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On N=2 strings and classical scattering solutions

arXiv:hep-th/9110075v1 30 Oct 1991ETH-TH/91-35August 1991On N=2 strings and classical scattering solutionsof self-dual Yang-Mills in (2,2) spacetimeAndrew ParkesTheoretische Physik,ETH-H¨onggerberg,8093 Z¨urich,Switzerland.ABSTRACTOoguri and Vafa have shown that the open N=2 string corresponds to self-dualYang-Mills (SDYM) and also that, in perturbation theory, it has has a vanishingfour particle scattering amplitude.We discuss how the dynamics of the threeparticle scattering implies that on shell states can only scatter if their momentalie in the same self-dual plane and then investigate classical SDYM with the aimof comparing exact solutions with the tree level perturbation theory predictions.In particular for the gauge group SL(2,C) with a plane wave Hirota ansatz SDYMreduces to a complicated set of algebraic relations due to de Vega. Here we solvethese conditions and the solutions are shown to correspond to collisions of planewave kinks.

The main result is that for a class of kinks the resulting phase shifts arenon-zero, the solution as a whole is not pure gauge and so the scattering seems non-trivial. However the stress energy and Lagrangian density are confined to stringlike regions in the space time and in particular are zero for the incoming/outgoingkinks so the solution does not correspond to physical four point scattering.

1. IntroductionRecently Ooguri and Vafa [1,2] have revived interest in string theories with twolocal worldsheet supersymmetries.

Such string theories were investigated early inthe days of dual models [3], and were found to have a critical dimension of two.However only comparatively recently was it realised that these are two complexdimensions [4]. Thus, they naturally live in a real space-time with a (2,2) signa-ture.

A string in (1,1) real dimensions cannot have any transverse oscillations andso it is entirely natural that it does not give rise to the usual infinite tower of mas-sive states. For a string in (2,2) dimensions it might be expected that transversemodes do occur, but for the N=2 string these turn out to correspond to local N=2supersymmetry transformations and so do not give rise to physical particles, andso the spectrum contains only a finite number of particles.

It turns out that theyare all massless, bosonic and scalar (see [4,5, 6,1,2] and references therein for moredetails).Ooguri and Vafa investigated the string scattering amplitudes in order to iden-tify the field theory corresponding to the particles in the spectrum. From the closedN=2 string three particle scattering they found that the scalar can be interpretedsimply as the K¨ahler potential of a Ricci flat complex manifold.

Such manifoldscan equivalently be described as self-dual gravity (SDG). This is in line with thefact that if one tries to couple the world sheet action to the geometry of the space-time then the only way to do this and preserve the N=2 supersymmetry is tocouple it only via the K¨ahler potential.

It was also shown that open strings withChan-Paton group factors attached at their edges gave rise to self-dual Yang-Mills(SDYM) theories.The four particle string amplitude usually consists of products of gamma func-tions and so there will be a infinite number of values for the Mandelstam s, t andu variables that will will lead to a pole in the scattering amplitude. For N=0 andN=1 strings these poles just correspond to the massive particles in the spectrum.However for N=2 strings there is no such infinite tower of massive states and so2

there is a potential loss of unitarity. The solution is simply that the kinematicfactor in front of the four particle amplitude vanishes whenever the external par-ticles are on shell.

This fact relies on a special identity concerning any three nullvectors in a (2,2) space-time [2, but see also 7]. Similarly it is believed that allthe higher point connected on-shell amplitudes also vanish.

This gives an almosttrivial solution to the problem of finding amplitudes that are dual with respectto the Mandelstam variables. This is certainly a very remarkable property for aquantum field theory to have and this paper is inspired by the hope of sheddingsome light on this property.Since there are no higher mass states to integrate out it follows that the fieldtheory (SDYM or SDG) from the N=2 string theory is not just a low energy theoryand so at tree level the theories should be equivalent.

When loops are included thenthey differ because the scalars act like Liouville modes with non-standard rules forpath integrals and also because modular invariance changes regions of integrationin the loops. Even at tree level the string approach has the advantage that theperturbation theory combines all connected field theory Feynman diagram into asingle string diagram.

However given the difficulty of a non-perturbative approachto string theory it is much easier to deal with exact solutions for the field theory.Hence in this paper we look again at classical SDYM in light of the results fromthe string theory.It is well known that classical SDYM is an integrable system. For the quantumversion of integrable systems in (1,1) dimensions we would usually expect to havean infinite number of conserved charges that prohibit any particle production andalso for the S matrix to factorise into products of S matrices for 2 →2 scattering.However, the string theory seems to predict that SDYM in (2,2) has almost theexactly opposite properties; that there is no (2→2) scattering and that there isparticle production from the (1→2) S-matrix contribution.Of course, we willignore the infrared problems that would occur in a careful discussion of scatteringof massless states.3

We observe that the kinematical properties of the string theory results of [1]suggest that the system exhibits self-dual null plane decoupling, meaning thatphysical particles only interact, at the S-matrix level, if their momenta lie in thesame selfdual null plane. Specifically for a massless particle with momentum k incoordinates (yz¯y¯z) we define ω = kz/k¯y and this parametrises the self-dual nullplane in which it lies.

