distinguished classical integrable fieldtheorieswhichadmitsbothanultralocalandanon-ultralocalformulation, related tooneanotherthroughagaugetransformation. Itsultralocaldescriptionservesasthe b starting pointinthe approach of [BLZ1,BLZ2,BLZ3,BHK]forquantising thetheory usingthequantuminversescatteringmethod. Asrecalledabove, withinthisapproach non-localquantumg-(m)KdV Hamiltonians canbeassociatedwithelements ofK (O) 0 and the QQ-system is obtained from corresponding relations in K (O) established in 0 [FH]. In other wordbs, the bottom left arrow of the diagram in (1.2) starts from the quantisatioen of g-(m)KdV theory in its ultralocal formulation. By contrast, treating classical g-KdV as a non-ultralocal theory enables one to regard it as a classical affine 1 Gaudin model . The proposal of [FF2] to then quantise g-KdV theory by viewing it b as a classical affine Gaudin model led to the conjectural description of its quantum b spectrum in terms of affine Lg-opers, corresponding to the top line of the diagram in b (1.2). Since the ultralocal and non-ultralocal formulations of classical g-KdV theory are related by a gauge transformation we expect that their respective quantisations b shouldagree. Inthissetting,thefactthatthework[MRV1,MRV2]makesthediagram b in (1.2) commutative can be seen as evidence of this. The goal of the present paper is to initiate a program for quantising non-ultralocal classicalintegrable fieldtheoriesandproposeaframework withinwhichtounderstand the massive ODE/IM correspondence for such models. Specifically, we introduce the notion of a classical dihedral (or real cyclotomic) affine g-Gaudin model associated with an arbitrary untwisted affine Kac-Moody algebra g. We then show that classical dihedral g-Gaudinmodelsdescribeageneralclassofclassicalnon-ultralocalintegrable e field theories, namely those whose Poisson bracket is as in (1.4) with R-matrix given e bythestandardsolutionoftheclassicalYang-Baxterequationonthe(twisted)double e loopalgebraG. Weillustratethisrelationbetween classicaldihedral g-Gaudinmodels and non-ultralocal classical integrable field theories on a wide variety of examples, listed in Table 1, including the principal chiral model on a real semisimple Lie group e G andthe Z -graded coset σ-models forany T Z as well assomeof their various 0 T ≥2 ∈ multi-parameter deformations introduced inrecent years [K1,K2,DMV3,Sfe, HMS1, DMV6]. Replacing the semisimple Lie algebra g by the Grassmann envelope of a Lie superalgebra, the present formalism also describes Z -graded supercoset σ-models T [Y, Mag, Vi1, KLWY] and various deformations recently constructed [DMV4, DMV5, HMS2]. It is interesting to note, in particular, that the examples of integrable field theories for which a massive ODE/IM correspondence has been formulated can all be recast as classical dihedral affine Gaudin models. Our proposal is therefore that the problem of quantising non-ultralocal integrable field theories and that of formulating anODE/IMcorrespondenceforsuchmodelscanbothbeaddressedwithinthecontext of quantisation of dihedral (affine) Gaudin models. 1Let usnote herethat it will also follow from §5.3, where we discuss affineeg-Toda field theory,that bg-mKdVtheory can beregarded as a classical cyclotomic affine Gaudin model. 