Contents
###### Abstract

We investigate the existence of solutions with 16 residual supersymmetries to Type IIB supergravity on a space-time of the form warped over a two-dimensional Riemann surface . The isometry extends to invariance under the exceptional Lie superalgebra . In the present paper, we construct the general Ansatz compatible with these symmetries, derive the corresponding reduced BPS equations, and obtain their complete local solution in terms of two locally holomorphic functions on , subject to certain positivity and regularity conditions. Globally, are allowed to be multiple-valued on and be holomorphic sections of a holomorphic bundle over with structure group contained in . Globally regular solutions are expected to provide the near-horizon geometry of 5-brane and 7-brane webs which are holographic duals to five-dimensional conformal field theories. A preliminary analysis of the positivity and regularity conditions will be presented here, leaving the construction of globally regular solutions to a subsequent paper.

2016 June 1

Warped in Type IIB supergravity I

Local solutions

Eric D’Hoker, Michael Gutperle, Andreas Karch, Christoph F. Uhlemann

Department of Physics and Astronomy

University of California, Los Angeles, CA 90095-1547, USA

Department of Physics

University of Washington, Seattle, WA 98195-1560, USA

dhoker@physics.ucla.edu; gutperle@physics.ucla.edu;

akarch@uw.edu; uhlemann@uw.edu

## 1 Introduction

Gauge-gravity duality, namely the equivalence between a quantum field theory in dimensions and a gravitational theory in dimensions via holography, has become one of the cornerstones of modern theoretical physics. The best understood examples are provided by the AdS/CFT correspondence and involve conformal field theories (CFTs) with a large number of fields and their anti-de Sitter (AdS) dual space-times with large degrees of supersymmetry. Maldacena’s original paper [1] contained examples of such equivalences in and and many more dual pairs have been uncovered in these dimensions since then. However, what is almost entirely missing to date is any well understood example of the AdS/CFT correspondence in dimension , namely between a 4+1 dimensional CFT and its 5+1 dimensional dual AdS space-time.

One reason that is special is the absence of maximally supersymmetric theories, in which the 16 Poincaré supersymmetries of a 4+1 dimensional CFT would be enhanced by 16 conformal supersymmetries to the maximum allowed number of 32 supersymmetries. The complete classification of superconformal algebras [2] indeed shows that is singled out and does not support a maximally supersymmetric CFT. Unlike its lower dimensional cousins, supersymmetric Yang-Mills theory with maximal Poincaré supersymmetry in 4+1 dimensions is not a conformal field theory at the origin of its moduli space. Instead, the theory is believed to flow in the ultraviolet (UV) to the 5+1 dimensional theory, a theory which itself features prominently in AdS/CFT (see for example [3]).

What does exist, however, are supersymmetric CFTs (SCFTs) in 4+1 dimensions with 8 Poincaré supersymmetries which are enhanced by 8 conformal supersymmetries to a total of 16 supersymmetries. The superconformal algebra in this case is based on the exceptional Lie superalgebra . This Lie superalgebra has a real form whose maximal bosonic subalgebra is [4], which may be viewed as the direct sum of the conformal algebra in 4+1 dimensions [4] and an R-symmetry. Explicit field theory examples realizing this algebra have been uncovered in [5, 6]. They are based on supersymmetric gauge theories which flow to a strongly coupled CFT in the UV, as long as the number of matter fields coupled to the gauge field remains sufficiently small. The upper bound on the number of matter fields depends on the gauge group and on the representations of the matter content and can generically be determined from the 1-loop running of the gauge coupling [6]. Since these supersymmetric theories allow for a large limit, one may expect that they will possess a holographic dual.

String theory realizations of these 4+1 dimensional CFTs via brane embeddings are known and can serve as a natural starting point to construct their duals. For example, 4+1 dimensional SCFTs based on the gauge group , with hypermultiplets in the fundamental and anti-symmetric tensor representations of , were realized in [5] via a Type IIA string theory construction involving a stack of D4 branes near a collection of planes and branes. A Type IIA supergravity solution based on this construction was given in [7] and generalized to quiver gauge groups in [8]. While some interesting questions can be addressed in this geometry, its construction leads to singularities, because the presence of the O8 planes forces the dilaton to blow.

