We recall the open-closed string map of Seiberg and Witten [22], which we recast in the following form: (g+B)μν=(Gμν+Θμν)-1,(1)where (g,B), (G,Θ) are respectively closed string and open string fields. The metrics g, G are of course symmetric, B, Θ are their antisymmetric counterparts and B=0 implies Θ=0, and vice versa. Our approach is to interpret the metric of the original solution as the open string metric G, add a deformation parameter Θ, then generate a new metric g and B-field. This map works for generic supergravity solutions, not necessarily coset spaces, for example the Schwarzschild solution. For spacetimes with U(1)2 isometry our method reduces to T-duality shift T-duality (TsT) transformations [40], but it is more generally applicable.

The NS sector of supergravity comprises, in addition to g and B, a scalar dilaton Φ. Moreover, when further extended to generalized supergravity, one encapsulates the modification of usual supergravity in a single one-form X [20]: X≡dΦ+iIB+I,(2)where I is the one-form related to the Killing vector; setting I=0, we recover usual supergravity. The NS sector of generalized supergravity is hence characterized by (g,B,Φ,I). We note that the B-field is specified up to the Λ-gauge transformation, B→B+dΛ and (1) and (2) are written in a particular Λ-gauge, while X, which appears in the EOMs of generalized supergravity, is Λ-gauge invariant [20,30]. However, this leaves the residual symmetry of shifting Φ by a constant, Φ→Φ+Φ0, without changing B.

Having specified how g, B are generated, we turn our attention to Φ and I. The dilaton transformation follows from the observation that there is a well-known T-duality invariant [3,4], e-2δΦdetgμν=detGμν,(3)where δΦ is the dilaton shift, modulo the constant Φ0. This ensures our prescription is not at odds with TsT. Lastly, the status of the final solution, being either supergravity or generalized supergravity, may be read off from the divergence of Θ with respect to the original metric [31], Iμ=∇νΘνμ.(4)The origin of this equation can be explained in terms of a consistency condition arising from the Λ-gauge invariance of D-brane actions [31]. For TsT transformations, Θ is a constant [30], so that I=0 and the final solution is a bona fide supergravity solution. This completes our description of the NS sector transformation.

We turn attention to the RR sector. The standard treatment in T-duality, or any frame change, is that there is a Lorentz transformation acting on a bispinor constructed from the RR field strengths Fn [41–43]. Here we adopt a novel approach, which makes the role of Θ manifest.

We recall the Page forms [32,33],²

We employ the notation B2=B∧B, etc., and later A⌟B=1p!Aμ1…μpBμ1…μpνp+1…νq for p-form A and q-form B with q≥p.

It is well known that the Page charges, integrals of Page forms over compact cycles, can be quantized [33]. In particular, in AdS/CFT the quantized charges correspond to ranks of the gauge groups in the dual gauge theory. Now, recall that Θ is a deformation parameter, which we can continuously set to zero. Since the Page charges can only change discretely, they can not depend on Θ, leading to the conclusion that they are invariant. Since the cycles do not change, this implies that the corresponding Page forms are indeed invariant. This invariance of the non-zero Page forms constitutes the basis of a consistent treatment of the RR sector.

In our proposal, the initial open string data is completed by specifying the Page forms. Since originally the B-field was absent, this implies there exist given Page forms in the new solution that satisfy Q2n+1=F2n+1,(7)where n is determined by the original nonzero field strengths F2n+1. The remaining Page forms are generated through descent by contracting Θ, Q2(n-p)+1=(-1)pp!Θp⌟Q2n+1.(8)Given (4), for each p, (8) is a solution to (6). Unraveling the Page forms using the generated B-field, one arrives at the final expression for the RR field strengths. It is worth emphasizing again that all information about the deformation is encoded in Θ.

Putting the NS and RR sectors together, one finds a prescription for writing down the deformed geometry solely on the basis of a knowledge of Θ. Let us recapitulate the key steps: (1)

Invert matrix G-1+Θ to determine g, B.

Here we illustrate our prescription with the geometry AdS2×S2×T6, which corresponds to the near-horizon of intersecting D3-branes. The initial supergravity solution is, ds2=(-dt2+dz2)z2+dζ2+sin2ζdχ2+ds2(T6),F5=(1+*10)12z2dt∧dz∧(ωr-ωi),(9)where we define the three-forms ωr-iωi=Ω3, with Ω3 being the complex (3, 0)-form on the torus. Observe that both the B-field and dilaton are zero.

We consider the following ansatz for the deformation Θtz=Θ1(t,z),Θζχ=Θ2(ζ,χ).(10)This ansatz honors the direct-product structure of the geometry, leaving us the task of solving for two functions. Note, we have not assumed that Θ is an antisymmetric product of Killing vectors from the outset, instead this is forced upon us by the EOMs as we now show.

Following our recipe, we arrive at a new solution to generalized supergravity, which is fully determined modulo Θi,i=1, 2. To give a flavor of the output, suppressing the torus, we record (g,B,Φ), ds2=z2(-dt2+dz2)z4-Θ12+dζ2+sin2ζdχ21+Θ22sin2ζ,B=Θ1z4-Θ12dt∧dz-Θ2sin2ζ1+Θ22sin2ζdζ∧dχ,e2Φ=e2Φ0z4(z4-Θ12)(1+Θ22sin2ζ).(11)We have presented the complete solution in Supplemental Material [44]. Note, B-field is pure gauge, so we could set it to zero, if desired.

When solving for Θi, it is most effective to initially recall that I is Killing and solve the Killing equation, ∇μIν+∇νIμ=0. This determines Θi up to eight integration constants: Θ1=c1tz+c2z(t2-z2)+c3z+c4z2,(12)Θ2=c5cosχ+c6sinχ+c7cotζ+c8sinζ.(13)As will be clear soon, modulo the c4, c8 terms that are forced to vanish, Θ has already been determined as a linear combination of antisymmetric products of Killing vectors.

