Physics Diary

Generalised Geometry, Non-commutativity, and Emergence

The project I am working on for my dissertation has to do with the notion of emergent de Sitter space. Of course an ongoing problem in string theory concerns whether asymptotic de Sitter spacetime can exist as a solution, and needless to say this question serves as one motivation for the research. With what appears to be the collapse of KKLT (this is something I will write about, as from my current perspective, the list of complaints against KKLT have not yet seemed to be satisfactorily answered), this academic year I wanted to start picking at the question of perturbative string de Sitter vacua from a different line of attack (or at least explore the possibility). Often, for instance, we approach de Sitter constructions by way of a classical supergravity approach with fluxes, non-geometry, or KKLT-like constructions which add quantum effects to stablise the moduli. One could also look at an alternative to compactification altogether and invoke the braneworld formalism. But, as it is, I’ve not been entirely satisfied with existing programmes and attempts. So the question over the autumn months, as we approached the winter break, concerned whether there was anything else clever that we can think of or take inspiration from. I’m not comfortable in divulging too much at this time, not least until we have something solid. Having said that, in this post let’s talk about some of the cool and fun frontier mathematical tools relevant to my current research.

For my project the focus is on a number of important concepts, including generalised and non-commutative geometry. Within this, we may also ask questions like whether spacetime – and therefore geometry – is emergent. Sometimes in popular talks, one will hear the question framed another way: ‘is spacetime dead?’ But before getting ahead of ourselves, we may start with a very well known and familiar concept in string theory, namely T-duality. Indeed, one motivation to study generalised geometry relates to T-duality, particularly as T-duality expresses how a string experiences geometry. For example, one will likely be familiar with how, in string theory, if we consider the propagation of a string in spacetime in which one spatial dimension is curled up into a circle, the idea is that when we compactify a dimension (in this case on a circle) we modify the string mass spectrum. Less abstractly, take some 10-dimensional string theory and then compactify on a circle $S^{1}$ of radius $R$. The string moves along the circle with the momenta quantised such that p = n / R (n \in \mathbb{Z}). When compactifying the 10th dimension we obtain for the compactified direction, \displaystyle X_{(s)}^{d} (\tau, \sigma + 2\pi) = X_{(s)}^{d}(\tau, \sigma) + 2 \pi \omega R, where we now have winding modes. This is because, as one will learn from any string theory textbook, the string winds around the circle with coordinate X. We can thus write the statement \delta X = 2\pi R m (m \in \mathbb{Z}). In this basic example T-duality is the statement R \rightarrow \frac{\alpha^{\prime}}{R} with n \longleftrightarrow m. The winding modes that appear are of course a deeply stringy phenomenon. And what is interesting is the question of the generalisation of T-duality. Moreover, how might we think of string geometry in such a way that T-duality is a natural symmetry? Generalised geometry was largely motivated by this duality property, such as in the work by Nigel Hitchin. The basic mathematical statement is that the tangent bundle TM of a manifold M is doubled in the sum of the tangent and co-tangent bundle TM \oplus T \star M. In this formalism we also replace the Lie bracket with a Courant bracket, which we may write as something of the form [X + \xi, Y + \eta]_{C} = [X, Y] + L_{X} \eta - L_{Y}\xi - \frac{1}{2} d(i_{X} \eta - i_{Y}\xi) such that X \xi, Y + \eta \in \Gamma (TM \oplus T \star M). In physics, there is also motivation to ask about the geometry of spacetime in which strings propagate. For instance, the existence of winding modes and the nature in which T-duality connects these winding modes to momentum hints that perhaps the fundamental geometry of spacetime should be doubled. This idea serves as one motivation for the study and development of Double Field Theory, which is something the great Barton Zwiebach has been working on in recent years and which uses the SO(d,d) invariant formalism (see his lecture notes).

Additionally, in these areas of thinking, one will often also come across notions like non-geometry or fuzzy geometry. Sometimes these words seem used interchangably, but we should be careful about their meaning. For instance, non-geometry possess a number of characteristics that contribute to its formal definition, one being that it refers strictly to non-Riemannian geometry. Furthermore, we are also speaking of non-geometry as non-commutative geometry [X_{i}, X_{J}] \approx \mathcal{O}(l_{s}) as well as non-associative geometry [X_{i}, X_{J}] X_{k} \approx (l_{s}). One of many possible ways to approach the concept in this regard is to think quantum mechanically. If General Relativity is a very good approximation at long distances, in which we may think of smooth and continuous manifolds; at the smallest scale – such as the string scale – there are important hints that our typical understanding of geometry breaks down.

We will spend a lot of time on this blog discussing technicalities. For now, I just want to highlight some of the different formalisms and tools. In taking a larger view, one thing that is interesting is how there are many similarities between non-commutative and non-associative algebra and generalised geometry, fuzzy geometry, and finally ideas of emergence and a generalised quantum mechanics, although a precise formulation of their relation remains lacking. But this is the arena, if you will, which I think we might be able to make some progress.

As for my research, the main point of this post is to note that these are the sorts of formalisms and tools that I am currently learning. The thing about string theory is that it allows for is no sharp distinction between matter and geometry. Then to think about emergent space – that spacetime is an emergent phenomena – this infers the idea of emergent geometry, and so now we are also starting to slowly challenge present comforts about such established concepts as locality. When we think about emergent geometry we might also think of the structure of perturbative string vacua and ultimately about de Sitter space as a solution that escapes the Swampland. There is a long way to go, but right now I think in general there is an interesting line of attack.

For the engaged reader, although dated the opening article by Michael Douglas in this set of notes from the 2002 summer school at the Clay Mathematics Institute may serve an engaging introduction or overview. A basic introduction to some of the topics described in this post can also be found for instance in this set of notes by Erik Plauschinn on non-geometric backgrounds. Non-commutative (non-associative) geometry is covered as well as things like doubled geometry / field theory. Likewise, I think this paper on non-associative gravity in string theory by Plauschinn and Ralph Blumenhagen offers a fairly good entry to some key ideas. Dieter Lüst also has some fairly accessible lecture notes that offer a glance at strings and (non)-geometry, while Mariana Graña’s lecture notes on generalised geometry are a bit more detailed but serve as a basic entry. Then there are Harold Steinacker’s notes on emergent geometry from matrix models and on non-commutative geometry in relation to matrix models. Finally, there are these lecture notes by Maxim Zabzine on generalised complex geometry and supersymmetry. This is by no means comprehensive, but these links should at least help one get their feet wet.

Maybe in one of the next posts I will spend some time with a thorough discussion on non-commutativity or why it is a motivation of Double Field Theory to make T-duality manifest (and its importance).

**Cover Image: Study of Curve Folding [http://pr2014.aaschool.ac.uk/EMERGENT-TECHNOLOGIES/Curved-Folding-Workshop]

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