A much more general brane realization of these 4+1 dimensional CFTs can be given via brane webs [9, 10] in Type IIB string theory. In these brane webs, D5 branes are suspended between NS5 branes, giving a 4+1 dimensional version of the construction pioneered in [11] for 2+1 dimensional gauge theories. The brane webs can realize the gauge theory at any point on its moduli space, as well as in the presence of relevant deformations such as mass terms or a finite bare coupling. In the limit when all the branes essentially lie on top of one another the webs do realize large classes of the 4+1 dimensional SCFTs of [6], most naturally those based on gauge theories, possibly with couplings to matter in the fundamental representation.111Alternatively these SCFTs can be realized in terms of M-theory compactifications on singular Calabi-Yau (CY) 3-folds [6]. For the special case of toric CYs, this construction is T-dual to the web [12]. In any case, the realization in terms of branes serves as a more natural starting point for holographic considerations, where field theory and AdS gravity are seen as two equivalent descriptions of a given brane setup. D7-branes can be added to the webs [13], thereby slightly expanding the set of SCFTs that can be realized. As we will see, the most general supergravity Ansatz respecting the symmetries of the web does allow for a non-trivial axion winding which is required to accommodate D7-branes. When no D7-branes are present, the webs form a special sub-class for which the axion winding vanishes.

It is our goal to construct Type IIB supergravity solutions which are holographically dual to the 4+1-dimensional CFTs realized via webs in the large limit. This will require solving the difficult problem of obtaining fully localized solutions for the corresponding intersecting branes. Examples where such solutions can give rise to warped AdS spaces were given in [14, 15]. Our approach will be to use the general Ansatz for the Type IIB fields consistent with the symmetries, reduce the BPS equations to this Ansatz, and then solve the BPS equations explicitly.

Earlier attempts at using the BPS equations appeared in [16, 17, 18], where reduced BPS equations were obtained, but not generally solved. A Type IIB T-dual configuration of the D4/D8 solution in Type IIA [7] was used to test these equations. This T-dual solution is even more troublesome than the original Type IIA solution. In addition to the singularity caused by the presence of the O8 plane on the IIA side, one now has an additional singularity due to T-dualizing a subgroup of the isometry associated with an internal sphere. The circle that is being T-dualized shrinks to zero size at the poles of the sphere, giving rise to a new singularity in the Type IIB solution. Nevertheless, this T-dual to the D4/D8 system provides a useful check on the BPS equations obtained in [16, 17, 18] and we will use it for a similar purpose.

In the present paper, we will construct the general Ansatz in Type IIB consistent with the symmetries, derive the reduced BPS equations, and construct their general local solutions in terms of two locally holomorphic222Throughout, we will use the terminology locally holomorphic to include the possibility that may be multiple-valued on , and/or have poles or other singularities on the boundary of . functions on a Riemann surface . To connect these local supergravity solutions to the CFTs originating from the brane system requires that we impose the necessary physical regularity conditions on the supergravity fields of the solutions. For this purpose, must be compact, with or without boundary, and geodesicly complete. The conditions on needed to guarantee the proper Minkowskian signature of the metric are local on and given by the following inequalities,

 0 < −|∂wA+|2+|∂wA−|2 0 < |A+|2−|A−|2+B+¯B (1.1)

where is the derivative on with respect to a local holomorphic coordinate , and is defined, up to an additive constant, by the relation . As a result, is also locally holomorphic. If has a boundary then the two inequalities of (1) must hold strictly in the interior of , and become equalities on the boundary of .

The group acting linearly on the doublet leaves the conditions (1) invariant and induces the standard duality transformations on the supergravity fields. Additionally, the supergravity solutions are invariant under constant shifts for constants satisfying , which form the additive group isomorphic to . The duality symmetry of Type IIB string theory allows us to consider supergravity solutions with identifications under , namely with non-trivial axion winding number. Mathematically, the problem then becomes to obtain holomorphic sections of a holomorphic bundle over with structure group contained in , subject to the positivity conditions of (1).

