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

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.

AdS/CFT and the cosmological constant problem – Kyriakos Papadodimas

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.