# 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.

# Generalised supergravity and the dilaton

I spent sometime in the early autumn months thinking about the cosmological constant problem (CC). This was actually secondary, because my primary note taking focused more on S-duality and manifestly duality invariant actions, non-perturbative corrections, and the dilaton. But my supervisor, Tony, has spent a lot of time thinking about this problem, with one of his big ideas being vacuum energy sequestering, so naturally there is motivation whenever we get the chance. There has also been some renewed interest in the CC problem in the context of generalised double sigma models and double field theory. In general, there is a lot of interesting cosmology to be investigated here.

I’m currently drafting a post on the CC problem from the view of string theory. This will hopefully provide the reader with a thorough introduction. But as a passing comment in this short note, it suffices to say that the role of the CC in string theory is generally mysterious. In standard textbook analysis, one sees that the mystery starts with the massless sector contribution, with the dilaton central to the discussion; but the mysteriousness comes further into focus once the role of dual geometry is investigated and the peculiar change of the CC under duality transformation. Intuitively, I am inclined to think that a piece of the picture is missing.

One idea I find interesting to play with involves adding extra fields. Another idea people play with is redefining the dilaton. An example comes from a breakthrough paper by Tseytlin and Wulff [1].

Admittedly, I wasn’t aware of this paper until my early autumn investigations. Within it, a 30 year old problem is solved using the Green-Schwarz (GS) formulation of supergravity theory. The short version is that, in the standard GS formulation of Type IIB string theory there is a problem with the number of degrees of freedom. The space-time fermions have 32 components. An on-shell condition reduces the degrees of freedom to 16, but it needs to be 8. It was later discovered that kappa-symmetry is present in the theory, which is a non-trivial gauge symmetry, and this symmetry may be used to reduce the remaining 8 degrees of freedom. However, issues remained in proving a number of associated conjectures – that is, until Tseytlin and Wulff formulated generalised type IIB SUGRA on an arbitrary background.

The key observation is that generalised SUGRA is equivalent to standard SUGRA plus an extra vector field. Furthermore, one of the characteristics is that, under generalised T-duality, there is a modification of the dilaton such that a non-linear term is added $\Phi \rightarrow \tilde{\Phi} = \Phi + I \cdot \tilde{x}$ [2]. I think this is quite interesting, and it is something I want to look at more deeply in the future.

Although the context of the calculation is completely different to my own investigations, it is worth noting that this generalised Type IIB theory can be obtained from double field theory. Perhaps not surprisingly, I have seen some pin their hopes that generalised SUGRA could contribute to solving the cosmological constant problem (and potentially also give de Sitter vacua). That seems premature, from my vantange; but in any case it is an interesting bit of work by Tseytlin, Wulff, and others.

References

[1] Tseytlin, A.A., Wulff, L., \textit{Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations}. (2016). [arXiv:1605.04884 [hep-th]].

[2] Tseytlin, A.A., et al, Scale invariance of the $\eta$-deformed $AdS5 \times S5$ superstring, T-duality, and modified type II equations. (2016). [arXiv:1511.05795 [hep-th]].

# Papers: Holography and the cosmological constant problem, plus a new publication from Zwiebach

[latexpage]

Non-perturbative de Sitter vacua via $alpha^{prime}$ corrections – Barton Zwiebach

Barton Zwiebach has a new paper out. It’s rather lovely.

The paper was written for, and submitted to, the Gravity Research Foundation 2019 Awards for Essays on Gravitation. Zwiebach’s contribution was awarded second place. However, given the content and general parameters of the first place paper, I believe Zwiebach’s contribution could have easily been given the top prize. (Granted, I have my stringy biases).

As for the paper itself, the premise is both subtle and interesting. The stage is set in the context of two-derivative supergravity theories, in which a large – indeed, infinite – number of higher-derivative corrections correlate with the parameter $alpha^{prime}$. One should note that $alpha^{prime}$ is indeed the dimensionful parameter in string theory.