Particles with different ω values would not see each otherat the S-matrix level, suggesting that the theory is not fully four dimensional. Ifvalid in all cases then this would presumably have strong effects on the allowedS-matrices in these theories.

This is rather reminiscent of the decoupling of leftand right movers in two dimensional conformal field theory. Thus, it is of interestto investigate whether the stringy results of [1] are of general application.One obvious proviso is that the string theory predictions are only a perturbativeresult and so could be misleading.Furthermore, even the perturbation theoryprediction of the lack of connected 2 →2 scattering has the potential problem thatthe calculation only makes sense when it is non-singular, that is, when stu ̸= 0.Hence the calculation does not directly rule out contributions of the form δ(s)or similar.Normally we would exclude such a possibility by analyticity of theS-matrix, but it is not so clear that this is also true for integrable theories or for(2,2) dimensions.In particular for an integrable theory we would expect there to be solutionswhere the scattering does not induce a change of momentum but only a phase shift.Thus, if the momenta of the external particles are ki with i = 1, 2, 3, 4 and k2i = 0,Σiki = 0 then we will be particularly interested in the case for which k3 = −k1and k4 = −k2.

This has (k1 + k3)2 = 0 and so corresponds to a singular set ofFeynman diagrams (or a string theory calculation at the boundary of moduli spacewhere two of the vertex insertions coincide on the world sheet) and so these maynot be reliable. Classically we expect this to correspond to the case in which thefields only depend on two of the four coordinates.

For example it is known thatunder certain conditions SDYM is equivalent to the two-dimensional KdV equation[8], and this certainly has solutions that display non-trivial soliton scattering. The4

relevance of such “phase shift only” solutions is that k1 and k2 will generally not liein the same self-dual null plane. If such scattering is truly relevant to SDYM thenthe decoupling of these planes would not occur, thus contradicting the perturbativeresults.

Hence in this paper we take a (very restricted) look for classical solutionsof SDYM that correspond to 2 →2 scattering or to 1 →2 scattering and attemptto clarify the situation with respect to classical scattering.We emphasise that when talking of the triviality, or otherwise, of the “classicalS-matrix” it is important to specify the type of waves that are colliding and whetherthey are to be considered as physical. As pointed out in [7], based on the exampleof the KP system, it can well be that finite size packets do not scatter but thatplane waves might still suffer a non-trivial phase shift.

In an integrable theory theinfinite number of conserved charges might seem to restrict scattering but for aplane wave the infinite extent of the wave front could lead to infinite values forthese charges thus invalidating the conservation laws.Dimensional reduction of SDYM is of interest in its own right because many ofthe usual integrable systems can be generated in this fashion (for example [8,9]).One can hope that SDYM is then a master integrable system for many others.However the emphasis here is not to generate new reductions but to see what thereductions say about SDYM itself.A slightly discordant note on the N=2 string is the redundancy of having aplethora of fermions on the world-sheet, and yet the theory contains no spacetimefermions. Also, since we have a (1,1) worldsheet embedded into a (2,2) space-timethen the transverse modes must be pure gauge in order to explain the lack of highermass states.

Hence we need to identify the nilpotent translations obtained by asupersymmetry transformation on the worldsheet with a non-nilpotent translationin the space-time coordinates and it seems unnecessary to have to face this problem.This reinforces the suggestion in [2]) that the theory has a formulation in terms ofsome purely bosonic extended object.In chapter 2 we discuss the peculiarities of kinematics in a (2,2) space-time. In5

chapter 3 we review the J-formulation of SDYM, in which the Yang-Mills fields arewritten as derivatives of a pre-potential[10, 11, 12]. The pre-potential J is a scalarand seems to be the natural outcome of the N=2 string.

The J-formulation is alsoa natural generalisation of the WZNW model in two dimensions. For the reasongiven above we are particularly interested in the case in which J depends on onlytwo coordinates η1, η2 for which it is trivial to see that the system reduces to chiralmodels with Wess-Zumino term in (1,1) dimensions.

In certain cases the systemhas obvious exact generic solutions and we relate these to the string predictionsfor three particle scattering.In chapter 4 we consider the special case in which the gauge group is SL(2,C).This group has the advantage that a Gauss decomposition for the group elementleads to Yangs equations which are derivable from an action. Using the tree levelFeynman rules we verify that the connected four point contributions sum to zero forgeneric external null momenta.

Also Yangs equations are homogeneous in the fieldsand de Vega exploited this to search for scattering solutions. With a fractionalHirota ansatz he reduced Yangs equations to a large set of algebraic equations,and was able to produce some scattering solutions.Here we show that with agood change of variables these equations have a reasonably simple general solutionand so we obtain a wide class of classical scattering solutions.

The solutions weobtain correspond to kink-kink collisions. They are non-trivial in the sense thatthe collision does indeed generate a phase shift for the kinks, but in order to getthe phase shift we are forced to impose conditions on the asymptotic states whichmean precisely that they have have zero value of the relevant stress energy tensor.In chapter 5 we conclude with a discussion of our results and their implicationsalong with a few extra observations.6

2. Kinematics in (2,2) space-timeConsider a complex four dimensional space with coordinates xα = (xµ, x¯µ)where xµ = (y, z), x¯µ = (¯y, ¯z), and with a complex metricds2 = gαβdxαdxβ = 2gµ¯µdxµdx¯µ = 2(dyd¯y −dzd¯z)(2.1)There are two simple ways to recover a real (2,2) space-time.

