## Periodic Reporting for period 1 - NCDIFFGEO (Models of noncommutative differential geometries)

Reporting period: 2016-02-01 to 2018-01-31

A standard technique in physics and engineering is to replace continuous geometric backgrounds by discrete approximations (lattice or graph), so that systems become more calculable. It has become clear that this can be done systematically as a case of Noncommutative geometry (NCG), namely one where differentials and functions do not commute. NCG, or `quantum' geometry, is also expected to arise from Quantum Gravity (QG), the physical regime at which gravitational and quantum interactions are equally strong; this occurs at either very high energies, or at very small distances, commonly referred to as the Planck scale. The fundamental assumption is that QG effects at the Planck scale modify the structure of space-time itself, leading to noncommutativity of spacetime coordinates, which are `quantised' in a similar manner as position and momenta in quantum mechanics. Such modification of spacetime introduces uncertainty or `fuzziness' at the Planck scale and also introduces other gravitational and cosmological effects that could ultimately be empirically verified. NCG allows us to model these quantum gravitational corrections in an effective description without full knowledge of QG itself. As such it has its own internal structure as a more general notion of classical geometry and could be used in many other situations where classical geometry breaks down; it amounts to as a general quantisation scheme for geometry itself.

The project builds on a specific body of knowledge accumulated over the last 25 years related in part to the notion of `quantum groups' or Hopf algebras. These fully emerged in the 1980s out of quantum integrable systems as a new more general notion of symmetry but they can also feature as `quantum symmetries' of quantum spacetimes. The mathematical theory of `noncommutative Riemannian geometry' was also developed recently. This was originally motivated by the wish to include the NCG of quantum groups themselves (just as classical geometry was driven on part by the geometry of Lie groups) but, as a general scheme, it potentially applies much more widely. The aim of the project was to obtain and study a new generation of quantum spacetime models and their noncommutative differential geometries in this context.

The project builds on a specific body of knowledge accumulated over the last 25 years related in part to the notion of `quantum groups' or Hopf algebras. These fully emerged in the 1980s out of quantum integrable systems as a new more general notion of symmetry but they can also feature as `quantum symmetries' of quantum spacetimes. The mathematical theory of `noncommutative Riemannian geometry' was also developed recently. This was originally motivated by the wish to include the NCG of quantum groups themselves (just as classical geometry was driven on part by the geometry of Lie groups) but, as a general scheme, it potentially applies much more widely. The aim of the project was to obtain and study a new generation of quantum spacetime models and their noncommutative differential geometries in this context.

Several results were obtained early, from quantum group techniques. Some quantum groups can be related to others by a 'Drinfeld twist' which changes the algebraic structure but up to a certain categorical equivalence. One can even twist a classical group to a quantum one as an approach to quantisation of both the group and any space on which it acts. We showed how this could be applied to quantum spacetimes so as to be able to compute physical effects such as modifications to the wave operator and dispersion relations for the propagation of fields in such a background. A different, more mathematical, work applied twisting to construct examples of Hopf algebroids, more general than quantum groups and potentially more widely applicable.

At the same time a significant amount of training was undertaken in the techniques of `noncommutative Riemannian geometry', centering around the notion of a `bimodule connection'. A novel feature of the algebraic framework is that we do not need to work over real or complex numbers but can do geometry over any field. We took this nearly to the extreme by working over the field F_2 of 2 elements {0,1}, which we called `digital NCG'. Even the classification of digital NCGs in the affine case where the `co-ordinate algebra' is just polynomials in n<4 variables turned out to be very interesting. The differential calculi on such algebras turned out to be constructed by n-dimensional commutative algebras. We also began the classification of digital NCGs in the finite case where the `coordinate algebra' is a finite-dimensional algebra over F_2.

The work was disseminated in the form of 4 papers on the arXiv preprint repository with three of these already published in regular research journals and the fourth under review by a journal. The work was also disseminated extensively by the MSC fellow in the form of 9 talks at conferences and seminars. These included a very large `new frontiers in physics' conference among others. The MSC fellow also helped organise a conference in Poland on a theme related to the project and helped to run the regular `Quantum Algebras' seminar at the host.

