Flowing from $AdS_5$ to $AdS_3$ with $T^{1,1}$

We construct supersymmetric domain wall solutions of type IIB supergravity that interpolate between $AdS_5\times T^{1,1}$ in the UV and $AdS_3\times\mathbb{R}^2\times S^2\times S^3$ solutions in the IR. The $\mathbb{R}^2$ factor can be replaced with a two-torus and then the solution describes a supersymmetric flow across dimensions, similar to wrapped brane solutions. While the domain wall solutions preserve $(0,2)$ supersymmetry, the $AdS_3$ solutions in the IR have an enhanced $(4,2)$ superconformal supersymmetry and are related by two T-dualities to the $AdS_3\times S^3\times S^3\times S^1$ type IIB solutions which preserve a large $(4,4)$ superconformal supersymmetry. The domain wall solutions exist within the $N=4$ $D=5$ gauged supergravity theory that is obtained from a consistent Kaluza-Klein truncation of type IIB supergravity on $T^{1,1}$; a feature driving the flows is that two $D=5$ axion like fields, residing in the $N=4$ Betti multiplet, depend linearly on the two legs of the $\mathbb{R}^2$ factor.


Introduction
Type II string theory possesses AdS 3 × S 3 × S 3 × S 1 solutions which are supported by magnetic three-form fluxes threading the S 3 factors as well as electric three-form flux threading the S 3 × S 3 × S 1 factor [1][2][3][4][5]. While these solutions have been known for a long time the dual field theory, which preserves a large (4,4) super conformal symmetry, remains elusive. A detailed discussion of some of the issues is presented in [6] and we note that a recent proposal for the dual field theory appears in [7]. In this paper we will discuss some new results on these type II solutions, using a rather indirect approach.
Starting with the AdS 3 × S 3 × S 3 × S 1 solutions of type IIB string theory with the magnetic and electric three-form fluxes in the RR sector, we can carry out two T-dualities on two circles to obtain other type IIB solutions with an AdS 3 factor. If we choose one of the two circles to be the explicit S 1 factor and the other to be a diagonal of the two Hopf fibres of the S 3 ×S 3 , then we obtain the AdS 3 ×T 2 ×S 2 ×S 3 solutions with non-trivial RR five-form and three-form fluxes as well as NS threeform flux that were first found in [8]. Here we will show that, as solutions of type IIB supergravity, these solutions preserve (4, 2) superconformal symmetry and not just the (0, 2) superconformal symmetry that was guaranteed from the original construction of [8].
A principal result of this paper is the construction of type IIB supergravity domain-wall solutions that interpolate between the AdS 5 × T 1,1 solution in the UV and approach these AdS 3 × T 2 × S 2 × S 3 solutions in the IR. The flow solutions preserve (0, 2) Poincaré symmetry which is enhanced to (4,2) superconformal symmetry at the IR fixed point. The supergravity solutions are constructed directly in type IIB supergravity. However, they can also be constructed in an N = 4 D = 5 gauged supergravity theory that can be obtained as a consistent KK truncation of type IIB supergravity on T 1,1 [9][10][11] (extending [12][13][14][15][16][17]). This perspective is helpful in identifying the deformations of the N = 1 SCFT that are needed to flow to the AdS 3 fixed points.
As is well known, the AdS 5 × T 1,1 UV fixed point is dual to an N = 1 SCFT in D = 4 that arises on D3-branes sitting at the apex of the conifold [18]. Our constructions can be viewed as a variation of wrapped-brane solutions [19] (see [20] for a review and [21][22][23] for recent constructions with AdS 3 factors), with the D3-branes wrapping a T 2 and sitting at the apex of the conifold with particular deformations switched on. In particular, an important ingredient is that there are two axion like fields in the Betti multiplet of the gauged-supergravity which are linear in the T 2 directions. This mechanism for preservation of supersymmetry differs from the usual one of activating R-symmetry currents, related to the spin connection of the cycle being wrapped, and also the constructions of [24,25] where there are magnetic fluxes threading a T 2 . In the most general solutions that we construct here, though, there is also a magnetic flux of the Betti vector field threading the T 2 factor.
We also analyse the flux-quantisation for the AdS 3 × T 2 × S 2 × S 3 fixed point solutions. This turns out to be somewhat subtle due to the presence of Page charges.
While the quantisation of Page charges have been discussed before [26][27][28][29], analysing our solutions reveals some new issues, which will also arise in the context other classes of solutions. We explain our prescription for quantising the Page charges and use this to obtain the central charge of the dual SCFT.
The above discussion focussed on solutions that flow from AdS 5 × T 1,1 to AdS 3 × T 2 × S 2 × S 3 . However, if one does not compactify two spatial dimensions then one has solutions flowing from AdS 5 × T 1,1 to AdS 3 × R 2 × S 2 × S 3 . Such solutions may have interesting applications in the context of applied AdS/CFT, where there has been various studies on the emergence of AdS 3 solutions after switching on magnetic fields, including examples preserving supersymmetry [24,25]. Our solutions, utilising axions, provide an alternative approach 1 to hitting such fixed points. It is also worth commenting that our type IIB domain wall solutions share some similarities with supersymmetric solutions of D = 11 supergravity that interpolate between AdS 4 × Q 111 in the UV and supersymmetric AdS 2 × R 2 × S 2 × S 2 × S 2 × S 1 solutions in the IR [36]; a difference, however, is that those flows were driven by electric and magnetic baryonic fluxes.
The plan of the rest of the paper is as follows. In section 2 we describe the supersymmetric domain wall solutions in the simplest setting and then generalise them to a one-parameter family of flows in section 3. We briefly conclude in section 4. We have three appendices. In appendix A we review the charge quantisation of the AdS 3 × S 3 × S 3 × S 1 solutions. In appendix B we demonstrate that the fixed point solutions preserve (4, 2) supersymmetry and in appendix C we discuss some additional aspects of the quantisation of Page charges. 1 Axions/massless fields that are linear in either null or spatial coordinates have been used in applied AdS/CFT in other works including [30][31][32][33][34][35].

