Stringy Things

Double Field Theory: The Courant Bracket

1. Introduction

In this post we are going to briefly and somewhat schematically discuss the appearance of the Courant bracket in Double Field Theory (DFT), following [1]. The point here is mainly to set the stage, so we jump straight into motivating the Courant bracket. In the next post, we will then study the B-transformations from the maths side and the C-bracket, following and expanding from [1] and others, with an emphasis in the end on how all of this relates to T-duality and strings.

What follows is primarily based on a larger collection of study notes, which I will upload in time.

2. Motivating the Courant Bracket

To understand the appearance of the Courant bracket in DFT, one way to start is by considering some general theory with a metric {g_{ij}(X)} and a Kalb-Ramond field (i.e., an antisymmetric tensor field) {b_{ij}(X)}, where {X \in M}. The symmetries and diffeomorphisms of {g_{ij}(X)} are generated by vector fields {V^{i}(X)}, where {V \in T(M)} with {T(M)} being the tangent bundle. As for {b_{ij}(X)}, the transformations are generated by one-forms {\xi_{i}(X)}, where {\xi^{i}(X) \subset T^{\star}(M)} with {T^{\star}(M)} being the cotangent bundle. We may combine {V^{i}(X)} and {\xi^{i}(X)} as a sum of bundles, such that (dropping indices) {V + \xi \in T(M) \oplus T^{\star}(M)}.

With these definitions, the opening question now is to ask, ‘what are the gauge transformations?’ To make sense of this, consider the following gauge parameters,

\displaystyle \delta_{V + \xi} g = \mathcal{L}_{V} g

\displaystyle \delta_{V + \xi} b = \mathcal{L}_{V} b + d\xi \ \ \ (1)

Here {\mathcal{L}} is the Lie derivative. Furthermore, note given that {V} generates diffeomorphisms, in (1) we get the Lie derivative in the direction of {V}. Also notice that {\xi} does not enter the gauge transformation of {g}; however, for the gauge transformation of {b}, we do have a one-form {\xi} and so we can take the exterior derivative. We should also note the following important properties of {\mathcal{L}}. For instance, when acting on forms the Lie derivative is,

\displaystyle \mathcal{L}_{V} = i_{V}d + di_{V} \ \ \ (2)

Where {iV} is a contraction with {V}. We’re just following the principle of a contraction with a vector times the exterior derivative. It is also worth pointing out that {\mathcal{L}} and the exterior derivative commutate such that,

\displaystyle \mathcal{L}_{V}d = d\mathcal{L}_{V} \ \ \ (3)

There are also some other useful identities that we are going to need. For instance, for the Lie algebra,

\displaystyle  [\mathcal{L}_{V_1}, \mathcal{L}_{V_2}] = \mathcal{L}_{[V_1,V_2]} \ \ (4)

Where {[V_1,V_2]} is just another vector such that {[V_1,V_2]^{k} = V_{1}^{p}\partial_{p}V_{2}^{k} - (1,2)}.

And, finally, we have,

\displaystyle  [\mathcal{L}_{X}, i_{Y}] = i_{[X,Y]} \ \ (5)

Now follows the fun part. Given the transformation laws provided in (1), we want to determine the gauge algebra. To do this, we must compute in reverse order the gauge transformations on the metric {g} and the {b}-field. For the metric we evaluate the bracket,

\displaystyle  [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \delta_{V_2 + \xi_2} \mathcal{L}_{V_1}g - (1,2)

\displaystyle = \mathcal{L}_{V_1}\mathcal{L}_{V_2}g - (1,2)

Using the identity (4) we find,

\displaystyle  [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \mathcal{L}_{[V_1, V_2]} g \ \ \ (6)

For the {b}-field we have,

\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]b = \delta_{V_2 + \xi_2} (\mathcal{L}_{V_1}b + d\xi_{1}) - (1,2)

\displaystyle = \mathcal{L}_{V_1}(\mathcal{L}_{V_2}b + d\xi_2) - (1,2)

\displaystyle = \mathcal{L}_{[V_1, V_2]} + d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1} \ \ \ (7)

It turns out that this bracket satisfies the Jacobi identity, although it is not without its problems because, as we will see, there is a naive assumption present in the above calculations. In the meantime, putting this aside until later, the idea now is to compare the above with (1) and see what ‘pops out’. Notice that we find,

\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}] = \delta_{[V_1,V_2]} + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} \ \ \ (8)

In which we have discovered a bracket defined on {T(M) \oplus T^{\star}(M)},

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}] = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2} \xi_{1} \ \ \ (9)

On the right-hand side of the equality we see a vector field in the first term and a one-form given by the final two terms. This Lie bracket is reasonable and, on inspection, we seem to have a definite gauge algebra. Here comes the problem allude a moment ago: there is a deep ambiguity in (9) in that we cannot, however much we try, determine unique parameters in our theory. Notice,