If we impose (xµ)∗=x¯µ then we still have complex coordinates and so call this a C1,1 slice; the metrichas a manifest holomorphic U(1,1) symmetry. Alternatively we can impose that xµand x¯µ are independent real coordinates, we call this a R2,2 slice.

For the sake ofcomparison with a real (3,1) space time we could also consider an R3,1 slice in which¯y = y∗but z ∈R and ¯z ∈R. Given momenta kα = (kµ, k¯µ) = (ky, kz, k¯y, k¯z), weuse the non-symmetric scalar product< p|k >:= gµ¯µpµk¯µ = pyk¯y −pzk¯z(2.2)The symmetric scalar product is p · k := gαβpαkβ =< p|k > + < k|p >.

Wewill later consider vectors ki and it is convenient to define kij :=< ki|kj >, sij :=kij + kji and cij := kij −kji.It turns out that all the interactions in the string theories or in SDYM or SDGare simple functions of the kij; even one-loop corrections do not seem to affectthis conclusion because the kinematic factors are similar to the tree level [13]. So,before proceeding to the specific case of SDYM, we can first discuss some generalkinematical properties of the on-shell amplitudes.For any 2x2 matrix M it is trivially true thatMµ[¯µMν ¯νMσ¯σ] = 0(2.3)where the square brackets mean antisymmetrisation.

So if we take Mµ¯µ = gµ¯µ andcontract the free indices with three arbitrary vectors ki we obtain a cubic identity7

in kij. For the particular case that the vectors are null we obtaink12k23k31 + k21k32k13 = 0(2.4)This “three nulls identity” is valid in the four complex dimensional space and so isalso valid for any restriction to a real four-dimensional space time.

If the real slicehas a (4,0) signature then there are no non-zero null vectors and so (2.4) is empty.If the real slice is (3,1) then this is essentially the same as the identity found in [7]and shown to be relevant to the consistency of multisoliton scattering in SDYMwith SL(2,C) gauge group. When the real slice has signature (2,2) then as shownin [1] it is responsible for the vanishing of the kinematic factor in the four particleon-shell string amplitude.If we consider on-shell scattering of three massless particles with momenta ki,then conservation of momentum implies that the external momenta span a nullplane.

The bivector k1αk2β −k1βk2α that defines the null plane can be either anti-self-dual or self-dual. The anti-self-dual case means that ki = (pi, piΩ, ¯pi, ¯pi¯Ω) withΩ¯Ω= 1, whilst for a self-dual plane we have ki = (pi, ¯piω, ¯pi, pi¯ω) with ω¯ω = 1.However for the case of the anti-self-dual plane we see that kij = 0 and so thereis no interaction in the theories under consideration.

In other words, if we writethe momenta of all particles in the form (pi, ¯piωi, ¯pi, pi¯ωi) then particles can onlyinteract with each other via the on-shell three leg vertex if they have the sameωi value and so lie in the same self-dual null plane. If we convert the momentato operators on some field φ we get (∂z −ω∂¯y)φ = (∂¯z −¯ω∂y)φ = 0 and thiscorresponds to what we shall call an “ω-ansatz”φ = φ(y + ¯ω¯z, ¯y + ωz)(2.5)In a R3,1 slice there are no null planes but only null lines and so cannot deal withscattering in this way.8

The three nulls identity (2.4) gives rise to the following identities, valid for anymomenta ki i = 1, . .

. , 4 with kii = 0 and Σiki = 0k13k42s13+ k12k43s12= 0k13k24s13+ k12k34s12+ k14 = 0(2.6)c13c24s13+ c12c34s12+ s14 = 0(2.7)and these will be used to compute the four particle scattering.Finally consider the kinematics of the four massless particle on-shell ampli-tude.

For this paragraph only, let us suppose that we have a real diagonal metric(−, +, +, ±), and that s12 ̸= 0. Then after rescaling and rotations of momenta thegeneric situation is that k1 = (1, 1, 0, 0), k2 = (1, −1, 0, 0), k3 = (−1, p1, p2, 0) andk4 = (−1, −p1, −p2, 0) where p21 + p22 = 1 and so s12 = −4, s13 = 2 + 2p1, ands14 = 2 −2p1.

The sign of g33 is irrelevant and so we don’t see the difference inthis case between a (2,2) and a (3,1) signature. The point here is simply that ina (2,2) signature there exist reasonable, non-parallel, sets of physical momenta forthe on-shell three and four particle amplitudes and so these amplitudes can havedirect physical significance.3.

SDYM and the J formulationIn order to set up SDYM in a complex space we take a complex gauge groupGC, and take Fαβ = [∇α, ∇β] with ∇α = ∂α+Aα. The (anti)-self-duality conditionis that Fαβ = −12 det(gµ¯µ)ǫαβγδF γδ where ǫyz¯y¯z = +1.

