Doubled diffeomorphisms and the generalised Ricci curvature

I was asked a question the other week about the idea of doubled diffeomorphisms, such as those found in double field theory. A nice way to approach the concept is to start with dualised linearised gravity [1]. That is to say, we start with a theory considering only the field $h_{ij}(x^{\mu}, x^a, \tilde{x}_a)$ . This field transforms under normal linearised diffeomorphism as

$\delta h_{ij} = \partial_i \epsilon_j + \partial_j \epsilon_i \ \ (1)$

and, under the dual diffeomorphism as

$\tilde{\delta} h_{ij} = \tilde{\partial}_i \tilde{\epsilon}_j + \tilde{\partial}_j \tilde{\epsilon}_i. \ \ (2)$

Now, take the basic Einstein-Hilbert action

$S_{EH} = \frac{1}{2k^2} \int \ \sqrt{-g} \ R, \ \ (3)$

and expand to quadratic order in the fluctuation field $h_{ij}(x) = g_{ij} - \eta_{ij}$ . Just think of standard linearised gravity with the following familiar quadratic action

$S^2_{EH} = \frac{1}{2k^2} \int \ dx \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij}]. \ \ (4)$

This is the Feirz-Pauli action and it is of course invariant under (1). But we want a dualised theory. The naive thing to do, for the field $h(x, \tilde{x})$ , is to add a second collection of tilde dependant terms. In comparison with (4), we also update the integration measure to give

$S^2_{EH} = \frac{1}{2k^2} \int \ dx d\tilde{x} \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h \\ + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij} + \\ \frac{1}{4} h^{ij} \tilde{\partial}^2 h_{ij} - \frac{1}{4} h \tilde{\partial}^2 h \\ + \frac{1}{2} (\tilde{\partial}^i h_{ij})^2 + \frac{1}{2} h \tilde{\partial}_i \tilde{\partial}_j h^{ij}]. \ \ (5)$

If you decompose $x, \tilde{x}$ such that $h_{ij} (x)$ no longer depends on $\tilde{x}$ , then this action simply reduces to linearised Einstein gravity on the coordinate space $x^a.$ Similarly, for the dual theory.

When the doubled action (5) is varied under $\tilde{\delta}$ , the second line is invariant under (2). However, the first line gives

$\tilde{\delta} S = \int [dx d\tilde{x}] [h^{ij} \partial^2 \tilde{\partial}_i \tilde{\epsilon}_j + \partial_i h^{ij} (\partial^k \tilde{\partial}_{k})\tilde{\epsilon}_j \\ - h \partial^2 \tilde{\partial} \tilde{\epsilon} + h(\partial_i \tilde{\partial}^i)\partial_j \tilde{\epsilon}^j \\ + \partial_i h^{ij} \partial^k \tilde{\partial}_j \tilde{\epsilon}_k + (\partial_j \partial_j h^{ij})\tilde{\partial} \tilde{\epsilon}. \ \ (6)$

As one can see, the terms on each line would cancel if the tilde derivatives were replaced by ordinary derivatives. Rearranging and grouping like terms, and then relabelling some indices we find

$\tilde{\delta} S = \int [dx d\tilde{x}] \ [(\tilde{\partial}_j h^{ij})\partial^k (\partial_i \tilde{\epsilon}_k - \partial_k \tilde{\epsilon}_i) \\ + (\partial_i \partial_j h^{ij} - \partial^2h) \tilde{\partial} \tilde{\epsilon} \\ + (\partial^i h_{ij} - \partial_j h)(\partial \tilde{\partial})\tilde{\epsilon}^j. \ \ (7)$

For this to be invariant under the transformation $\tilde{\delta}$ we have to cancel each of the terms. In order to cancel the variation, new fields with new gauge transformations are required. For the first term, a hint comes from the structure of derivatives, namely the fact we have a mixture of tilde and non-tilde derivatives. The Kalb-Ramond b-field mixes derivatives in this way, and, indeed, for the first term to cancel we may add $b_{ij}$ . We denote this inclusion to the action as $S_b$

$S_b = \int [dx d\tilde{x}] \ (\tilde{\partial}_j h^{ij})\partial^k b_{ik}, \\ with \ \ \tilde{\delta}b_{ij} = - (\partial_i \tilde{\epsilon}_j - \partial_j \tilde{\epsilon}_i). \ \ (8)$

The second term can similarly be killed upon introduction of the dilaton $\phi$ . It takes the form

$S_{\phi} = [dx d\tilde{x}] (-2) (\partial_i \partial_j h^{ij} - \partial^2 h) \phi, \ \ \text{with} \ \ \tilde{\delta}\phi = \frac{1}{2}\tilde{\partial} \tilde{\epsilon}. \ \ (9)$

This is quite nice, if you think about it. It is not the full story, because in the complete picture of double field theory we need to add more terms and their are several subtlties. In the naive case of dualised linearised gravity, we find in any case that linearised dual diffeomorphisms for the field $h_{ij}$ requires, naturally and perhaps serendipitously, a Kalb-Ramond gauge field and a dilaton – i.e., the closed string fields for the NS-NS sector.

