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Geometrical origin of integrability for Liouville and Toda theory

arXiv:hep-th/9303094v1 17 Mar 1993UCLA/92/TEP/36Geometrical origin of integrability for Liouville and Toda theory⋆Kenichiro Aoki and Eric D’Hoker†Department of PhysicsUniversity of California Los AngelesLos Angeles, California 90024–1547ABSTRACTWe generalize the Lax pair and B¨acklund transformations for Liouville andToda field theories as well as their supersymmetric generalizations, to the case ofarbitrary Riemann surfaces. We make use of the fact that Toda field theory arisesnaturally and geometrically in a restriction of so called W–geometry to ordinaryRiemannian geometry.

This derivation sheds light on the geometrical structureunderlying complete integrability of these systems.⋆Invited talk presented at the 877th meeting of the American Mathematical Society, USC,November 1992 and at the YITP workshop “Directions on Quantum Gravity”, Kyoto,November 1992. Research supported in part by the National Science Foundation grantNSF–PHY–89–15286.† Electronic mail addresses: aoki@physics.ucla.edu, dhoker@uclahep.bitnet.

1. Introduction.It has long been known that certain 1 + 1 dimensional classical field theoriesare completely integrable.

This property has been characterized in a wide varietyof ways, including the existence of an infinite number of conserved charges, theexistence of a B¨acklund transformation, the existence of a Lax pair, the solvabilityby the inverse scattering method and so on.† Perhaps the most fundamental char-acterization of all is that the field equations for the system arise as the flatnesscondition on a certain connection or gauge field. If this is so, then a Lax pair mayalways be deduced, and from it a B¨acklund transformation, the inverse scatteringsolution and infinite numbers of conserved charges.

Given an equation though, itis in general far from clear how to conclude whether a given system is a flatnesscondition on some connection. Many attempts at finding such an algorithm havebeen made, but at present it is unclear that any useful procedure indeed exists.Short of a decisive test on complete integrability, one may proceed from theopposite direction, and trace back the existence of a Lax pair to the geometricalcontext in which the completely integrable system arises.

It has long been suggestedthat all these integrable systems are special cases of the self–duality equations onfour dimensional gauge fields, equations that are known to be completely integrable[2]. Beautiful as this connection may be, it is also perhaps too general, and giveslittle clue as to why specifically any system arises as a reduction of the self–dualityequations.In some recent work, it was shown that Liouville theory [3] on a general back-ground geometry, arises as a constant curvature condition, which in turn is relatedto a flatness condition on an SL(2, IR) connection [4] as discovered in the groupmanifold approach [5], as well as in topological field theory considerations [6,7].The corresponding Lax pair is given by the parallel transport equation under thisSL(2, IR) connection [4].

This derivation is easily generalized to the case of N = 1super–Liouville theory, where the relevant gauge group is OSp(1, 1) [4].A natural extension to include the case of Toda field theory coupled to an ar-bitrary background geometry, requires the extension of 2–dimensional Riemanniangeometry to W–geometry [8], in which in addition to the spin 2 metric, additionalhigher spin fields are coupled. Toda field theory, here arises as a natural reduc-tion of general W–geometry to ordinary 2–d Riemannian geometry, and the flatconnection naturally arises as the Maurer–Cartan form on higher rank groups,generalizing SL(2, IR), or its supersymmetric generalizations [9].We shall review and slightly extend the above results in these lectures.† For some of the original work and standard reviews, see [1].2

2. Liouville Theory and 2–d Riemannian GeometryThe classical Liouville equation, on an arbitrary background geometry, deter-mines the Weyl factor that scales an arbitrary 2–d metric to a constant curvaturemetric; and follows from the Weyl transformations properties of the curvature.

