Twistor Actions for Integrable Systems

Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions.


Introduction
Many integrable systems can be obtained from dimensional reductions of the self-dual Yang-Mills equations. This is a powerful organizing principle in the theory of integrable systems [1][2][3]. Solutions of the self-dual Yang-Mills equations can be reformulated as holomorphic vector bundles on twistor space. Thus many integrable systems can be realized as reductions of holomorphic vector bundles on twistor space. Some of the features of integrable systems that look mysterious on spacetime have natural and geometrical interpretations on twistor space. For instance, it is often useful to organize the data of an integrable system by introducing a formal, complex valued "spectral parameter." In the twistor formulation, the spectral parameter is just one of the dimensions of twistor space.
A basic object in the theory of integrable systems is the Lax operator. A Lax operator is a Lie algebra valued 1-form, L, obeying the flatness condition dL`L^L " 0. The flatness condition encodes the equations of motion of an integrable system. The Lax operator can from twistor space as the parameter space of real twistor lines. In the last subsection, we describe the action of infinitesimal spacetime conformal transformations on twistor space.

Twistor Space
Twistor space is a bundle of complex structures on spacetime. An almost complex structure on R 4 is a linear map on each tangent space that squares to´1. For example, the actions of on R 4 define almost complex structures. In this example, we also have IJK "´1. This fact together with I 2 " J 2 " K 2 "´1 implies I, J, and K all anticommute. It follows that paI`bJ`cKq 2 "´pa 2`b2`c2 q . (2.2) This means the linear combination I " aI`bJ`cK is an almost complex structure for each triple pa, b, cq with a 2`b2`c2 " 1. We thus obtain a two-sphere of almost complex structures on R 4 . Let ξ be the usual holomorphic coordinate on the northern patch of the two-sphere. A convenient parametrization for this two-sphere of almost complex structures is I " 1 1`ξξ`p 1´ξξqI`pξ`ξqJ`ipξ´ξqK˘. (2. 3) The almost complex structures (2.3) have two properties that distinguish them from other almost complex structures on R 4 . First, if g is the flat Euclidean metric then I˚g " g. Second, it is always possible to find an oriented orthonormal basis of the form pe 1 , Ie 1 , e 3 , Ie 3 q. So these almost complex structures are compatible with the standard metric and orientation.
It is not too hard to see that are no other almost complex structures on R 4 that are compatible with the standard metric and orientation. To see why, first observe that a linear map on tangent space with j˚g " g must be in SOp4q. Asking for the existence of an oriented orthonormal basis of the form pe 1 , je 1 , e 3 , je 3 q implies all of the almost complex structures we are considering are SOp4q conjugate (since they all look the same in the standard oriented basis).
This conjugacy class of almost complex structures will be a quotient of SOp4q. To find the quotient, we need to compute the stabilizer. In other words, we need to pick an almost complex structure, j, and find all M P SOp4q with It is convenient to pick j " K and represent the matrices with 2ˆ2 blocks. The equation to be solved becomes˜0´1 (2.5) In this equation, 0, 1, A, B, C, and D are regarded as 2ˆ2 matrices. The solutions are the matrices`A B B A˘"`1 0 0 1˘b A``0 1 1 0˘b B, which we may as well call A`iB. The fact that the matrix we started with was in SOp4q implies A`iB P Up2q.
So the set of all almost complex structures on R 4 compatible with the standard metric and orientation is SOp4q{ Up2q » S 2 . This is the two-sphere of almost complex structures (2.3) we started with.
These almost complex structures are actually complex structures. Let px 1 , x 2 , x 3 , x 4 q be Cartesian coordinates on R 4 . For each ξ P C, define a pair of complex coordinates on Then pdλ, dµ) is an I-holomorphic basis for (the complexification of) each cotangent space of R 4 . So the almost complex structures we are discussing are in fact complex structures. None of these complex structures are more natural than the others. One of the key ideas of twistor theory is to avoid making a choice of complex structure by studying the bundle of all metric and orientation compatible complex structures. Thus the twistor space of R 4 is (as a real manifold) Z has a natural complex structure. The complex structure on R 4 is I, which is now a function of the S 2 coordinate. The one forms are holomorphic forms on twistor space if dξ is holomorphic. So we endow S 2 with the opposite of the standard complex structure. Let ζ " iξ. Then pλ, µ, ζq is a holomorphic chart on the northern patch of twistor space (ζ P C). Let r ζ be the holomorphic coordinate on the southern patch of S 2 . Define Then p r λ, r µ, r ζq is a holomorphic chart on the southern patch of twistor space ( r ζ P C). On the overlap region, This result can be summarized by saying that Z is the holomorphic vector bundle Op1q ' Op1q Ñ CP 1 .