We are now only left with one term, which is the one with curious structure on the third line in (7). To kill this term, we can observe that the gauge parameter $\tilde{\epsilon}$ satisfies the constraint $\partial \cdot \tilde{\partial} = 0$ derived from the level matching condition. This constraint says that fields and gauge parameters must be annihilated by $\partial \tilde{\partial}$ , and it is fairly easy to find in an analysis of the spectrum in closed string field theory.

So that is one way to attack the remaining term. But what is also interesting, I think, is that it is possible to accomplish the same goal by adding more fields to the theory. This is a non-trivial endeavour, to be sure, as the added fields would need to be invariant under $\delta$ and $\tilde{\delta}$ transformations. Ideally, one would likely want to be able to generalise the added fields to the formal case of the duality invariant theory. But it presents an interesting question.

***

From the perspective of string field theory, double field theory wants to describe a manifestly T-duality invariant theory (we talked about this in a number of past posts). The strategy is to look at the full closed string field theory comprising an infinite number of fields, and instead select to focus on a finite subset of those fields, namely the massless NS-NS sector. So DFT is, at present, very much a truncation of the string spectrum.

As a slight update to notation to match convention, for the massless fields of the NS-NS sector let’s now write the metric $g_{ij}$ , with the b-field $b_{ij}$ and dilaton $\phi$ the same as before. The effective action of this sector is famously

$\displaystyle S_{NS} = \int dX \sqrt{-g} e^{-2\phi} [R + 4(\partial \phi)^2 - \frac{1}{12}H^2] + \text{higher derivative terms}. \ \ (10)$

As one can review in any string textbook, this action is invariant under local gauge transformations: diffeomorphisms and a two-form gauge transformation. The NS-NS field content transforms as

$\displaystyle \delta g_{ij} = L_{\lambda} g_{ij} = \lambda^{k} \partial_k g_{ij} + g_{kj}\partial_i \lambda^k + g_{ik}\partial_i \lambda^k,$

$\displaystyle \delta b_{ij} = L_{\lambda} b_{ij} = \lambda^k \partial_k b_{ij} + b_{kj}\partial_i \lambda^k + b_{ik}\partial_i\lambda^k,$

$\displaystyle \delta \phi = L_{\lambda} \phi = \lambda^k \partial_k \phi. \ \ (11)$

We define the Lie derivative $L_{\lambda}$ along the vector field $\lambda^i$ on an arbitrary vector field $V^i$ such that the Lie bracket takes the form

$\displaystyle L_{\lambda} V^i = [\lambda, V]^i = \lambda^j \partial_j V^i - V^j \partial_j \lambda^i. \ \ (12)$

For the Kalb-Ramond two-form $b_{ij}$ , the gauge transformation is generated by a one-form field $\tilde{\lambda}_i$

$\displaystyle \delta b_{ij} = \partial_i \tilde{\lambda}_j - \partial_j \tilde{\lambda}_i. \ \ (13)$

One way to motivate a discussion on doubled or generalised diffeomorphisms in DFT is to understand that what one wants to do is essentially generalise the action (10). This means that at any time we should be able to recover it. The generalised theory should therefore possess all the same symmetries (with added requirement of manifest invariance under T-duality), including diffeomorphism invariance.