TheLiouville equation∆ˆgφ + Rˆg + µ2e2φ = 0(2.1)is equivalent toRg + µ2 = 0gmn = e2φˆgmn(2.2)where µ2 is a real constant.Two dimensional geometry is parameterized by† a frame (or zweibein) ea =dξmema and a U(1) spin connection ω = dξmωm. Torsion and curvature are definedbyT a = dea + eb ∧ωǫba = 12dξn ∧dξmTmnaR = dω = 12dξn ∧dξmRmn(2.3)Covariant derivatives acting on tensors of weight n are defined byDa(n) ≡eam(∂m + inωm)so that the Laplacian on scalars takes the form∆g = −2D¯zDz(0)Weyl transformations are defined byema = eφˆema,ωm = ˆωm + ǫmp∂pφ(2.4)under which torsion and curvature transform asT a = eφ ˆT aRg = Rˆge−2φ + ∆gφ(2.5)Zero torsion and constant curvature are equivalent to Liouville’s equation (2.1),as can be seen from (2.2).

We now show that these condition arise as a flatnesscondition on an SL(2, IR) valued connection.† Einstein indices are denoted by m, n, · · · and frame indices by a, b · · ·. Frame indices splitup into complex conjugates under U(1) frame rotations a = (z, ¯z) where δz¯z = δ¯zz = 1,ǫzz = −ǫ¯z ¯z = i.

The metric is given by gmn = emaenbδab and the Gaussian curvature byRg = 12ǫmnRmn.3

Consider the algebra G, with generators J3, Jz and J¯z, satisfying[J3, Jz] = Jz;[J3, J¯z] = −J¯z;[Jz, J¯z] = −2λzλ¯zJ3(2.6)The fundamental representation can be written in terms of 2 × 2 Pauli matrices:J3 = 12σ3;Jz = λz2 (σ1 + iσ2);J¯z = λ¯z2 (−σ1 + iσ2)(2.7)We define a G–valued gauge field, constructed from the frame ea and the U(1) spinconnection ω by assembling these fields as followsA = −iωJ3 + ezJz + e¯zJ¯z(2.8)The curvature form is easily evaluatedF = dA + A ∧A = (−iR −2λzλ¯zez ∧e¯z)J3 + T zJz + T ¯zJ¯z(2.9)Setting F = 0 yields T a = 0 and constant curvature as in (2.2) with µ2 = −2λzλ¯z.Notice that this algebra in SU(2) for µ2 < 0, SL(2, IR) for µ2 > 0 and the 2–dimensional Euclidean group E(2) for µ2 = 0.Geometrically, the significance of the flatness condition F = 0 is as follows.When F = 0, A is the Maurer-Cartan form on G, parameterized by the underlyingRiemann surface. This form always exists globally on the surface.

Upon projectiononto the symmetric space G/U(1), the U(1) generator becomes the spin connectionon the coset space, and the generators complementary to U(1) become the frameon the coset space.In fact, more generally, higher dimensional conditions forsymmetric spaces can be expressed as flatness condition as well [10].By construction, a flatness condition F = 0 is the integrability condition ona system of first order differential equations, namely the equations for paralleltransport. This provides us right away with the correct expressions for the Laxpair.

Introducing e.g. a G–doublet field ψ, we consider the system of first orderlinear differential equations in ψ(∂m + Am) ψ = 0ψ = ψ1ψ2!

(2.10)where Am is given in (2.8).This system is integrable when Fmn = 0, which,as shown above, is precisely equivalent to Liouville’s equation. This can be seeneven more clearly by first performing the Weyl rescaling (2.4) on (2.8), yielding4