8 BENOÎTVICEDO Non-ultralocal field theory Divisor D σ Autg ∈ Principal chiral model (PCM) 2 0+2 PCM with WZ-term 2 k+·2 ,·k∞ R× e Yang-Baxter (YB) σ-model (iη·)+2 ·∞, η ∈R id >0 YB σ-model with WZ-term (k+iA)+2 ·∞, k R∈×, A R >0 ·∞ ∈ ∈ bi-Yang-Baxter σ-model eiϑ +ei(ψ+π) + , ϑ,ψ ]0, π[ ∞ ∈ 2 Z -graded coset σ-model 2 1+ T · ∞ σ = T, qq--ddeeffoorrmmaattiioonn((qq∈=R1)) p+eiϑp−+1∞+, ϑ,∈p]0,]Tπ0,[1[ |T|∈Z≥2 | | ∞ ∈ Affine Toda field theory 2 0+2 σ Coxeter · ·∞ Table 1. Examples of dihedral affine Gaudin models associated with an untwisted affine Kac-Moody algebra g. To end this introduction we motivate the defienition of classical dihedral g-Gaudin models by considering the simpler case where g is replaced by a finite-dimensional complex semisimple Lie algebra g. The datum for a (classical) g-Gaudin model with e irregular singularities can be described by a divisor D on P1, i.e. a formal sum of a e finitesubsetofpoints z = z N onP1 weighted by positiveintegers n Z for each x z. We furthe{r ir}eis=tr1i∪ct{∞att}ention in this introduction to the casxe∈whe≥re1 n = 1fora∈llx z forsimplicity. Thealgebraofobservablesoftheclassicalg-Gaudin x ∈ model is then given by the N-fold tensor product S(g)⊗N of the symmetric algebra S(g) on g. The classical quadratic Hamiltonians Hcl, i = 1,...,N of the model are i given by the same expressions as the quantum Hamiltonians H in (1.1) but regarded i as elements of S(g)⊗N. They can be obtained from the Lax matrix L(z), defined by the expression N I dz L(z)dz = a Ia(j), z z ⊗ j j=1 − X as the spectral invariants Hcl = res L(z),L(z) dz where the inner product is taken i zih i over the first tensor factor, i.e. the auxiliary space. Now let σ Autg be an automorphism of g whose order divides T Z and pick ≥1 ∈ ∈ aprimitive Tth-rootof unity ω−1 C×. Thesebothinduce actions ofthe cyclicgroup Γ := Z on g and P1, respectively∈. The quadratic Gaudin Hamiltonians of a classical T cyclotomic g-Gaudin model are similarly obtained from the same spectral invariants but using the Lax matrix defined by N 1 I dz L(z)dz = αˆ a Ia(j), T z z ⊗ j=1α∈Γ (cid:18) − j(cid:19) XX where αˆ denotes the action of α Γ on g-valued meromorphic differentials defined by combining the action on g with t∈he pullback on differentials over P1. This Lax matrix hastheΓ-equivarianceproperty σL(z) = ωL(ωz),whereσ actsontheauxiliaryspace. If we are also given an anti-linear automorphism τ Autg of g which preserves the ∈ eigenspaces of σ then we obtain an action of the dihedral group Π := D of order 2T 2T on g. Promoting also the action of Γ on P1 to an action of Π by adding complex ON INTEGRABLE FIELD THEORIES AS DIHEDRAL AFFINE GAUDIN MODELS 9 conjugation z z¯, we can define the Lax matrix of the classical dihedral g-Gaudin 7→ model by an expression similar to the above but replacing the sum over Γ by a sum over Π. Specifically, we should now take the tensor product over R rather than C and use dual basis elements of the realification of g so that we set N 1 I dz iI dz L(z)dz = αˆ a Ia(j)+αˆ − a iIa(j) . (1.5) 2T z z ⊗ z z ⊗ j=1α∈Π (cid:18) − j(cid:19) (cid:18) − j (cid:19) ! XX By construction, this Lax matrix is Π-equivariant in the sense that σL(z) = ωL(ωz) and τL(z) = L(z¯) where σ and τ both act on the auxiliary space. Inordertodescribeintegrable fieldtheories onthecircleoneshouldreplace ginthe above discussion by an untwisted affine Kac-Moody algebra g. Concretely this means replacing the dual bases Ia and I of g in the expression for the Lax matrix (1.5) a { } { } by dual bases Iea and Ia of g and working in a suitableecompletion of the tensor product. We w{ill}demon{stera}te that in this affine setting the Lax matrix (1.5), or its generalisation to other divisors D, reproduces the Lax matrices and twist functions e of all the integrable field theories in Table 1. The plan of the article is as follows. We begin in §2 by recalling some basic results about (anti-)linear automorphisms on finite-dimensional Lie algebras and affine Kac- Moody algebras. In §3 we construct a direct sum of Takiff algebras