In the generalised metric formulation [2] the DFT action reads

$\displaystyle S_{DFT} = \int d^{2D} X e^{-2d} \mathcal{R}, \ \ (14)$

where

$\displaystyle \mathcal{R} \equiv 4\mathcal{H}^{MN}\partial_M \partial_N d - \partial_M \partial_N \mathcal{H}^{MN} \\ - 4\mathcal{H}^{MN}\partial_{M}d\partial_N d + 4\partial_M \mathcal{H}^{MN} \partial_N d \\ + \frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL} - \frac{1}{2} \mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}. \ \ (15)$

This action is constructed [2] precisely in such a way that it captures the same dynamics as (10). Here $\mathcal{H}$ is the generalised metric, which combines the metric and b-field into an $O(D,D)$ valued symmetric tensor such that

$\displaystyle \mathcal{H}^{MN}\eta_{ML}\mathcal{H}^{LK} = \eta^{NK}, \ \ (16)$

where $\eta$ is the $O(D,D)$ metric. We spoke quite a bit about the generalised metric and the role of $O(D,D)$ in a past post (see this link also for further definitions, recalling for instance the T-duality transformation group is $O(D,D; \mathbb{R})$ , which is discretised to $O(D,D; \mathbb{Z})$ . If $O(D,D)$ is broken to the discrete $O(D,D;\mathbb{Z})$ , then one can interepret the transformation as acting on the background torus on which DFT has been defined). Also note that in (15) $d$ is the generalised dilaton. In the background independent formulation of DFT [5], $e^{-2d}$ is shown to be a generalised density such that the dilaton $\phi$ with the determinant of the undoubled metric $g = \det g_{ij}$ on the whole space is combined into an $O(D,D)$ singlet $d$ establishing the identity $\sqrt{-g}e^{-2\phi} = e^{-2d}$ . We’ll talk a bit more about this later.

There are a number of important characteristics built into the definition of the generalised Ricci (15). Firstly, it is contructed to be an $O(D,D)$ scalar. One can show that the action (14) possesses manifest global $O(D,D)$ symmetry

$\displaystyle \mathcal{H}^{MN} \rightarrow \mathcal{H}^{LK}M_{L}^{M}M_{K}^{N} \ \ \text{and} \ \ X^{M} \rightarrow X^{N}M_{N}^{M}, \ (17)$

where $M_{L}^{K}$ is a constant tensor which leaves $\eta^{MN}$ invariant such that

$\displaystyle \eta^{LK} M_{L}^{M} M_{K}^{N} = \eta^{MN}. \ \ (18)$

Importantly, given $O(D,D)$ extends to a global symmetry, we may define this under the notion of generalised diffeomorphisms. Unlike with the supergravity action (10), which is invariant under the gauge transformations (11) and (12), in DFT the metric and b-field are combined into a single object $\mathcal{H}$ . So the obvious task, then, is to find a way to combine the diffeomorphisms and two-form gauge transformation in the form of some generalised gauge transformation. This is really the thrust of the entire story.

To see how this works, as a brief review, we define some doubled space $\mathbb{R}^{2D}.$ To give a description of this doubled space, all we need to start is some notion of a differential manifold with the condition that we have a linear transformation of the coordinates $X^{\prime} = hX$ , where $h \in O(D,D)$ (similar to the transformation we defined in the post linked above). We will include the generalised dilaton $d$ and we also include the generalised metric $\mathcal{H}$ , although we can keep this generic in definition should we like. For $\mathcal{H}$ we require only that it satisfies the $O(D,D)$ constraint $\mathcal{H}^{-1} = \eta \mathcal{H} \eta$ , where, from past discussion, one will recall $\eta$ is the $0(D,D)$ metric. It transforms $\mathcal{H}^{\prime}(X^{\prime}) = h^{t}\mathcal{H}(X)h$ . We now have everything we need.

Definition 1. A doubled space $\mathbb{R}^{2D}(\mathcal{H},d)$ is a space equipped with the following:

1) A positive symmetric $2D \times 2D-\text{matrix}$ field $\mathcal{H}$ , which is the generalized metric. This metric must satisfy the above conditions and transform covariantly under $O(D,D).$

2) A generalised dilaton scalar $d$ , which is a $2D$ scalar density such that $d = \phi - \frac{1}{2} \ln \det h$ (we’ll show this in a moment).

a) The generalised dilaton is related to the standard dilaton as already described above.

With this definition, we can then advance to define the notion of an $O(D,D)$ module, generalised vectors and vector fields, and so on. To keep our discussion short, the point is that in defining an $O(D,D)$ vector we may combine from before the vector $\lambda^i$ and one-form $\tilde{\lambda}_i$ as generalised gauge parameters

$\displaystyle \xi^M = (\tilde{\lambda}_i, \lambda^i). \ \ (19)$

One can see how this is done in [2,3]. In short, the combination of the gauge transformations into the general gauge transformation with parameter $\xi^M$ is defined under the action of a generalised Lie derivative. The result is simply given here as

$\displaystyle \mathcal{L}_{\xi}A_M \equiv \xi^P \partial_P A_M + (\partial_M \xi^P - \partial^P \xi_M)A_p,$

$\displaystyle \mathcal{L}_{\xi}B^M \equiv \xi^P \partial_P B^M + (\partial^M \xi_P - \partial_P \xi^M)B^p. \ \ (20)$

From this definition, where, it should be said, $A$ and $B$ are generalised vectors, we can eventually write the generalised Lie derivative of $\mathcal{H}$ and $d$ .