the following Lax pair for Liouville theory on a general 2–d background geometry,given by frame ˆema and U(1) spin connection ˆωm:∂m −iˆωmJ3 −iǫmp∂pφJ3 + eφˆemzJz + eφˆem¯zJ¯zψ = 0(2.11)By identification with the Lax pair for Toda field theory on flat space-time, we seethat λ = λz/λ¯z plays the role of a spectral parameter, which is always real for realµ2. This parameter emerges in a natural group theoretical way within this context.From the Lax pair, we recover the B¨acklund transformation in a well–knownway by introducing the field σ : ψ2 = eσψ1 and we get∂mσ + iˆωm + iǫmp∂pφ = λ¯zˆem¯ze(φ−σ) + λzˆemze(φ+σ)(2.12)The integrability condition on this system viewed as an equation for σ is preciselythe Liouville equation (2.1), and if viewed as an equation for φ, the integrabilitycondition is a linear equation in σ:∆ˆgσ = ˆǫmn∂m(ˆǫnpˆωp)(2.13)The Lax pair and B¨acklund transformations reduce to the ones for flat backgroundgeometry, and again allow for a complete explicit solution of (2.1) [3].The flatness condition, and the equation for parallel transport may be derivedfrom an action principle.S =Ztr N F(2.14)where N is an auxiliary field in the (co) adjoint representation of the algebra.

Uponeliminating N z and N ¯z, torsion is set to zero, and the remaining action coincideswith the one proposed for 2–d gravity by Jackiw and Teitelboim [6].3. Two–dimensional Dilaton Gravity.The SL(2, IR) connection of (2.8) may be generalized by addition of a field “a”multiplying a generator I that commutes with all J’s [11]:A = −iωJ3 + ezJz + e¯zJ¯z + aI(3.1)and we postulate the following structure relations for G, satisfying the Jacobiidentity:[J3, Jz] = Jz;[J3, J¯z] = −J¯z;[Jz, J¯z] = µ2J3 + λI;[Ja, I] = 0(3.2)For µ2 ̸= 0, one may redefine the generator J3 by µ2 ˜J3 = µ2J3 + λI so that thealgebra is easily recognized as GL(2, IR) or U(2) in the compact case.

However for5

the two dimensional Poincar´e algebra (µ2 = 0), such a redefinition is not possible,and we have a non–trivial central extension of this Poincar´e group. The curvatureof A is:F =−iR + µ2ez ∧e¯zJ3 + T zJz + T ¯zJ¯z +da + λez ∧e¯zI(3.3)The field equations F = 0 have the following interpretation: for µ2 = 0, theinteresting case, the geometry is flat (R = 0) Riemannian (T = 0) and the oneform field “a” obeys an interesting equation as well.

Under a Weyl transformationas in (2.4), these equations reduce to those of 2–dimensional dilaton gravity [12].4. Toda Field theory and Two–Dimensional W–geometryThe study of W–geometry has begun only recently, and there are a number ofdifferent formulations which are presumably equivalent.

W–geometry, also calledW–gravity in physics literature, may be defined in terms of an action principle, inanalogy with ordinary two–dimensional gravity [7]S =Ztr N FF = dA + A ∧A(4.1)but now the fields A, F and N assume values in a more general Lie algebra G.We define a Lie algebra G (not necessarily finite dimensional) by the followingChevalley relations[hi, hj] = 0,[hi, x±αj] = ±kijx±αj,[xαi, x−αj] = δijhi(4.2)Here the generators of the Cartan subalgebra H are denoted by hi and the positive(negative) simple roots by xαi (x−αi) for i = 1, · · ·, r(≡rank G). To find all theroots, one successively commutes the simple roots, using the constraints of theJacobi identity and the Serre relationsAd(1−kij)xαixαj = 0(4.3)Equivalently, given the set of all roots ∆, one has[xβ, xγ] = Nβγxβ+γ if xβ+γ ∈∆(4.4)and Nβγ = 0 otherwise.

As a result, any root is a linear combination of the rsimple roots, with integer coefficients which are all positive (negative) for positive(negative) roots: γ = Piγiαi. The height η of a root is defined byη(γ) =Xiγi(4.5)It vanishes on the Cartan subalgebra, and equals 1 on any simple (positive) root.6

To define topological W–geometry, we introduce a general G–valued connection[13,9]A =Xiωihi +Xγ∈∆eγxγ(4.6)Here, ωi are the components of the Abelian connection with gauge group H, and eγare the generalizations of the frame on the Riemann surface. As a generalization ofordinary Riemannian geometry, W–geometry contains 2–d Riemannian geometry;this is easily seen from the fact that any semi–simple Lie algebra has an SU(2)subalgebra.

