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Suresh Govindarajan, Philip Nelson and Eugene Wong

arXiv:hep-th/9110062v1 21 Oct 1991UPR–0477TSemirigid GeometrySuresh Govindarajan, Philip Nelson and Eugene WongPhysics DepartmentUniversity of PennsylvaniaPhiladelphia, PA 19104 USAWe provide an intrinsic description of N-super Riemann surfaces and TN-semirigidsurfaces. Semirigid surfaces occur naturally in the description of topological gravity aswell as topological supergravity.

We show that such surfaces are obtained by an integrablereduction of the structure group of a complex supermanifold. We also discuss the super-moduli spaces of TN-semirigid surfaces and their relation to the moduli spaces of N-superRiemann surfaces.9/91

1. IntroductionSemirigid surfaces have been shown [1][2] to provide a geometric framework to describe2d topological gravity and supergravity.

For example, in the simplest theory the dilaton aswell as the puncture equations have been proven using the semirigid formalism [3][4]. In thispaper, we provide an intrinsic or coordinate invariant definition of semirigid super Riemannsurfaces (SSRS) as well as ordinary super Riemann surfaces (SRS).

The discussion ofSRS is a natural extension to similar discussions provided in [5] and applied in [6] forthe case of N = 1 SRS and in [7] for N = 2; the framework follows Cartan’s theory ofG-structures. (For an introduction to G-structures, see for example [8][9][6][10].) We showthat these structures subject to some conditions called “torsion constraints” are integrable,which relates our intrinsic definition to the coordinate dependent definitions.We will first discuss the various definitions and illustrate G-structures via two exam-ples in sect.

2. We also find the appropriate group G for superconformal and semirigidsurfaces and the corresponding torsion constraints.

Sect. 3 deals with showing that theG-structures we impose are integrable provided the constraints are satisfied.

Briefly theresults are as follows. If we begin with a complex supermanifold, then N-SRS have noessential torsion constraints, generalizing Baranov, Frolov, and Schwarz [11], who consid-ered N = 1.1 We will refer to semirigid surfaces with N-supersymmetry as “topologicalN-SRS,” or TN for short.

TN = 0 surfaces have a rather trivial essential constraint whileTN = 1 surfaces have several. Both in the usual and in the topological case the category ofsurfaces with appropriate G-structures, integrable in the sense we will specify, is equivalentto the corresponding category of surfaces with appropriate patching data.

(Actually wewill limit ourselves to proving this for N ≤3 and TN ≤1 to keep the algebra simple.) Inparticular there are no second-order conditions for flatness, just as for ordinary N = 0 con-formal structures.

Throughout this paper we will consider only untwisted superconformaland semirigid structures, since our focus is primarily on local properties. The integrabilityresults we prove will also apply to the study of twisted surfaces.We should comment on the relation of this work to [1][2].

In these papers the coordi-nate definition of semirigid surfaces was used. The interpretation of such surfaces as havinga special G-structure was crucial for finding the right patching maps, but no attempt wasmade to prove the equivalence of the two approaches, i.e.

the theorem that every integrableG-structure gave a semirigid surface. That is what we do here.1 This generalization was asserted in the appendix to [12].

The constraints found in [13] anddiscussed in [6] arise when we begin with a real supermanifold.1

2. G-structures on manifolds and supermanifoldsWe begin by stating the problem, then recall the general idea of G-structures withsome examples.2.1.

Patch definition of SRS and SSRSOne way of defining SRS or SSRS is to cut a supermanifold into patches, put coor-dinates on them and sew them back together with transition functions given by supercon-formal or semirigid coordinate transformation. Let us begin with SRS.

Generalizing theN = 1 superconformal transformation [14][11][15][12], we start with C1|N and define fori = 1, . .

., NDi =∂∂θi + gijθj ∂∂z(2.1)where gij = δij. We impose the condition that {Di} transform linearly among themselves(not mix with∂∂z ) under a superconformal coordinate transformation (z, θi) →(ez, eθi).This condition resembles the one for a complex manifold, where the good coordinate trans-formations do not mix the ∂zi with the ∂¯zi.

Thus,Di = F jieDj;F ji= Dieθ j,(2.2)where eDi =∂∂˜θi + gij ˜θj ∂∂˜z and F is some invertible matrix of functions. It follows that thesuperconformal transformations are those for whichDiez = gjkeθjDieθk.

(2.3)An N-superconformal surface is then just a supermanifold patched together from pieces ofC1|N related by N-superconformal transition functions.Semirigid surfaces (or SSRS) are patched together by restricted superconformal tran-sition functions. The restriction imposed is that θ+ be global, where θ± ≡1√2(θ1 ± iθ2).For instance, to obtain the TN = 0 semirigid coordinate transformations, we start withN = 2 superconformal coordinate transformations and impose eθ+ = θ+.

This restrictiontogether with (2.3) fixes the coordinate transformations on the rest of the coordinates.Such restricted coordinate transformations then provide the transition functions to build aTN = 0 SSRS [1]. One can similarly obtain TN = 1 semirigid coordinate transformationsfrom N = 3 superconformal coordinate transformations by the same method.2

Although this method of deriving SRS and SSRS is adequate for doing physics, thereare at least two features that are buried in them. One would like to classify the super-conformal or semirigid coordinate transformations as being coordinate transformationswhich preserve some geometrical object.

This object is not obvious using the above patchconstruction. In addition, to find the superconformal or semirigid moduli space, one wouldlike to have a coordinate invariant definition of SRS or SSRS so that it is clear thatdeformations of their structure are not artifacts of coordinate transformations.

This is ofinterest when one studies the moduli space of these surfaces, where one’s interest is to finddeformations which cannot be undone by allowed coordinate transformations.We will provide such an invariant description in the sequel by means of G-structures.To prove that the patch definition is equivalent to the intrinsic definition (i.e. the one usingG-structures), we will show that a manifold constructed by the above patching functionsimplies a G-structure.To invert this correspondence and so establish equivalence wewill ask whether every G-structure arises by this construction.

In general this last steprequires that the given G-structure be “integrable,” a concept whose meaning we will recallin the following examples. We will find the appropriate integrability conditions in sect.

2.3and show that they really do lead to an equivalence between the patch and G-structuredefinitions.While this is not too difficult for TN = 0, it does require some work forTN = 1, i.e. for topological supergravity.2.2.

Two examplesIn this subsection, we will illustrate G-structures and the question of their integrabil-ity [9][16]. We will also demonstrate how one obtains coordinate transformations whichpreserve the G-structure chosen.

This enables us to relate this definition to the patchdefinition once integrability is proved.Suppose we are given a smooth manifold.Then its tangent space can be locallyspanned by a field of frames {ea}. However, there are in general no global frames.

