# AdS 3 × w (S 3 × S 3 × S 1 ) solutions of type IIB string theory

Abstract: We analyse a recently constructed class of local solutions of type IIB supergravity that consist of a warped product of AdS 3 with a seven-dimensional internal space. In one duality frame the only other non-vanishing fields are the NS 3-form and the dilaton. We analyse in detail how these local solutions can be extended to give infinite families of globally well-defined solutions of type IIB string theory, with the internal space having topology S 3 × S 3 x S 1 and with properly quantized 3-form flux.

## Summary (2 min read)

### 1 Introduction

- Starting with the work of [1], general characterisations of the geometries underlying such solutions, using G-structure techniques [2, 3], have been achieved for various d and for various amounts of supersymmetry [4]–[23].
- The solutions are again locally determined by a six-dimensional Kähler metric and a choice of a closed, primitive (1, 2)-form on the Kähler space.
- After two T-dualities on the two-torus it was also shown that these explicit solutions give type IIB AdS3 solutions with non-vanishing dilaton and RR three-form flux only.
- Furthermore the authors need to check that the three-form flux is properly quantised.

### 2.1 The local solutions

- The authors start with the explicit class of AdS3 solutions of section 4.3 of [26].
- After a further S-duality transformation the authors obtainAdS3 solutions with only NS fields non-vanishing, but they will continue to work with the above solution.
- The two generators are clearly the two copies of S3, at a fixed point on the other copy.

### 2.5 A quotient of M6 and integral three-forms

- Such as the suitably normalised RR three-form, has integral periods.the authors.
- 7We thank Dominic Joyce for suggesting this approach.the authors.the authors.

### 3.1 Electric and magnetic charges

- The authors next turn to the magnetic three-form charge.
- In the previous subsection the authors gave a prescription for performing such integrals by instead calculating integrals on submanifolds of the quotient space M̂6.
- This will illuminate and confirm many of their observations about the topology in the previous section.
- In the present case, however, there is a much simpler way to impose flux quantisation.
- Here the authors are not distiguishing between Φ and Π∗Φ. where G(3) is the three-dimensional Newton’s constant and RAdS3 is radius of the AdS3 space.

### 3.2 Computing periods using coordinate patches

- This provides a nice cross-check on various calculations carried out so far.
- Recall from section 2.5 that instead of considering the circle bundle L over M5 with total space M6 the authors should consider the circle bundle L̂ = L (p+q)q with total space M̂6 = M6/Z(p+q)q.
- This is because the azimuthal coordinate φ degenerates at the poles of the base S2.
- Consider first the overlap of U1N with U2N .

### 3.3 Taking the limit Q→ 0

- In particular, the solutions, with the internal manifold having topology S3 × S3 × S1, and all other parameters fixed, are smoothly connected with each other as Q varies and they all have the same central charge.
- While this may in fact be the correct interpretation, it seems difficult to draw this conclusion based on the results of this paper.
- The authors also need to check what happens to the global identifications they have made on the coordinates, and also the quantisation of the flux.
- The problem arises in taking the limit of the v circle fibration.
- In this limit the supergravity approximation is clearly breaking down and one needs to analyse string corrections before any definite conclusions can be drawn.

### 4 More general identifications

- In this section the authors will generalise the class of solutions that they have already constructed.
- Similarly imposing p = −q is not compatible with (2.22). of the last two sections into larger families.
- The authors will not need the explicit details.
- After completing the square in the metric on M6 the authors obtain an expression for what will become the globally defined angular one-form corresponding to the v′ circle fibration and it has the form dv − Av′Dψ − k(y)Dw′ (4.6) for some smooth function k(y) that can easily be determined.
- Note that, in general, the solutions of this section are not exactly marginal deformations of those in section 3: for example, in section 3 the authors saw that the magnetic three-form flux quantum numbers were constrained to be of the form (3.36), whereas here there is no such constraint.

### 5 Final Comments

- The authors have analysed in detail some local supersymmetric AdS3 solutions of type IIB supergravity, first found in [26], that have non-vanishing dilaton and RR three-form flux.
- The authors have shown that the parameters can be chosen and coordinates identified in such a way that the solutions extend to give rich classes of globally defined solutions of the form AdS3 ×w (S3 × S3 × S1) with properly quantised flux.
- The authors have shown that the solutions depend on continuous parameters and are hence dual to continuous families of SCFTs in two spacetime dimensions with (0, 2) supersymmetry.
- This will lead to larger families of solutions that would be worth exploring.
- Looking at equation (A.16) of [38] the authors see that the β-deformation activates a non-trivial dilaton and three-form.

### B More on the topology of M5

- This in turn defines a circle bundle with Chern number a, using (2.34).
- One can do similar computations for the three-submanifolds E1 and E2, with similar conclusions.
- 17Changing the sign of the last term leads to type IIB bubble solutions, as explained in [26].
- When Q 6= 0, one will also need to check the flux quantisation conditions and this will require generalising the techniques that the authors have used in this paper.

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^{1}, Imperial College London

^{2}, King's College London

^{3}, University of Oxford

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