# Twistor Parametrization of Locally BPS Super-Wilson Loops

Authors: Cristian Vergu
Journal: Advances in Mathematical Physics
 (2017)

---

## Abstract

We consider the kinematics of the locally BPS super-Wilson loop inN=4 super-Yang-Mills with scalar coupling from a twistorial point of view. We find that the kinematics can be described either as supersymmetrized pure spinors or as a point in the product of two super-Grassmannian manifolds G2∣2(4∣4)×G2∣2(4∣4). In this description of the kinematics the scalar–scalar correlation function appearing in the one-loop evaluation of the super-Wilson loop can be neatly written as a sum of four superdeterminants.


---

## Body

## 1. Introduction

The dual superconformal symmetry is one of the more surprising and unexpected results arising from studies of scattering amplitudes inN=4 theory. It started as an observation (see [1]) of an inversion symmetry property of integrals appearing the perturbative expansion. From there it grew into a full superconformal symmetry, as shown in [2], and finally into an infinite-dimensional Yangian in [3].The Yangian symmetry is at the core of the duality between polygonal super-Wilson loops and scattering amplitudes; in particular it contains both the conformal group acting on scattering amplitudes and the dual conformal group acting on super-Wilson loops.Initially the duality was between polygonal Wilson loops and MHV scattering amplitudes (see [4–8]). Later it was extended beyond MHV for scattering amplitudes and to polygonalsuper-Wilson loops (see [9, 10]). However, it has proven challenging to actually use this big symmetry to constrain the functional form of scattering amplitudes or of super-Wilson loops, mainly because of IR/UV divergences (see however [11–16] for progress in this area).For the super-Wilson loops the divergences arise because of the cusps in the contour. Reference [17] asked the question of whether it was possible to avoid the cusps (and the divergences) while still preserving the Yangian symmetry. They found that this is possible but, besides the coupling to the gauge field A, they had to add a coupling of the scalars ϕ as well, in a way which would correspond to the dimensional reduction of a light-like Wilson loop in ten-dimensional N=1 super-Yang-Mills theory. For a SU⁡(N) gauge theory, this super-Wilson loop reads(1)W=1Ntr⁡Pexp⁡∫Cdtp·A+θ˙·Aθ+θ¯˙·Aθ¯+q·Φ,where p=x˙+(1/2)θ˙θ¯-(1/2)θθ¯˙, q is a six-dimensional vector such that p2-q2=0, A is the Grassmann even gauge superfield, Aθ and Aθ¯ are the Grassmann-odd components of the gauge superfield, and Φ is the scalar superfield.Such Wilson loops were studied soon after the formulation of the AdS/CFT correspondence, in [18, 19]. Their supersymmetrization was studied in [20].The authors of [17] found that the Yangian symmetry is present in one-loop order for the first few orders in the Grassmann expansion. Later, [21] studied the superconformal symmetry, κ symmetry, and finiteness of these operators while [22] studied the Yangian symmetry to one-loop order but to all orders in the Grassmann expansion.In [21] the scalar–scalar two-point function was computed with the result. (Here and in the following we adopt a matrix notation, which allows us to omit all the indices. The index contraction is just matrix multiplication or matrix trace. In this notation x is a 2×2 matrix, θ is a 2×4 matrix, and θ¯ is a 4×2 matrix. The conventions are similar to the ones in [21, 22].)(2)Φab1Φ¯cd2∝1+θ¯21x21+-,-1θ21ca1+θ¯21x21+-,-1θ21dbdet⁡x21+-,where point i has coordinates (xi,θi,θ¯i) and θ12=θ1-θ2, x12+-=x1+-x2-+θ1θ¯2, and x±=x∓(1/2)θθ¯.This result, while not difficult to obtain, does not make the superconformal symmetry manifest. Ideally one would like to find a set of variables with simple superconformal transformations, which can be used as building blocks for superconformal and Yangian invariants (in fact, we actually only need invariance up to total derivatives). This is what we set out to do in this paper.We describe the kinematics by two(2∣2)×(2∣4∣2) supermatrices V and V~ and two (2∣4∣2)×(2∣2) supermatrices U and U~, such that(3)VU~=0,V~U=0,sdet⁡V~·=sdet⁡U;·-1,sdetV·=sdet⁡U~;·-1,where the · can be replaced by any (2∣4∣2)×(2∣2) supermatrix such that all the superdeterminants make sense. The V and V~ transform by a right action of the superconformal group and are defined up to a left action by (2∣2)×(2∣2) supermatrices. The same holds for U and U~ but with left and right interchanged.Using these variables, the scalar–scalar correlation function can be written as(4)q1·Φ1q¯2·Φ¯2∝sdet⁡V1U2+sdet⁡V1U~2+sdet⁡V~1U2+sdet⁡V~1U~2.We hope that this parametrization, besides making the superconformal symmetry manifest, will be useful in finding the Yangian invariants as well, along the lines of [23–25].