The embedding of 2–d Riemannian geometry into W–geometry is notunique in general, and has to be specified [13].eαi = Piez;e−αi = Pie¯z;eγ = 0 if |η(γ)| ≥2;PiiXjωjkji −ρiω= 0(4.7)Under the maximal embedding Pi = ρi = 1 for all i, the spin of the additional fieldsis simply related to the height of the corresponding root by spin(eγ) = |η(γ)| + 1.†The geometrical interpretation is that A is a connection in the bundle G withstructure group H over the manifold G/H. The latter is always K¨ahler, so thatthe field contents of W–geometry may be viewed as resulting from embedding aRiemann surface into a K¨ahler manifold G/H, or equivalently from dimensionalreduction of the manifold G/H to a Riemann surface.

Analogous embedding prob-lems and their relation to Toda systems were considered in [14].The field strength of the connection A may be recast in terms of the framefield eγ and connection ωi,F =Xidωihi+12Xγ∈∆eγ∧e−γ[xγ, xγ]+Xγ∈∆deγ+Xi,jωikijγj∧eγ+12Xγ′,γ′′γ′+γ′′=γeγ′∧eγ′′Nγ′γ′′(4.8)The first line on the right of (4.8) contains all the curvature terms in the Cartansubalgebra, whereas in the second line we have the contributions proportional tothe roots. The dynamics of W–gravity governed by action (4.1) yields F = 0,and this condition yields constant curvature–type conditions from the generatorsin the Cartan subalgebra, and torsion like conditions from the roots.

Note howeverthat the torsion–type conditions are non–linear in the frame fields. W–geometry† The addition of 1 results from the conversion of Einstein indices into frame indices, analogousto isospin–spin transmutation.7

is analogous to supergeometry in which torsion is not generally zero, but certaincomponents are (covariantly) constant.Toda field theory corresponds to a very natural special case of the general gaugefield (4.6), in which the generators corresponding to simple positive and negativeroots only are retained, together with those in the Cartan subalgebra, and all otherare set to zero [9]. This condition is a generalization of the maximal embedding ofordinary 2–d Riemannian geometry but where now we allow for arbitrary complexscale factors φi:eαi = exp(φi)eze−αi = exp(φi)e¯ziXjωjkji = ωρi + eaǫabDbφi(4.9)for i, j = 1, · · ·, r and eγ = 0 when |η(γ)| ≥2.

That this Ansatz is consistent canbe shown from the expression for the curvature in this case:F =Xie¯z∧ez h−Dzωi¯z −D¯zωiz + exp(2φi) −Tz¯zaωiahi + eφiTz¯zzxαi + eφiTz¯z ¯zx−αii(4.10)Thus for r unknown fields φi, we have r equations, together with the torsionconstraints of the two dimensional background geometry. These r fields satisfy theToda field equations [15] on a two dimensional geometry with frame ea and U(1)spin connection ω:∆gφi +Xjexp(2φj)kji + Rgρi = 0(4.11)Recall that the Toda equations are Weyl or conformal invariant for any underlyingLie algebra:gmn →gmne2σ;φi →φi + σ(4.12)However, when G is a Kac–Moody algebra, the Cartan matrix has rank r −1and there is an eigenvector with zero eigenvalue, denoted by ni.

