Cosmological constant, the duality symmetric string, and Atkin-Lehner symmetry

I was going through one of my notebooks and I came across a page with several comments on old papers by Arkady Tseytlin [1] and Gregory Moore [3], respectively. The notes must have been written last autumn at the start of the academic year, because it was around this time my supervisor and I were talking about the cosmological constant problem. In the referenced papers, two interesting approaches to the CC in string theory are presented.

Let’s start with Tseytlin. We’ve discussed in the past Tseytlin’s formulation of the duality symmetric string for interacting chiral bosons, so I direct the reader to that entry for a background introduction. Jumping straight to the point, what we find in the final sections of [2] is that, upon computing the 3-graviton amplitudes, the following 3-graviton interaction is obtained

$\displaystyle S_3 = \int d^D x_{+} d^D x_{-} [h_{\alpha \beta} (h_{\lambda \rho}\partial_{+ \alpha} \partial_{- \beta} h_{\lambda \rho} + 2\partial_{+ \alpha} h_{\lambda \rho}\partial_{- \rho}h_{\beta \lambda})], \ \ (1)$

where $\partial_{\pm \mu} \equiv \partial / \partial x^{\mu}_{\pm}$ and $h_{\mu \nu} \equiv H_{(\mu \nu)} (x_{+}, x_{-})$ . When (1) is written in terms of doubled coordintes $(x, \tilde{x})$ the low-energy effective theory takes the form

$\displaystyle S_3 = \int d^D x d^D \tilde{x} [R_3 (\partial) - R_3 (\tilde{\partial})], \ \ (2)$

where $\partial_{\mu} = 1/\sqrt{2} (\partial_{+ \mu} + \partial_{-\mu}) = \partial / \partial x^{\mu}$ and $\tilde{\partial}_{\mu} = 1 / \sqrt{2} (\partial_{+ \mu} - \partial_{-\mu} = \partial / \partial \tilde{x}_{\mu}$ . The 3-graviton term $R_3 (\partial)(R_3(\tilde{\partial}))$ in the expansion of the scalar curvature for the metric $G_{\mu \nu} = \delta_{\mu \nu} + h_{\mu \nu}$ with $h_{\mu \nu}(x, \tilde{x})$ can be written

$\displaystyle R_3 (\partial) = 1/4 h_{\mu \nu} \partial^2 h_{\mu \nu} - 1/4 h_{\alpha \beta}(h_{\lambda \rho} \partial_{\alpha} \partial_{\beta} h_{\lambda \rho} + 2\partial_{\alpha} h_{\lambda \rho}\partial_{\rho}h_{\beta \lambda}) + ..., \ \ (3)$

$\displaystyle \equiv R_2 + R_3 + ..., \ \ (4)$

with $\partial_{\mu} h_{\mu \nu}= 0$ and $h^{\mu}_{\mu} = 0$ .

As we then see in [2], in the case $\tilde(\partial)_{\lambda} h_{\mu \nu} = 0$ it follows (2) reduces to the standard Einstein vertex. But as Tseytlin also notes, there is a contradiction in the structure of (2) owed to the presence of the minus sign. What happens is that, if $R_3(\partial)$ and $R_3 (\tilde{\partial})$ are replaced for the full Einstein scalars, the corresponding linearised equations for $h_{\mu \nu}$ contains the difference of $\partial^2$ and $\tilde{\partial}^2$ which does not match the mass-shell condition $(\partial^2 + \tilde{\partial}^2)H_{\mu \nu} = 0$ . To remedy this, the full off-shell generalisation of (2) is considered

$\displaystyle S_{Eff} = \int d^D x d^D \tilde{x} \sqrt{g(x,\tilde{x})} \sqrt{\tilde{g}(x, \tilde{x})} [R(g, \partial) + R(\tilde{g}, \tilde{\partial}) + ...], \ \ (5)$

which I think is fair to say is a quite famous result. Take particular notice of the structure of this effective action. For me, I could stare at it for lengths of time; it is one of my current favourite results in the context of duality symmetric string theory and I have several thoughts about it. In fact, some of my ongoing research is focused on thinking more broadly about the geometric structure of the full 2D-dimensional space, and I think there is still quite a bit left to be said about potential insight offered in (5).