## 2. Kinematics

For a2n×2n antisymmetric complex matrix A, there is a decomposition A=UΣUT, where U is a unitary matrix and Σ is a block-diagonal antisymmetric matrix with 2×2 blocks:(5)Σ=0c100⋯⋯-c1000⋯⋯000c2⋯⋯00-c20⋯⋯⋮⋮⋮⋮⋱⋯0000cn⋯000-cn0.Moreover, we can choose the ci to be real and positive.A similar decomposition exists for real antisymmetric matrices, for which the unitary matrices in the decomposition above are replaced by orthogonal matrices.In the following we considerqab, which is a 4×4 antisymmetric complex matrix (therefore we set n=2 in the general discussion above). We denote by q¯ the complex conjugate matrix. Its decomposition reads q¯=U∗ΣU†.The matrixq can be identified with a six-dimensional vector. We denote the norm of this vector by q2. The matrices q and q¯ are related by a constraint qq¯=-q214. Using the decomposition above, we find that this implies q2=c12=c22 and, since c1 and c2 are real and positive, c1=c2. We denote the common value by c.The final constraint we impose is the duality conditionqab=(1/2)ϵabcdq¯cd. It implies that det⁡U=1; therefore we have U∈SU(4) while initially we had U∈U(4).In conclusion,q=UΣUT with U∈SU(4) and(6)Σ=0c00-c000000c00-c0,and c>0.Givenq, there are several matrices U which yield the same q. For example, if PΣPT=Σ, U and U′=UP give the same q. Such matrices P form a group, which is Sp(4,C). If we also want to preserve the unitarity conditions we should take P∈Sp(4,C)∩U(4). This is a real form (the group Sp(2) can be also defined as the subgroup of GL(2,H) which preserves the Hermitian form 〈x,y〉=∑i=12x¯iyi, where xi,yi∈H and x¯i is the quaternionic conjugate of xi) of Sp(4,C), called Sp(2) (or sometimes USp(4)).This means thatq is parametrized by the length q2=c2 and by the coset SU(4)/Sp(2). Using the isomorphisms SU(4)≃Spin(6), Sp(2)≃Spin(5), SO(5)≃Sp(2)/Z2, and SO(6)≃SU(4)/Z2, the coset is seen to be a five-dimensional sphere(7)SU4Sp2≃Spin6Spin5≃SO6SO5≃S5.A conformal transformationg can be seen as an element of SU(2,2). We write(8)g=ABCD,where A, B, C, and D are 2×2 matrices. The block decomposition of the inverse is(9)g-1=A~B~C~D~.The condition thatg∈SU(2,2) is(10)ABCD01-10A†C†B†D†=01-10,which implies that(11)A~B~C~D~=D†-B†-C†A†.The complexified and compactified Minkowski space is theG2(4) Grassmannian of two-planes through the origin of C4. A point on the Grassmannian can be described as a 4×2 matrix of rank two modulo the right action of GL(2). In some appropriate coordinate patch, a representative of this coset can be written as x1. The 2×2 matrix x encodes the space-time coordinates. In (1,3) Lorentzian signature x is Hermitian.A conformal transformationg, described by a 4×4 matrix above, acts on C4, the space on which the Grassmannian is defined, by left multiplication. If we perform such a transformation followed by a right multiplication by GL(2) to go back to the canonical form, we find the transformation law(12)x1⟶ABCDx1~Ax+BCx+D-11=x′1.Therefore, x→x′=(Ax+B)(Cx+D)-1.Let us also consider a dualG2(4) Grassmannian, which is the set of two-planes through the dual C4 to the one previously considered. A point in this Grassmannian can be represented by a 2×4 matrix, modulo a left action by GL(2). The conformal group acts on the right, by multiplication by the inverse. A representative of this coset can be written as 1-x. Under conformal transformations we have(13)1-x⟶1-xA~B~C~D~~1-A~-xC~-1-B~+xD~=1-x′.We therefore have two ways to write the transformations of x: x→x′=(Ax+B)(Cx+D)-1 and x→x′=(-xC~+A~)-1(xD~-B~). These two ways are related by Hermitian conjugation.For a2×2 matrix A we define A¯=(det⁡A)A-1=-ϵATϵ. This operation has the following properties: AB¯=B¯A¯, (A-1)¯=(A¯)-1, det⁡A=det⁡A¯, and A¯¯=A.The2×2 matrix p is Hermitian since x is Hermitian. Therefore, p can be diagonalized by a unitary matrix v:(14)p=vc100c2v†.The unitary matrixv can be restricted to be in SU(2) (in fact, even in PSU(2)) and we will do so from now on. Since det⁡v=1 and v is 2×2, we have v-1=v†=-ϵvTϵ and also(15)pϵ=v0c1-c20vT,p¯ϵ=v0c2-c10vT,p¯=vc200c1v†,which shows that (pϵ)T=-(p¯ϵ). If we denote by p2 the norm of the momentum, then p2=c1c2=det⁡p=det⁡p¯ and p0=1/2tr⁡p=1/2tr⁡p¯.Givenp, v is not uniquely determined. In fact, p can be parametrized by c1, c2 and a coset S2≃SU(2)/U(1). In other words, the momentum is parametrized by its zero component, its norm, and the direction of its space-like projection.