In orderto obtain a global structure, we define an equivalence class of frames. The equivalencerelation is given by a group G of matrices whose elements act on the frames, that is,{K ba eb} is defined to be equivalent to {ea}, where K is a function with values in G.Without any extra structure beyond smoothness, all we can say about the matrices Kab isthat they belong to the group GL(n, R).

However, with additional structures, the structuregroup can be reduced to a subgroup of GL(n, R). The structure group can be thought ofas the local symmetry group of a physical theory defined on the manifold.

In general, not3

all manifolds admit a reduction of structure group due to possible global obstructions [9]2.Also, there are geometrical structures like connections and projective structures that arenot G-structures. What we will see in this paper is that SRS and SSRS as defined insect.

2.1 do arise as reductions of the structure group of a supermanifold.We first consider a smooth manifold with additional structure provided by a metricg = gabea ⊗eb,(2.4)where ea is the dual to the frame ea. Since a metric provides information about the lengthof a vector, it selects out from the classes of frames {ea} acted on by elements of the groupGL(n, R) those that are orthonormal, that is, gab = δab.

The structure group that acts onthe family of orthonormal frames is the group O(n) leaving δab invariant. Thus we have areduction of structure group from GL(n, R) to O(n) imposed by the additional structure,the metric.

Conversely, given a reduction of structure group to O(n), it induces a metricon the manifold: we simply substitute any good frame into (2.4). Like the metric, theimposition of a G-structure on a manifold is an intrinsic concept.

Note that the morestructures one imposes, the smaller the class of good frames. For example, imposing inaddition an orientation lets us restrict further to the class of oriented orthonormal frames;these are related by the smaller group SO(n).For our second example consider the case of a 2n-dimensional manifold M endowedwith an almost complex structure, specified by a tensor J similar to the metric.Thetensor is given at a point P by JP : TP M →TP M everywhere satisfying J2P = −I.

Whendiagonalized, J splits the complexified tangent TcM into holomorphic (with eigenvalue i)and antiholomorphic (with eigenvalue −i) tangent spaces. We can use J to define goodframes {ea, e¯a} as those for which ea are +i eigenvectors and e¯a are the complex conjugatesof ea, a = 1, .

. ., n. ThenJ = i(ea ⊗ea −e¯a ⊗e¯a).

(2.5)J thus selects out from the class of frames related by GL(2n, R) a smaller class related byGL(n, C), since J is invariant only under GL(n, C) transformation of frames. Conversely,given a reduction of structure group to GL(n, C), which gives us the class of good frames{ea, e¯a}, we can obtain J by substituting any good frame in (2.5).

Thus an almost complex2 We will not consider such obstructions because they are not relevant in establishing theequivalence between the patch and intrinsic definitions.4

structure is nothing but a GL(n, C) structure, an equivalence class of frames {ea, e¯a} whereany two frames are related by a complex matrix of the forme′ae′¯a=A00¯A eae¯a. (2.6)¯A is the complex conjugate of the invertible matrix A.We have given a coordinate invariant characterization of a G-structure.

But sometimesit is convenient to use coordinates. Since a G-structure makes sense even locally, let usfirst consider the problem of specifying one on an open set U of Rn.

For any choice ofcoordinates {xa} on U we first choose a standard frame given by some universal rule.For example in Riemannian geometry we choose ˆe{x}a=∂∂xa . (We will choose a morecomplicated standard frame in the superconformal and semirigid cases.) If we begin witha different set of coordinates {ya}, in general the two frames ˆe{x}a, ˆe{y}ado not agree.However if we arrange for them to agree modulo a G-transformation then they do definethe same G-structure.

This happens whenˆe{y}a|P = K(P) ba ˆe{x}b|P(2.7)for some function K in G. Since G is a group, the set of all coordinate transformationsy(x) defined by (2.7) is a group too; we call it the group of G-coordinate transformations,or simply the “good” transformations.Thus one way to specify a G-structure on a manifold M is to present an atlas of coor-dinate charts Uα with coordinates xα all related on patch overlaps by G-transformations.Let us illustrate the above discussion with our two examples. In Riemannian geometrythe only coordinate transformations preserving the standard frame up to O(n) are the onespreserving the standard metric, i.e.

the rigid Euclidean motions. For the almost-complexstructure example things are more interesting.

Given a choice of real coordinates {ua, va},a = 1, . .

., n we let za = ua+iva and take the standard frame to be ˆe{z}a=∂∂za , ˆe{z}¯a=∂∂¯za .Let {wa, ¯wa} be another complex local coordinate with standard frame {∂wa, ∂¯wa}. Onthe overlap, let w and z be related by a coordinate transformation wa = wa(zb, ¯zb) so that∂za∂¯za= M∂wa∂¯wa,whereM =∂zawb∂za ¯wb∂¯zawb∂¯za ¯wb.

(2.8)For w and z to be complex coordinates for the same complex structure, we need M to beof the form (2.6). This means that the “good” coordinate transformations preserving thecomplex structure are holomorphic maps.5

More generally, a manifold obtained by patching together coordinate charts by a classof G-transformations gets a G-structure. Clearly if we replace each local coordinate xaαby yaα = ψα(xbα) where ψα is itself a G-transformation, we determine exactly the sameG-structure.We would also like to show the converse: a manifold equipped with a G-structurecan always be constructed from a set of “good” transition functions.

In fact this converseis not always true. To find out when it is so, we introduce coordinate patches on themanifold with the G-structure.

We seek coordinates {xα} on a local patch Uα such thatthe standard frame {ˆe{xα}a} determines the given G-structure.Since a G-structure isgiven by an equivalence class of good frames we are thus seeking a local coordinate whosestandard frame belongs to the same equivalence class as the given {ea}. If we can find sucha coordinate system, we then call the G-structure integrable.

However, this is in general notpossible unless the frames belonging to the G-structure satisfy certain constraints. Afterall, {xa} contains only n =dimM degrees of freedom, while the given {ea = eµa∂µ} has n2minus the dimension of G. This counting also makes it clear that different G-structuresimpose different integrability constraints.

For instance, we will see that the superconformalstructure does not need any such conditions while the semirigid case needs some first orderconstraints. Of course there is more to do than just count conditions.

The statementthat a set of local constraints on a G-structure really does suffice to find local coordinatesinducing that structure is called an integrability theorem.Let us illustrate these ideas in the two examples given above. For the case of Rie-mannian geometry, G = O(n), it turns out that a G-structure is integrable iffits Riemanncurvature tensor R vanishes (see for example [17]).

That is, if R ≡0 in the neighborhoodof a point, then there exist local coordinates (called inertial) such that the metric is in thestandard form g = δabdxa⊗dxb. Comparing this metric with the one specified by the givenO(n)-structure g = δabea ⊗eb, we see that the frames are related by ea = Kab∂∂xb , whereK ∈O(n).

Thus the frame defining the G-structure ea is G-equivalent to the standardframe of some coordinates, which is what we called integrability earlier. Notice that theintegrability condition is given by constraining the curvature, a function involving up tosecond order derivatives of the original frame.