$\displaystyle \mathcal{L}_{\xi} \mathcal{H}_{MN} = \xi^P \partial_P \mathcal{H}_{MN} + (\partial_M \xi^P - \partial^P \xi_M)\mathcal{H}_{PN} + (\partial_N \xi^P - \partial^P \xi_N)\mathcal{H}_{MP},$

$\displaystyle \mathcal{L}_{\xi}(e^{-2d}) = \partial_M(\xi^M e^{-2d}). \ \ (21)$

What we see is that, indeed, the generalised dilaton, which we may think of as an $O(D,D)$ singlet, transforms as a density. This means we may think of it as a generalised density. It can also be shown that the Lie derivative of the $O(D,D)$ metric $\eta$ vanishes and therefore the metric is preserved.

What we want, for the purposes of this post, is the generalised Lie derivative of the generalised scalar curvature (15). What we find is that, indeed, it transforms as a scalar provided that the definition of (15) includes the full combination of terms.

$\displaystyle \mathcal{L}_{\xi} \mathcal{R} = \xi^M \partial_M \mathcal{R}.$ (22)

Or, looking at the action (14) as a whole, the subtlety is that the generalised dilaton forms part of the integration measure. The action does not possess manifest generalised diffeomorphism invariance in the typical sense that we might think about it, but it is constructed precisely in such a way that

$\displaystyle \mathcal{L}_{\xi}(e^{-2 d})\mathcal{R} = \partial_I (\xi^{I} e^{-2d}\mathcal{R}) \ \ (26)$

vanishes in the action integral (due to being a total derivative). So we find (14) does indeed remain invariant.

As a brief aside, from the transformations of the generalised metric and the dilaton, we can define an algebra [4]

$\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}] = \mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2} - \mathcal{L}_{\xi_2} \mathcal{L}_{\xi_1} = \mathcal{L}_{[\mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2}]_C}, \ \ (23)$

where we find first glimpse at the presence of the Courant bracket. Provided the strong $O(D,D)$ constraint of DFT is imposed

$\displaystyle \partial_N A_I \partial^{N} A^J = 0 \ \forall \ i,j, \ \ (24)$

then the Courant bracket governs this algebra such that

$\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}]^{M}_{C} = \xi_{1}^{N}\partial_{N}\xi_{2}^{M} - \frac{1}{2}\xi_{1N}\partial^{M}\xi_{2}^{N} - (\xi_1 \leftrightarrow \xi_2). \\ (25)$

An important caveat or subtlety about this algebra is that it does not satisfy the Jacobi identity. This means that the generalised diffeomorphisms do not form a Lie algebroid. But nothing fatal comes from this fact for the reason that, whilst we may like to satisfy the Jacobi identity, the gauge transformation leaves all the fields invariant that fulfil the strong $O(D,D)$ constraint.

In closing, recall that DFT starts with the low-energy effective theory as a motivation. It is good, then, that a solution of (24) is to set $\tilde{\partial} = 0$ giving (10). The Ricci scalar is the only diffeomorphism invariant object in Riemannian geometry that can be constructed only from the metric with no more than two derivatives. In DFT, we have an action constructed only from the generalised metric and doubled dilaton with their derivatives.

References

[1] Hull, C.M., and Zweibach, B., Double field theory. (2009). [arXiv:0904.4664 [hep-th]].

[2] Hohm, O., Hull C.M., and Zwiebach, B., Generalized metric formulationof double field theory. JHEP, 08:008, 2010. [arXiv:1006.4823 [hep-th]].

[3] Zwiebach, B., Double field theory, T-duality, and Courant brackets. [arXiv:1109.1782 [hep-th]].

[4] Hull, C.M., and Zwiebach, B., The gauge algebra of double field theory and courant brackets. Journal of High Energy Physics, 2009(09):090–090, Sep 2009. [arXiv:0908.1792 [hep-th]].

[5] Hohm, H., Hull, C.M., and Zwiebach, B., Background independent actionfor double field theory. Journal of High Energy Physics, 2010(7), Jul 2010. [arXiv:1003.5027 [hep-th]].

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