Real Twistor Lines
In the previous subsection, we constructed twistor space as the bundle of metric and orientation compatible complex structures on flat Euclidean R 4 . Replacing the flat metric, g, with a conformally related metric, Ω 2 g, gives the same twistor space. So Z is really associated to the conformal manifold pR 4 , rgsq, where rgs is the equivalence class of metrics conformally related to the flat metric. In this subsection, we will describe how to recover pR 4 , rgsq from Z. Global sections of Z " Op1q ' Op1q Ñ CP 1 are defined by holomorphic functions, f and r f , on the northern and southern patches of CP 1 , respectively, subject to f pζq " ζ r fˆ1 ζ˙( 2.14) on the overlap region. Expanding in power series forces the global sections to be linear. On the northern patch, they have the coordinate expression The parameter space of global sections is C 4 . Call these sections complex twistor lines. A real structure on a complex manifold is an antiholomorphic involution. Twistor space inherits a real structure from the fiberwise action of the antipodal map ζ Ñ´1{ζ. Comparing with (2.6) and (2.7) gives u " x 1`i x 2 and v " x 3`i x 4 . We thus recover R 4 as the parameter space of real twistor lines. Each point of twistor space lies on a unique real twistor line. This gives a nonholomorphic projection Z Ñ R 4 sending each point in Z to the corresponding line. The point pλ, µ, ζq lies on the line with This projection defines a nonholomorphic chart, pu,ū, v,v, ζ,ζq, on twistor space that is particularly well-suited for recovering spacetime physics.

Conformal Transformations
This subsection describes the action of infinitesimal spacetime conformal transformations on twistor space. The results are summarized in Table 1. The conformal group of R 4 is SOp5, 1q. R 4 is the parameter space of real twistor lines, so the action of conformal transformations on real twistor lines is immediate. The problem is to extend this to an action on twistor space itself.
Recall that a complex twistor line has the coordinate expression (2.15) The conformal structure on the moduli space of complex twistor lines, C 4 , is fixed by the requirement that two points in C 4 are null separated if and only if the twistor lines they represent intersect. This endows the moduli space of complex twistor lines with the conformal structure of the flat metric Real twistor lines haveû "´ū andv "v. Plugging these equations into (2.20) gives the expected flat conformal structure on R 4 . Infinitesimal conformal transformations are given by conformal Killing vectors. Let F be the space whose elements are pairs, pL, pq, where L is a complex twistor line and p is a point on L. There are many ways to lift a conformal Killing vector, X, to a vector field, X 2 , on F because there are many ways to choose the action of X 2 on p. However, in general, X 2 will not have a well defined push forward to twistor space along the projection pL, pq Ñ p because the action of X 2 on p can depend on L. The key to lifting the action of X to twistor space is to choose X 2 such that the push forward to twistor space is well defined.
Concretely, let be a conformal Killing vector of C 4 . An element of F is labeled by five parameters, pu, v,û,v, ζq. Let be one choice for the lift of X to F . We want to choose X ζ such that the push forward of X 2 to twistor space along pL, pq Ñ p is well defined. This projection, pL, pq Ñ p, is given by pu, v,û,v, ζq Ñ pλ, µ, ζq " pu`vζ, v`ûζ, ζq . (2.23) The fibers of this projection have tangent vectors So the push forward of X 2 to twistor space will be well defined if rX 2 , s " rX 2 , ms " 0 modulo linear combinations of and m. A short calculation using the conformal Killing equation gives wheremeans equality modulo linear combinations of and m, and Q is constant along and m. So setting X ζ " Q gives rX 2 , s -rX 2 , ms -0. This means X 2 has a well defined projection to twistor space, namely The components of X 1 are constant along and m and therefore functions of pλ, µ, ζq alone. X 1 is the lift of the conformal Killing vector X to twistor space. The conformal Killing vectors of R 4 and their lifts to twistor space are listed in Table 1.