We next study the Einstein equation, the EOM for the B-field and the dilaton EOM, where the first two equations lead to the same set of constraints: κ2=-c12+4c2c3,(14)κ2=c52+c62+c72,(15)c4=c8=0.(16)Here we have redefined the constant shift in the dilaton e2Φ0=1+κ2. Equation (14) recently appeared in [45]. Note that these EOMs split between the AdS2 and S2 spaces, but are connected via the constant dilaton shift. The dilaton EOM is satisfied given (14) and (15).

One can also check the EOMs involving the RR field strengths. Our descent procedure for the Page forms ensures that the EOMs are satisfied by construction and hence one finds no further constraints. Therefore, subject to the constraints, we have the most general solution for Θ. We now turn to the interpretation.

Let us begin with κ=0. Evidently, there is no real solution to (15), so this precludes a deformation of the two-sphere. In contrast, there is an allowed deformation of the AdS2 space with two free parameters. We can interpret both of these results from the perspective of the associated homogeneous CYBE, provided Θ is an r-matrix solution. To do so, let us label the six Killing vectors of the AdS2×S2 geometry as T1=-t∂t-z∂z,T2=-∂t,T3=-(t2+z2)∂t-2tz∂z,T4=∂χ,T5=-cosχ∂ζ+cotζsinχ∂χ,T6=sinχ∂ζ+cotζcosχ∂χ.(17)Constructing the most general r-matrix r=12rijTi∧Tj, and substituting the components of the r-matrix into the homogeneous CYBE corresponding to the sl(2)⊕su(2) algebra, we arrive at the constraints: r12r31=(r23)2,(r45)2+(r56)2+(r64)2=0.(18)Relabeling the components of the r-matrix, r12=-c3, 2r23=c1, r31=-c2, r45=-c5, r56=-c7, r64=-c6, it is easy to check that the (nonzero) r-matrix is r=Θ1∂t∧∂z.(19)Thus, when κ=0, (14) and (15) are essentially the homogeneous CYBE for the Lie algebras sl(2) and su(2), respectively. It is well known that there is a redundancy in the CYBE and r-matrix solutions are equivalent up to automorphisms. In the geometry, these correspond to coordinate changes and it is easy to check that under a special conformal transformation, one can set c1=c2=0, while under a translation, one can set c1=c3=0, leaving one parameter.

When κ≠0, namely when there is a constant dilaton shift, we find an apparent five-parameter class of deformations of AdS2×S2, the redundancy of which can be removed again by coordinate change leaving a single parameter κ. This corresponds to a solution to the modified CYBE with modification κ. Modulo a coordinate transformation, ρ=z2-1-t22z,cost˜=z2+1-t24t2+(z2+1-t2)2,r=cosζ,φ=χ,(20)where we have added tildes to differentiate new coordinates, we can recover the known solution in the literature [20] through the choice c2=-κ2, c3=-κ2, c7=-κ.

The important take-home lesson from this simple example is that the CYBE naturally emerges from the open-closed string map and the EOMs of generalized supergravity. Once the CYBE is imposed, we are guaranteed a new supergravity solution where Θ corresponds to the r-matrix. The well-known redundancy of the r-matrix under automorphisms corresponds to coordinate changes in the geometry.

To confirm that the previous analysis was no fluke, we repeat for another geometry, the Schwarzschild black hole, ds2=-(1-2mr)dt2+dr2(1-2mr)+r2(dζ2+sin2ζdχ2).In contrast to the previous example, Schwarzschild does not admit a coset construction and is unlikely to be an integrable σ-model background. Furthermore, for this example it is difficult to solve for Θ directly, so we choose Θ to an antisymmetric product of Killing vectors: Θtζ=-εcosχ+λsinχ,Θtχ=δ+cotζ(εsinχ+λcosχ),Θζχ=αcosχ-βcotζ+γsinχ.(21)Note, this corresponds to Θ=αT4∧T5+βT5∧T6+γT6∧T4+δT2∧T4+εT2∧T5+λT2∧T6,(22)where α, β, etc., are constants and we have employed (17). While this is ostensibly the same form as the r-matrix, an important distinction is that the coefficients are not fixed. Before proceeding, we remark that the original geometry is Ricci-flat with no RR sector.

As an initial consistency check on Θ, one confirms from (4) that I=βT4+γT5+αT6 is a valid Killing vector. We now repeat the same matrix inversion from the open-closed string map and substitute into the EOMs. Assuming non-zero coefficients, the EOMs are satisfied provided, 0=βε-δγ=αε-γλ=αδ-λβ,0=α2+β2+γ2.(23)The key observation now is that these equations are the same as the homogenous CYBE for the Lie algebra u(1)⊕su(2), in line with our expectations. It is worth stressing that our statement supergravity recovers the CYBE holds beyond strict coset geometries.

Indeed, the CYBE for this algebra involves selecting three generators from four, so we get precisely four equations, only three of which are independent. Here, without an RR sector, the constant shift in the dilaton makes no difference, so we cannot consider the modified CYBE. It is easy to see that α=β=γ=0, so the only permitted deformation involves δ, ε, λ with no constraint. However, here again we encounter a redundancy and two of these parameters can be removed using two-sphere rotations. The remaining single parameter deformation is equivalent to a TsT transformation of the original background in the (t,χ)-directions. This example has been chosen in order to illustrate the applicability of our method to noncoset geometries. Note, however, that nontrivial new solutions to generalized supergravity can be obtained as well, as has been shown in [34,35] for deformations of flat space and Bianchi cosmologies, neither of which are quite as striking as Schwarzschild, so we opted to present the latter.