We thus call this a second order constraint.The condition R = 0 implies flat space; thus integrability conditions are sometimes calledflatness conditions, even though they may be given by first order constraints in other cases.Instead of Riemannian geometry we can enlarge O(n) somewhat to the group of ma-trices with KtgK ∝g — the conformal group. The obstruction to flatness is now just a6

part of the Riemann curvature, namely the Weyl tensor [18]. An important case is twodimensions, where there is no Weyl tensor at all and every conformal (or C×)-structure isintegrable.In the case of an almost complex structure, the counterpart of the curvature is theNijenhuis tensor [19], given in terms of J byN (X, Y ) = [X, Y ] + J[JX, Y ] + J[X, JY ] −[JX, JY ].

(2.9)where X and Y are arbitrary vector fields. The integrability theorem [20] says if N ≡0,then there exists a local complex coordinate system {za} , i = 1, .

. ., n such that J is ofthe form (2.5) with the frames given by ˆe{z}a=∂∂za , ˆe{z}¯a=∂∂¯za .

Thus N = 0 becomes theflatness condition. It is however a first order condition unlike the O(n) case, since (2.9)clearly involves at most first derivatives of J.

As mentioned above, the “good” coordinatetransformations (those preserving J) are the holomorphic maps.Given an integrable G-structure on M, we can now return to the question of whetherit can be constructed via patching maps. On each coordinate patch choose a coordinateinducing the given G-structure.

Then on patch overlaps the chosen coordinates are re-lated by what we have called a “good” or G-transformation: xβ = φαβ(xα). Hence wecan construct M with its G-structure from patching coordinate charts with the “good”coordinate transformations.

Of course on each patch we have some freedom to redefinethe good coordinate xaα by some G-transformation yaα = ψα(xbα). This simply correspondsto replacing the {φαβ} by the equivalent family {ψα ◦ψαβ ◦ψ−1β } as discussed above.To summarize, given G and a choice of standard frames we may define a G-manifoldas a collection of patching G-transformations modulo the substitution {φαβ} 7→{ψα ◦φαβ ◦ψ−1β }, where ψα are themselves G-transformations.

Or we may define a G-manifoldas a smooth manifold with a collection of frames defined modulo G satisfying appropriateintegrability conditions. We have seen that these two definitions are equivalent once theappropriate integrability theorem is established.For the case of specifying the N ≥1 superconformal structure, a coordinate invarianttensor analogous to the metric g or the tensor J is not known.

However, one can still choosea group G and specify a G-structure by giving a frame defined up to transformations byelements of G. Without the analog of g or J, we cannot define a tensor like R or Nmeasuring the local obstruction to integrability. Thus, one has to find another way to givethe flatness condition for the case of superconformal structures or else prove that there is7

no such condition, that is, all G-structures are flat. The situation is similar for semirigidstructures.Let us once again use the case of an almost complex structure on a 2n dimensionalreal manifold to clarify how first-order flatness conditions can come about.

The flatnesscondition N = 0 can be replaced by a condition similar to the one used in the Frobeniusintegrability theorem, namely tab¯c = 0, where[eA, eB] = tABCeC(2.10)and A denotes either a, ¯a. In other words, the Lie bracket of the holomorphic tangentframes stays in the same subspace.

Conditions of this type are sometimes called “essentialtorsion constraints”[6].We now recall a general prescription [6] to obtain the torsion constraints with theabove example in mind and see that they are necessary conditions for integrability. In ourexamples the structure constants ˆtcab all vanish when we use the standard frame {ˆe{x}a}in [ˆe{x}a, ˆe{x}b] = ˆtcab ˆe{x}c. (More generally they will at least all be constants in the cases ofinterest.) Of course the same may not be true when we substitute some other equivalentframe {ea} to get tabc.

We obtain an arbitrary representative of the standard G-structureby letting an arbitrary function in G act on the standard frame. Those tabc that remainequal to ˆtcab clearly have the same values in any good frame.

Thus we have found someconditions on tabc which follow from the assumption that our frame is equivalent to somestandard frame. These conditions may be overcomplete; for example some may be relatedto others by Jacobi identities.In other words given a frame we have found some conditions which must be met if thecorresponding G-structure is to be integrable.

These “torsion constraints” are first orderconditions on any frame representing the given structure since the Lie bracket entering tcontains one derivative. If we find that they are also sufficient for flatness, then we have anintegrability theorem with only first order constraints.

This is the case for G = GL(n, C)since here the torsion constraints amount to the vanishing of the Nijenhuis tensor; it willalso be true for superconformal and semirigid geometry. (And as we have mentioned, forsuperconformal geometry there will be no essential torsion constraints at all.) Howeveras we have seen it is false for Riemannian geometry.

It is sometimes convenient to im-pose further G-invariant “inessential” torsion constraints corresponding to normalizationconditions [6], as we will recall below.We will now apply all these ideas to the cases of N superconformal and TN semirigidstructure.8

2.3. Intrinsic Definitions of SRS and SSRSWe now provide an intrinsic definition of N superconformal structures [12] general-izing [11][21].

Below we will propose a similar intrinsic definition of semirigid structures.Let ˆM be a complex supermanifold of dimension 1|N equipped with a holomorphic dis-tribution (subbundle of TM) E of dimension 0|N. Given ( ˆM, E), one can always define asymmetric bilinear B : E ⊗E →T /E, where T is the holomorphic tangent bundle.

Thebilinear is given by B(Ei, Ej) ≡[Ei, Ej] mod E, where [ , ] is the graded Lie bracket andEi ∈E. Following [11][21][12], we will call ( ˆM, E) an N-SRS if B is non-degenerate.A SRS can also be regarded as a reduction of the structure group on ˆM.

We simplydeclare a frame {E0, ⃗E } as “good” if E0 is even and Ei are an (odd) frame for the givenE. Then all good frames are related to one another by elements of a supergroup as follows: E′0⃗E′=a2⃗ω⃗0a↔M E0⃗E,(2.11)where a is an invertible even function, ⃗ω are odd functions, and↔M is an invertible matrixof even functions.

In order for the set of frames {E′i} to span the same distribution E as{Ei}, we have required the column ⃗0.We can always put a SRS in a more canonical form. The non-degeneracy conditionabove implies that the bilinear B is diagonalizable.

Thus we can always use a transforma-tion of the form (2.11) to get from a frame {E0, ⃗E } to a normalized frame with[Ei, Ej] = 2gijE0 mod E(2.12)where gij = δij. Such normalized good frames are then all related by a smaller group than(2.11), in which M is in the orthogonal group O(N, C).

This residual group we will callGN, and we will call a GN-structure an almost superconformal structure. Since we canalways pass to normalized frames, and the new frame is unique modulo the residual group,we find that an N-SRS in the above sense is precisely a reduction of the structure groupof ˆM to GN.