As a result, theparticular combination φ = Piniφi satisfies a free field equation∆gφ + Rgρini = 0(4.13)Elimination of this field results in breaking of the Weyl invariance for the remainingequation. In this way for example, one obtains the famous sine–Gordon equation8

for G = S ˆU(2). To see this, we definegmn = e−φ1−φ2ˆgmn;ϕ = 12(φ1 −φ2)(4.14)so that ϕ satisfies a generalization of the sine-Gordon equation to a general back-ground geometry.∆ˆgϕ + 4 sinh 2ϕ + Rˆg = 0(4.15)This equation is no longer Weyl invariant.The Lax pair is identified as the equation for parallel transport under theG–connection A in some representation of G [9]:(∂m + Am) ψ = 0(4.16)In terms of frame index notation we get a set of very simple expressions D(0)z+Xiωizhi +Xiexp(φi)xαi!ψ = 0 D(0)¯z+Xiωi¯zhi +Xiexp(φi)x−αi!ψ = 0(4.17)These equations now provide a Lax pair for Toda field theory on an arbitraryRiemann surface.

Spectral parameters arise as in Liouville theory. For G a Kac–Moody algebra, the field φ may be eliminated from the Lax pair as well.From the Lax pair, we construct the B¨acklund transformation [9] by passingfrom homogeneous coordinates ψ of a linear representation of G to inhomogeneouscoordinates of a non–linear realization of G. We shall now examine how this is donefor an arbitrary (finite–dimensional) representation µ with highest weight vectorµ, and highest weight |0; µ⟩[16].

All other weights are built by applying loweringoperators:|j1 · · · jp; µ⟩= x−αjp · · · x−αj1|0; µ⟩(4.18)where it is understood that |j1 · · · jp; µ⟩= 0 if the corresponding weight does notbelong to the weight diagram of µ. Application of Cartan generators and simple9

roots is straightforward:hj|j1 · · · jp; µ⟩= λ(p+1)j;µ|j1 · · · jp; µ⟩x−αj|j1 · · · jp; µ⟩= |j1 · · · jpj; µ⟩xαj|j1 · · · jp; µ⟩=pXq=1δj,jqλ(q)j;µ|j1 · · ·ˆjq · · · jp; µ⟩(4.19)Here the hat denotes omission andλ(q)j;µ ≡µj −q−1Xm=1kjjm(4.20)B¨acklund conjugate variables are defined as⟨0; µ|ψ|0; µ⟩expψj1···jp;µ = ⟨j1 · · · jp; µ|ψ|0; µ⟩(4.21)From sandwiching the Lax equations between the states ⟨j1 · · ·jp; µ| and |0; µ⟩, weget the B¨acklund transformations:D(0)zψj1...jp;µ −pXq=1φjq+ ipωz +rXi=1exp{φi + ψj1...jpi;µ −ψj1...jp;µ} −exp{φi + ψi;µ}= 0D(0)¯zψj1...jp;µ +pXq=1φjq+ ipωz +pXq=1λ(q)jq;µ exp{φjq + ψj1...ˆjq...jp;µ −ψj1...jp;µ} = 0(4.22)For SU(n) = G and µ the fundamental representation, the rank of the group isprecisely the dimension of the fundamental representation −1, so that the numberof Toda fields φi and the number of B¨acklund fields ψi is the same. In this case,we have a B¨acklund transformation in the usual sense.A few remarks are in order for the case of Kac–Moody algebras, when theCartan matrix has a zero eigenvalue.

In this case, equation (4.9) is not invertiblefor the U(1) connections ωi, and by the same token its validity requires already acondition on the fields φ. Denote ni the zero eigenvector of k, then we must have0 = ωρini + eaǫabDbφ(4.23)where φ was defined in (4.13).Equation (4.23) always implies (4.13), but thereverse is only true in general on compact surfaces, where the Laplacian is invertible10

up to a constant. Furthermore, condition (4.23) also imposes a restriction on thebackground geometry.

It is not in general true that ω is the curl of a single valuedscalar field φ. This requires that all its 1–cycles are trivial on the surface.