General set-up
We will construct supersymmetric solutions of type IIB supergravity [37,38] using the conventions given in [39]. We will consider solutions with trivial axion and dilaton and hence the R-R and NS-NS three-forms can be combined into the complex threeform G = −dB − idC, where B, C are both two-forms. The Bianchi identities for G and the self-dual five-form F , satisfying F = * F , are given by while the equation of motion for G can be written The Killing spinor equations take the form We begin by recalling the standard AdS 5 × T 1,1 solution [40] of type IIB supergravity. The metric and the self-dual five-form are given by 1 where we have defined These conditions are equivalent to Γ 5678 ε = −ε, Γ 5649 ε = −ε, corresponding to the conifold (the Calabi-Yau cone over T 1,1 ), combined with iΓ 0123 ε = −ε corresponding to putting a D3-brane at its apex. The four Poincaré Killing spinors can be written where∇ is the Levi-Civita connection on T 1,1 with coordinates y m .
We are interested in constructing supersymmetric domain walls that approach AdS 5 × T 1,1 in the UV, and flow to particular AdS 3 × M 7 solutions in the IR. The ansatz that we shall consider first is given by where λ is a constant and A, B, U, V are functions of ρ only. We will discuss the dual SCFT interpretation of this ansatz in section 2.3 and discuss a generalisation in section 3.
Observe that, by construction, the ansatz has a self-dual five-form, F = * F , and that both the Bianchi identities (2.1) and the equation of motion for G (2.2) are satisfied. To analyse the conditions for preservation of supersymmetry, (2.3) and (2.4), we use the orthonormal frame e 0 = e A dt, e 1 = e A dx, e 2 = e B dx 1 , e 3 = e B dx 2 , e 4 = dρ, (2.10) We will continue to impose the algebraic conditions (2.7) and we will also impose Γ 23 ε = iε or equivalently A calculation now shows that we can solve (2.12) provided that we choose ε = e A/2 ε 0 with ε 0 satisfying (2.8), and that the functions A, B, U, V satisfy the following coupled first order differential equations Since our ansatz satisfies F = * F , the Bianchi identities (2.1) and the equation of motion for G (2.2), we can conclude from the result in appendix D of [39], that any solution to these differential equations will also solve the type IIB Einstein equations and hence gives rise to a supersymmetric solution of type IIB supergravity preserving at least two supersymmetries.
We immediately recover the AdS 5 × T 1,1 solution (2.6) by setting λ = 0 and (2.14) When λ = 0, it is convenient to scale the coordinates x i and shift the function B by a constant to set λ = 2, without loss of generality. We then find another exact solution to (2.13) corresponding to a solution with an AdS 3 factor: Indeed, if we substitute this solution into the ansatz (2.9) and scale x i = 1 6 1/2 z i we find that the metric can be written as 1 L 2 ds 2 = 1 3 3/2 2 ds 2 (AdS 3 ) + dz 2 1 + dz 2 2 + ds 2 1 + ds 2 2 + 1 2 (dψ + P ) 2 , This solution was first found 3 in section 3.1.2 of [8]. Observe that the topology of the internal five-dimensional compact space is unchanged from that of T 1,1 , namely Thus the topology of the D = 10 solution is AdS 3 × R 2 × S 2 × S 3 , or, if we take x i (or equivalently the z i ) to parametrise a two-torus, AdS 3 × T 2 × S 2 × S 3 .
By construction this AdS 3 solution preserves (0, 2) Poincaré supersymmetry and this is supplemented by a further two supersymmetries to give (0, 2) superconformal symmetry. In fact this was already known from the construction in [8]. However, as we show in appendix B, and further discuss in section 2.5, the fixed point actually preserves an enhanced (4, 2) superconformal supersymmetry (i.e. twelve supersymmetries in total).