We will prove in sect. 3 that this reduction N ≤3 is always integrable.We would like to point out that E+ and E−(in a complex basis) in the N = 2 case arepreserved up to a multiplicative factor on a SRS because in this basis matrices in O(2, C)are diagonal.

Hence the distribution E is split into two line bundles. This is not true forN ≥3, a fact related to the existence of a nonabelian current algebra in the superconformalalgebra starting at N = 3.9

What are the “good” coordinate transformations for this superconformal structure?To answer this, and to make precise what we wish to prove in the integrability theorem, wemust specify the standard frames associated to a coordinate patch. We choose ˆE{z}0=∂∂z,ˆE{z}i= Di where z ≡(z, ⃗θ ) and Di are defined in (2.1).

We can then identify the N-superconformal coordinate transformations as those complex coordinate transformationsthat leave this structure unchanged along the lines similar to the discussion below (2.8).Then the “good” coordinate transformations preserving the standard G structure will takez to ez with ∂˜z⃗eD=a2⃗ω⃗0a↔M ∂z⃗D. (2.13)The set of coordinate transformations in the form of (2.13) are given by the N-supercon-formal transformations defined by (2.2)–(2.3).

As in the general analysis above, this leadsto a patch definition of super Riemann surfaces. Once the integrability theorem is provedin sect.

3 we thus have that every N-SRS in the above sense is also a SRS in the sense ofsect. 2.1.Next we turn to the semirigid case.

An almost TN-structure is obtained by reductionof the structure group from an (N + 2)-superconformal structure.Consider the set offrames spanning E, {Ei} = {E+, Er, E−} where E± =1√2(E1 ±iE2) and r = 3, . .

., N +2.The metric gij in this frame isgij =0⃗01⃗0egrs⃗01⃗00,(2.14)where egrs = δrs. The reduction from (2.11) is specified by the G-structure where now thegroup consists of matrices of the formK =a2ω−⃗ωω+01⃗Y−12Y egY t⃗0⃗0a↔M−aMegY t00⃗0a2.

(2.15)Here ⃗Y ,↔M have even elements, MegM t = eg, a is invertible and the ω are odd functions. Itcan be verified that matrices of type (2.15) form a supergroup GT N, which is a subgroupof GN+2.

In fact this structure group arises by a reduction from GN+2 by imposing extrastructure: we have chosen a 1d subbundle D−⊂E ⊂T . D−is not trivial; indeed we10

also choose a (parity-reversing) isomorphism D−∼= T /E. The good frames are those goodsuperconformal frames for which E−spans D−and corresponds to E0 mod E under thechosen isomorphism.

These frames are then all related by (2.15).The motivation for this construction is simple for TN = 0. Any kind of topolog-ical field theory should have a superspace formulation involving a global, spinless oddcoordinate for bookkeeping.

For us this coordinate will be θ+. For TN = 0 (2.15) saysthat “good” coordinate transformations take D+ to itself, and hence they also take θ+ toitself as desired.

For N > 0 this may not be so clear, but in fact (2.15) again ensures thatthe “good” TN-coordinate transformations are just N-superconformal transformationswhich keep θ+ fixed [2]. Note that the N-superconformal structure group is embeddedin that of TN semirigid geometry, GN ⊂GT N ⊂GN+2 by comparing with (2.11).

Thisis seen by setting ⃗Y = ω+ = ω−= 0. This is why the TN-coordinate transformationsinclude the N-superconformal group and give rise to topological supergravity.In sect.

4 we will find first order constraints which are sufficient flatness conditionsfor the existence of a coordinate system with the standard frames GT N-equivalent to theframes Ea defining the semirigid structure. Hence as in our general discussion a complexsupermanifold with an integrable GT N-structure is glued together by semirigid transitionfunctions, which recovers the patch definition of semirigid surfaces given in sect.

2.1.3. Superconformal IntegrabilityIn sect.

2.3 we defined an almost superconformal structure. We shall prove that thisreduction is always integrable for N = 3; there are no flatness conditions to impose in thiscase.

The cases N < 3 are much easier and can easily be obtained from our derivation.We expect N = 4 to be similar.We are given a distribution E which satisfies the non-degeneracy condition. As we havediscussed above we can always choose a frame {E0, ⃗E } with ⃗E spanning E and satisfying(2.12), or in the notation of (2.10)t0ij = 2gij ≡2δij,(3.1)and any two such frames are related by (2.11) with the matrix M orthogonal.

Indeed(2.11) shows that we have a lot of freedom with E0; modifying it by adding any linearcombination of the ⃗E does not change the superconformal structure. Given a normalized11

frame we can thus discard E0 and focus on ⃗E, regenerating E0 when needed by 12[E+, E−]or some other convenient variant.Recall that a complex structure has been given on the manifold and that {E0, ⃗E} areholomorphic. Hence in an arbitrary complex coordinate system with coordinates given byw and λi, we can represent {Ei} byEi = Mij∂j + αi∂w,(3.2)where ∂i ≡∂∂λi , ∂w ≡∂∂w and Mij and αi are holomorphic functions of w and λi.We would like to show that we can find a coordinate system in which {Ei} is GN-equivalent to the standard frame {Di}.

We shall proceed in four steps, order by order inthe odd coordinate λ.Step 1: We shall first find a coordinate system in whichEi = ∂i + O(λ). (3.3)Let M ji= m ji+ O(λ) and αi = αi0 + O(λ) in (3.2).

We make the following complexcoordinate transformation:eλi = λj[m−1]ji;ew = w.(3.4)Under coordinate transformation (3.4), we obtain thatEi = e∂i + αi0 e∂w + O(eλ).We can now drop the tildes for convenience. We make another complex coordinate trans-formationeλi = λi;ew = w + λrβr,(3.5)and obtainEi = e∂i + (βi + αi0)e∂w + O(λ).Choosing βi = −αi0, we obtain (after dropping the tildes again) (3.3).Step 2: Restoring λ terms in Ei, we haveEi = {δ ki + λrµkri }∂k + λrari∂w + O(λ2),(3.6)12

where we have introduced two functions µkriand ari. The normalization conditions (3.1)are easily seen to imply thata(ij) = δija0where a0 is some invertible function.

The antisymmetric part of aij can now be removedby a coordinate transformation of the formew = w + 12λsλtbst,while the trace bit can be set to one by a further transformation of the form ew = ew(w)with ∂ew∂w = a−10 . We will now use our freedoms to put µrik into more canonical form.Again we perform coordinate transformations.

Leteλi = λi + 12!λrλsρisr;ew = w,(3.7)where ρisr = ρi[sr] . In this coordinate systemEi = {δ ki + eλr(ρkri + µkri )}e∂k + eλi∂˜w + O(eλ2).

(3.8)We can also consider the GN-equivalent frame E′i = (KE)i, whereK ji = δ ji + λr(αjri + ξrδ ji ) + O(λ2). (3.9)Here αrij = αr[ij] is a generator of SO(3, C).