Ofcourse, if φ is not assumed to be single valued, then this restriction does not apply.For the Kac–Moody case, there is another Lax pair realization, which we shallnow discuss.† We start with an ordinary Lie algebra G of rank r, and Cartan matrixkij and we retain all the simple positive and negative roots, plus the highest positiveand negative roots, with all other roots zero:eαi = ez exp φieγ = ie¯z exp φe−αi = e¯z exp φie−γ = iez exp φ(4.24)where αi are the simple positive roots i = 1, · · ·, r and γ is the highest root:γ = Piγiαi. With this Ansatz, the torsion equations in (4.8) are still linear in theframe fields, and requireiXjωjkji = ωρi + eaǫabDbφi(4.25a)iXi,jωjkjiγi = −ω −eaǫabDbφ(4.25b)Here, since k is the Cartan matrix of a finite dimensional algebra, (4.25a) mayalways be solved for ωj.

Equation (4.25b) on the other hand represents a constraintfor the field φ.There are r curvature equations for the fields φi, which read:∆gφj −Rgρj +Xie2φikij +Xie2φβikij = 0(4.26)where we have defined the vector β from the highest root commutator:[xγ, x−γ] =Xiβihi. (4.27)The first order differential equation (4.25b) implies a second order equation, anal-† A very similar construction was discussed in [17].11

ogous to (4.26):∆gφ −Rg −Xi,je2φikijγj −Xi,je2φβikijγj = 0(4.28)Introducing the following r + 1 dimensional quantities:φI = (φI, φ)ρI = (ρi, 1)βI =βi, −1kIJ =kij −PℓkiℓγℓPℓβℓkℓj −Pℓ,mβℓkℓmγm(4.29)we may recast the Toda equation in the standard form∆gφI −RgρI +XJe2φJkJI = 0(4.30)However, the Cartan matrix k now has one eigenvector with zero eigenvalueXIβIkIJ = 0with all other eigenvalues positive. In fact, k is the Cartan matrix of the Kac–Moody algebra extension of G.Namely, 2kIJ = (αI · αJ)/(αI · αI), whereα0 ≡Pi γiαi.

Lax pair and B¨acklund transformations are readily deduced from(4.6) and (4.24), (4.25).5. Super–Liouville Theory.Two–dimensional N = 1 supergeometry is defined by† [18] a frame EM A anda U(1) connection ΩM.The coordinates are denoted (ξ, ξ, θ, θ) where ξ, ξ arecommuting and θ, θ are anticommuting.

We will work with functions of these vari-ables called “superfields”, which may be expanded in terms of its anticommuting†Einstein indices are denoted by M, N, · · ·, and run over the coordinates ξ, ¯ξ, θ, ¯θ; frameindices are denoted by A = (a, α), a = z, ¯z and α = ±. We have (γz)++ = (γ ¯z)−−= 1,ǫzz = −ǫ¯z ¯z = i and ǫ++ = −ǫ−−= i/2.12

coordinates. A superfield F(ξ, ¯ξ, θ, ¯θ), for instance, may be expanded asF(ξ, ¯ξ, θ, ¯θ) = f(ξ, ¯ξ) + θψ(ξ, ¯ξ) + ¯θ ¯ψ(ξ, ¯ξ) + θ¯θ ˆf(ξ, ¯ξ)Super–covariant derivatives on superfields of U(1) weight n are defined byD(n)A= EAM(∂M + inΩM)(5.1)Torsion and curvature forms are defined byT A = dEA + EB ∧ΩǫBA = 12Ec ∧EBTBCAR = dΩ= 12EB ∧EARAB(5.2)or in terms of structure relation:[DA, DB](n)± = TABCDC + inRAB(5.3)The standard N = 1 supergravity torsion constraints areTabc = Tαβγ = 0Tαβc = 2(γc)αβ(5.4)These constraints are left invariant under super Weyl transformations given byEM a = exp(Φ) ˆEM aEM α = exp12Φ ˆEM α + ˆEM a(γa)αβDβΦΩM = ˆΩM + ˆEM aǫab ˆDbΦ + ˆEM αǫαβ ˆDβΦ(5.5)In general torsion and curvature transform asTa = exp(Φ) ˆT aT α = exp(12Φ) ˆT αR+−= exp(−Φ)( ˆR+−−2i ˆD+ ˆD−Φ)(5.6)When R+−is set to a constant in (5.6), the second line becomes the Liouvilleequation on the field Φ, assuming that the torsion constraints (5.4) are satisfied onthe background geometry.13