In the above metric thisreduces toFµν = F¯µ¯ν = 0gµ¯µFµ¯µ = 0(3.1)Hence there exist independent group elements D and ¯D such thatAµ = D−1∂µDA¯µ = ¯D−1∂¯µ ¯D(3.2)Yang-Mills gauge transformations now correspond to D →Dg and ¯D →¯Dg withg ∈GC and so the gauge invariant quantities live on (GC ⊗GC)/GC. A natural9

representative for each orbit is the gauge invariant quantityJ := D ¯D−1(3.3)and the self duality condition reduces to∂¯y(J−1∂yJ) −∂¯z(J−1∂zJ) = gµ¯µ∂¯µ(J−1∂µJ) = 0(3.4)In particular this implies that gµ¯µ∂µ∂¯µ(ln det J) = 0 so det J is simply a free fieldand is of no dynamical interest. In the process of making the above gauge choices togo to the J formulation we have reduced the manifest Lorentz invariance, althoughof course gauge invariant quantities still must be fully Lorentz invariant.

Howeverwe do gain the semi-local GC ⊗GC symmetryJ →gL(x¯µ)JgR(xµ)(3.5)By definition J is gauge invariant, however in order to recover any Yang-Millsstructures from it we need to split first into D and ¯D and the arbitrariness ofthis split is the Yang-Mills gauge symmetry.Thus, gauge potentials and fieldstrengths are gauge dependent functions of the gauge invariant J. The symmetry(3.5) clearly leaves the Yang-Mills potentials invariant and so J is not uniquelygiven by the field strengths.

Also, given the field strengths it is a non-local processto find a representative J, however the J formulation could be considered as justas fundamental as the Yang-Mills formulation.The classical integrabilty of the system is revealed by fact that (3.4) is simplythe compatibility condition for the linear system ǫµνJ∂νΨ + λgµ¯µ∂¯µ(JΨ) ≡0 forall µ and all λ. From this an infinite number of conservation laws can be derivedin the standard fashion (see [12] and references therein).Ultimately we need to impose reality conditions on the coordinates and if wealso wish to finish with a real gauge group GR then we need to impose reality10

restrictions on D and ¯D and so on J. For the R2,2 slice we can trivially takeD, ¯D, J ∈GR so that Aα† = −Aα.For the C1,1 case we need Aµ† = −A¯µand so must take ¯D = (D†)−1 and this reality condition is preserved by gaugetransformations satisfying g†g = 1; hence the gauge invariants in this case live onGC/(GC)U where (GC)U is the subgroup of unitary elements of GC.

In this caseJ = DD† becomes a positive hermitian element of GC. The fundamental exampleis to take GR=U(N) and GC to be its complexification GL(N,C), and then J is apositive hermitian NxN matrix.

Even though C1,1 and R2,2 are of course triviallyrelated by coordinate changes the corresponding J fields are not the same, or evenin the same space in general.The equation of motion (3.4) shows that the J formulation is a dimensionalgeneralisation or complexification of the Wess-Zumino-Novikov-Witten (WZNW)model in (1,1) space-time dimensions and so one cannot expect directly to havea manifestly local and group invariant action. An action is feasible if we pick aparticular parametrisation for the group (see the next chapter) or if one includesextra coordinates [14].

However without an action we can obtain some Feynmanrules from (3.4). Presumably the properties of the N=2 string are true in anybackground that solves the equations of motion, but here we will only considerexpansions around trivial flat backgrounds.

Expanding about J = 1 gives a 1/p2propagator and vertices with arbitrary numbers of external legs but which are allproportional to c12If one now looks at four particle on-shell tree level scattering then the threediagrams have intermediate (squared) momenta s12,s13 and s23 and so we wouldnormally expect their differing momentum dependencies to be forbid any cancel-lation. However, due to the identity (2.7) the Feynman diagram with an apparents13 pole can be changed into an apparent s12 channel pole and so it is now reason-able that the three diagrams combine into one term linear in kij which can thancancel the contribution from the four point vertex.

It is then straightforward tocheck that the connected four point tree level diagrams do indeed sum to zero whenthe external legs are all on shell. This of course matches the expectation from the11

N=2 string theory [1,2]; but it is interesting to note that we do not need to usethe reality conditions at any time and so this is actually true in C4.For reasons given in the introduction we now dimensionally reduce SDYM, inthe same fashion as [7], by considering generalised null plane waves.J = J[η1, η2](3.6)where ηi = kiαxα = kiµxµ + ki¯µx¯µ and the ki are linearly independent and null.Using ∂i = ∂/∂ηi and cij = ǫijc12 then the self-duality condition (3.4) givess12[∂1(J−1∂2J) + ∂2(J−1∂1J)] + c12ǫij∂j(J−1∂iJ) = 0(3.7)The case of k2 = 0 is trivial: if k21 = 0 then J[η] is arbitrary. Hence any null planewave solves both the full non-linear equations and their linearised versions whichis unusual for a non-linear system.

If we were to attempt a classical version ofS-matrix theory perturbation theory then we would usually want to switch offtheinteraction at infinity for the in and out states. This switching offof the interactionis not very pleasing, but does not seem to be necessary for SDYM (or for SDG).If neither ki vanishes then (3.7) just the chiral model with Wess-Zumino term(CMWZ) model (1,1) space-time.