Together with (3.7) we see that we can shiftµ byµ →µrik + ρrik + αrik + ξrδik.To begin simplifying this we see we may without loss of generality use ρ to get µ = µ(ri)k,symmetric on the first two indices. A little algebra then shows that with an appropriatechoice of further ρ, α transformations we may take µ = µ(rik), and moreover using ξ wecan get µiij = 0.Step 3: Thus we haveEi = Di + λrµkri ∂k + 12λsλtǫtsu(Mkui ∂k + Θui∂w) + O(λ3),where again µ = µ(rik) and we have introduced the next order, coefficients Mkuiand Θui.Using our freedom to choose a convenient E0 we now takeE0 = 16Xi[Ei, Ei] + F ℓEℓ,(3.10)13

where F ℓis some function of w, λ of order λ. Imposing (3.1) to O(λ) now shows thatµ ≡0 and Θui ∝δui.

But this means that we may remove Θ altogether by the coordinatetransformation ew = w + λ3β, whereλ3 ≡16λsλtλuǫuts.Step 4: Thus we haveEi = Di + 12λsλtǫtsuMkui ∂k + λ3(si∂w + σ ℓi ∂ℓ)where si, σ ℓiare new sets of coefficients. There remain the freedom to make coordinatetransformations of the form eλi = λi + λ3Ki as well as SO(3, C) × C× frame rotations.One readily sees that this freedom suffices to make σ traceless symmetric, M = Mu(ik),si ≡0, and Miik = 0.We now make a convenient choice of F ℓin (3.10):E0 = 16Xi[Ei, Ei] −16(2λsǫsiuMℓui + λsλtǫtsiσiℓ)Eℓ.

(3.11)Then the condition (3.1) says σ ≡0, M ≡0. Thus we haveEi = Dias was to be shown.We close this section by remarking that superconformal integrability should be re-lated to the conformal flatness of an appropriate supergravity theory.

Indeed N = 3, 4supergravity theories have been constructed using conformal flatness as a principle [22].Perhaps the rather simple idea of superconformal geometry can shed some light on thestructure of these theories.4. Semirigid Integrability4.1.

TN=0 IntegrabilityWe now investigate the local integrability of semirigid structures. To begin supposewe have been given a TN = 0 (or “almost semirigid”) structure specified by a frame14

{E0, E+, E−} obeying (3.1). This is the same information as in the superconformal case,but now we do not consider two frames equivalent unless they are related by (2.15), i.e.E′0E′+E′−=c· · ·01000cE0E+E−.

(4.1)Thus to integrate the frame we have a harder job than in sect. 3: find local coordinatessuch that the standard frame equals the given one modulo GT N=0 ⊂GN=2 (4.1), not justmodulo GN=2 (2.11).We can again simplify the problem somewhat by noticing that E0 will take care ofitself once we put the ⃗E into the desired form.

Accordingly we take E0 =12[E+, E−],since this choice is still normalized correctly and is related to the given one by a GT N=0transformation (4.1).Following the procedure in sect. 2.2, we look for torsion constraints by taking astandard frame and applying an arbitrary transformation of the form (4.1): E+ = D+,E−= cD−where c is a function.

By the remark in the previous paragraph we then takeE0 = 12[E+, E−]. Computing tCABwe find that in addition to (3.1), preserved since wehave maintained the normalization condition, we also preserve various other elements of t,including in particulart+++= 0.

(4.2)Thus (4.2) is necessary for a frame to be integrable.Now suppose our given frame does satisfy (4.2). By the result of the previous sectionwe can at least find superconformal coordinates, i.e.

coordinates z = (z, θ±) such that{E0, ⃗E } is GN=2-equivalent to {∂z, ⃗D }. In particularDi = N ji EjN =ab−100ab∈SO(2, C)(4.3)for some invertible functions a, b.

We would now like to find another set of coordinatesez(z) witheDi = (K−1) ji EjK =100c. (4.4)Since z is not a good semirigid coordinate, ez(z) is not a semirigid transformation.

However(4.3)–(4.4) say z and ez are at least superconformal coordinates and so ez(z) will be a super-conformal transformation. But we know how ⃗D transform under the latter (eqn.

(2.2)).15

Putting it all together, given a, b in (4.3) we need to choose ez and c in (4.4) such that1c−1 b/a(ab)−1 D+D−= D+eθ+D−eθ−−1 D+D−. (4.5)In other words, while we can always adjust c to satisfy the second equation, we do needto find a superconformal transformation for which D+eθ+ = a/b.

As expected we see thatin general there is no solution. Imposing (4.2), however, tells that D+(a/b) = 0, whichensures that an appropriate function eθ+ exists.

To see that there is a ez with this eθ+, weneed to inspect the most general N = 2 superconformal coordinate transformation:ez = f + θ+tψ + θ−sτ + θ+θ−∂z(τψ)eθ+ = τ + θ+t + θ+θ−∂zτeθ−= ψ + θ−s −θ+θ−∂zψ,(4.6)where ∂zf = ts −τ∂zψ −ψ∂zτ. Thus, we haveD+eθ+ = t + 2θ−∂zτ −θ+θ−∂zt(4.7)and we can choose t, τ to match this to any chiral superfield a/b.4.2.

TN=1 IntegrabilityIn this subsection, we start with an N = 3 SRS endowed with the TN = 1 structuregiven by a frame {E0, E±, E3} normalized per (3.1), (2.14). As in the previous subsectionwe may discard E0 and replace it by E0 =12[E3, E3] without changing the semirigidstructure.Proceeding as before we get the torsion constraints by acting on the standard frame⃗D with3K =1x−x220a−ax00a2(4.8)where a is invertible.

Examining the commutators of Ei = K ji Dj we find that in additionto (3.1) we have (among other things)t+ij= 0 ,t3−−= 0. (4.9)3 Recall that in this basis the metric gij is antidiagonal.16

We will show that these necessary conditions are sufficient for integrability.Suppose then that we have a local frame ⃗E for E obeying (3.1) and (4.9) once weset E0 = 12[E3, E3]. Once again we can use the result in sect.

3 to choose superconformalcoordinates, so that Ei = (N −1)ijDj where N belongs to SO(3, C) × C×.Semirigidintegrability means that there exists a superconformal coordinate transformation ez(z) suchthat the given frame Ei is GT N=1-equivalent to the standard frame eDi:(K−1E)i = eDiwith K some matrix function of the form (4.8). Analogous to (4.5) this requires us to solveF = NKwhereFjk = Djeθk.

(4.10)In this equation we are seeking suitable ez(z) and K given N. Once again this is in generalimpossible until we impose the constraints (3.1), (4.9) on N.We will subdivide our task by writing K = K1K2 and choosing K1 to put ˆN = NK1into the form of a lower triangular matrix L (when N++ is invertible) or an “upper”triangular matrix U (when N++ is not invertible; see below) with unit determinant. Thisputs our problem (4.10) into standard form: F = LK2 or = UK2.