The supersymmetric flow
We would now like to construct, numerically, a supersymmetric flow from the AdS 5 × Here we have used the freedom to shift A by a constant in (2.13) to eliminate an integration constant. Notice that the expansion depends on one constant c 1 . We will comment on the dual D = 4 SCFT interpretation of this UV expansion in the next subsection.
We now consider the expansion about the AdS 3 fixed point (2.15). We find that as ρ → −∞ it is fixed by three integration constants, a 0 , s 1 and s 2 : where R = In figure 1 we have plotted the behaviour of the functions appearing in the supersymmetric domain wall solution.

D = 5 perspective and dual SCFT interpretation
The AdS 5 × T 1,1 solution is dual to an N = 1 d = 4 SCFT described in [18]. The We first observe that the domain wall flow solutions we have constructed are actually contained within an D = 5 N = 4 gauged supergravity theory arising from a consistent truncation of the Kaluza-Klein reduction on T 1,1 . Recall that there is a consistent KK truncation of type IIB on a generic five-dimensional Sasaki-Einstein space to a D = 5 N = 4 gauged supergravity with two N = 4 vector multiplets [12][13][14][15][16][17]. Expanding about the AdS 5 × T 1,1 vacuum these fields give rise to SU (2, 2|1) multiplets, consisting of the gravity multiplet, a hypermultiplet, a massive gravition multiplet and a massive vector multiplet. For the special case when SE 5 = T 1,1 , the D = 5 N = 4 gauged supergravity has an extra N = 4 "Betti" vector multiplet [9,10] (supersymmetry was discussed in [11]). Expanding about the AdS 5 × T 1,1 vacuum, the latter gives rise to a massless Betti vector multiplet, corresponding to the baryonic symmetry, as well as a Betti hypermultiplet. In fact the solutions that we have just constructed are actually solutions of a further truncation to an D = 5 N = 2 gauged supergravity theory in which one discards the fields associated with the massive gravitino multiplet and also the massless Betti vector multiplet (while keeping the Betti hypermultiplet) [9].
A key feature of the UV expansion (2.17) that is driving the supersymmetric flow , is that two D = 5 "axion" scalar fields (labelled by b Φ , c Φ in [9] and e 1 0 , e 2 0 in [11]) are equal to −λx 1 , λx 2 , respectively. These axions lie in the Betti hypermultiplet, which is identified with T r(W 2 1 − W 2 2 ) in the dual SCFT [9,41], and are dual to marginal operators with dimension ∆ = 4. The expansion (2.17) also has a free integration constant c 1 which corresponds to an operator of dimension ∆ = 6 acquiring an expectation value. This scalar operator lies in the massive vector multiplet, which is identified with T r(W 2 [9,41]. It is worth highlighting the novelty of using D = 5 axions in our construction. Indeed the standard way of obtaining an AdS 3 × R 2 solution utilises massless D = 5 vector fields carrying magnetic charges, with field strength proportional to vol(R 2 ) [24,25]. By contrast in the solutions that we have constructed both the R-symmetry vector field and the Betti vector field actually vanish identically 4 . The way in which the D = 5 supersymmetry is being preserved for the solutions can be easily obtained using the results of [11] 5 Finally, we note that in [42] a supersymmetric AdS 3 × R solution of N = 4 D = 4 gauged supergravity was constructed in which the D = 4 axion field is linear in the coordinate on R. This solution was further discussed in [1]. It would be interesting to investigate the dimensional reduction of the D = 5 gauged supergravity of [9,10] on a circle to D = 4 and the relationship between the two AdS 3 solutions.