Topological supergravity is based on the gauge (super) group OSp(1, 1) [19].It is convenient to rewrite its structure relations as[J3, JA] = −iǫABJB;[JA, JB]± = ΓABCJC + ΓAB3J3(5.7)where the structure constants are given byΓαβc = 2(γc)αβ;Γαbγ = −Γbαγ = µ(γb)αγ;Γz¯z3 = −Γ¯zz3 = −2µ2;Γ+−3 = Γ−+3 = 2µ(5.8)all other components vanish.We now introduce OSp(1, 1) valued gauge fields,decomposed onto frame and connection as follows:A = −iΩJ3 + EAJA(5.9)The curvature form decomposes as follows:F =T A + 12EC ∧EBΓBCAJA −iR + i2EC ∧EBΓBC3J3(5.10)From (5.6), it is now clear that F = 0 corresponds to the super–Liouville equationswith R+−= µ.To construct the Lax pair, we let OSp(1, 1) act on a triplet ψ = (ψ1, ψ2, ψ3)Ttransforming under the fundamental representation. The equations(∂M + AM) Ψ = 0(5.11)are integrable precisely when F = 0, which is just the super-Liouville equation,plus the N = 1 torsion constraints; thus (5.11) is the Lax pair for this system.

Toexhibit the super–Liouville field explicitly, we perform a super–Weyl transformation(5.5), and we obtain the following projections onto frame indices:ˆD(0)α −iˆΩαJ3 + Jα exp12Φ−i(γ5)αβ ˆDβΦJ3ψ = 0(5.12a)ˆD(0)a−iˆΩaJ3 + exp(Φ)Ja + (γa)αβ ˆDβΦ exp12ΦJα −iǫab ˆDbΦJ3ψ = 0(5.12b)By squaring the differential operator in (5.12a) and using the torsion constraints,one precisely recovers (5.12b), so that we just retain (5.12a) as the basic Lax pair.14

To construct the B¨acklund transformation, we introduce the inhomogeneouscoordinatesexp Σ = ψ2ψ1η =ψ3√ψ1ψ2(5.13)assuming that ψ1 and ψ2 are of even and ψ3 of odd grading; (5.12a) now decomposesas follows:ˆD(0)+ Σ + iˆΩ+ −ˆD+Φ =pλzη exp12Φ + 12ΣˆD(0)−Σ + iˆΩ−+ ˆD−Φ =pλ¯zη exp12Φ −12ΣˆD(0)+ η = −pλz exp12Φ + 12ΣˆD(0)−η = −pλ¯z exp12Φ −12Σ(5.14)The system of equations (5.14) is integrable provided Φ satisfies the super–Liouvilleequation, whereas Σ satisfies a linear equations, analogous to (2.13):2i ˆD(−1/2)+ˆD(0)−Σ = ˆD(1/2)−ˆΩ+ −ˆD(−1/2)+ˆΩ−(5.15)Lax pair (5.12a) and B¨acklund transformation (5.14) generalize those that wereknown on a background of two–dimensional flat space [20]. In turn, we see thatthe OSp(1, 1) Toda system on the plane with global N = 1 supersymmetry maybe consistently coupled to N = 1 supergravity.6.