But for the interests of the present post, we want to focus on an altogether different matter: the cosmological consant. To share something else that is interesting, in [1] perhaps a lesser known about ansatz is presented for the large distance effective gravitational action based on the effective theory (5). It takes the form

$\displaystyle \bar{S} = \frac{S}{V} = \frac{\int d^D x \sqrt{g} (R + L_M)}{\int d^D x \sqrt{g}}. \ \ (6)$

What we have here is a gravity plus matter system $\bar{S}$ that is given by the standard action $S$ divided by the volume $V$ of spacetime. How to make sense of it? Much of [1] is spent arriving at (6), and so I’ll spare the details as they are quite clear in that paper. The main idea, in summary, is that from (5) in which the coordinates are doubled at the Planck scale, one can essentially integrate out the dual coordinates $\tilde{x}$ (really, the dual coordinates are treated in Kaluza-Klein fashion and as such one sees that the integral over the dual coordinates decouples) so that, as a step to arriving at (6), an action is obtained for the standard curvature scalar $R$ that includes the dual volume $\tilde{V}$ that is the inverse of the usual volume. It looks like this

$\displaystyle \hat{S} \simeq \tilde{V} \int d^D x \sqrt{g} R + ..., \ \ (7)$

with

$\displaystyle \tilde{V} = \int d^D \tilde{x} \sqrt{\tilde{g}(\tilde{x})}. \ \ (8)$

What was really clever by Tseytlin resides in how, motivated by an earlier proposal by Linde, he saw that although some mechanism to solve the CC problem at the level of the Planck scale looked unlikely, one might be able to explain why the CC looked small through some modification of the low-energy effective gravitational action using a sort of nonlocality. He saw, quite rightly, such a possibility naturally emerges within the structure of duality symmetric string theory. However, as it stands, there are issues with radiative stability in this set-up, despite some claims in the literature. This was most recently explored in relation to vacuum energy sequestering. But despite these issues, among a number of other questions, I think there could still be something in the general line of thought; hence my interests in the target space of this theory.

***

The other paper [3] I started taking notes on was by another legend, Gregory Moore. One of the issues with the CC in string theory is the contribution to it by the massless sector. One can easily see this from an analysis of the standard string. But what Moore observes is how this contribution may be cancelled by a tower of massive states, such as by using the Atkin-Lehner symmetry for instance.

Atkin-Lehner (AL) symmetry is really quite neat. It originates from number theory and the study of modular forms, but there is some suggestion and deep hints that AL symmetry is present in string theory. Admittedly, I am not deeply familiar with this topic and have merely flagged this paper as interesting for when I have some time to go back and think about the CC. But from my understanding is that, given the fact that the string path integral can be viewed as an inner product of modular forms over some moduli space, then in the case of certain backgrounds the moduli space can be seen to exhibit AL symmetry.

In short, the motivation for Moore is to look for any kind of enhanced albeit hidden symmetry (for instance, in parameter space). In the expansion of the trace for a complete set of stringy states, the one-loop path integral can be interpreted as an inner product of left and right-moving wave-functions $Z = \langle \Psi_R \vert \Psi_L \rangle$ . From a stringy point of view, it is argued that the vanishing of the cosmological constant in our universe could then be interpreted from understanding why $\Psi_R$ and $\Psi_L$ are orthogonal. Naturally, Moore turns to heterotic theory. He finds that the one-loop string cosmological constant vanishes in non-trivial non-supersymmetric backgrounds when viewing the path integral as an inner product of orthogonal wave-functions.

But from what I understand, there are issues with the construction in [3], for example when applied in the case of four-dimensional spacetime. There is also another paper that I am aware of on twisted modular forms, but I have not read it. That said, I would like to understand AL better and also the issues faced in [3]. It is a very interesting paper. Given time with a return to thinking about the CC, it would be a fun to properly work through. For that reason I share it here.

References

[1] A. A. Tseytlin. Duality-Symmetric String Theory and the Cosmological-Constant Problem. Phys. Rev. Lett. 66 (1991), 545-548. doi:10.1103/PhysRevLett.66.545. url: https://inspirehep.net/literature/299973

[2] A. A. Tseytlin. Duality symmetric closed string theory and interacting chiral scalars. Nucl. Phys. B 350 (1991), 395-440. doi:10.1016/0550-3213(91)90266-Z.

[3] G. Moore. Atkin-Lehner Symmetry. Nucl. Phys. B293 (1987) 139. url: https://lib-extopc.kek.jp/preprints/PDF/1987/8705/8705249.pdf.

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