To us the most interesting case is when the mo-menta lie in the same null plane and so can correspond to on-shell three particlescattering. This means that s12 = 0 and we see that the corresponding reductionof SDYM is not an evolution equation.

This is related to the previous observationthat any null wave is a solution. To see this heuristically recall that an evolutionequation will have an initial surface on which the field and some finite number ofderivatives are specified, but one can (at least locally) pick a direction perpendicu-lar to this and since this direction is also null the solution can depend arbitrarily onit and hence cannot be specified by a finite number of initial values.

More specifi-cally we have two cases: If the ki lie in an anti-self-dual plane then c12 = 0 and so12

J is totally arbitrary compared to a string prediction that there is no scattering.Whilst if the ki form a self-dual plane then we haveǫij∂i(J−1∂jJ) = 0(3.8)which corresponds to a two dimensional (topological) theory with only the WZterm and no kinetic term. The generic solution of this “pure WZ” system is thatJ = J[f(xα)] where J ∈GC and f ∈R are arbitrary.

The fact that we do notobtain evolution equations for the cases which correspond to the kinematics ofthe string or SDYM derived three point vertex suggests that we need to be verycareful when talking of the implications for scattering in the theory. In particularit is obviously possible, but not very meaningful, to write down functions J andf that look like any number of solitons “scattering” into any number of others;scattering of two into one is allowed but not at all special.In light of the above relation between pure WZ models and self-dual systems webriefly mention some work of Park [15].

With some rearrangement we can rewrite[15] in the notation of this paper and in a more symmetric fashion. Consider atensor field with components Gµ¯µ(xα) and ¯G¯µµ(xα) and define the vector fieldsAµ := Gµ¯µǫ¯µ¯ν ∂∂x¯ν¯A¯µ := ¯G¯µµǫµν ∂∂xν(3.9)So each Aµ generates diffeomorphisms (not necessarily area preserving) on theself-dual null plane xµ =constant.

Similarly each A¯µ gives diffeomorphisms on theself-dual null plane x¯µ= constant. Introduce the derivative ∇α = ∂α + eAα wheree is a coupling constant and the field strength operator Fαβ = [∇α, ∇β].

We thenenforce the operator condition thatFµν = F¯µ¯ν = 0for alle(3.10)The O(e) term givesǫ¯µ¯ν∂¯µGρ¯ν = 0ǫµν∂µ ¯G¯ρν = 0∀ρ, ¯ρ(3.11)with solution that Gµ¯µ = ∂¯µFµ and ¯G¯µµ = ∂µ ¯F¯µ for some functions F, ¯F. In13

particular this forces the diffeomorphisms to be area preserving on the correspond-ing planes (here our emphasis differs form [15]). To match Park we now forceGµ¯µ = ¯G¯µµ so that we can regard it as a metric on the space-time (this stageseems ad hoc and perhaps instead we should allow coupling to an antisymmetrictensor).

Then (3.11) gives Gµ¯µ = ∂µ∂¯µΩand so this metric is K¨ahler. With theseconditions on Gµ¯µ the O(e2) terms of (3.10) have two consequences.

FirstlyG¯µµFµ¯µ = 0for alle(3.12)so the field strength F is “anti-self-dual for all e” with respect to the metric Gµ¯µ.Secondly det Gµ¯µ is forced to be a constant and so Gµ¯µ defines a Ricci flat K¨ahlermetric on the spacetime, meaning that we have SDG. Hence SDG in (2,2) space-time is almost SDYM with a gauge group which consists of diffeomorphisms actingon self-dual null planes embedded in the spacetime, but with the extra conditionsthat the self-duality holds true even under constant rescalings of the gauge fieldsand that the resulting metric is symmetric.The relation to pure WZ models is simply that Fµν = 0 for all e, has thesolution Aµ = D−1∂µD with ǫµν∂µ(D−1∂νD) = 0 which is a pure WZ model onthe xµ-plane for a fixed point in x¯µ (and similarly for the barred quantities).As discussed in [2] the N=2 string theories seem to be almost topologicaltheories.

It is also well known that the Yang-Mills stress energy tensor vanishesfor SDYM and this would be a sign that the theory is topological were it not forthe fact that the self-duality condition itself uses a metric. So it is interestingto note that the system discussed above does not any reference to a metric onthe space-time but only assumes a complex structure; that is, we only needed theconditions Gµ¯µ = ¯G¯µµ and the tensors ǫµν and ǫ¯µ¯ν for (3.10).

Instead the metricarises from the parameters of a gauge transformation. Also, it is reminiscent ofWittens work on (2,1) gravity [16] that the usual Yang-Mills perturbation theoryaround Aµ = ¯A¯µ = 0 corresponds to expanding about Gµ¯µ = 0 and not aroundthe usual flat metric.