One of the challenges since the 1980s has been to achieve a complete description of these higher-derivatives. It turns out a rather interesting approach may be taken by way of duality covariant ‘stringy’ field variables, as opposed to directly supergravity field variables.

From a quick readthrough, the idea presented in Zwiebach’s paper is to drop all dependence on spatial coordinates, such that only time dependence remains. The time dependent ansatz enables a general analysis on cosmological, purely time-dependent backgrounds on which all the duality invariant corrections relevant to these backgrounds may be classified. Leveraging the duality group $O(d,d, R)$, an $O(d,d, R)$ invariant action is constructed of the form

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} – frac{1}{8}tr(dot{S}^{2}) ]

Then, Zwiebach includes in the two-derivative action all of the $alpha^{prime}$ corrections. Around this point, the work of Meissner is referenced. Admittedly, I will have to review Meissner’s work moving future. Meanwhile, I rewrite the general duality invariant action from the paper as it is quite nice to look at,

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} + sum_{k=1}^{infty}(alpha^{prime})^{k+1}c_{k}tr(dot{S}^{2k}) + multi-traces ]

This action, it turns out, encodes the complete $alpha^{prime}$ corrections. Moreover, these corrections are for cosmological backgrounds. The aim is thus to classify the higher-derivative interactions, and then derive general cosmological equations to order of $alpha^{prime}$. Upon deriving these equations of motions, the key point to highlight is how one can then go on to construct non-perturbative de Sitter solutions.

Perhaps it remains to be said that de Sitter vacua are a hot topic in string theory. In the end, Zwiebach finds a non-perturbative de Sitter solution – that is, non-perturbative in $alpha^{prime}$, which is very cool.

After only a few minutes with this paper, I am eager to dig a bit deeper. Needless to say, it is intriguing to think more about the string landscape in the context of its study.

This 2011 paper by Kyriakos Papadodimas is on the cosmological constant problem in the context of the dual conformal field theory. The preprint version linked is several years old, and so I should preface what follows by stating that I have not yet had a chance to review all of the papers that directly proceed it. Indeed, this paper ends with the allusion of a follow-up, and I look forward to reading it.

Another comment for the sake of historical context, this paper on the CC problem precedes by almost two-years the publication of two important papers by Papadodimas and Suvrat Raju, the focus of which is on the black hole interior and AdS/CFT.

Now, as for the paper linked above, one of the key questions entertained concerns whether holography theory can teach us anything new about the cosmological constant problem. Of course the CC problem – or what is sometimes dramatically referred to as the vacuum catastrophe – represents one of the big questions for 21st century physics. It is one which Prof. Padilla and I sometimes talk about.

In approaching Papadodimas’ paper, the following question may be offered: is it possible that Nature’s mechanism for apparent fine-tuning may reside in a fundamental theory of emergent fields?

In the first section or two, Papadodimas builds an analogue of c.c. fine-tuning in terms of 4-point functions. We come to understand that, in the CFT, 4-point functions can more easily be translated into correlation functions. Using Witt diagrams, which are like Feynman diagrams, one takes the external points to infinity – or, in this case, to the conformal boundary of AdS. (The basics of this approach will most certainly be studied in my series of blog posts on string theory, beginning with the collection of notes on Conformal Field Theory). The advantage, moreover, is that the 4-point functions become Witt diagrams, and, then, through the AdS/CFT correspondence, one can relate these to the correlators on the boundary.

So this is the plan, which, Papadodimas argues, will enable the study of fine-tuning in the bulk in terms of fine-tuning within the dual CFT.

In proceeding his study, Papadodimas gives quite a few comments as to how we might think of fine-tuning as it manifests in the dual CFT. The picture here is really quite interesting; because if, as noted above, we can express fine-tuning in terms of graviton correlation functions, what I have so far failed to mention is that these correlation functions of gravitons are duel correlation functions of the stress-energy tensor on the boundary.