In the simplest case where the Riemann surface is compact and has no boundary, the second inequality in (1) becomes trivial since the arbitrary constant in can always be chosen to satisfy the inequality. The associated mathematical problem then also simplifies, and may be formulated directly in terms of a holomorphic bundle of one-forms satisfying the first relation in (1). In this paper, we will provide a preliminary analysis into the existence of such global solutions but leave a detailed investigation for future work.

We close the introduction with some remarks on the relation of this work to other investigations into half-BPS solutions to Type IIB, M-theory, and six-dimensional supergravities on space-times built as products of an space and one or several spheres warped over a Riemann surface . In each case, the isometries of the space and the sphere factors are used to reduce the BPS equations to a complicated set of non-linear partial differential equations on , which can be solved exactly in terms of harmonic or holomorphic data on . This strategy was employed successfully to the construction of a large variety of novel supergravity solutions in different contexts. Type IIB supergravity duals to four-dimensional super-Yang Mills theory were found in the presence of a planar interface [19] giving supersymmetric Janus field theories [20, 21] and in the presence of Wilson loops [22]. M-theory duals to field theories in three and six dimensions were found in the presence of various defect operators in [23, 24, 25]. Finally, six-dimensional supergravity duals were found to two-dimensional conformal field theories with string junctions in [26].333See also [27, 28, 29, 30, 31] for related work on these systems using a variety of different strategies.

The unifying principle of this strategy was explained in [32] as follows. The integrability conditions on the BPS equations produce Bianchi and field equations. With enough supersymmetry, all the Bianchi and field equations may be obtained as integrability conditions on the system of BPS equations, which therefore play a role somewhat analogous to that of a Lax pair for integrable systems. Upon reduction to warped over , the BPS equations reduce to equations for functions on and genuinely become a set of Lax equations for a system that must therefore be integrable in the classic sense. The conformal invariance of these systems lies at the origin of their solvability in terms of harmonic and holomorphic data on a Riemann surface .

We note that close cousins to the half-BPS solutions to Type IIB supergravity obtained here are the half-BPS solutions on a space-time of the form warped over a Riemann surface . The superconformal algebra is now a different real form of the exceptional Lie superalgebra , this time with maximal bosonic subalgebra , and 16 supersymmetries as well. The two problems are related just as duals to Wilson loops in [22] are related to the planar interface solutions in [19].

### 1.1 Organization

The outline of this paper is as follows. In section 2 we will review the basics of Type IIB supergravity and introduce a suitable Ansatz. In section 3 we use the decomposition of the supersymmetry generators onto Killing spinors of and to reduce the BPS equations for the reduced supergravity fields to a system of algebraic and partial differential equations on the surface , which we partially solve. In section 4 we solve the reduced BPS equations completely and obtain the most general local solutions to the BPS equation in terms of holomorphic data. In section 5 we summarize the expressions for all the supergravity fields in terms of the holomorphic data, analyze their behavior under symmetry of Type IIB supergravity, obtain the regularity conditions, recover the singular T-dual of the D4/D8 system, and conclude the section with arguments in favor of monodromy of the holomorphic data. We conclude with a discussion in section 6. In appendix A, a basis for the Dirac-Clifford algebra adapted to our Ansatz is presented, while the geometry of Killing spinors is reviewed in appendix B. Details of the derivation of the BPS equations is in appendix C, of the Bianchi identities in appendix D, and of the expressions for the supergravity fields in terms of holomorphic data is in appendix E.

## 2 Type IIB supergravity and AdS6×S2×Σ Ansatz

In this section, we provide a brief review of the Type IIB supergravity fields, Bianchi identities, field equations and BPS equations, and their duality symmetry, and go on to construct the general Ansatz for the bosonic fields of the solutions we seek to construct. As laid out in the introduction, the bosonic symmetries of the Ansatz are completely determined by the superconformal algebra.