Flux quantisation and central charge
In this section we analyse the flux quantisation for the supersymmetric domain wall solutions that we just constructed, assuming that we have compactified the two spatial directions labelled by x i . The flux quantisation involves Page charges and is somewhat subtle; some additional details are presented in appendix C. We will also obtain the central charge of the d = 2 SCFT dual to the IR AdS 3 solution.
Note that so far we have been working in the Einstein frame with φ = 0. In this subsection, we view the metric as being in the string frame and, furthermore, we redefine our R-R fields F → g s F so that we are using similar conventions to [6] although we will not set 2πl s = 1 as they do.
We begin by assuming that the x i have period 2πd i , and parametrise a T 2 . The topology of internal space is then T 2 ×S 2 ×S 3 . The S 2 ×S 3 is realised as a circle fibration over an S 2 1 × S 2 2 base space. A positive orientation on S 2 × S 3 is given by Dψ ∧ vol 1 ∧ vol 2 . A smooth three-manifold, S 3 1 , that can be used to generate H 3 (S 2 × S 3 , Z) is provided by the circle bundle restricted to the S 2 1 factor on the base space. We can also choose S 3 2 , defined to be the circle bundle restricted to the S 2 2 factor on the base space, with opposite orientation. To find a smooth manifold that can be used to generate H 2 (S 2 × S 3 , Z) we consider any smooth manifold S on the base that represents the cycle Since the circle bundle is trivial over S, there is a section s and we can uses s(S) to generate H 2 (S 2 × S 3 , Z). A to N = 4 D = 5 gauged supergravity of [9,10] but not the further truncation to N = 2 of [9], which have similar structure for the axion fields and, amongst other features, also have a magnetic field for the Betti vector field. 5 See equation (28)- (31) and especially (33) of [11]. The connection terms in (33) are potentially problematic, but our solutions have A = τ = A = 0 and we note that dA is related to the linear axions via (15) of [11].
more detailed discussion is presented in appendix C; here we record the values of the following integrals:

Flux quantisation
We begin with the three-form flux quantisation. We have and we note that both of these are globally defined and closed three-forms. We then demand that Now we turn to the five-form. The relevant terms are which is globally defined since the one-form Dψ is. Recall that the Bianchi identity for the five-form is given by dF − H ∧ dC = 0. We will demand that a corresponding Page charge should be quantised. Specifically we demand that for any five-cycle Σ 5 . As we will see there are some subtleties in imposing this condition. Furthermore, as will be clear from the subsequent discussion, the subtleties are not removed by having the Page charges defined by integrating, instead, the five- There are two five-cycles to consider. For Σ 5 = S 2 × S 3 the gauge-dependent terms involving the two-form B do not contribute and we find The delicate case to consider is the five-cycle that is the product of the T 2 with the generator of H 3 (S 2 × S 3 ). Recall that for the latter we can consider S 3 1 which is the circle bundle over S 2 1 at any fixed point on S 2 2 . We can also consider S 3 2 which is the circle bundle over S 2 2 at any fixed point on S 2 1 , but with opposite orientation. We first calculate that These differ because F is not closed and hence does not define a cohomology class. We now need to consider a suitable gauge for the two-form B. It does not seem possible to find a single gauge-choice for B that is well defined as a two-form for an arbitrary three manifold representing H 3 (S 2 × S 3 ). However, it is possible to find a gauge for a specific representative. In particular, if we integrate over S 3 1 we can choose the gauge . We will discuss this more carefully below. We then calculate The quantisation condition that we will impose is given bȳ With this condition we see that the Page charge (2.25) when Σ 5 = T 2 × S 3 1 and Σ 5 = T 2 × S 3 2 , with the gauge-choices for B given above, are equal toN and -N , respectively. While these are quantised, one might be concerned that they are not equal given the two choices of Σ 5 are homologous and that the integrand is closed.
The key point is that the integrand is not a differential form since it changes under gauge-transformations, which we make precise below. 6 Recall that the central charge of the d = 4 SCFT dual to AdS 5 × T 1,1 is given by a = (N 2 /4)π 3 /(vol(T 1,1 ).
Note that the condition (2.29) is equivalent to the statement that the product of the three-form fluxes is constrained to be an even number: (2.30) Let us now elaborate a little on the gauge choices for B that we made above. We first introduce four coordinate patches U N N , U N S , U SN , U SS , each isomorphic to R 4 × S 1 , to cover S 2 × S 3 . We take U N N to consist of the northern hemispheres of the two S 2 's on the base as well as a coordinate ψ N N with period 4π. Next, U N S is the northern hemisphere of S 2 1 and the southern hemisphere of S 2 2 on the base, as well as a coordinate ψ N S with period 4π, and similarly for the rest. Now we know that the one-form Dψ ≡ dψ + P is globally defined and we have (2.31) On the overlaps of the patches we have which shows that we have a good circle bundle: e.g.
(and we note that the factors of 1/2 are present because ψ has period 4π).
For the five manifold T 2 × S 2 × S 3 we can consider four coordinate patches, isomorphic to T 2 × R 4 × S 1 , labelled in the same way. In particular, as we will see, the T 2 essentially just comes along for the ride. Now we consider the gauge for the two-form B given in the N N patch by which is clearly well defined in U N N . We see that it is also well defined in U SN , after using (2.32). Thus, it makes sense to integrate this over S 3 1 which lies in the union of these two patches 7 , giving the result in (2.28). Observe that if we instead move to U N S then we have Observe that we have defined S 3 1 here to be sitting at a fixed point on the northern hemisphere of the second two-sphere. Now the first term on the right hand side is well defined in this patch, but the last term isn't. However, moving to this patch we can employ a gauge-transformation on the two-form given by To see this is well defined, we recall that the definition of the integrality of the threeform curvatureH of a gerbe 8 connection (or "curving")B is given by  Similarly, if we consider the gauge for the two-form B given by we see that it is well defined in U N N and also on U N S . Hence this is something that can be integrated on the manifold S 3 2 (sitting at a point in the northern hemisphere of the first two sphere) leading to the result given in (2.28). To see that this is a well-defined gerbe connection we can calculate the difference between B (1) and B (2) on, say, the N N patch. We find which is a good gauge-transformation since ψ N N has period 4π.

