## Final Report Summary - T-DUALITIES (Beyond Abelian T-duality)

Almost 30 years after the discovery of T-duality in string theory, it remains relevant. Strings, being spatially one-dimensional objects, "see" spacetime differently from particles, since they have the freedom to wind on compact, periodic directions. At the heart of T-duality, one finds the observation that the spectrum of the string is invariant under the interchange of winding and momentum degrees of freedom. This duality is Abelian and a direct analogue of Kramers-Wannier duality in the two-dimensional Ising model, where the duality allows one to identify the critical temperature above which ferromagnetism disappears. This project concerns generalisations of T-duality to non-Abelian and fermionic isometries, which are poorly understood. This is especially the case when it comes to the interpretation of the transformation in the context of gauge/gravity duality, a conjectured equivalence between gauge theories and gravity theories.

This project has achieved four results related to non-Abelian and fermionic T-duality, three of which may be described as significant. The remaining one is more at the level of an observation.

The first result, published in Class.Quant.Grav. 32 (2015) no.3, 035014, studied supersymmetry (a mathematical symmetry between particles) breaking in the context of T-duality. At the time of publication, the understanding was that the Kosmann Lie-derivative accounted for the breaking of supersymmetry in supergravity, but its precise relation was not explicit in the literature. Inspired by this observation, I revisited the case of Abelian T-duality in supergravity to see how the Kosmann derivative emerges. I confirmed that the supersymmetry conditions before and after T-duality are related up to the Kosmann derivative, and provided it vanishes, supersymmetry is preserved.

We subsequently extended this analysis to the simplest example of non-Abelian T-duality, an SU(2) transformation, finding again that the supersymmetry conditions before and after the transformation are related through the Kosmann derivatives with respect to the SU(2) factor. This makes it clear that supersymmetry will be preserved if the Kosmann derivative vanishes, thus backing up earlier claims in the non-Abelian T-duality literature. As a by-product of our analysis, we were able to derive the transformation rule for the RR sector from first principles, which had appeared in the literature largely without derivation.

The second significant result was published in Phys.Rev. D94 (2016) no.10, 106006. Based on analogy with AdS/CFT duality in the context of a duality between the geometry AdS5 x S5 and N=4 super Yang-Mills, where hidden symmetries in scattering amplitudes in Yang-Mills theory can be explained in terms of a self-dual mapping of the geometry AdS5 x S5, it was expected that the geometry AdS4 x CP3 is also self-dual. For a geometry to be self-dual, it should be possible to perform a series of bosonic and fermionic T-dualities, so that it returns to its original form. The idea is then that this symmetry would explain the symmetries documented in scattering amplitudes in the AdS/CFT dual theory.

In the paper, we focused on the fermionic T-duality and enumerated all the possible transformations one could perform. We identified 4 candidate sets, only one of which had been previously analysed, showing that in each case one encountered a singularity in a scalar field called the dilaton. I believe our presentation is clear and we have not confused the picture by trying to analyse bosonic T-dualities. Fundamentally, we find that the fermionic T-duality transformation, as initially proposed by Berkovits & Maldacena, is not general enough to allow self-duality of AdS4 x CP3. As a result, sermonic T-duality applied to AdS4 x CP3 does not explain hidden symmetries in scattering amplitudes.

The third publication, JHEP08(2015)121, studied non-Abelian T-duality for the geometries AdS3 x S3 x T4 and AdS3 x S3 x S3 x S1. While most of the results hinged on the latter geometry, for the former, we noticed that a non-Abelian T-dual initially found by Sfetsos and Thompson actually fell into a class of geometries that had been classified by Nakwoo Kim and collaborators in 2007 (later by me 2010). There was no known example in this class, but supersymmetry permits it. It turned out that the non-Abelian T-dual fits in this class. In the same paper, we studied properties of the field theories dual to non-Abelian T-duality and extensively studied supersymmetry. The results of this paper led to the spin-off, PhysRevD.93.086010, which was inspired by our work on non-Abelian T-duality.

In work in the final stages of this project, I have been studying Yang-Baxter deformations of AdS5 x S5. We recall that strings moving in AdS5 x S5 are described by a two-dimensional sigma-model, which is integrable, in the sense that it has an infinite number of conserved charges. One can deform the sigma-model to a so-called Yang-Baxter sigma-model, which also preserves integrability. It has been conjectured that the Yang-Baxter deformation is equivalent to non-Abelian T-duality, and this connection has been demonstrated for a number of examples.