We will in the followingconcentrate on the case when N++ is invertible, prove the integrability theorem and thencomment on the other case.We organize the proof into four steps. First, we will show that NK1 can be put intothe form of L. Then we impose the semirigid essential torsion constraints on L (recall theconstraints are GT N=1-invariant).

With this done, we will substitute L into (4.10), andsolve for the eθ+ component of the superconformal coordinate transformation in terms of theunconstrained superfield components of L just as in sect. 4.1.

The rest of the components,eθ3 and eθ−, can always be made to satisfy (4.10) by choosing K2 appropriately. Finally,we will show by construction that there really does exist a superconformal coordinatetransformation with the required eθ+.The torsion constraint (3.1) implies that N belongs to SO(3, C) × C×, meaningNgN t ∝g.

In particular we haveN+−= −(N+3)22N++andN3−= N3+2( N+3N++ )2 −N+3N33N++. (4.11)17

(NK1)+3 is set to zero by choosing K1 in (4.8) withx1 = −a1N+3N++ . (4.12)Substituting (4.11) and (4.12) into the expressions (NK1)+−and (NK1)3−, they toovanish.Furthermore, a1 is chosen so that NK1 has unit determinant, that is, a1 =(det N)−13 .

Thus, NK1 by construction is a lower triangular matrix given byL =b00y10−y22b−yb1b. (4.13)We can now let eEi = (K−11 E)i = (L−1D)i.

While we have used our GT N=1 freedomto put L into the standard form (4.13), still further restrictions come when we imposethe torsion constraints (4.9). Since these torsion constraints are by construction GT N=1-invariant, we can impose them on eEi.

The constraints give respectively(L−1)−k Dk(L−1)−lLl3 = 0and(4.14){(L−1)(ik Dk(L−1)j)l −gij(L−1)3k Dk(L−1)3l} Ll+ = 0. (4.15)Substituting (4.13) into (4.14) and (4.15), we obtain the following four constraints on thetwo independent matrix elements b and y of the matrix L:D+b = 0,(4.16)D+y + D3b = 0,(4.17)bD−b −2bD3y −yD+y = 0,and(4.18)y22 D+y + byD3y −b2D−y = 0.

(4.19)In appendix A, we show that under this set of torsion constraints we obtain a unique oddsuperfield Ωsatisfyingb = D+Ω,y = D3Ω,and gij(DiΩ)(DjΩ) = 0. (4.20)We are now ready to show that there exists a superconformal coordinate transforma-tion and suitable K2 for which F = LK2.

That is,D+eθ+D+eθ3D+eθ−D3eθ+D3eθ3D3eθ−D−eθ+D−eθ3D−eθ−=bbx2−bx222yx2y + a2−x2(a2 + x2y2 )−y22b−yb (a2 + x2y2 )1b (a2 + x2y2 ),(4.21)18

where a2 and x2 are the independent elements of K2. Taking the determinant of bothsides of (4.21), we see that we have to choosea2 = det (F)13 .

(4.22)As for x2, we will choose it so that bx2 = D+eθ3. Eqn.

(4.20) then shows that the firstcolumn of equations (4.21) are satisfied when we identify eθ+ as Ω. One can show that theremaining five components of the matrix equation (4.21) are then satisfied by the use of thesuperconformal conditions (2.3).

These turn into two sets of readily applicable relationsFgF t = g(det F)23(4.23)and the set of equations where we replace F by F −1, since F −1 is also a superconformaltransformation.Finally, the question is if there exists an N = 3 superconformal coordinate transfor-mation with eθ+ given by the function Ω. The answer is yes; details are given in appendixB.

The point is that from the superconformal conditions ez can be expressed in terms ofthe components of the transformation of eθi, i = +, 3, −. The only requirement left for thecoordinate transformation to be superconformal is that the eθi satisfy the superconformalconditions among themselves.

In appendix B, we have expanded ez and eθi in components.There are four even and four odd components in each of the superfields. We set out witheθ+ given, namely Ω, and there are sixteen degrees of freedom in the components of eθ3and eθ−to choose to satisfy the internal superconformal conditions.

The superconformalconditions among eθi are linear in the components of eθ3 and eθ−and there are sixteen suchequations. We have shown in the appendix that indeed a solution exists.

If all the evencomponents of eθ+ are invertible, then we use all sixteen degrees of freedom to solve thesixteen equations. If one or more even components of eθ+ are noninvertible, then the lin-ear matrix equations become singular and it implies that there are more variables thanequations.

Thus, there exists a family of solutions.When N++ is not invertible, from the fact that N belongs to SO(3, C) × C×, weimmediately obtain that N−+, N+−, and N33 are invertible. Since N−+ is invertible, wecan choose elements in K1 so that (NK1) takes the formU = NK1 =−y22b−yb1by10b00.

(4.24)19

All the essential torsion constraints are the same as before with the roles of D+ and D−interchanged. We again wish to find a superconformal coordinate transformation F sothat (4.10) is satisfied.

We then have b = D−eθ+ and y = D3eθ+. The rest of the proofis analogous to the previous case with the roles of the superfield components switchedbetween the untilded + and −components and a sign change for the tilde componentsalong with interchanging the + and −components (e.g.

s−→s+ and eψ−→−eψ+).5. Moduli space of semirigid surfacesThere exists a natural projection from the moduli space of TN-semirigid surfaces tothat of N-SRS [1][2].

We will show that this is the case for TN = 0, 1. This can beeasily extended for the case of arbitrary N. As explained earlier, an N-SRS is obtainedby patching together pieces of C1|N by means of N-superconformal transformations:zα = fαβ(zβ; ⃗m, ⃗ζ )(5.1)where z = (z, ⃗θ ) and ⃗m (⃗ζ ) are the even (odd) moduli.

Following [1][3] we obtain aug-mented N-superconformal transformations by introducing a new global odd variable θ+and promoting all the functions given above to be arbitrary functions of θ+ in addition toz. Now an augmented N- superconformal surface is obtained by patching together piecesof C1|N+1 by means of the augmented superconformal transformations.

An augmented N-superconformal surface still has a distinguished distribution ¯E of dimension 0|N spannedby ⃗D. This is seen by checking that under augmented superconformal transformations, ¯Eis preserved.The group of augmented N-superconformal transformations is isomorphic to TN-semirigid transformations.This has been proved for the cases of TN = 0 [1] and forTN = 1, 2 [2].

Since we may represent any SSRS by a collection of semirigid patchingfunctions, we can apply this isomorphism to obtain an augmented SRS and vice versa.4This isomorphism implies that the moduli spaces of TN-SSRS and augmented N-SRSare identical. Hence, it suffices to study the moduli space of augmented SRS.4 Note however that as a complex supermanifold the TN-surface is of dimension 1|N + 2while the corresponding augmented N-SRS is of dimension 1|N + 1.