Toda Theory for Supergroups and N = 1 Supergeometry.We shall now be interested in coupling Toda theory for supergroups G [21]to Riemannian geometry and N = 1 supergeometry. We begin by considering aG–valued connection:A =XiΩihi +Xγ∈∆Eγxγ(6.1)with curvature equations:F =XidΩihi −12Xγ∈∆E−γ ∧Eγ[xγ, x−γ]+Xγ∈∆dEγ +Xi,jΩikijγj ∧Eγ −12Xγ=γ′+γ′′Eγ′ ∧Eγ′′Nγ′′γ′xγ(6.2)We shall again postulate the equation F = 0 for G–gravity.15

Two classes of supergroups G must be distinguished [22]. First, we have su-pergroups for which all simple positive roots may be chosen to have odd grad-ing.

In the Kac classification, the finite dimensional group with this property areA(n, n −1), B(n −1, n), B(n, n), D(n + 1, n), D(n, n) and D(2, 1; α).For thesegroups, Toda field theory in flat space is N = 1 supersymmetric and we shall seethat in these cases, Toda field theory may be coupled to N = 1 supergravity in aninvariant way.The second class of supergroups are those for which at least one simple rootmust have even grading, i.e. all the others.

Corresponding Toda field theories inflat space are not N = 1 supersymmetric, and they cannot be coupled to N = 1supergravity in an invariant way.We shall couple them here only to ordinarygravity.For supergroups in the first group, we use the convention that all simple rootsare of odd grading andNαi,αj = N−αi,−αj = 2We recover N = 1 super Toda theory by settingEαi = exp 12ΦiE+ + (D+Φi)EzE−αi = exp 12ΦiE−+ (D−Φi)E¯zEαi+αj = exp 12(Φi + Φj)Ez × 2i ̸= j1i = jE−αi−αj = exp 12(Φi + Φj)E¯z × 2i ̸= j1i = jEγ = 0 if |η(γ)| ≥3. (6.3)and whereEA, Ωis an arbitrary two dimensional supergeometry defined in (5.1).The torsion type equations are solved byiXjΩjkji = 12Ωρi + EAJABDBΦi(6.4)with the supercomplex structure JAB defined byJAB = δAB × +iA = z, +−iA = ¯z, −(6.5)The flatness condition F±± = 0 evaluated on (6.3) and (6.4) reduces to the tor-sion constraints of ordinary supergravity and F+−= 0 yields the Toda equation,16

coupled to a general N = 1 supergravity background:D−D+Φi +Xjexp(Φj)kji −i2R+−ρi = 0(6.6)The other components of FAB = 0 are automatically satisfied using this Todaequation, the Jacobi identity and the torsion constraints. The Lax pair is easilywritten down in terms of ± components: D(0)+ +XiΩi+hi +Xiexp12Φixαi!Ψ = 0 D(0)−+XiΩi−hi +Xiexp12Φix−αi!Ψ = 0(6.7)Notice that these supergroups always contain the basic supergroup OSp(1, 1).In the case of supergroups for which all simple roots cannot be chosen oddgrading, we may couple Toda field theory only to ordinary gravity.

In this case, itis still convenient to use the notation of supergravity and we embed gravity intosupergravity by setting the gravitino and auxiliary fields to zero in Wess–Zuminogauge:D(n)+= ∂θ + θD(n)z−i2θ¯θωz∂¯θ;D(n)−= ∂¯θ + ¯θD(n)¯z−i2θ¯θω¯z∂θ(6.8)The gauge field is of the following formA+ =XiΩi+hi +Xαi∈∆oddsexp12Φixαi +Xαi∈∆evensθ exp12ΦixαiA−=XiΩi−hi +Xαi∈∆oddsexp12Φix−αi +Xαi∈∆evens¯θ exp12Φix−αi(6.9)Where ∆odds, ∆evensdenote the odd and even simple roots respectively. Analogousto the previous case, the zero curvature condition reduces to the Toda equationcoupled to gravityD−D+Φi +Xαj∈∆oddsexp{Φj}kji +Xαj∈∆evensθ¯θ exp{Φj}kji −i2R+−ρi = 0(6.10)17

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