Central charge
These flux quantisation conditions we have just derived are valid for the entire domain wall flow solution. We can also calculate the central charge of the d = 2 (0, 2) SCFT 8 As we will see in appendix C the essential aspects of these arguments don't really involve gerbes but more familiar gauge-connections.
that is dual to the AdS 3 solution (2.15). We use the standard formula where R AdS 3 is the AdS 3 radius and G 3 is the effective 3d Newton's constant. Our D=10 Lagrangian in the string frame is of the form and a calculation leads to where the second expression arises from (2.30).

T-duality and enhanced supersymmetry
It was pointed out in [8] that, locally, the is related, after two T-dualities on the T 2 , to the well known AdS 3 × S 3 × S 3 × S 1 solution of type IIB that is dual to a d = 2 SCFT with large (4, 4) supersymmetry [1] (see [6] for a detailed discussion). To see this we carry out two T-dualities along the directions z 1 , z 2 (as in (2.16)) using, for example, the formulae in appendix B of [8].
After then introducing rescaled coordinates and defining we obtain where If α i are periodic coordinates with period 4π then (2.45) is the metric on a round, unit radius three-sphere and we have the standard AdS 3 × S 3 × S 3 × S 1 solution of type IIB supergravity, which we we briefly review in appendix A.
Observe that (2.43) implies that Thus, locally, starting with the AdS 3 × S 3 × S 3 × S 1 solution we can obtain the AdS 3 × T 2 × S 2 × S 3 solution by carrying out a T-duality on the S 1 factor, generated by ∂z, and on the diagonal of the two U (1) Hopf fibres generated by 1 2 (−∂ α 1 + ∂ α 2 ). Recalling that the AdS 3 × S 3 × S 3 × S 1 solution preserves 16 supersymmetries (8 Poincaré and 8 superconformal), this suggests that the AdS 3 × T 2 × S 2 × S 3 solution will preserve more than the obvious 4 supersymmetries. Indeed the explicit Killing spinors for the AdS 3 ×S 3 ×S 3 ×S 1 solution, in a D = 11 incarnation, were constructed in [3] and the corresponding superisometry algebra was found. Using the arguments in section 7 of [3] one can determine the Killing spinors which are left invariant under the action of the Lie derivative with respect to the Killing vector generating the diagonal U (1) on the S 3 × S 3 factor. We find that this action preserves all eight Killing spinors given by equation (48) of [3] and four of the eight given by equation (47) of [3] (in particular satisfying the projection (1 + Γ 121 2 )ε = 0 in the notation of that paper). We have verified this counting by a direct construction of the Killing spinors for the type IIB solutions in appendix B.
A more careful examination of the global aspects of the T-duality will be left to future work. Note that the relevant AdS 3 × S 3 × S 3 × S 1 solution is fixed by two integers, Q D1 , Q D5 and the central charge is given by c = 3Q D1 Q D5 , which can be compared with the second expression in (2.41). However, the first expression in (2.41) suggests that orbifolds of AdS 3 × S 3 × S 3 × S 1 might need to be considered.

A more general class of flows
In this section we construct a more general class of flows interpolating between AdS 5 × T 1,1 and a one-parameter family of AdS 3 × R 2 × S 2 × S 3 solutions found in [8]. The flows again preserve (0, 2) supersymmetry and we show in appendix B that the AdS 3 fixed point solutions preserve (4, 2) superconformal symmetry.
Specifically, here we consider an ansatz for the type IIB fields given by with A, B, U 1 , U 2 , V, f all functions of ρ, and λ, Q are constants. The interpretation within the dual D = 4 SCFT will be discussed below. One can check that the five-form is self-dual and that the Bianchi identities (2.1) are satisfied. We find that if we again write the Killing spinors as ε = e A/2 ε 0 , demand that they satisfy the projections (2.7), (2.11) and (2.8), then the Killing spinor equations (2.3),(2.4) lead to the following system of first order differential equations (3. 2) The first five equations come from (2.3) while the sixth comes from (2.4). One can show that these equations imply that the equation of motion for the complex threeform G, given in (2.2), is satisfied.
Notice that if we set U 1 = U 2 = U and f = Q = 0 we recover the equations (2.13) that we had in the last section 9 . In particular, the AdS 5 × T 1,1 solution is obtained via (2.14). The set of equations (3.2) also admits the following one parameter family of AdS 3 solutions After scaling x i = 2 1/2 λ3 1/2 (1−4Q 2 ) 1/4 z i the resulting type IIB solution can be written as