Chern-Simons Action
The Penrose-Ward correspondence relates integrable systems to holomorphic vector bundles on twistor space. Under this correspondence, the Lax operator becomes a p0, 1q connection, A, on twistor space obeying the partial flatness condition Here and in what follows, we are dropping the wedge product symbol for brevity. Equation (3.1) is the equation of motion of the Chern-Simons action where Ω is a p3, 0q form. There is no holomorphic p3, 0q form on twistor space so we will choose, in some sense, the next best thing, Ω has second order poles at ζ " 0 and ζ " 8 (recall 2.13). To get a sensible theory, we need boundary conditions on A at the poles of Ω.

Lorentz invariance
The poles of Ω break 4d Lorentz invariance. However, it turns out that Ω is compatible with 2d Lorentz invariance. So the Chern-Simons action we are discussing is a reasonable starting point for 2d integrable models. Let us elaborate on this point in a simple example. Consider the nonholomorphic twistor coordinates pu,ū, v,v, ζ,ζq defined by (2.18). Recall u " x 1`i x 2 and v " x 3`i x 4 and suppose we dimensionally reduce along x 2 and x 4 to get a 2d theory with spacetime coordinates x 1 and x 3 . The spacetime Killing vector lifts to the twistor space vector field (see Table 1) After dimensional reduction, X is the generator of the 2d Lorentz group (which in this example is just SOp2q). On twistor space, X 1 has no component along B ζ , so it acts trivially on the poles of Ω.
Thus Ω is compatible with 2d Lorentz invariance in this simple example. All of the other examples we will discuss work similarly.

Boundary conditions
The first integrable model we consider is the 2d principal chiral model (PCM), one of the simplest integrable sigma models. To describe the boundary conditions for this model, first define w " x 1`i x 3 and s w " x 1´i x 3 . The chart pw , s w , x 2 , x 4 , ζ,ζq is convenient for dimensional reduction. Reducing along x 2 and x 4 gives a 2d theory with coordinates w and s w .
We need to ensure that Ω CSpAq has no poles at ζ " 0 and ζ " 8.
The required boundary conditions are We also require gauge transformations to vanish at ζ " 0 and ζ " 8. These conditions ensure that Ω CSpAq is smooth 3 . We assume further that A is trivial on real twistor lines. This is a standard part of the Penrose-Ward correspondence. It is a natural condition on A because real twistor lines correspond to points in spacetime and we are ultimately interested in recovering spacetime fields. Note that s Bw " s B s w " 0 on real twistor lines, so this assumption guarantees the existence of a Lie group valued function,σ, such that Aζ "σ´1Bζσ . (3.9) σ is ambiguous up to left multiplication,σ Ñ gσ, by a Lie group valued function, g " gpu,ū, v,vq. To fix this ambiguity, setσ " id at ζ " 8. Then the value ofσ at ζ " 0 defines a Lie group valued function, σ. We assume without loss of generality thatσ is independent of arg ζ 4 .

Solution
For dimensional reduction, we drop all dependence on x 2 and x 4 . So σ " σpw , s w q and σ " pw , s w , |ζ|q. The solution of Fζ w " Fζ s w " 0 that is compatible with the boundary conditions is A "σ´1 s Bσ`σ´1A 1σ , (3.10)

11)
A clearly satisfies the boundary conditions (3.7)-(3.8). A and A 1 are gauge equivalent, so it is enough to check the equations of motion Fζ w " Fζ s w " 0 for A 1 . The poles at ζ "˘i look like they might spell trouble, but s Bw " pζ´iq and s B s w " pζ`iq, so Fζ w " Fζ s w " 0. The Lax operator is defined to be The p0, 1q forms appearing here are s Bw " 1 3 Note that s Bw s B s w dζ "´i 2 p1`ζζq´2ζ 2 p1`ζ 2 qΩ is itself a smooth form 4 After a gauge transformation, A Ñ g´1 s Bg`g´1Ag, and Aζ " pσgq´1Bζpσgq. Our boundary conditions require g " id at ζ " 0 and at ζ " 8, but g is otherwise arbitrary. So we can assumeσ is a independent of arg ζ.
L is a family of one-forms on spacetime labeled by a parameter, ζ, the spectral parameter. L is defined using A 1 rather than A because the Lax operator is simplest in the gauge with Aζ " 0. In this gauge, the Chern-Simons equations of motion Fζ w " Fζ s w " 0 imply that A w and A s w are independent ofζ. The Chern-Simons equation of motion B w A 1 s w´B s w A 1 w`r A 1 w , A 1w s " 0 implies the flatness condition dL`L 2 " 0 (3.14) for the Lax operator. The equations of motion of the PCM can be obtained by expanding (3.14) in powers of ζ. The existence of the Lax formulation implies the integrability of the PCM.
Reinserting A into the Chern-Simons action gives the action of the PCM. It is conve-