The missing θ−carries noinformation, though it was crucial to get superfield formulas in [1].20

The moduli of the augmented superconformal surfaces are obtained by replacing themoduli of the superconformal surfaces by functions of θ+ in (5.1), that isma 7→ema + θ+ ˆmaζµ 7→eζµ + θ+ˆζµ(5.2)where we have introduced extra odd(even) moduli, ˆma (ˆζµ) and placed tildes on ema (eζa)to avoid confusion with the ma (ζa) on the original space. Hence, given any family ofN-SRS, we obtain a family of augmented N-SRS with twice as many parameters.

Welack global information regarding the moduli space of augmented superconformal surfaces.For example, we do not know if any of the new even coordinates ˆζµ are periodic. But wecan easily argue that infinitesimally, (5.2) spans the full tangent to the moduli space whenwe vary em, ˆm, eζ, ˆζ.

First, we note that deformations of any augmented SRS involvesmall changes in the patching maps. These are generated by vector fields Vαβ on Uα ∩Uβwith no θ+ component (this follows from the global nature of θ+).

Expanding Vαβ in apower series in θ+, we get two identical copies of the deformation space of N-SRS, withopposite parity. Furthermore, given V A(z, ⃗θ, θ+) = vA(z, ⃗θ ) + θ+νA(z, ⃗θ ) with V θ+ = 0,the vector field {vAαβ} generates infinitesimal deformations in the moduli em and eζ and thevector field {νAαβ} generates infinitesimal deformations in ˆm and ˆζ from (5.2).A projection down to the moduli space of N-SRS corresponds to forgetting the newmoduli introduced, that is, given a point with coordinates ( ema, ˆma, eζµ, ˆζµ) in the aug-mented moduli space, we project down to the point with coordinates (ma = ema, ζa = eζa)in the moduli space of SRS.

We would like to show that the projection is natural.Let us now discuss projections in general. We wish to define a map Π from a spacecM to M. Let (exa, ˆxa) be a set of coordinates near eP onˆM and xa be coordinates nearP on M. We can define a projection Π by taking xa(P) = exa( eP) or in other wordsΠ∗(xa) = exawhich we refer to as the “forgetful” map.

Unfortunately, the definition of Π depends on thechoice of coordinates. Let (eya = eF a(exb, ˆxb ), ˆya = ˆF a(exb, ˆxb )) be another set of coordinatesnear eP.

Also, let ya = F a(⃗x) be a new coordinate near P. The new coordinates will definethe same map Π as the old ones only ifeya ≡eF a(exb, ˆxb ) = F a(exb). (5.3)21

Of course arbitrary coordinates for cM will not be related to (exa, ˆxa ) by (5.3). But ifcM has some natural class of coordinates all related by (5.3) then we do obtain a globalprojection Π.

We will now see that semirigid moduli space does have such a natural classof coordinates.Begin with the case of TN = 0 following the discussion in [1]. A Riemann surface isobtained by patching together pieces of C1 by means of the transition functionzα = fαβ(zβ, ma),where ma are complex coordinates on the moduli space of complex dimension (3g −3).We now obtain a class of augmented Riemann surfaces parametrized by ( ema, ˆma) usingthe augmented transition functionszα = fαβ(zβ, ema + θ+β ˆma),= fαβ(zβ, ema) + θ+β ∂afαβ(zβ, ema) ˆma;θ+α = θ+β,(5.4)where a point in the moduli space of augmented Riemann surfaces has coordinates ( ema,ˆma).

Coordinates obtained in this way are not the most general ones, and indeed we willnow show that they are all related by the special class of maps (5.3).Let na(ma) be a new set of coordinates on the moduli space of ordinary Riemannsurfaces. We obtain the patching function parametrized by na by means of the followingidentification:ˇfαβ(zβ,⃗n) ≡fαβ(zβ, ⃗m(⃗n)).

(5.5)The corresponding family of augmented Riemann surfaces is again given by the rule (5.2):zα = ˇfαβ(zβ, ena + θ+ˆna)= ˇfαβ(zβ, ena) + θ+β ∂a ˇfαβ(zβ, enb)ˆnaθ+α = θ+β. (5.6)Comparing (5.4) and (5.6) using (5.5) shows that the two sets of coordinates on the modulispace of TN = 0 surfaces are related by the transition functionena = ena( emb),ˆna = ∂na∂mbm=emˆmb,22

which is not only of the form (5.3) but in fact split. Hence in particular the projectionfrom augmented N = 0 surfaces to ordinary ones is natural, and as we have already seenthat this gives the desired projection from TN = 0 surfaces to N = 0.For the case of TN = 1, the situation is similar.Let ( ema,ˆma, eζµ, ˆζµ) be thecoordinates of a point in the moduli space of augmented N = 1 SRS and (ena, ˆna, eφν, ˆφν)be the coordinates of the same point on another patch.

Following similar arguments as forTN = 0, we obtainena = ena( emb, eζν),ˆna = ∂na∂mb ˆmb + ∂na∂ζν ˆζν,eφµ = eφµ( emb, eζν),ˆφµ = ∂φµ∂mb ˆmb + ∂φµ∂ζν ˆζν,which is again of the form (5.3) and hence the “forgetful” map is again natural. Thiscan be seen to hold for the case of arbitrary TN since the only property which makes thetransition function split is the global nature of θ+.

Thus, there exists a natural projectionfrom the moduli space of TN-SSRS to the moduli space of N-SRS. The significanceof this result is that [1] it means we can use string-theory methods to get a measure onthe big space, then integrate it over the fibers of this projection to get a measure on thesmaller space, namely the moduli space of N-superconformal surfaces, which is where theobservables of topological gravity should live.6.

ConclusionIn this paper, we have provided an intrinsic definition of N-SRS and TN-SSRSwhich appeared naturally in (super)gravity and topological (super)gravity respectively.The intrinsic definitions are given in the context of G-structures. It is straightforward todefine superconformal or semirigid G-structure from the coordinates given in the patchdefinition of SRS or SSRS.

Much of our analysis was devoted to showing how one canrecover the patch definition given a G-structure on a manifold. That is, we first obtainedthe necessary torsion constraints where needed and showed that the almost G-structure isintegrable under such conditions.Moreover, we have shown that there exists a natural projection from the moduli spaceof TN-SSRS to that of N-SRS.

Since a field theoretical realization of topological TN-gravity can yield an integration density on the moduli space of TN-SSRS, the natural23

projection allows us to integrate along the fibers of the projection and obtain an integrationdensity on the moduli space of N-SRS. If there are non-trivial observables, then the fieldtheory provides for us cohomology classes on the moduli space, thus probing its topology.This procedure has been used for the case TN = 0 in [3][4]; it would be interesting to seewhat topologies one can probe for TN ≥1 cases.We would like to thank J. Distler, B. Ovrut and S-J.