## 3. Pure Spinor Parametrization

Let us now solve the following equation inλ:(16)λp¯ϵ-qλ∗=0.Using the decomposition of p¯ and q worked out above we can reduce this equation to(17)ξ0-c2c10+cϵ00ϵξ∗=0.This equation needs to be solved in the 4×2 matrix ξ. Given a solution of this equation we can find λ=Uξv†.Whenc1c2=c2, which just corresponds to a 10D light-likeness condition, something special happens. Indeed, consider the explicit form ξij with i=1,…,4, j=1,2. Then,(18)ξ0-c2c10+cϵ00ϵξ∗=c1ξ12+cξ21∗-c2ξ11+cξ22∗c1ξ22-cξ11∗-c2ξ21-cξ12∗c1ξ32+cξ41∗-c2ξ31+cξ42∗c1ξ42-cξ31∗-c2ξ41-cξ32∗.If we set(19)ξ11=c1cξ22∗=cc2ξ22∗,ξ21=-c1cξ12∗=-cc2ξ12∗,ξ31=c1cξ42∗=cc2ξ42∗,ξ41=-c1cξ32∗=-cc2ξ32∗,then the left-hand side vanishes. If we want to see the transformation(20)ξ⟼ξ0-c2c10+cϵ00ϵξ∗,as a real-linear operator acting on a 16-dimensional real space (the space of ξ), then this linear operator has a real 8-dimensional kernel. Therefore, it must have a real 8-dimensional image.If we fix the normalization ofξ by |ξ12|2+|ξ22|2+|ξ32|2+|ξ42|2=c2 and as a consequence |ξ11|2+|ξ21|2+|ξ31|2+|ξ41|2=c1, then we have(21)ξ†ξ=c100c2,and therefore(22)λ†λ=p,where we recall that λ is a 2×4 matrix. The hermiticity of p is manifest.Next we define(23)ξ~=c2αξ11c1αξ12c2αξ21c1αξ22c2βξ31c1βξ32c2βξ41c1βξ42,with(24)α=1cξ322+ξ422ξ122+ξ222,β=-1cξ122+ξ222ξ322+ξ422.The ξ and ξ~ satisfy some useful identities(25)ξ†ξ~=0,∗ξ∗ϵξ†=ξ~ϵξ~T,where ∗M computes the Hodge dual of a 4×4 antisymmetric matrix M; that is, (∗M)ab=(1/2)ϵabcdMcd.We can also easily compute that(26)ξ~†ξ~=c200c1,ξϵξT+ξ~ϵξ~T=cϵ00ϵ.Using the definitions(27)λ=Uξv†,λ~=Uξ~v†,and the identities(28)vTϵv=ϵ,∗U∗MU†=U∗MUT,valid for v∈SU(2), U∈SU(4), and M antisymmetric, we can show that(29)λ†λ~=0,∗λ∗ϵλ†=λ~ϵλ~T,p=λ†λ,p¯=-ϵλTλ∗ϵ,q=λϵλT+λ~ϵλ~T,q¯=λ∗ϵλ†+λ~∗ϵλ~†.The constraints in the first two lines above are precisely the pure spinor constraints in a 4D-reduced form. They were presented in this form by Berkovits and Fleury in [26].We can now do the construction in the opposite direction; we start with two2×4 matrices λ and λ~ satisfying the pure spinor constraints; we compute p, p¯, q, and q¯ by the equations above and show that they satisfy the correct constraints. A similar construction was done by Howe in [27].Clearlyq and q¯ are antisymmetric and also q¯=q∗ and q¯=∗q. To check that the other constraints are satisfied by q and q¯ we need the following identity:(30)MN+∗N∗M=12tr⁡MN14,where M and N are 4×4 antisymmetric matrices. This identity can be shown by explicit computation.There are four terms in the productqq¯:(31)qq¯=λϵλTλ∗ϵλ†+∗λ∗ϵλ†∗λϵλT+λϵλT∗λϵλT+λ~ϵλ~T∗λ~ϵλ~T.The last two terms vanish by antisymmetry while the first two terms combine in a trace by using (30). We obtain(32)qq¯=12tr⁡λϵλTλ∗ϵλ†14=-det⁡λ†λ14=-det⁡p14.The constraints forp are automatically satisfied.