17)
A 0 obeys the Chern-Simons equation of motion s BA 0`A 2 0 " 0, and A 3 1 " 0. The Chern-Simons functional reduces to The first three terms do not contribute to the action. The calculations are similar in all three cases, so we will only bother to discuss the first term. The integral to be evaluated iś The integrand is The p0, 3q form appearing here is Let ζ " ρe iϕ . Then the integral over ϕ vanishes becauseσ is independent of ϕ in these coordinates. Similar calculations show that the second and third terms in (3.18) do not contribute to the action either. The only nonzero contribution to the action comes froḿ 1 2πi This is a boundary term and the only contribution is from the pole at ζ " 0. The trace is The only contribution is from the pole at ζ " 0, whereσ " σ. The result is The field σ " σpw , s w q is a function of w " x 1`i x 3 and s w " x 1´i x 3 only. Integrating over x 2 and x 4 gives the action of the 2d PCM 5 .

Dimensionally Reduced Gravity
Dimensionally reducing general relativity with respect to a time translation and a rotation gives the Breitenlohner-Maison (BM) model [14,17]. It is an SLp2, Rq{SOp2q coset sigma model. The solution space includes black holes. For example, the Kerr metric is stationary and axisymmetric, so it is a solution of the BM model. The BM model is defined on a flat Euclidean half-plane. In the black hole spacetime, the half-plane is a slice at fixed time and azimuthal angle. In this subsection, we derive the Lax operator and action of the BM model from the Chern-Simons action (3.2)-(3.3) on twistor space.
As before, consider the nonholomorphic chart pu,ū, v,v, ζ,ζq on twistor space. Introduce a new chart, pw,w, θ, t, ζ,ζq, by setting u " re iθ , v " z`it , w " z`ir . (3.30) We are going to dimensionally reduce with respect to θ and t. Write To ensure that Ω CSpAq is nonsingular, we choose the boundary conditions The requirement that A is trivial on real twistor lines fixes Aζ "σ´1Bζσ . As before, setσ " id at ζ " 8 and let σ be the value ofσ at ζ " 0. For dimensional reduction, assume σ " σpw,wq andσ "σpw,w, |ζ|q are independent of θ and t. To get the BM model, further assume that σ andσ are positive definite and symmetric (σ " σ T ) SLp2, Rq matrices. Define z " e´i θ ζ. The boundary conditions and the Fz w " Fzw " 0 equations of motion imply The Lax operator is Getting the equations of motion of the BM model from the Lax operator requires some care. The spacetime Killing vectors, X t " B t and X θ " B θ , lift to on twistor space (see Table 1). The orbits of X 1 t and X 1 θ are the surfaces with w " µ`λ{ζ " 2z`rˆ1 z´z˙" constant . This fixes the position dependence of the spectral parameter, z, to be The equations of motion of the BM model are obtained by computing dL`L 2 " 0 using (3.41) and expanding in powers of z. Reinserting (3.35)-(3.37) into the Chern-Simons action produces four possible terms, As before, the only nonzero contribution comes from the fourth term. The final result is the spacetime action Integrating over t and θ gives the BM model with spacetime coordinates w " z`ir and w " z´ir.

Lorentzian Signature
So far we have been working in Euclidean signature. In the remainder of this section, we briefly discuss Lorentzian signature.
Complexified spacetime, C 4 , is the moduli space of complex twistor lines. Euclidean spacetime, R 4 , is the moduli space of real twistor lines, with respect to the real structure (2.16). To get Lorentzian signature spacetimes, we need to change the real structure. The new real structure is pλ, µ, ζq Ñˆμ ζ ,λ ζ , 1 ζ˙.