### 2.1 Type IIB supergravity review

The bosonic fields of Type IIB supergravity [33, 34] consist of the metric , a one-form and gauge connection representing the axion-dilaton field strengths, a complex three-form field strength , and a self-dual five-form field strength. The fields satisfy the following Bianchi identities,

 0 = dP−2iQ∧P 0 = dQ+iP∧¯P 0 = dG−iQ∧G+P∧¯G 0 = 8dF(5)−iG∧¯G (2.1)

The field strength is required to be self-dual,

 F(5)=∗F(5) (2.2)

The field equations are given by,

 0 = ∇MPM−2iQMPM+124GMNPGMNP 0 = ∇PGMNP−iQPGMNP−PP¯GMNP+23iF(5)MNPQRGPQR 0 = RMN−PM¯PN−¯PMPN−16(F2(5))MN (2.3) −18(GMPQ¯GNPQ+¯GMPQGNPQ)+148gMNGPQR¯GPQR

The fermionic fields are the dilatino and the gravitino , both of which are complex Weyl spinors with opposite 10-dimensional chiralities, given by , and . The supersymmetry variations of the fermions are,

 δλ = i(Γ⋅P)B−1ε∗−i24(Γ⋅G)ε δψM = DMε+i480(Γ⋅F(5))ΓMε−196(ΓM(Γ⋅G)+2(Γ⋅G)ΓM)B−1ε∗ (2.4)

where is the charge conjugation matrix of the Dirac-Clifford algebra.444Our convention for the signature of the 10-dimensional space-time metric is ; the Dirac-Clifford algebra is defined by the relations ; and the charge conjugation matrix is defined by the relations and . We will use the convention that repeated indices are to be summed; complex conjugation will be denoted by bar for functions and by star for spinors; and we will use the notation for the contraction of any antisymmetric tensor field of rank and the -matrix of the same rank. The BPS equations are obtained by setting .

The Bianchi identities (2.1) for the field strengths can be solved in terms of a complex scalar ; a complex 2-form potential , and a real 4-form potential . The fields and are expressed as follows,

 P=f2dB f2=(1−|B|2)−1 Q=f2Im(Bd¯B) (2.5)

while the fields and are conveniently expressed in terms of and with the help of the complex field strength ,

 G = f(F(3)−B¯F(3)) F(5) = dC(4)+i16(C(2)∧¯F(3)−¯C(2)∧F(3)) (2.6)

The scalar field is related to the complex scalar and the axion , and dilaton by,

 B=1+iτ1−iττ=χ+ie−2ϕ (2.7)

The expectation value of is related to the string coupling constant.

### 2.2 Su(1,1) duality symmetry

Type IIB supergravity is invariant under symmetry. This symmetry leaves the Einstein frame metric as well as the 4-form invariant, acts on the field by Möbius transformations, and acts on the 2-form and its complex conjugate by a linear transformation,

 B → Bs=uB+v¯vB+¯u C(2) → Cs(2)=uC(2)+v¯C(2) (2.8)

with and . In this non-linear realization of on , the field takes values in the coset , and the fermions and transform linearly under the isotropy gauge group with composite gauge field . The transformation rules for the field strengths are [33],

 P → Ps=e2iθP Q → Qs=Q+dθ G → Gs=eiθG (2.9)

where the phase is defined by,

 eiθ=(v¯B+u¯vB+¯u)12 (2.10)

The symmetry will serve as a useful guide to organize the holomorphic data in our local solution. As is well-known, the invariance of Type IIB supergravity under the continuous group is reduced in Type IIB string theory to invariance under the discrete S-duality symmetry, due to the charge quantization of non-perturbative one-branes, five-branes and D-instantons. In the construction of supergravity solutions, we will always allow for the continuous symmetry.