\displaystyle \delta_{V + \xi}b = \mathcal{L}_{V}b + d\xi

\displaystyle = \mathcal{L}_{V + (\xi + d \sigma)} \ \ \ (10)

The point being that the ambiguity of the one-form {\xi} is so up to some exact {d\sigma}. To put it another way, if we change {\xi} by {d\sigma}, we’re not actually changing anything at all. We would just get {\mathcal{L}_{V}b + d(\xi + d\sigma)} where, when the exterior derivative hits {d\sigma} we simply get nothing. So, given that {\xi} is ambiguous up to some exact {d\sigma}, in a sense what we have is a symmetry for a symmetry. In other words, the present construction is not sufficient.

What we can do to correct the situation is analyse the mistake in (7). Let us, for instance, look at the right-hand side of the summation sign in this equation,

\displaystyle d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1}) = d(di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)

The logic follows that the first term {di_{V_1}\xi_{2}} is killed by {d}. It doesn’t make any contribution and its coefficient is just {1}. The trick then is to see, without loss of generality, that we may change the implicit coefficient {1} in front of {di_{V_1}\xi_{2}}. It turns out, the coefficient that we can use is {1 - \frac{\beta}{2}},

\displaystyle  = d((1-\frac{\beta}{2} di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)

\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \frac{1}{2}\beta di_{V_1}\xi_{2}) - (1,2)

\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d [i_{V_1}\xi_{2} - i_{V_2}\xi_{1}]) \ \ \ (11)

What we end up achieving is the construction of a much more general bracket,

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d(iV_{1}\xi_{2} - iV_{2}\xi_{2}) \ \ \ (12)

What is so lovely about this result might at first seem counterintuitive. It turns out, as one can verify, for {\beta \neq 0}, we do not satisfy the Jacobi identity. So at first (12) may not seem lovely at all! But it makes perfect sense to consider cases of non-vanishing {\beta}. In mathematics, the case for {\beta = 1} was introduced by Theodore James Courant in his 1990 doctoral dissertation [5], where he studied the bridge between Poisson geometry and pre-symplectic geometry. The idea here is to forget about the Jacobi identity – consider its loss an artefact of field theory with anti-symmetric tensors and gravity – and impose {\beta = 1}. When we do this what we obtain is indeed the famous Courant bracket. That is, given {\beta = 1}, the case of maximal symmetry is described by,

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta=1} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2} d(iV_{1}\xi_{2} - iV_{2}\xi_{2} \ \ \ (13)

Although the Jacobi identity does not hold, one can show that for {Z_{i} = V_{i} + \xi_{i}, i = 1,2,3}, the Jacobiator assumes the form,

\displaystyle [Z_1, [Z_2,Z_3]] + \text{cyclic} = dN(Z_1, Z_2, Z_3)

Which is an exact one-form. This gives us a first hint that the unsatisfied Jacobi identity does not provide inconsistencies, because exact one-forms do not generate gauge transformations.

But why {\beta = 1}? Courant argued that the correct value of {\beta} is in fact {1} because, as he discovered, there is an automorphism of the bracket. This means that if do an operation on the elements, it respects the bracket. This automorphism is, moreover, an extra symmetry known in mathematics as a B-transformation. What follows from this is, I think, actually quite special. Given the Courant bracket is a generalisation of the Lie bracket, particularly in terms of an operation on the tangent bundle {T(M)} to an operation on the direct sum of {T(M)} and the p-forms of the vector bundle, what we will discuss is how the B-transformation in mathematics relates in a deep way to what in physics, especially string theory, we call T-duality (target- space duality). This is actually one of the finer points where mathematics and physics intersect so wonderfully in DFT.

In the next post we’ll carry on with a discussion of the B-transformation and then also the C-bracket, finally showing how everything relates.

References

[1] Zwiebach, B. (2010). ‘Double Field Theory, T-Duality, and Courant Brackets’ [lecture notes]. Available from [arXiv:1109.1782v1 [hep-th]].

[2] Hohm, O., Hull, C., and Zwiebach, B. (2010). ‘Generalized metric formulation of double field theory’. Available from [arXiv:1006.4823v2 [hep-th]].

[3] Hull, C. and Zwiebach, B. (2009). ‘Double Field Theory’. Available from [arXiv:0904.4664v2 [hep-th]].

[4] Hull, C. and Zwiebach, B. (2009). ‘The Gauge Algebra of Double Field Theory and Courant Brackets’. Available from [arXiv:0908.1792v1 [hep-th]].

[5] Courant, T. (1990). ‘Dirac manifolds’. Trans. Amer. Math. Soc. 319: 631–661. Available from [https://www.ams.org/journals/tran/1990-319-02/S0002-9947-1990-0998124-1/].

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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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