Rey for useful conversations.This work was supported in part by NSF grant PHY88-57200, DOE contract DOE-AC02-76-ERO-3071 and by the A. P. Sloan Foundation.Appendix A. Solving the torsion constraintsWe will show that under the constraints (4.16) to (4.19), we can find a unique Ωtorepresent b and y by D+Ωand D3Ωrespectively and satisfying the constraintD+ΩD−Ω= −12(D3Ω)2.

(A.1)Anticipating Appendix B we note that if Ω= eθ+ then (A.1) is one of the superconformalconditions involving only eθ+ in (4.23) when F −1 is used.We will now go through the constraints (4.16) to (4.19) and show how we get Ω.Eqn. (4.16) implies that b = D+Ωwhere Ωis an odd superfield.

To see that is possible,one way is to expand both b and Ωin components θi and constrain b by (4.16). Thenit is straightforward that equating b and D+Ωturns to algebraic equations between theircomponents, thus solving for the components of Ωin terms of that of b.

However there isa residual freedom Ω→Ω′ = Ω+ ω where D+ω = 0 leaves b = D+Ωinvariant. We willmake use of this degree of freedom to make y = D3Ω.

Substituting b = D+Ωinto (4.17),it implies that y = D3Ω+ B, where D+B = 0. Here we will use the freedom in choosingΩ′ to cancel B, that is, D3ω = −B.

This is possible because both ω and B are annihilatedby D+, and in components, it means solving two algebraic equations and two first orderlinear differential equations in the components of ω in terms of that of B. There is stilla little freedom left in Ω′, namely Ω′′ = Ω′ + φ, where D+φ = D3φ = 0.

In components,this means φ = θ−φ−, and φ−is a constant. Again, this constant will be used later on.24

Dropping the primes, we now have b = D+Ωand y = D3Ω; substituting both into (4.18)and (4.19), we obtainD3ΩD+D3Ω= −D+ΩD+D−Ω(A.2)and−12(D3Ω)2 D3D+Ω+ D+ΩD3Ω∂zΩ+ (D+Ω)2 D3D−Ω= 0. (A.3)Eliminating D3ΩD+D3Ωin (A.3) by (A.2), we get−12D3ΩD−D+Ω+ D+ΩD−D3Ω= 0.

(A.4)Equations (A.2), (A.3) and (A.4) can be rewritten asDi[D−Ω+ 12(D3Ω)2D+Ω] = 0,(A.5)where i = +, 3, −respectively. This implies that whatever is inside the square bracket canat most be some arbitrary constant.

This constant can be cancelled by the remaining freeconstant φ. By construction, D3 and D+ annihilate φ, and D−φ = φ−.

Thus, φ−will bechosen to cancel the arbitrary constant, and we are left with (A.1).Appendix B. N=3 superconformal coordinate transformationIn this appendix, we will give the conditions for the N = 3 superconformal coordinatetransformation. We will then show that there exists an N = 3 superconformal coordinatetransformation when eθ+ is given subject to (A.1).

This is needed in the proof of N = 3semirigid integrability.We expand the superconformal transformation in components. Letez = f + θ+φ+ + θ3φ + θ−φ−+ θ3θ+ ef+ + θ+θ−ef + θ−θ3 ef−+ θ+θ−θ3 eφ(B.1)andeθi = λi + (mt)ijθj + (Γt)ijgjk 12ǫkℓmθℓθm + eℓiθ+θ−θ3,(B.2)where i = +, 3, −andmij =t+n+s+tnst−n−s−,Γij =eτ+eν+eψ+eτeνeψeτ−eν−eψ−,(B.3)25

λi =τνψ,eℓi =etenes,(B.4)the metric g =001010100, and ǫ−3+ = 1.The superconformal conditions can be compactly written asmgmt = g(mgmt)33,(B.5)(mgΓt)ij = gij(mgΓt)33 + ǫijk(g−1mg∂zλ)k,(B.6)(mgeℓ)i = (Γg∂zλ)i −14ǫijk[2ΓgΓt + (∂zm)gmt −mg∂zmt]jk,(B.7)∂zf = (mgmt)33 + (∂zλ)tgλ,(B.8)(φ)i = (mgλ)i,(B.9)( ef)i = (Γgλ)i,and(B.10)eφ = eℓtgλ −(mgΓt)33. (B.11)There are two things that one notices from (B.5) to (B.11).

One is that m belongs toSO(3, C)×C×, thus only four matrix elements are independent. The rest can be expressedin terms of the four independent variables.

The other observation is that the componentsof ez of the transformation are expressed in terms of the components of eθi given by (B.8)to (B.11). Thus (B.5) to (B.7) are internal superconformal conditions that have to besatisfied by eθi.Our problem is that we are given the components of eθ+, and we wish to see thatthere exists a superconformal coordinate transformation with this eθ+ by choosing thecomponents of eθ3,−to satisfy the internal superconformal conditions.

Let us work withthe case when N++ is invertible. This implies that b = D+eθ+ is also invertible and henceso is t+.

From the lowest component of (A.1), when Ωis identified with eθ+, we havet2 = −2t+t−. Thus even though eθ+ is handed to us, we know that t is not independentof t+ and t−.

We will take t+ and t−as two independent elements of m. There are twoleft, and we will choose them to be s−and n. For now, the only constraint we put on s−and n is that they are invertible. This gives m a invertible determinant.

The rest of thefive entries of m are expressed in terms of t+, t−, s−and n by (B.5). Since eθ+ is givento us, we now have, in addition to s−and n, the rest of the six elements of Γ, the lowest26

and highest components of eθ3 and eθ−to choose to satisfy the eight conditions in (B.6) andthree in (B.7). Since t+, s−and n are invertible, we invert them in (B.6) to solve for ∂zν,∂zψ, eψ+, eν+,−and eν, thus satisfying six of the eight conditions of (B.6).

The two variableseψ−and eψ have coefficients t−. t−is given to us and it may vanish.

If it does not, thenwe can invert it and choose eψ−and eψ to satisfy the last two conditions. If t−vanishesthen by (B.5) and by (A.1), we conclude that n2−= −2s−t−= 0, t−= t = et = eτ−= 0and eτ = ∂zτ = keτ+, where k is some even function.

Under these circumstances, the twoconditions become vacuous. Similarly, we invert t+ and n in (B.7) to solve for the highestcomponents of eθ3 and eθ−, en and es respectively, thus leaving one condition to be satisfied.The problem is that we cannot choose et to satisfy this equation, but one can see that ift−is invertible, then we can invert n−and choose ∂zen to satisfy this condition.

If t−vanishes, then this condition becomes vacuous. Thus, all superconformal conditions canbe satisfied given eθ+ and we are able to complete the rest of the superconformal coordinatetransformation.27

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