Riemannian Lie algebroids and harmonic maps

I have placed a preprint on the arXiv ‘Killing sections and sigma models with Lie algebroid targets’ (arXiv:1506.07738 [math.DG]). In the paper I recall the notion of a Riemannian Lie algebroid, collect the basic theory and proceed to define Killing sections.

Lie algebroids are a generalisation of the tangent bundle of a manifold. The mantra here is that whatever you can do on a tangent bundle you can do on a Lie algebroid. This includes developing a theory of Riemannian geometry on them.

The notion of a Riemannian metric on a Lie algebroid is just that of a metric on the underlying vector bundle. There is no compatibility condition or anything like that. So, as all vector bundles can be equipped with metrics, all Lie algebroids can be given a metric. The interesting fact is that the Lie algebroid structure allows you to build the theory of Riemannian geometry in exactly the same way as you would on a standard Riemannian manifold. A Lie algebroid with a metric are known as Riemannian Lie algebroids.

In particular we have a good notion of torsion (which is generally missing) and have the notion of a Levi-Civita connection. Moreover, we have the fundamental theorem that says that such a connection is uniquely defined by metric compatibility and vanishing torsion, just as we have in the classical case. All the formula generalise directly with little fuss.

This all begs the question of developing general relativity on a Lie algebroid. Indeed one can formulate the Einstein field equations in this context, see [1]. The geometry here is clear and very neat, the applications to the theory of gravity are less clear.

Killing sections
Something I noticed that was generally missing in the literature was the notion of a Killing section of a Riemannian Lie algebroid. Such a section is a natural generalisation of a Killing field on a Riemannian manifold; they represent infinitesimal isometries. In the paper I show how the basic idea generalises to Riemannian Lie algebroids giving the notion of a Killing section. Moreover, I show how the common ways of expressing the notion of a Killing field directly generalise to Lie algebroids.

Sigma models and harmonic maps
With the above technology in place, I then look at the theory of sigma models that have a Riemannian Lie algebroid as their target. I took the work of Martinez [2] on classical field theory on Lie algebroids, and applied it to this class of theories. The basic idea is that the fields of such a theory are Lie algebroid morphisms from the tangent bundle of our source manifold to a Lie algebroid target. Equipping both the source and target with a metric allows us to build a model in exactly the same way as a standard sigma model on the space of maps between two Riemannian manifolds. The critical points of the Lie algebroid sigma model are seen to be a generalisation of harmonic maps.

I show, as expected, that the infinitesimal internal symmetries of the Lie algebroid sigma model are described by the Lie algebra of Killing section.

After thoughts
Non-linear sigma models represent a large class of models that have found applications in high energy physics, string theory and condensed matter physics. From a mathematical perspective, sigma models provide a strong link between differential geometry and field theory. In this work, I do not attempt to find such applications of the Lie algebroid sigma model, I focus on the differential geometry. However, studying such models seems very natural and hopefully useful.

[1] M. Anastasiei & M. Girtu, Einstein equations in Lie algebroids, Sci. Stud. Res. Ser. Math. Inform. 24 (2014), no. 1, 5-16.

[2] E. Martinez, Classical field theory on Lie algebroids: variational aspects, J. Phys. A: Math. Gen. 38 (2005) 7145.

Quantum gravity

The subject of a quantum theory of gravity is interesting, technical and very difficult. However, there are three basic principles that we expect such a theory to obey.

Creating a full quantum theory of gravity seems to be out of our reach right now. String theory comes close, but the full theory here is not understood. Loop quantum gravity also offers a good picture, but again technicalities spoil achieving the goal.

I am no expert in quantum gravity, but I thought it maybe interesting to outline three basic ‘rules’. The full quantum theory of gravity should be:

  1.  Renormalisable (maybe not perturbatively) or finite.
  2. Background independent.
  3. Reducible to general relativity (plus small corrections) in a sensible classical limit.

As a warning, I will not be too technical here, but will use some standard language from quantum field theory.

The standard methods of quantum field theory are to expand the theory about some fixed configuration, usually the vacuum, and consider small fluctuations about this reference configuration. However, in doing so some techniques are needed to remove the appearance of infinite values of things you would like to measure in the lab. These methods are collective known as ‘perturbative renormalisation’. For example, we know that the quantum theory of electrodynamics can be handled properly using these methods.

However, general relativity as described by Einstein is not amenable to methods of perturbative renormalisation. Well, this is true if we want a full theory. What one can do is consider quantum general relativity as an effective theory. That is we accept that at some energy scale the theory will breakdown, but as long as we are not at that scale the theory is okay. By adding a ‘cut-off’ we can understand quantum general relativity using Feynman diagrams to ‘one-loop’ and calculate graviton scattering amplitudes and so on.

Interestingly, there is some evidence that general relativity or something close to it is nonperturbatively renormalisable; this is known as asymptotic safety. With no details, the idea is that quantum general relativity is not ‘sick’ and well-defined, just not as a perturbative theory like quantum electrodynamics. This is fascinating as it means that a proper quantum theory of gravity may not be a theory of gravitons after all! Recall that small ripples in the electromagnetic field are quantised and understood to be photons. Maybe it is not really possible to describe quantum gravity in a similar way where small ripples in space-time are quantised.

Alternatively, a full theory of quantum gravity could be finite. That is we can employ perturbative methods, but do not need renormalisation techniques. Amazingly, we know of supersymmetric Yang-Mills theories that are finite. Moreover, superstring theory is also finite (I am unsure as to how rigours the proof are here, but the string community generally accept this as fact). It maybe possible that the full theory of quantum gravity is finite from the start. This suggests that looking at supersymmetric theories of gravity is a good idea, but by no means the only thing one can think about.

In short, any full quantum theory of gravity must allow us to calculate things we can hope to measure.

Background independence
This means that the theory should not depend on any chosen background geometric fields. In particular, this is taken to mean that the theory should not require some chosen background metric.

String theory as it stands fails on this. However, string theory is usually employed using perturbation theory and so some classical background is chosen, often 10-d flat space-time.

Loop quantum gravity seems better in this respect, but it has other problems.

In short, any full quantum theory of gravity should not require us to fix the geometry (and maybe topology) from the start.

Reduce to general relativity
General relativity has been so successful in describing classical gravitational phenomena. It is tested to some huge degree of accuracy and so far no deviations from it’s predictions have been found. General relativity is a good theory within the expected domains of validity.

Thus, any quantum theory of gravity must in some classical limit reduce to general relativity, up to small corrections. These quantum corrections must be small enough as not to be seen already in astrophysics and cosmology.

If a quantum theory of gravity cannot be shown to reduce to general relativity in some limits (there maybe several ways of doing this) then we cannot be sure that we really have a quantum theory of gravity.

Today we know that string theory gives us general relativity + small corrections. In essence this is because the spectra of closed string theory contains a spin-2 boson, via rather general arguments we know that this has to be the graviton and the field equations are essentially the Einstein field equations. (Remember this is all in perturbation theory).

Recovering general relativity from loop quantum gravity has yet to be done. This I would say is a sticking point right now.

In short, any full quantum theory of gravity must reproduce the phenomena of general relativity is some classical limit(s).