We seek a general Ansatz in Type IIB supergravity with the following symmetry group,

 SO(2,5)×SO(3) (2.11)

The factor requires the geometry to contain , while the factor requires , so that our space-time is given by,

Here stands for the remaining two-dimensional space over which the product is warped. In order for the above space to be a Type IIB supergravity geometry, must carry an orientation as well as a Riemannian metric, and is therefore a Riemann surface, possibly with boundary. This -invariant Ansatz for the metric can be written as,

where and are functions of only. We introduce an orthonormal frame,

 em = f6^emm=0,1,2,3,4,5 ei = f2^eii=6,7 ea a=8,9 (2.14)

where and respectively refer to the orthonormal frames for the spaces and with unit radius, and is an orthonormal frame for the metric on . In particular, we have,

 ds2AdS6 = η(6)mn^em⊗^enη(6)=diag(−+++++) ds2S2 = δij^ei⊗^ej ds2Σ = δabea⊗eb (2.15)

By -invariance, the fields and are given as follows,

 P=paea G=gaea∧e67 Q=qaea F(5)=0 (2.16)

where . The components , and are complex. Note that the Bianchi identity for the five-form field (2.1) is automatically satisfied with this Ansatz.

## 3 Reducing the BPS equations

The residual supersymmetries, if any, of a configuration of purely bosonic Type IIB supergravity fields are governed by the BPS equations of (2.1). Our interest is in purely bosonic field configurations which preserve 16 independent supersymmetries given by the Ansatz of the preceding section. It will turn out that any such configuration automatically solves the Bianchi and field equations, and thus automatically provides a half-BPS solution to Type IIB supergravity.

In this section, we will reduce the BPS equations to the Ansatz by decomposing the supersymmetry parameter of (2.1) onto the Killing spinors of . We will expose the residual symmetries of the reduced BPS equations, and solve those reduced equations which are purely algebraic in the supersymmetry spinor components. This will produce simple algebraic expressions for the metric factors in terms of the spinors. The remaining reduced BPS equations will then gradually be solved for the remaining bosonic fields as well as for the residual supersymmetries in subsequent sections. The strategy employed here is very similar to the one used in [19] and so our discussion will closely follow that work.

### 3.1 Killing spinors

The Killing spinor equations on are,555The decomposition of the 10-dimensional Dirac-Clifford matrices under the reduction to the Ansatz, and the details of the Killing spinor equations, are relegated to appendices A and B respectively.

 (^∇m−12η1γm⊗I2)χη1,η2α = 0 (^∇i−i2η2I8⊗γi)χη1,η2α = 0 (3.1)

where and stand for the covariant spinor derivatives respectively on the spaces and with unit radius. Recall that , , and are all frame indices. The spinors are 16-dimensional, and the parameters and can take the values . The solutions to these equation are 4-fold degenerate for each value of , and this degeneracy will be labeled by the index . The chirality matrices act as follows,

 (γ(1)⊗I2)χη1,η2α = χ−η1,η2α (I6⊗γ(2))χη1,η2α = χη1,−η2α (3.2)

The way these equations should be understood is as follows. We begin with , and pick a basis for the four-dimensional vector space of spinors for fixed such that the action of and are diagonal. Then, we can simply define the basis for for the remaining three values of by the action of the chirality matrices above.

Since appears in the fermion variations of (2.1), we also need to understand how to express the complex conjugate spinor in this basis. If satisfies (3.1), then by complex conjugating the entire first equation and using , we conclude that satisfies the same equation, with the same values of . Proceeding analogously for the second equation, we conclude that also satisfies the same equation, with the same values of . As a result, we must have the following linear relation,

 (B−1(1)⊗B−1(2))(χη1,η2α)∗ = 4∑β=1Mη1,η2αβχη1,η2β (3.3)

for some matrix for each pair .

We will now show that one may choose a basis for the Killing spinors in which . Iterating the complex conjugation, we conclude that , for all values of . Specializing first to , we have a single matrix satisfying . Now every such matrix may be rotated to the identity by a general linear complex-valued matrix , using the relation . An easy way to construct is as follows. An arbitrary invertible complex matrix in may be written as an exponential, of a complex matrix . Given , the matrix is not unique. The condition requires to be real-valued. Thus, we choose the solution to the equation , and the relation for may be diagonalized as follows,

 (B−1(1)⊗B−1(2))(χ+,+α)∗ = χ+,+α (3.4)

for . For the other values of we use (3.1) to express in terms of ,

 χ+,+α = (γ(1)⊗I2)χ−,+α (3.5) = (I6⊗γ(2))χ+,−α = (γ(1)⊗γ(2))χ−,−α

Using the fact that commutes with , while anti-commutes with , we find,

 (B−1(1)⊗B−1(2))(χη1,η2α)∗ = η2χη1,η2α (3.6)

for all values of . Since this decomposition is now canonical in terms of the degeneracy index , we will no longer indicate it explicitly.