(However, since the conditions are to be enforced for all ethe standard perturbation theory will not apply. )14

For completeness, we note that for SDG expanding around a flat metric givesthe Plebanski equationgµ¯µ∂µ∂¯µφ + ǫµνǫ¯µ¯ν(∂µ∂¯µφ)∂ν∂¯νφ = 0(3.13)If we then dimensionally reduce as we did for SDYM by imposing φ = φ(η1, η2)with k11 = k22 = 0 we gets12∂1∂2φ + k12k21[(∂21φ)∂22φ −(∂1∂2φ)2] = 0(3.14)Then s12 = 0, k12 ̸= 0 forces(∂21φ)∂22φ −(∂1∂2φ)2 = 0(3.15)which has the generic solution that either φ = φ(aη1 + bη2) where a and b arearbitrary constants, or that φ = (η2 −˜η2)Φ[(η1 −˜η1)/(η2 −˜η2)] + b where ˜ηi,b arearbitrary constants and Φ is an arbitrary function. Again these do not behave asscattering solutions.4.

SDYM with gauge group SL(2,C)This has been extensively studied previously; the motivation was either toimpose reality conditions and so obtain self-dual SU(2) solutions [17,11], or toexploit the fact that it is a complex group and so can have self-dual solutions in a(3,1) metric [7]. We study it simply because it is the simplest non-trivial case.

Themost straightforward approach to SL(2,C) is based on the Gauss decompositionJ = exp gρ 0010!exp gS 100−1!exp g¯ρ 0100!= 10gρ1! egS00e−gS!

1g¯ρ01! (4.1)where S, ρ, and ¯ρ are independent complex fields.

The equations of motion from15

(3.4) are thengµ¯µ(−∂µ∂¯µS + ge2gS∂µρ∂¯µ¯ρ) = 0gµ¯µ∂µ(e2gS∂¯µ¯ρ) = gµ¯µ∂¯µ(e2gS∂µρ) = 0(4.2)which follow from the Lagrangian [10]L = gµ¯µ(∂µS∂¯µS + e2gS∂µρ∂¯µ¯ρ)(4.3)To put this in context we note that dimensional reduction, as in the previous chap-ter, would give a manifestly local action for the WZNW model in two dimensions,but of course manifest group invariance has been lost. From the Lagrangian weobtain a stress-energy tensorTµν = ∂µS∂νS + e2gS∂µρ∂ν ¯ρT¯µ¯ν = ∂¯µS∂¯νS + e2gS∂¯µρ∂¯ν ¯ρT¯µµ = Tµ¯µ = ∂µS∂¯µS + e2gS∂µρ∂¯µ¯ρ(4.4)the equations of motion (4.2) imply that ∂αTαβ = 0.

We see that Tαβ is symmetric,but not covariant because of the way in which ρ and ¯ρ appear. This T is of coursedifferent from the stress tensor from the Yang-Mills action.If we only wish to work at tree level then we can ignore any ghosts that arisefrom the gauge fixing of the action from (4.3), and so it is easy to obtain theFeynman rules.

In particular, after dropping total derivatives, the quadratic part ofthe Lagrangian is simply L0 = −12gαβ(∂αS∂βS +∂αρ∂β ¯ρ) giving 1/p2 propagators.The vertex with a ρ of momentum k1, a ¯ρ of momentum k2 and n legs of S comeswith a factor (2g)nk12. It is now straightforward to sum the connected tree-leveldiagrams with four on-shell external legs.

Then< ρ(k1)¯ρ(k2)¯ρ(k3)ρ(k4) > ∝(2g)2k13k42s13+ k12k43s12< ρ(k1)S(k2)S(k3)¯ρ(k4) > ∝(2g)2k13k24s13+ k12k34s12+ k14(4.5)Putting kii = 0 and Σiki = 0 and using (2.6) shows that in any case for whichthe calculation is non-singular then the answer is zero. It is interesting that this16

happens despite the fact that the map from the variables λa of the previous chapterto S, ρ, ¯ρ is very non-linear. Non-linear changes of field variables in this case donot affect the conclusion that the connected four particle S matrix is zero.

We alsonote that ρ and ¯ρ appear only quadratically in the action so that in principle wecould integrate them out, however this does not seem to be directly useful.We now want to exhibit a class of exact solutions by extending the work of deVega [7]. Without loss of generality we set g = 1 and put S = −ln φ so that nowJ = 1φ 1¯ρρφ2 + ρ¯ρ!

(4.6)and the self-duality equations reduce to Yangs equationsgµ¯µ(φ∂µ∂¯µφ −∂µφ∂¯µφ + ∂µρ∂¯µ¯ρ) = 0gµ¯µ(φ∂¯µ∂µρ −2∂µρ∂¯µφ) = 0gµ¯µ(φ∂µ∂¯µ¯ρ −2∂µφ∂¯µ¯ρ) = 0(4.7)We shall immediately impose dimensional reduction, as in the last chapter, byinsisting that we have functions of η1 and η2 only.Then one trivial solution,extending a solution of de Vega, is that φ = 1, ρ = ρ1(η1) + ρ2(η2) and ¯ρ =c(k12ρ1 −k21ρ2) + b where b, c are arbitrary constants and ρ1, ρ2 are arbitraryfunctions. To obtain non-trivial solutions we exploit the bilinearity of the equationsby making the Hirota style ansatzφ = F∆ρ = N∆¯ρ =¯N∆(4.8)where∆= ∆0 + ∆1eη1 + ∆2eη2 + ∆12eη1+η2F = F0 + F1eη1 + F2eη2 + F12eη1+η2N = N0 +N1eη1 +N2eη2 +N12eη1+η2¯N = ¯N0 + ¯N1eη1 + ¯N2eη2 + ¯N12eη1+η2(4.9)Substituting this ansatz into Yangs equations and equating to zero the coeffi-cients of the different powers of eη1 and eη2 gives a set of 15 complicated equations17

in the 16 unknown coefficients. Hence, the system will indeed have solutions andde Vega produced a solution for the special case in which F12 = 0.