(3.44)
This is an antiholomorphic involution on twistor space and therefore a real structure. Complex twistor lines are given by gives the split signature metric on R 2,2 . This becomes R 1,1 after dimensional reduction. Unlike the real structure (2.16) introduced earlier, the new real structure has fixed points. The fixed points have |ζ| " 1 and λ "μζ. The space of fixed points is Z R " R 2ˆS1 .
Let u " t`iz and v " re iθ . Define x˘" t˘r. The chart px`, x´, z, θ, ζ,ζq is convenient for dimensional reduction. Dimensionally reducing along z and θ gives the Lorentzian version of the BM model. The sigma model coordinates are x˘.
The Chern-Simons action is the same as before, The boundary conditions are Reinserting A into the Chern-Simons action gives the 2d action This action governs the dimensional reduction of general relativity to two spacetime dimensions in the case for which the 2d spacetime has Lorentzian signature.

Chern-Simon-Matter Action
Dimensional reducing N " 1 supergravity from four dimensions to two dimensions gives a supersymmetric version of the BM model. The physical degrees of freedom are described by an SLp2, Rq{SOp2q coset sigma model coupled to a pair of matter fermions. Nicolai [17] found a Lax operator for this model. In this subsection, we derive the Lax operator and action of the Nicolai model from a Chern-Simons-matter action on twistor space. The Chern-Simons-matter action is defined by coupling (3.2)-(3.3) to a pair of fermions supported on branes in twistor space. The Chern-Simons part of the action is This has the solution (3.35)-(3.37) A "σ´1 s Bσ`σ´1A 1σ , (4.2) σ is a positive definite, symmetric SLp2, Rq matrix. It is convenient to write σ " U T U , (4.4) where U is an SLp2, Rq-valued field. Note σ T " pU T U q T " U T U " σ. Of course, U is not unique. Indeed, we can rotate U by δU "´hU , (4.5) where h is an sop2q-valued field. Under this sop2q action, because h T "´h. The Lie algebra, g " slp2, Rq, decomposes as where h " sop2q and k is the orthogonal complement of sop2q in g. Note where Q P h and P P k. Under the action of (4.5), Q transforms as an SOp2q gauge field, δQ " s Bh . (4.10) In the next subsection, we are going to couple Q to a pair of fermions with an SOp2q global symmetry.

Coupling to Fermions
First, consider the free fermion action 1 4 ż pδ z"i ΩΩq s χ I γ w B w χ I . (4.11) χ I " χ I pw,wq are a pair of Majorana fermions (I " 1, 2). The delta function, δ z"i , localizes the integral to a brane at z " e´i θ ζ " i in twistor space. The integrand is independent of t and θ, so we could integrate out those directions and obtain a 2d action. The fermions have an SOp2q global symmetry, with infinitesimal action ( P R) So we can couple the fermions to Q, the SOp2q gauge field introduced in the previous subsection. We do this by promoting B w Ñ D w , where (4.14) The SOp2q charge is set equal to 3{2 to get the Nicolai model. We have suppressed an sop2q Lie algebra index on Q w . In this formula, Q w is the component of Q w with respect to the sop2q basis element`0 1 1 0˘.
We define a Chern-Simons-matter action by coupling the Chern-Simons action (4.1) to a pair of fermions supported on branes at z "˘i:

Dimensionally Reduced Supergravity
To begin, it is helpful to rewrite the uncoupled solution (4.2)-(4.3) in terms of P and Q, since Q is what couples to the fermions. First, use the basic identity p s Bσqσ´1 "´2U T P pU T q´1 to eliminate σ. Then make a gauge transformation to eliminate factors of U T . Let A 2 " pU T q´1 s BU T`p U T q´1A 1 U T . The result is In the coupled theory, A 2 becomes where r A is a sum of fermion bilinears. We expect r A P sop2q because the fermions couple to A through Q P sop2q. We also assume r Aζ " 0, and write r A " r A w s Bw`r Aw s Bw, because otherwise A would be nontrivial on real twistor lines.
Note that A and A 2 are gauge equivalent but only A (and not A 2 ) obeys the boundary conditions (3.32)-(3.33). Therefore, in the next subsection, when we compute the dimensionally reduced action, it is important that we reinsert A (and not A 2 ). However, in the present section, we are going to compute equations of motion. The equations of motion are the same in either gauge (at least up to boundary terms which, in the present work, we ignore). So we will compute the equations of motion using A 2 .
The terms containing jw are iz pz`iqpz´iq 2 Tr`pB zσ qσ´1pB w σqσ´1U T jwpU T q´12 iz pz`iqpz´iq 2 Tr`pB zσ qσ´1U T jwpU T q´1pB w σqσ´1˘(4.34) When Bz acts on z{pz´iq 2 or on i{pz`iq, we obtain a term proportional to Tr`pU T q´1pB w σqσ´1U T jw˘"´2 TrpP w jwq " 0 , (4.35) which vanishes because P is in the orthogonal complement of sop2q. So we can pull z{pz´iq 2 and i{pz`iq outside of the Bz's. The resulting expression sums to zero because the trace is cyclic. The terms in (4.33) containing j w work similarly. This eliminates four of the quadratic fermion terms from the Chern-Simons action. The remaining three sum to a total derivative, We used the fact that A 0 is pure gauge to set s BA 0`A 2 0 " 0. It is not immediately obvious that this integral vanishes because there could be contributions from the poles at z " 0, z " 8, and z "˘i. The pole at z " 0 does not contribute because it gives terms proportional to TrpP w jwq " TrpPwj w q " 0. The pole at z " 8 does not contribute becausê σ " id there. The poles at z "˘i are more tricky. Nothing we have said so far forces them to vanish. However, we can eliminate them by demanding thatσ " id at z "˘i. This is legal because the only boundary conditions on the gauge transformations are at z " 0 and z " 8. We are going to make this choice, and set (4.36) to zero. It is an interesting problem, which we leave for the future, to understand if the dimensionally reduced action could have additional quadratic fermion terms in a different gauge.

Result
The 2d action picks up quadratic fermion terms from the fermion part of the twistor action (4.15), 1 4 ż pδ z"i ΩΩq s χ I γ w D w χ I`1 4 ż pδ z"´i ΩΩq s χ I γwDwχ I . (4.37) Combining these terms with (4.32) gives the action of the Nicolai model up to (and including) quadratic fermion terms: S " 4 ż du dv dū dvˆTr P w Pw´i 2 s χ I γ µ D µ χ I˙, (4.38) where µ " w,w. The coordinates are u " re iθ , v " z`it and w " z`ir. The integrand is independent of t and θ, so we can integrate over those directions and get a 2d action.

Comments on Quartic Fermion Terms
In our calculation, the only quartic fermion term comes from 1 2πi ż Ω TrpA 2 s BA 2 q . (4.39) The integrand contains z pz`iq 2 Trˆσ´1U T j w pU T q´1σBzˆz pz´iq 2σ´1 U T jwpU T q´1σ˙˙s Bw s Bw dζ (4.40) and a similar term with w andw interchanged. Distributing Bz using the product rule gives terms proportional to Trpj w jwpBzσqσ´1q " 0 and Trpjwj w pBzσqσ´1q " 0, which vanish because j w , jw P sop2q (recall that Pauli matrices satisfy Trpσ a σ b σ c q " 2i abc ). We also get a term containing Bzrzpz`iq´2s. This term looks divergent but it gives a finite contribution because it is multiplied by s Bw s Bw " pz`iqpz´iq.
The end result is´ż du dv dū dv Trpj w jwq . This is not the quartic fermion term of the Nicolai action. Recall that the quartic fermion terms of the supergravity model are fixed by supersymmetry and we have not imposed supersymmetry in the twistor setup. It might be that we need to impose supersymmetry on the twistor side to get the correct quartic fermion terms in the action. The story at the level of the equations of motion is simpler. Nicolai [17] has shown that the flatness condition for the Lax operator correctly reproduces the equations of motion, including quartic fermion terms, despite the fact the Lax operator (4.23) only has quadratic fermion terms. In other words, the quadratic fermion terms in the Lax operator and the integrability of the model completely fix the quartic fermion terms in the equations of motion. It would be interesting to find a similar principle at the level of the action. Can the quartic fermion terms in the action be fully fixed by the quadratic fermion terms in the action and integrability? We leave this interesting open question for the future.