The supersymmetric flows
We now discuss the domain wall solutions that interpolate between AdS 5 × T 1,1 in the UV and this one-parameter family of AdS 3 × R 2 × S 2 × S 3 solutions. There is an expansion about the AdS 5 × T 1,1 solution that involves three integration constants c i , in addition to Q and the deformation parameter λ. The UV expansion analogous to (2.17) is rather long so we shall not write it out explicitly. The key feature is that as ρ → ∞ three integration constants c i appear, schematically, as The UV expansion that we had before, given in (2.17), is obtained by setting c 2 = c 3 = 0. We will discuss the holographic interpretation of the c i below.
We now set λ = 2. Fixing 0 ≤ Q < 1/2, our UV expansion has three integration constants and our IR expansion has four. On the other hand our system of differential equations (3. which is associated with a particular relevant operator with ∆ = 2 being switched off (see below). In the next subsection we will argue that flux quantisation implies a rationality condition on Q.
The more general ansatz (3.1) that we are using for the supersymmetric flows is again contained within the N = 4 D = 5 gauged supergravity obtained from a consistent Kaluza-Klein truncation on T 1,1 [9][10][11]. As before, the two D = 5 axion scalar fields in the Betti hypermultiplet, dual to ∆ = 4 operators, are equal to −λx 1 , λx 2 , respectively, and this deformation is driving the flow. We next note that the field strength of the massless vector field lying in the Betti vector multiplet (labelled d(a Φ 1 ) in [9] and r 2 in [11]) is of the form λ(f + Qλ)dx 1 ∧ dx 2 . This reveals that the Q-deformation corresponds to switching on a magnetic field for the massless gauge-field dual to the ∆ = 3 current associated with the baryonic U (1) symmetry.
The three constants c i in (3.6) are related to various operators in the dual d = 4 SCFT acquiring expectation values, which can be deduced from the results of [9,11,14]. The constant c 1 is again associated with the scalar in the massive vector multiplet which is dual to an operator of dimension ∆ = 6. Similarly, the constant c 2 is associated with the scalar in the Betti vector multiplet (labelled w in [9]) dual to an operator of dimension ∆ = 2. Finally, the constant c 3 is associated with the massive one-forms (labelled b 1 , c 1 in [9]) appearing in the massive gravitino multiplet, dual to operators with dimension ∆ = 5.

Flux quantisation and central charge
Repeating the steps in section 2.4 (with λ = 2) we find that the quantisation of the three-form flux leads to the same results, namely Similarly, integrating the five-from flux on the five-cycle as before.
The calculation of the Page charge for the five-cycle Σ 5 = T 2 × S 3 , however, exhibits some new features. We follow the same prescription that we deployed in section 2. To carry out the integral (2.25) over T 2 × S 3 1 and T 2 × S 3 2 we use the gauge choices: respectively, and we obtain the two quantisation conditions respectively.
Since f is a function of ρ we obviously cannot satisfy (3.12) throughout the whole flow. We can however, demand that the flux is properly quantised at the AdS 5 boundary, where f → 0, and also at the AdS 3 fixed point, where f = −2Q, for suitable choices of rational Q. This would place additional constraints on the product Q N 5 Q D5 generalising (2.30). The Page charge would then change along the radial flow, reminiscent of the flows in [43]. Further exploration of the Page charges will be left for future work.
By following a similar calculation as in section 2.4.2, we find the central charge for the AdS 3 fixed point solutions is given by

T-duality and supersymmetry
Starting with (3.5) we introduce rescaled coordinates (3.14) and then carry out two T-dualities along the directionsz 1 ,z 2 using, for example, the formulae in appendix B of [8]. Making the further change of ordinates we obtain where, as before, . When α i have period 4π this the general AdS 3 × S 3 × S 3 × S 1 type IIB solution reviewed in appendix A.