Here we pointout that the equations resulting from the above ansatz can be solved directly forrather general conditions. In order to do this we use a change of variables suggestedby the asymptotic properties of the ansatz.

Thus,η2 →−∞⇒φ(η1, η2) →φ−(η1) = F0 + F1eη1∆0 + ∆1eη1(4.10)and so the φ field is a kink of height f1 = F1/∆1 −F0/∆0. However it is 1/φ andnot φ that occurs in χ and so we also look at1φ−(η1) = ∆0 + ∆1eη1F0 + F1eη1 ≡∆1F1−∆0F0g(η −ln F0F1) + ∆0F0(4.11)where g(η) := (1 + eη)−1 describes the shape (since it is independent of the coef-ficients we only really get one type of wave in this ansatz) and so 1/φ−is a kinkwith phase ln(F0/F1).

Similarly,η2 →∞⇒φ(η1, η2) →φ+(η1) = F2 + F12eη1∆2 + ∆12eη1(4.12)and so 1/φ+ has phase ln(F2/F12). Hence the phase shift between these two limitsis ln(F0F12/F1F2).

Furthermore the phase depends only on the denominator and soonly on F. In particular this means that 1/φ = ∆/F, ρ/φ = N/F and ¯ρ/φ = ¯N/Fall suffer the same phase shift. From this it follows that it is natural to introducethe variable f12 := F12F0 −F1F2.

The phase shift is zero ifff12 = 0. Thus wechange to variables that describe the heights of the kinks for φ,ρ and ¯ρ by settingfor i=1,2Fi = ∆i F0∆0+ fiNi = ∆iN0∆0+ ni¯Ni = ∆i ¯N0∆0+ ¯ni(4.13)and to variables suggested by the form of the phase shift for 1/φ byF12 = F1F2 + f12F0∆12 = ∆1∆2 + δ12∆0N12 = N1N2 + n12N0¯N12 =¯N1 ¯N2 + ¯n12¯N0(4.14)18

We assume that ∆0∆1∆2F0N0 ¯N0 ̸= 0, so that this change of variables is non-singular. If k12 = −k21 or k12 = 0 or k21 = 0 then we have one of the casesconsidered in the previous chapter, so we now also assume that k12k21(k12 +k21) ̸=0.

Then the eη1+η2 terms of Yangs equations immediately yieldδ12 = ∆20F 20f12 + ∆1∆2∆20F 20n1k12¯n2 + n2k21¯n1k12 + k21n12 = N20∆20δ12 + 2∆1∆2N0F0n1k12f2 + n2k21f1k12 + k21−∆1∆2n1n2¯n12 =¯N20∆20δ12 + 2∆1∆2¯N0F0f1k12¯n2 + f2k21¯n1k12 + k21−∆1∆2¯n1¯n2(4.15)The exp(η1 + 2η2) and exp(2η1 + η2) terms will give equations for f12 and the fullgeneral solution then depends on whether any of f1, . .

. , ¯n2 are zero.

However, inorder to obtain a solution with f12 ̸= 0 we find that we must impose the constraintsf21 + n1¯n1 = 0f22 + n2¯n2 = 0(4.16)and we then obtain a solution of Yangs equations as long asf12 = −∆1∆2(2f1f2 + n2¯n1 + n1¯n2)k12k21(k12 + k21)2(4.17)In this generic solution, besides the kij we have the 12 parameters ∆0, ∆1, ∆2, F0,N0, ¯N0, f1, f2, n1, n2, ¯n1 and ¯n2 subject to the two constraints (4.16). Howeverwe can trivially set ∆0 = 1, and by shifting the coordinate origin we could set∆1 = ∆2 = 1 and so there are really only 12-2-3=7 true parameters to this generalsolution.

Also the overall scale leaves AY M unchanged (see expressions for A givenby de Vega) and so might not be considered a true parameter.With this solution it turns out that (φ2+ρ¯ρ)/φ is also of the form of the Hirotaansatz and so effectively we have simply imposed such an ansatz directly on thecomponents of J.19

In order to interpret the solution we can look at the behaviour of the stressenergy tensor of (4.4).We find that for η2 →−∞we have that T becomesproportional to the constraints (4.16)and so for this solution generically T willbecome zero at ηi = ±∞. Hence in the above solution to Yangs equations thestress energy generically vanishes at long distance from the region where the kinksare colliding.

In this respect is is more like an instanton solution of the underlying 2dimensional chiral model. The particular solution that de Vega found is the specialcase F12 = 0 and this is a degenerate case in which the stress tensor spreads outinfinitely in one particular direction.If we impose reality conditions to try to get an SU(2) solution then we in-evitably find we are forced to trivialise the solution.