### 3.2 Decomposing onto Killing spinors

An arbitrary 32-component complex spinor may be decomposed onto the above Killing spinors as follows,

 ε=∑η1,η2=±χη1,η2⊗ζη1,η2 (3.7)

where is a complex 2-component spinor for each , and the 4-fold degeneracy has not been indicated explicitly. As a supersymmetry generator in Type IIB, the spinor must be of definite chirality , which places the following chirality requirements on ,

 γ(3)ζ−η1,−η2=−ζη1,η2 (3.8)

The charge conjugate spinor is given by,

 B−1ε∗=iB−1(1)⊗γ(2)B−1(2)⊗B−1(3)∑η1,η2(χη1,η2)∗⊗ζ∗η1,η2 (3.9)

Since anticommutes with , we obtain,

 B−1ε∗ = iI8⊗γ(2)⊗I2∑η1,η2(B−1(1)⊗B−1(2)(χη1,η2)∗)⊗(B−1(3)ζ∗η1,η2) (3.10)

It will be convenient to denote the result as follows,

 B−1ε∗ = ∑η1,η2χη1,η2⊗⋆ζη1,η2⋆ζη1,η2=−iη2σ2ζ∗η1,−η2 (3.11)

As in [19] we will use the matrix notation introduced originally in [29] in order to compactly express the action of the various matrices on . Defining with , the identity matrix and with the standard Pauli matrices, we can write,

 (τ(ij)ζ)η1,η2≡∑η′1,η′2(τi)η1η′1(τj)η2η′2ζη′1η′2 (3.12)

### 3.3 Symmetries of the reduced BPS equations

Using the decomposition of into Killing spinors and the streamlined notation of the matrices we can finally write down the BPS equations in a reduced form. The reduced dilatino equation is,

 0=−4paγaσ2ζ∗+gaτ(03)γaζ (3.13)

while the reduced gravitino equations take the following form,

 (m) 0=−i2f6τ(21)ζ+Daf62f6γaζ−116gaτ(03)γaσ2ζ∗ (3.14) (i) 0=12f2τ(02)ζ+Daf22f2γaζ+316gaτ(03)γaσ2ζ∗ (a) 0=(Da+i2^ωaσ3−i2qa)ζ+316gaτ(03)σ2ζ∗−116gbτ(03)γabσ2ζ∗

The derivative is defined with respect to the frame , so that the total differential takes the form , while the -connection with respect to frame indices is .

Before we move on to solving these equations, let us briefly look at their symmetries. The axion/dilaton field transforms non-linearly under of Type IIB supergravity and takes values in the coset . Global transformations on the fields are accompanied by local gauge transformations, given in (2.2), and which induce the following symmetry transformations on the fields of the reduced BPS equations,

 U(1)q ζ→eiθ/2ζ (3.15) qa→qa+Daθ pa→e2iθpa ga→eiθga

The reduced BPS equations are also invariant under the following discrete symmetries,

 ζ → Iζ=−τ(11)σ3ζ ζ → Jζ=τ(32)ζ (3.16)

which leave all the bosonic fields invariant. Both and commute with . Finally, complex conjugation is naturally combined with , and we have,

 ζ → Kζ=eiθτ(22)σ2ζ∗ qa → Kqa=−qa+2Daθ pa → Kpa=e4iθ¯pa ga → Kga=−e2iθ¯ga (3.17)

The chirality requirement of Type IIB restricts the spinor to the subspace,

 Iζ=−τ(11)σ3ζ=ζ (3.18)

In the next subsection, we will investigate the restrictions to the eigen-spaces of and imposed by the reduced BPS equations. The symmetries commute with one another, so that we may diagonalize them simultaneously, and restrict to any one of their common subspaces.