Final comments
We have constructed a novel class of type IIB supergravity solutions, preserving (0, 2) supersymmetry, that interpolate between AdS 5 × T 1,1 in the UV and a class of AdS 3 ×T 2 ×S 2 ×S 3 solutions in the IR. The IR solutions preserve (4, 2) superconformal supersymmetry and are related, locally, by two T-dualities to the well known AdS 3 × S 3 × S 3 × S 1 solutions. It would be interesting to establish in more detail how this T-duality works globally. We examined the quantisation of Page charges for the AdS 3 × T 2 × S 2 × S 3 solutions, finding some novel features. In particular, it does not seem possible to have the connection two-form B of the gerbe be well defined as a two-form on an arbitrary five-manifold, representing the homologically non-trivial five cycles, on which one wants to integrate in order to get the Page charge. However, for specific choices of the five-manifolds, we can find an appropriate connection, related by gauge transformations, so that it is well defined. Furthermore, we found that a placing a constraint on the three-form fluxes ensured that the Page charges obtained in different gauges were all integers. It would be helpful to investigate this in more detail as similar issues will arise in other supergravity solutions with fluxes. One approach, in the present setting, is to try and make a precise connection with the globally realised T-duality.
It is known that classical type IIB string theory on AdS 5 × T 1,1 is not integrable [44]. However, it seems likely that it will be integrable on the AdS 3 × T 2 × S 2 × S 3 solutions we have discussed here (for related discussion see [45][46][47]). It would be interesting to confirm this and also to investigate how the integrability emerges along the RG flow.
Another direction for further study would be to investigate whether the solutions that we have constructed here can be generalised to solutions that flow from more general AdS 5 × SE 5 solutions, where SE 5 is a five-dimensional Sasaki-Einstein solution.
It is reasonable to expect that if we choose SE 5 to be one of the Y p,q spaces [48] then there will be flows to the various AdS 3 × T 2 solutions found in section 4 of [8] (global aspects of the T-dual solutions are discussed in [49]). These flow solutions might be difficult to construct explicitly, however, because unlike the case we have considered in this paper, there is not a known consistent KK truncation on Y p,q analogous to the one on T 1,1 . These solutions would relate four dimensional superconformal field theories to two-dimensional superconformal field theories with (0, 2) supersymmetry, complementing other such examples [19,[21][22][23][24][25][50][51][52][53].
A The AdS 3 × S 3 × S 3 × S 1 solution Consider the standard AdS 3 ×S 3 ×S 3 ×S 1 solution [1] (see also [6]) which is supported by RR fluxes. In the conventions of section 2.4 it can be written ds 2 =L 2 ds 2 (AdS 3 ) + r 2 1 ds 2 (S 3 1 ) + r 2 2 ds 2 (S 3 2 ) + dy 2 , where ds 2 (S 3 i ) are the standard round metrics on three-spheres, y ∼ = y + ∆y and, in the notation of (3.16), with 0 ≤ Q < 1/2. Observe that r 2 1 + r 2 2 = r 2 1 r 2 2 . We next quantise the flux. For the electric flux we have For the magnetic flux we have Observe that we have which agrees with (2.17) of [6] (they have set 2πl s = 1), which shows that the radius of the circle is fixed by the RR charges gQ. Using (2.39),(2.40) we calculate the central charge as again agreeing with (2.20) of [6].

B Enhanced supersymmetry
In this appendix we show that the AdS 3 × R 2 × S 2 × S 3 solutions given in (3.3) have a (4, 2) superconformal supersymmetry. We set L 2 = 3 3/2 /(1 − 4Q 2 ) 1/2 and use the orthonormal frame e µ = √ 2ē µ , µ = 0, 1, 4 , whereē µ is an orthonormal frame for a unit radius AdS 3 . From (2.4) we deduce that We thus write The Killing spinor equations (2.3) then take the following form. For the µ = 0, 1, 4 components we havē where∇ ≡ē µ∇ µ is the Levi-Civita connection on a unit radius AdS 3 . For the 2,3 components we get For the 5,6 components we get For the 7,8 components we get Finally, for the 9 component we get By examining the integrability conditions for the two equations involving derivatives with appesct to z 1 , z 2 , (B.6), we deduce that and also that ∂ z i ε 1 = 0. Examining (B.9) we also see that ∂ ψ ε 1 = 0. From (B.5) we can deduce that∇ and the integrability conditions for the second line implies As a consequence we conclude from (B.6) that ∂ z i ε 2 = 0.
It is now convenient to further decompose After projecting out the equations using (1 ± iΓ 23 )/2 and (1 ± iΓ 56 )/2 we deduce the following general solution. Firstly, ε 1 and ε + 2 are given by and The d a only depend on the coordinates on AdS 3 and should satisfy Secondly, the parameter ε − 2 takes the simple form Again d 3 only depends on the coordinates on AdS 3 and now should satisfy To solve (B.18), (B.21) we use the following frame and coordinates for AdS 3 : to obtain where α ± 1 , α ± 2 and α ± 3 satisfy the projections (B.17) and (B.20), respectively, and in addition We see that α + 3 parametrises the (0, 2) Poincaré supersymmetry that is preserved throughout the whole flow of the domain wall solutions (recall (2.7) and (2.11)). The α − a parametrise an enhancement of the Poincaré supersymmetry to (4,2). The remaining six supersymmetries, labelled by α − 3 and α + a , parametrise the superconformal supersymmetries.
As noted in the text, using the results of [3] we can deduce that the superisometry algebra is of the form D (2, 1|α) × G, where G ⊂ D(2, 1|α) and has a bosonic subalgebra given by SL(2, R) × U (1) 2 . This could be verified using the explicit Killing spinors that we have constructed, but we shall not do that here.