for g attached to thefinitesubsetz P1 asaquotientofadirectsumofloopalgebrasofg,anddescribe its dual space in te⊂rms of certain g-valued meromorphic differentials on P1. The main e section is §4 where we define classical dihedral g-Gaudin models and establish their e relation to ageneral family of non-ultralocal integrable field theories. The Lax matrix e is defined as the canonical element of the dual pair constructed in §3. Finally, §5 is e devotedtoadetailedconstruction ofimportantnon-ultralocal integrable fieldtheories as dihedral g-Gaudin models. We collect in an appendix some facts about dual pairs and our conventions on tensor index notation. e Acknowledgements. The author thanks Sylvain Lacroix for a careful reading of the draft and many useful comments and suggestions. 2. Real affine Kac-Moody algebras Let T Z . We denote the dihedral group of order 2T by ≥1 ∈ Π := D = s,t sT = t2 = (st)2 = 1 . 2T h | i Let Γ := s Π be the cyclic subgroup of order T, which is normal in Π. We refer to h i ⊂ elements of Γ as orientation preserving and to elements of the coset Γt as orientation reversing. Given a complex Lie algebra a, we let Auta denote the group of all linear and anti- linear automorphisms of a. The subgroup Auta of linear automorphisms is normal of index 2. We denote by Aut a the subset in Auta of all anti-linear automorphisms of − a so that Auta = Auta Aut a. − ⊔ For any χ Aut a we can identify Aut a with the coset χAuta. − − ∈ 10 BENOÎTVICEDO 2.1. Finite-dimensional Lie algebras. Let g be a finite-dimensional complex Lie algebra and σ Autgbea linear automorphism whoseorder divides T,i.e. such that ∈ σT =id. Fix a primitive Tth-root of unity ω and let T−1 g = g(j),C (2.1) j=0 M be the decomposition of g into the eigenspaces g(j),C := x g σx = ωjx of σ. { ∈ | } Let τ Aut g be an anti-linear involutive automorphism of g, namely such that − ∈ τ2 = id, and let g := x g τx = x denote the corresponding real form of g. Its 0 complexification g0 R{C i∈s na|turally i}somorphic to g. We shall assume ⊗that each of the eigenspaces g(j),C for j ZT is τ-stable, i.e. ∈ τg(j),C = g(j),C. (2.2) It follows from this, and using the property ω¯ = ω−1, that (σ τ)2 = id. We use the property (2.2) to define the real subspaces g(j) :=g(j),C g0 fo◦r each j ZT so that ∩ ∈ T−1 g = g (2.3) 0 (j) j=0 M Note that σ preserves the real subspace g only if 2j = 0 in Z . Indeed, given any (j) T x g we have σx = ωjx and τx = x but τ(σx) = σ−1(τx) = ω−jx = ω−2jσx. (j) ∈ Note that σk τ Aut g defines an anti-linear involutive automorphism of g for − each k Z . Int◦rod∈uce the corresponding real forms of g by T ∈ g := gσk◦τ = x g σkτx = x . (2.4) k { ∈ | } The notation reflects the fact that the case k = 0 gives back the original real form g . 0 We note that for each k Z , the anti-linear involutive automorphism σk τ clearly T also preserves the eigens∈pace g(j),C for each j ZT. For any p ZT the ◦anti-linear ∈ ∈ map ω−kpσk τ is also an involution (but in general not an automorphism). We shall ◦ make use of the corresponding real subspaces g := gω−kpσk◦τ = x g σkτx = ωkpx . (2.5) k,p { ∈ | } In this notation we have g = g . We shall also use the notation g for any p Z, k k,0 k,p ∈ which will be understood to mean gk,pmodT. By virtue of the relations σT = τ2 = (σ τ)2 = id satisfied by the automorphisms ◦ σ and τ, we have an action of the dihedral group Π on the complex Lie algebra g by linear and anti-linear automorphisms. That is, we have a group homomorphism r : Π֒ Autg, α r (2.6) α −→ 7−→ defined by rs := σ, rt := τ. Suppose, moreover, that g is equipped with a non-degenerate invariant symmetric 0 bilinear form , : g g R. (2.7) 0 0 h· ·i × −→