## 4. Supersymmetrization

The infinitesimal supersymmetry generator is represented by the supermatrix(33)m=0θ000θ¯000,where we have adopted a (2∣4∣2)×(2∣4∣2) block writing. The corresponding finite supersymmetry transformation is(34)M=exp⁡m=1θ12θθ¯01θ¯001.In fact, we can combine the supersymmetry transformations with translations in a single supermatrix(35)M=1θx-01θ¯001,M-1=1-θ-x+01-θ¯001,where x±=x∓(1/2)θθ¯.The supersymmetrized versions of the bosonic quantities can be obtained by acting with the transformation matrixM. For example, we find(36)M001=x-θ¯1,001M-1=1-θ-x+.These quantities are supersymmetric but not superconformal. Under superconformal action they transform as elements of a coset; that is, we need to include a right action by the stability group in order to preserve their form.We want to supersymmetrize the bosonic variables discussed in the previous section. Using their transformation properties under Lorentz andR-symmetry, it is easy to identify the bosonic part which should be supersymmetrized. We find(37)00λ001⟶M00λ001=U=θλx-λθ¯01,0λ†0100⟶0λ†0100M-1=V=0λ†-λ†θ¯1-θ-x+.This defines U to be a (2∣4∣2)×(2∣2) supermatrix and V to be a (2×2)×(2∣4∣2) supermatrix. These quantities transform covariantly under supersymmetry transformations (U under a left action and V under a right action), but under general superconformal transformations they require a compensating transformation by a (2∣2)×(2∣2) supermatrix (on the right for U and on the left for V).The spinorλ~ has the same transformation properties as λ so we can define U~ and V~ by replacing λ→λ~ in the definition of U and V, respectively.Here are the possibleVU products which yield (2∣2)×(2∣2) supermatrices:(38)VU~=0,V~U=0,VU=p000,V~U~=λ~†λ~000.The identities in the first two lines above are consequences of the constraints λ†λ~=0 and λ~†λ=0 described previously. We will describe later how the remaining pure spinor constraints arise.The productViUj between such supermatrices corresponding to different points in superspace is(39)ViUj=λi†λj-λi†θ¯ij-θijλj-xij+-,where θij=θi-θj and xij+-=xi+-xj-+θiθ¯j.Under superconformal transformations all these quantities transform homogeneously and get multiplied by a left and a right super-matrix. This implies that the super-determinant is covariant under superconformal transformations.