The greater goal – or ideal outcome – would be to study fine-tuning on the boundary, particularly how it is resolved. What we come to learn is how the large N expansion, the concept of which one may be familiar from their study of QFT, plays an important role. Moreover, one of the interesting suggestions is how the $frac{1}{N}$ suppression of correlators, which relates to the large N gauge theories in the t’Hooft limit, gives us early hints of fine-tuning.

Now, one of the questions I had early on pertained to the comment that it would not be difficult to split up a correlator into a sum of terms, and then expose this broken correlator to some interpretation of fine-tuning. That is to say, nothing is stopping us from breaking up the correlator into a sum of terms and then pointing at this object and saying, ‘see we have fine-tuning!’. But Papadodimas is a smart guy, and he already knows this.

To be clear, and to summarise what I am talking about, part of the strategy explained by the author early on includes expanding “the correlators into sums of terms, each of which corresponds to the exchange of certain gauge invariant operators in intermediate channels i.e. into a sum of conformal blocks. While the sum of all these contributions is suppressed by the expected power of 1/N, individual terms in the sum can be parametrically larger, as long as they cancel among themselves” (p.2). These cancelations are what we’re looking for, because, the interpretation of this paper is that they are in a way an expression of bulk fine-tuning in the dual CFT.

Papadodimas quickly extinguished my one concern, however; because we are not arbitrarily breaking up the correlator. He recognised that one may perform such an act artificially, which wouldn’t have much meaning. Instead, he searches for and performs an canonical procedure. I found this very interesting to follow. What we end up with is an expansion in conformal blocks that looks like this,

[ langle tilde{T}(x_{1})tilde{T}(x_{2})tilde{T}(x_{3})tilde{T}(x_{4})rangle_{con} = sum_{A} mid C_{TT}^{A}mid^{2}G_{A}(x_{1}, x_{2}, x_{3}, x_{4}) ]

Then, following a brief review of holography, some comments on hierarchy vs. fine-tuning, and also a study of important cutoffs, the idea of a sort of dual picture emerges, in which the physical description may appear natural in one picture – such as in the $frac{1}{c}$ expansions discussed in the paper – and then finely tuned in the second picture.

There is too much to summarise in such a small space. So I will just pick up on how, in the dual CFT, I found it intriguing that, should the arguments hold, the double operator product expansion of the correlator written above displays partial sums over the conformal blocks that cancel. As alluded early, in the context of the particular CFT, the interpretation is that the conformal blocks appear fine-tuned. But the mechanism behind this exhibited fine-tuning is not fine-tuning as I read it, such that it becomes clear that in large N gauge theories this expansion of the conformal block is natural. The naturalness of the expansion is consistent with what we know of the large $frac{1}{N}$ expansion in the t’Hooft limit.

Papadodimas closes with the explanation that in the context of a CFT with a large central charge, there is a notable c.c. problem in the holographic dual. As I focused on, it is found in this CFT a semblance of fine-tuning in the correlator blocks of the $frac{1}{N}$ expansion. However this may be interpreted, the interesting idea is that: “this fine-tuning may be visible when the correlators are expressed in terms of the exchange of conformal primaries, it may disappear if the correlators are written in terms of the fundamental fields of the underlying QFT” (p.32).

If this is in any way true, it raises some very intriguing questions and raises some very interesting potential pathways for future study, the likes of which Papadodimas spells out in explicit terms. The suggestion of the use of a toy theory, as opposed to a non-perturbative study, makes me think of the SYK model. It will be curious to follow-up on the papers that directly follow.

One last closing comment: I am excited to say that Papadodimas will be one of the lecturers at a special engagement I will be attending this summer at the University of Madrid. The engagement, SIFTS 2019, comprises of two weeks of discussion, review, study and brainstorming on issues in string theory (and fundamental physics broadly). Papadodimas will be lecturing on the black hole interior and AdS/CFT. It shall be brilliant.