In addition the MSC fellow undertook outreach activities such as speaking at the event `Women count' at the host during International Women's week, and at a Career Advice event for PhD students also at the host. On training, the MSC fellow trained for and acquired a certificate of Higher Education Academy (ADEPT). As a result of her activities the MSC fellow was offered and accepted a UK academic, ending the grant 4 months early. Therefore the grant can also be considered highly successful in its career development and training elements.

At the same time a significant amount of training was undertaken in the techniques of `noncommutative Riemannian geometry', centering around the notion of a `bimodule connection'. A novel feature of the algebraic framework is that we do not need to work over real or complex numbers but can do geometry over any field. We took this nearly to the extreme by working over the field F_2 of 2 elements {0,1}, which we called `digital NCG'. Even the classification of digital NCGs in the affine case where the `co-ordinate algebra' is just polynomials in n<4 variables turned out to be very interesting. The differential calculi on such algebras turned out to be constructed by n-dimensional commutative algebras. We also began the classification of digital NCGs in the finite case where the `coordinate algebra' is a finite-dimensional algebra over F_2.

The work was disseminated in the form of 4 papers on the arXiv preprint repository with three of these already published in regular research journals and the fourth under review by a journal. The work was also disseminated extensively by the MSC fellow in the form of 9 talks at conferences and seminars. These included a very large `new frontiers in physics' conference among others. The MSC fellow also helped organise a conference in Poland on a theme related to the project and helped to run the regular `Quantum Algebras' seminar at the host.

In addition the MSC fellow undertook outreach activities such as speaking at the event `Women count' at the host during International Women's week, and at a Career Advice event for PhD students also at the host. On training, the MSC fellow trained for and acquired a certificate of Higher Education Academy (ADEPT). As a result of her activities the MSC fellow was offered and accepted a UK academic, ending the grant 4 months early. Therefore the grant can also be considered highly successful in its career development and training elements.

The research undertaken significantly advanced the state of the art. Regarding the twisting work, while the twisting of quantum groups and algebras (including differential algebras) on which they act was known for some years, extending this to the construction of actual quantum geometries in a useful way was demonstrated here for the first time, including for example dispersion relations for wave propagation. These techniques could be particularly useful in the future not only for more models but as a way to construct non-associative models (where differentials and functions multiply nonassociatively together; this seems to be indicated by the physics at the Poisson level due an anomaly or obstruction to the associative theory). Similarly, while quantum groupoids were known for some years as generalisation of quantum group, it was shown how one can usefully construct them by twisting. Finally, while the general formulation of `noncommutative Riemannian geometry' has been developed in recent years (much of it as the host) its specialisation to F_2 was relatively new; there was one previous work on a particular 4-dimensional algebra A_2 over this field as an example in another context, but no comprehensive study or classification of digital Riemannian geometries. Note that this goes much further even than lattice approximations since the values themselves of any quantities are {0,1} and hence the entire geometrical state space is finite, which could allow exact (albeit digital) computations of QG (and help to give a flavour of some of the features of that). We also showed how to build such objects using digital AND and EXOR gates as shown in the graphic. This innovative work could one day lead, for example, to `digital quantum computers' which in turn could potentially realise some of the benefits of quantum computers within standard digital electronics.

Neither experimental QG nor actual digital or quantum computing were in the scope of this theoretical project but these are two potential impacts in the longer term. Similarly we did not envisage immediate societal or socio-economic impact but in the longer term both topics have the potential to open the door to dramatic new technologies. Both interact with other EU supported scientific projects such as the COST network on quantum spacetime and EU investments in quantum technology.

Neither experimental QG nor actual digital or quantum computing were in the scope of this theoretical project but these are two potential impacts in the longer term. Similarly we did not envisage immediate societal or socio-economic impact but in the longer term both topics have the potential to open the door to dramatic new technologies. Both interact with other EU supported scientific projects such as the COST network on quantum spacetime and EU investments in quantum technology.