C Page charge quantisation
We will discuss the essential aspects of the quantisation of Page charges that we employed in the bulk of the paper in a simplified setting. We suppose that we have a manifold with topology S 2 × S 3 which is presented as a circle bundle fibred over S 2 1 × S 2 2 exactly as for T 1,1 : where c i are non-zero constants which won't be important, and and ψ has period 4π. We note that Dψ is a globally defined one-form. A positive orientation on S 2 × S 3 is given by Dψ ∧ vol 1 ∧ vol 2 . Topology: A smooth manifold S 3 1 that can be used to generate H 3 (S 2 × S 3 , Z) is provided by the circle bundle restricted to the S 2 1 factor on the base space. We can also choose the circle bundle restricted to the S 2 2 factor on the base space, with opposite orientation, which we call S 3 2 . Observe that Dψ ∧ (vol 1 − vol 2 ) is closed (since d(Dψ) = vol 1 + vol 2 ) and hence when it is integrated over a three-cycle, it will only depend on the homology class of the cycle. We have: To find a smooth manifold that can be used to generate H 2 (S 2 ×S 3 , Z) we consider any smooth manifold S on the base that represents the cycle [S] = [S 2 2 ] − [S 2 1 ]. Since the circle bundle is trivial over S, there is a section s and we can uses s(S) to generate H 2 (S 2 × S 3 , Z). We can take, for example, θ 1 = θ 2 and −φ 1 = φ 2 at fixed ψ. In particular we have It can be helpful to explicitly identify the Poincaré duals of the above generators.
We could imagine that we have carried out a dimensional reduction on the T 2 , for example. The advantage of doing this is that the ambiguities in defining Page charges will just involve gauge-transformations of U (1) gauge-connections rather than gerbes. We consider, therefore, the following globally defined fluxes with dF 2 = dG 2 = 0 and dF 3 = F 2 ∧ G 2 . We will assume that F 2 and G 2 are the curvature two-forms for two U (1) connections with integer Chern numbers. Thus we demand that k, l ∈ Z so that that we have the quantisation conditions: 1 2π s(S) F 2 = k, 1 2π s(S) G 2 = −l . (C.8) If we write F 2 = dA 1 , a natural Page charge to consider quantising is (C.9) For definiteness we define S 3 1 and S 3 2 to sit at a fixed point on the northern hemisphere of the other two-sphere. We can calculate which differ because F 3 is not closed. We now introduce two gauge connections given by A Being connections with cohomologically non-trivial field strengths, these cannot be globally defined one-forms. However, they should patch together using U (1) gauge transformations. Let us first consider A 1 . It is clearly defined on the N N patch. It is also well defined on the SN patch after using (C.6). Next, moving to the N S coordinate patch we get something that is well defined up to a U (1) gauge transformation: (dψ N S + (1 − cos θ 1 )dφ 1 + (1 + cos θ 2 )dφ 2 ) + kdφ 2 , = − k 4 (dψ N S + (1 − cos θ 1 )dφ 1 + (1 + cos θ 2 )dφ 2 ) − ie −ikφ 2 d(e ikφ 2 ) , (C.12) where we recall that φ 2 has period 2π. Moving to SS is similar. Thus A 1 is a U (1) gauge connection for F 2 . Furthermore, we observe that A 1 is a globally well defined one-form on S 3 1 since for a fixed point on the northern hemisphere patch of the S 2 2 we can switch to the SN patch in a regular manner using (C.6).
Similar comments apply to A which shows that they are related by a good U (1) gauge transformation, since ψ has period 4π. Thus A 1 is also U (1) gauge connection for F 2 and, in contrast to A 1 , is now a well defined one-form on S 3 2 . We can now calculate: , (C.14) and hence Now F 3 − A 1 ∧ G 2 is closed, so we may naively have thought that these should be equal. However, A 1 is connection, so F 3 − A 1 ∧ G 2 is not a three-form and does not define a cohomology class. If we demand that kl = 2N withN ∈ Z then both of these are integers.
In essence this is the flux quantisation procedure that we have adopted in the main text. An open issue, which we leave for the future, is to determine what happens if we choose other smooth three-manifolds Σ to represent H 3 . What are the conditions for there to exist a connection one-form, related to A (i) by a gauge transformation, which is well defined on Σ and, when it does exist, is the corresponding Page charge always an integer timesN ? We believe that similar issues will arise in other contexts.