## 5. Correlation Functions

Using the expression forq and q¯ in terms of pure spinor components, we find(40)tr⁡q1q¯2=-2det⁡λ2†λ1+det⁡λ~2†λ1+det⁡λ2†λ~1+det⁡λ~2†λ~1.Supersymmetrically we expect the answer for the correlation function of two scalar superfieldsq1·Φ(1)q¯2·Φ¯(2) to be obtained by replacing(41)det⁡λ2†λ1det⁡x12⟶sdet⁡λ2†λ1-λ2†θ¯21-θ21λ1-x21+-=sdet⁡V2U1,where(42)sdet⁡MNPQ=det⁡M-NQ-1Pdet⁡Q=det⁡Mdet⁡Q-PM-1N.This is similar to the result obtained in [28], but it differs in some details. In the next section we will discuss the relation to harmonic superspace in more detail.Explicitly, we find(43)sdet⁡V2U1=det⁡λ2†λ1+λ2†θ¯21x21+-,-1θ21λ1det⁡x21+-=-12tr⁡λ2†1+θ¯21x21+-,-1θ21λ1ϵλ1T1+θ¯21x21+-,-1θ21Tλ2∗ϵdet⁡x21+-,where we have used the identity det⁡x=-1/2tr⁡xϵxTϵ, valid for all 2×2 matrices.Adding up all the four contributions reproducesq1=λ1ϵλ1T+λ~1ϵλ~1T and q¯2=λ2∗ϵλ2†+λ~2∗ϵλ~2†.Other superconformal covariants can be formed by joining together two(2∣4∣2)×(2∣2) supermatrices to form a (2∣4∣2)×(2∣4∣2) super-matrix. We define(44)sdet⁡U1;U2=sdet⁡x1-θ1λ1θ2λ2x2-θ¯1λ1λ2θ¯21001,where we have permuted the columns in order to make it possible to compute the superdeterminant (in order to be able to compute the superdeterminant the supermatrix should be a linear map between spaces of the same structure; that is, the sequence of integers describing the dimensions of the even and odd spaces should be the same for the domain of definition and for the image). In general permuting rows or columns of different parity does not preserve the value of the superdeterminant, but in the expression above we will simply take the right-hand side to be definition of the sdet⁡(U1;U2) notation.The easiest way to compute this superdeterminant is to apply a supertranslation which preserves it:(45)sdet⁡x1-θ1λ1θ2λ2x2-θ¯1λ1λ2θ¯21001=sdet⁡00-θ12λ2-x12+-0λ1λ2-θ¯121001.Next, we apply the identity(46)sdet⁡MNPQ=sdet⁡M-NQ-1Psdet⁡Q-1,where M, N, P, and Q are themselves supermatrices (see [29, page 104, Theorem 3.4] for a proof). This yields(47)sdet⁡U1;U2=sdet⁡00-θ12λ20λ1λ2-100-x12+--θ¯12=det⁡x12+-det⁡λ1,1+θ¯12x12+-,-1θ12λ2.Finally, we need to compute the4×4 determinant in the denominator. For a 4×4 matrix M=(λ,μ), where λ, μ are 4×2 matrices, we have(48)det⁡M=ϵabcdMa1Mb2Mc3Md4=ϵabcdλa1λb2μc1μd2=14ϵabcdλϵλTabμϵμTcd=-12tr⁡∗λϵλTμϵμT.Using this formula and applying the constraint ∗(λϵλT)=λ~∗ϵλ~†, we find(49)sdet⁡V~1U2=sdet⁡U1;U2-1.This identity is the supersymmetric version of the second pure spinor constraint. Since this holds for arbitrary U2, the dependence on U2 can be eliminated and then the identity reads(50)sdet⁡V~·=sdet⁡U;·-1,where · can be replaced by any (2∣4∣2)×(2∣2) supermatrix.Let us check that the expression for the correlation function(51)q1·Φ1q¯2·Φ¯2∝sdet⁡V1U2+sdet⁡V1U~2+sdet⁡V~1U2+sdet⁡V~1U~2has the required symmetry properties. For example, since q·Φ=q¯·Φ¯, we should find that q1·Φ(1)q¯2·Φ¯(2)=q¯1·Φ¯(1)q2·Φ(2). In Section 6 we show that sdet⁡(ViU~j)=sdet⁡Xij with Xij=Xi-Xj and therefore(52)sdet⁡ViU~j=sdet⁡Xij=sdet⁡-Xji=sdet⁡-VjU~i=sdet⁡VjU~i.Moreover,(53)sdet⁡ViUj=sdet⁡U~i;Uj-1=sdet⁡Uj;U~i-1=sdet⁡V~jU~i.The required symmetry follows from these two identities.