### 3.4 Restricting to a single subspace of J

We will assume that does not vanish identically. We now use the dilatino equation to derive a first set of bilinear relations. Multiply the dilatino equation to the left by and choose the -matrix so that the flux part vanishes,

 gaζtTτ(03)σ2γaζ=0 (3.19)

Since is symmetric for , this condition will require to satisfy the condition that the product is anti-symmetric, which has the following solutions,

 T∈T={τ(01),τ(11),τ(31),τ(20),τ(22),τ(23)} (3.20)

The equation implied on is then given by the complex conjugate of,

 ¯paζ†Tγaζ=0 (3.21)

When , we can draw from this equation only an orthogonality relation. To obtain a full vanishing condition, we make further use of the chirality condition, and obtain,

 ¯paζ†Tτ(11)γaσ3ζ=0 (3.22)

We obtain such a relation when both and belong to , which is the case for only a single pair, namely . As a result, we have the equivalent relations,

 ζ†τ(20)γaζ=ζ†τ(31)γaζ=0 (3.23)

Next, we analyze the gravitino equations. We multiply equations and of (3.14) on the left by for , and obtain cancellation of the last term when is antisymmetric (which is the same condition we had for the dilatino equation),

 0 = −i2f6ζ†Tτ(21)σpζ+Daf62f6ζ†Tσpγaζ 0 = 12f2ζ†Tτ(02)σpζ+Daf22f2ζ†Tσpγaζ (3.24)

In view of (3.23), the second term will cancel when and , so that we obtain the following relations from the remaining cancellation of the first term,

 ζ†τ(01)σpζ = 0p=0,3 ζ†τ(22)σpζ = 0 (3.25)

and their chiral conjugates, obtained by using the chirality condition ,

 ζ†τ(10)σpζ = 0p=0,3 ζ†τ(33)σpζ = 0 (3.26)

Next, we use the general result of [19] that the bilinear equation is solved by projecting onto a subspace with the help of a projection matrix that anti-commutes with . This result was established for the case of 2-dimensional , which is in fact the case also here by reduction. Thus, we must find a projector which commutes with and with the following properties,

 [P,τ(11)σ3]={P,τ(01)σp}={P,τ(22)σp}=0 (3.27)

The solutions to these conditions are and , possibly multiplied by a factor of . These four possibilities are pairwise equivalent under the chirality relation. Now the projector precisely corresponds to the symmetry , so imposing a restriction on the spinor space by this operator is the only consistent restriction. Therefore, we will impose,

 τ(32)ζ=νζν=±1 (3.28)

which solves all the above bilinear relations. One must pick one value of or the other in the projection.

Imposing the chirality relation (3.18), , as well as the projector (3.28) we just derived, we may solve for the relations between the components of . Denoting the components by , which take values , the components labels the -matrix basis, while labels the chirality basis in which is diagonal. We then have two independent complex-valued components which we denote by and , and which are defined as follows,

 ¯α = ζ+++=−ζ−−+=−iνζ+−+=+iνζ−++ β = ζ−−−=+ζ++−=−iνζ−+−=−iνζ+−− (3.29)

To reduce the equations to a basis of complex frame indices , we will use the following conventions,

 ez=12(e8+ie9) e¯z=12(e8−ie9) γ¯z=12(γ8−iγ9)=(0010) (3.30)

In particular, we have,

 δz¯z=2δz¯z=12 (3.31)

It will also be convenient to have the following results of ,

 γzzσ2=−γ¯z¯zσ2=iσ1 (3.32)

### 3.5 The reduced BPS equations in component form

Using the solution (3.29) we found for the projection condition on the preserved supersymmetries, the reduced dilatino equations become,

 4ipzα−gzβ = 0 4ip¯z¯β+g¯z¯α = 0 (3.33)

The algebraic gravitino equations are,

 12f6¯α+Dzf62f6β−i16gzα = 0 −12f6β+D¯zf62f6¯α+i16g¯z¯β = 0 ν2f2¯α+Dzf22f2β+3i16gzα = 0 ν2f2β+D¯zf22f2¯α−3i16g¯z