We have observed that for a large class of Yang-Baxter deformations that there is a corresponding open string metric that is undeformed and all the information on the Yang-Baxter deformation is encoded in the non-commutativity parameter of the open string. This makes the connection between twists of the conformal algebra, giving rise to non-commutative spacetimes, and Yang-Baxter deformations. In addition, we realised that a so-called unimodularity condition, which picks out valid supergravity solutions, can be translated into the requirement that the non-commutative parameter be divergence-free. The work is now on the arXiv, 1702.02861 and we have submitted it to Physical Review Letters, a high-impact journal.

This project has achieved four results related to non-Abelian and fermionic T-duality, three of which may be described as significant. The remaining one is more at the level of an observation.

The first result, published in Class.Quant.Grav. 32 (2015) no.3, 035014, studied supersymmetry (a mathematical symmetry between particles) breaking in the context of T-duality. At the time of publication, the understanding was that the Kosmann Lie-derivative accounted for the breaking of supersymmetry in supergravity, but its precise relation was not explicit in the literature. Inspired by this observation, I revisited the case of Abelian T-duality in supergravity to see how the Kosmann derivative emerges. I confirmed that the supersymmetry conditions before and after T-duality are related up to the Kosmann derivative, and provided it vanishes, supersymmetry is preserved.

We subsequently extended this analysis to the simplest example of non-Abelian T-duality, an SU(2) transformation, finding again that the supersymmetry conditions before and after the transformation are related through the Kosmann derivatives with respect to the SU(2) factor. This makes it clear that supersymmetry will be preserved if the Kosmann derivative vanishes, thus backing up earlier claims in the non-Abelian T-duality literature. As a by-product of our analysis, we were able to derive the transformation rule for the RR sector from first principles, which had appeared in the literature largely without derivation.

The second significant result was published in Phys.Rev. D94 (2016) no.10, 106006. Based on analogy with AdS/CFT duality in the context of a duality between the geometry AdS5 x S5 and N=4 super Yang-Mills, where hidden symmetries in scattering amplitudes in Yang-Mills theory can be explained in terms of a self-dual mapping of the geometry AdS5 x S5, it was expected that the geometry AdS4 x CP3 is also self-dual. For a geometry to be self-dual, it should be possible to perform a series of bosonic and fermionic T-dualities, so that it returns to its original form. The idea is then that this symmetry would explain the symmetries documented in scattering amplitudes in the AdS/CFT dual theory.

In the paper, we focused on the fermionic T-duality and enumerated all the possible transformations one could perform. We identified 4 candidate sets, only one of which had been previously analysed, showing that in each case one encountered a singularity in a scalar field called the dilaton. I believe our presentation is clear and we have not confused the picture by trying to analyse bosonic T-dualities. Fundamentally, we find that the fermionic T-duality transformation, as initially proposed by Berkovits & Maldacena, is not general enough to allow self-duality of AdS4 x CP3. As a result, sermonic T-duality applied to AdS4 x CP3 does not explain hidden symmetries in scattering amplitudes.

The third publication, JHEP08(2015)121, studied non-Abelian T-duality for the geometries AdS3 x S3 x T4 and AdS3 x S3 x S3 x S1. While most of the results hinged on the latter geometry, for the former, we noticed that a non-Abelian T-dual initially found by Sfetsos and Thompson actually fell into a class of geometries that had been classified by Nakwoo Kim and collaborators in 2007 (later by me 2010). There was no known example in this class, but supersymmetry permits it. It turned out that the non-Abelian T-dual fits in this class. In the same paper, we studied properties of the field theories dual to non-Abelian T-duality and extensively studied supersymmetry. The results of this paper led to the spin-off, PhysRevD.93.086010, which was inspired by our work on non-Abelian T-duality.

In work in the final stages of this project, I have been studying Yang-Baxter deformations of AdS5 x S5. We recall that strings moving in AdS5 x S5 are described by a two-dimensional sigma-model, which is integrable, in the sense that it has an infinite number of conserved charges. One can deform the sigma-model to a so-called Yang-Baxter sigma-model, which also preserves integrability. It has been conjectured that the Yang-Baxter deformation is equivalent to non-Abelian T-duality, and this connection has been demonstrated for a number of examples.

We have observed that for a large class of Yang-Baxter deformations that there is a corresponding open string metric that is undeformed and all the information on the Yang-Baxter deformation is encoded in the non-commutativity parameter of the open string. This makes the connection between twists of the conformal algebra, giving rise to non-commutative spacetimes, and Yang-Baxter deformations. In addition, we realised that a so-called unimodularity condition, which picks out valid supergravity solutions, can be translated into the requirement that the non-commutative parameter be divergence-free. The work is now on the arXiv, 1702.02861 and we have submitted it to Physical Review Letters, a high-impact journal.