## 6. Harmonic Superspace

In this section we drop the reality conditions and complexify all coordinates.The results in the previous section can also be presented in the language of harmonic superspace, but we need to usetwo copies. The relevant superspace is the G2∣2(4∣4) super-Grassmannian introduced by Howe and Hartwell in [30]. The super-Grassmannian G2∣1(4∣2) was introduced even earlier for N=2 superconformal field theories by Rosly and Schwarz (see [31]). See also the book [32] by Manin, for a more mathematical treatment.For a4×4 antisymmetric matrix, the action of the Hodge dual can be written as(54)∗MN-NTP=PϵNϵ-ϵNTϵM.Next assume that we can gauge-fix(55)λ=1y†,λ†=1y.Then,(56)λϵλT=ϵϵy∗y†ϵy†ϵy∗,∗λ∗ϵλ†=yTϵy-yϵ-ϵyTϵ=λ~ϵλ~T,where(57)λ~=-y1,λ~†=-y†1.The equality ∗(λ∗ϵλ†)=λ~ϵλ~T relies on the identity yϵyT=yTϵy, valid for 2×2 matrices. This implies(58)q=ϵ+yTϵyϵy∗-yϵy†ϵ-ϵyTy†ϵy∗+ϵ,q¯=ϵ+y†ϵy∗ϵy-y∗ϵyTϵ-ϵy†yTϵy+ϵ,p=1+yy†.Using this gauge-fixed form of the variablesλ and λ† we find(59)V=01y-θ¯′-yθ¯″1-θ′-θ″-x+,where we split the θ and θ¯ in 2×2 blocks θ=(θ′,θ″) and θ¯=θ¯′θ¯″. By an SL(2∣2) left action, the V super-matrix can be transformed into a canonical form(60)V~01y-θ¯′-yθ¯″10θ′y-θ″-θ′θ¯′-θ′yθ¯″-x+≡0110X,where(61)X=y-θ¯′-yθ¯″θ′y-θ″-x-12θ′θ¯′+12θ″θ¯″-θ′yθ¯″.A similar analysis can be done for theU~ and its canonical form contains the same components as X. We have(62)ViU~j=Xi-Xj=Xij.It is important to notice that the product in the left-hand side can be written as a difference. These differences are manifestly invariant under a subgroup of the superconformal group, which acts by translations on X. This is one advantage of this parametrization of the harmonic variables.In [28], Howe and West expressed the correlation function ofprojections of scalar superfields as a super-determinant of Xij. More explicitly, instead of being contracted with the vector q, in their work the superfield Φ was contracted with λϵλT, so the second term λ~ϵλ~T did not appear. For our purposes, it is essential that we keep all the contributions since the super-Wilson loop contains contractions q·Φ for generic q while in [28] the superfield Φ is always contracted with a rank two matrix.In order to add these contributions, we follow the same idea as above and, after puttingV~ and U in canonical forms, we are led to introduce X~ as(63)X~=-y†y†θ¯′-θ¯″-θ′-θ″y†-x+12θ′θ¯′-12θ″θ¯″+θ″y†θ¯′.These are coordinates on a second G2∣2(4∣4) super-Grassmannian.So far we have shown that two of the terms in the scalar–scalar superfield correlation function can be written in a simple way in terms of harmonic coordinates:(64)sdet⁡V1U~2=sdet⁡X12,sdet⁡V~1U2=sdet⁡X~12.The other two contributions sdet⁡V1U2 and sdet⁡V~1U~2 donot have such a simple form. In particular, they can not be written in terms of differences of coordinates. To see this it is easiest to notice that VU≠0.The subgroup which acts by translating the variablesX has appeared before in [33, Section 3.1] by Berkovits and Maldacena on fermionic T-duality. The main result of this paper is that the type IIB AdS5×S5 string background is preserved by a sequence of T-dualities in the four space-time directions and in the eight fermionic directions corresponding to a set of anticommuting chiral supercharges. Moreover, this invariance under T-duality implies Yangian symmetry of the super-Wilson loop (see also [34]).As shown in [22] the nonchiral super-Wilson loop with a coupling to scalars is also Yangian invariant to one-loop order. Is it possible to show Yangian invariance to all orders in the ’t Hooft coupling by applying T-duality as in the chiral case? Berkovits and Maldacena also point out in Sections 3.1 and 3.2 of [33] that there isanother set of T-dualities which also preserve the AdS5×S5 background. The directions in which these T-dualities are performed are parametrized by X but can also correspond to X~. We should emphasize that X and X~ do not transform in the same way and in fact translations of X and of X~ do not commute.An obvious guess is that one needs to perform the T-dualities corresponding toX and then the T-dualities corresponding to X~, or the other way around. At weak coupling and for the chiral case, the effect of T-dualities was to pass to the dual space pi→xi-xi+1, qi→θi-θi+1 and one may hope to introduce a dual space for the nonchiral superspace in a similar way. The strong-coupling situation is more confusing. For example, in the chiral case one starts with some D3-branes which after four T-dualities along the space-time directions become D(-1) instantons, which are the vertices of the super-Wilson loop. In the nonchiral case we can perform some T-dualities along four space-time directions in X and then four T-dualities along the four space-time directions in X~. These space-time directions, however, differ by nilpotent quantities. It should be interesting to understand what kind of brane configuration gets generated by such a sequence of T-dualities.



---

*Source: 4852015-2017-12-31.xml*