For example, in the C1,1 casewe must impose ρ∗= ¯ρ and φ∗= φ; whence also n∗i = ¯ni and f∗i = fi and then(4.16) forces fi = ni = ¯ni = 0 and in particular f12 = 0. The reality conditionsare homogeneous and so it is not unreasonable to have tried the Hirota ansatz inthis case.

There is no problem if we want SL(2,R) or SU(1,1) as we simply take areal solution or solve (4.16)with n∗i = −¯ni. Perhaps one point worth mentioningwith respect to SU(2) solutions is that in a (2,2) signature the B¨acklund transfor-mations in [18] now preserve the reality conditions and so take SU(2) solutions toSU(2) solutions.

They no longer alternate between SU(1,1) and SU(2) solutions ashappens in a (4,0) signature.An obvious question is whether this can be repeated for other gauge groups.The main step is the use of an Hirota ansatz and this will only be useful forhomogeneous equations. Since (3.4)are homogeneous in the matrix elements of Jfor any G then we expect that GL(n) could be treated in this way.

However, aswe just saw, this will not usually work for subgroups because the extra conditionsare usually not homogeneous and so not suitable for the Hirota ansatz. Note thesesolutions are just scattering in a general chiral model, which is integrable, and soin principle all solutions are known, but in practise are not trivial.20

5. Conclusion and open questionsIn this paper we briefly reviewed the kinematics in (complexified) space-timewith a (2,2) signature.

We found a simple way to obtain the three-nulls identitythat is vital to calculations of on-shell Feynman diagrams. We then briefly coveredthe J formulation of SDYM, its perturbation theory and dimensional reduction toa two dimensional chiral model with Wess-Zumino term.As pointed out in [1] a dominant feature of the perturbation theory is that theon-shell connected amplitudes vanish for four or more external legs.

This leavesonly three particle scattering and then the momentum dependence of the verticeshas the implication that two particles (physical states) can only interact if theirmomenta are in the same self-dual null plane. This only applies to the S-matrix butnot to Greens functions but does seem to imply that the effective dimensionalityhas been reduced by one.

It would be interesting to classify all possible actionsfor which the S-matrix has the same properties. For example we might observethat the action (4.3) is a non-symmetric non-linear sigma model, and so insteadwe could generalise toL = Gab[φ]∂µφa∂¯µφbgµ¯µ(5.1)Enforcing the vanishing of the appropriate Feynman diagrams will presumablythen have some geometric meaning for the non-symmetric metric on the spaceparametrised by the φa.

Presumably including fermions will be related to SDYMwith supergroups or to some supersymmetric extension of the self-duality relation:It should be noted that a tree level diagram with all bosonic external legs cannothave any internal fermion lines and so the bosonic sector of any such generaltheory should already possess the property that only the three point function isnon-trivial. The S-matrix predicted by the perturbation theory is very simple, andif this structure is still present non-perturbatively then it would be very interestingto find the consequences for the S-matrix (compare factorisation of the S-matrixin two dimensions).21

A classical equivalent of particle momenta being in the same self-dual plane isthat the fields depend only on the two coordinates of the plane (or more generallyto be non-linear superpositions of such fields on different planes) In this case SDYMreduces to the equation of motion from a chiral model in (1,1) spacetime with theWess-Zumino term only. Such a “pure WZ” model is classically exactly solvableand topological.

It is also not an evolution equation and so this leads us to questionthe validity of the perturbation theory prediction for the scattering in the theory. (This also means that SDYM is on the edge of being topological [2] and so wemight expect some of the physical observables to be non-local and measured byappropriate Wilson lines.

)In order to see whether these properties hold non-perturbatively in the couplingconstant (but still at tree level) we were lead to consider exact classical solutionsof SDYM. In particular there are certainly some exact solutions which correspondto 2 to 2 scattering with no momentum exchange but a non-zero phase shift.

Forexample the theory can be reduced to the KdV equation or a (1,1) chiral model.In this case the momenta would not necessarily lie on the same self-dual plane andso the factorisation considered above would be destroyed (and the perturbationtheory would have been very misleading). To clarify this we extended some workof de Vega [7] on the scattering of plane waves in SL(2,C).Using the Hirota ansatz and Gaussian decomposition of the SL(2,C) field wewere able to solve the SDYM equations for the case of a generalised plane wave.The solution described the collision of kinks in the J field, and consisted of non-trivial scattering in the sense that the kinks suffered a non-zero phase shift fromthe collision.

However the restrictions necessary to obtain such a solution were alsoprecisely sufficient to force the in and out waves to have vanishing stress-energytensor and Yang-Mills field strengths. That is, the stress tensor (meaning the onefrom the action for Yangs equations not the one from the Yang-Mills action) isnon-zero only in the region where the kinks are colliding.

On the positive side,this does not seem to correspond to 2 to 2 scattering of physical states and so willnot affect the above S-matrix properties. The negative side is that there is still the22

possibility that there is no scattering of physical states.In our SL(2,C) solution we have a two dimensional surface of non-zero energyembedded in the spacetime which is reminiscent of a string theory again. It wouldbe interesting to investigate whether such solutions have a stringlike behaviour inwhich strings beget strings along the lines suggested in [19].Acknowledgements: I would like to thank C. Devchand for useful discussions.

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