The problem of dark energy in string theory
Cum Vain is already the new guru of string theory; he replaces his doctoral thesis director. Edward Witten Father of the swampland idea, which extends the landscape, leads the search for an extensive network of conjectures that differentiate both spaces of solutions from the theory. Several suggest that our Sitter-type spacetime (dS), with a positive cosmological constant, is found in the swampland; in the landscape there are only solutions with a space-time of anti-de Sitter type (AdS), with negative cosmological constant, or Minkowski type, without cosmological constant. To describe the dark energy Vafa proposes to look for models of quintessence in the landscape; as well as exploring the swampland with new eyes until finding islands of isolated landscape that avoid the most restrictive conjectures.
Every revolution begins with a crisis. Perhaps the third revolution of string theory - which was born 50 years ago - is approaching. Many media have polemicized with the ideas of Vafa, offering capciosos headlines. "Dark energy might be incompatible with string theory," we read in the popular Quanta Magazine; despite healing in health with a "could be", these headlines confuse many laymen. Others bring out the debate among experts on this subject, such as the one starring Arthur Hebecker against Thomas Van Riet and Vafa in a summer workshop at the Simons Center (Univ. Stony Brook, New York); Hebecker commented that these ideas are "very dangerous" for string theorists because in Europe they could lead to distrust and cuts in funding. But this wave of pessimism could be the prelude to renewed optimism.
The conjectures that characterize the difference between swamplad and landscape have their positive side. The Spanish Luis Ibáñez and his collaborators are using them to make generic predictions about the phenomenology of low energy string theory. A very promising line of which we will have a good summary next month in Madrid, during the workshop "Views over the Swampland", IFT UAM-CSIC, 19-21 Sep 2018. There will again coincide Vafa, Van Riet and Hebecker, but We will surely enjoy a much more optimistic vision led by Ibáñez.
The articles of Vafa origin of all this commotion are Prateek Agrawal, Georges Obied, ..., Cumrun Vafa, "On the Cosmological Implications of the String Swampland," arXiv: 1806.09718 [hep-th] (already quoted 24 times), and Georges Obied , Hirosi Ooguri, ..., Cumrun Vafa, "De Sitter Space and the Swampland," arXiv: 1806.08362 [hep-th] (already quoted 28 times); as well as Ulf H. Danielsson, Thomas Van Riet, "What if string theory has not Sitter vacuums?" arXiv: 1804.01120 [hep-th] (already quoted 22 times).
In the optimistic line led by Ibáñez I recommend the video of his talk "Particle Physics Constraints from WGC," 2018 Simons Summer Workshop, 08 Aug 2018 [video], and his articles Luis E. Ibanez, Victor Martin-Lozano, Irene Valenzuela, " Constraining Neutrino Masses, the Cosmological Constant and BSM Physics from the Weak Gravity Conjecture, "Journal of High Energy Physics 2017: 66 (Nov 2017), doi: 10.1007 / JHEP11 (2017) 066, arXiv: 1706.05392 [hep-th], and Eduardo Gonzalo, Luis E. Ibáñez, "The Fundamental Need for SM Higgs and the Weak Gravity Conjecture," arXiv: 1806.09647 [hep-th], among others.
There has been a lot of echo in media and blogs: Clara Moskowitz, "String Theory May Create Far Fewer Universes Than Thought," Scientific American, 30 Jul 2018; Natalie Wolchover, "Dark Energy May Be Incompatible With String Theory," Quanta Magazine, August 9, 2018; Peter Woit, "Theorists with a Swamp, not Theory," NEW, 13 Aug 2018, "On Status of KKLT," NEW, 16 Aug 2018; Luboš Motl, "Vafa, quintessence vs. Gross, Silverstein," TRF, 27 Jun 2018, "Quintessence is a form of dark energy," TRF, 10 Aug 2018, "Deep thinkers build conjectures upon conjectures upon 5+ more floors," TRF , 15 Aug 2018; among others.
[PS 29 Aug 2018] A good book that discusses the current situation of phenomenology and cosmology in string theory, including the problems associated with the KKLT model and the dS universe in the last chapter, is Mariana Graña, Hagen Triendl, "String Theory Compactifications, "Springer (2017) [74 pp.]. [/]
Theoretical cosmology was born with the Einstein equation of general relativity, because with Newtonian gravitation, or with a post-Newtonian extension compatible with special relativity, it is not possible to rigorously describe the entire universe in an intrinsic way. Sometimes we forget that we can not do cosmology in string theory since we ignore the equation of string theory (its non-perturbative formulation). After 50 years of intense work we only have approximations (perturbative formulation) to the interactions of many fundamental objects of the theory, but not yet of all (for example, this formulation is still sought for the dynamics of the M5 branes in theory M) . Therefore, in string cosmology (as in loop quantum cosmology) cosmological ideas inspired by string theory (or loop quantum gravity) are used, but rigorous theoretical cosmology can not be done (at least as this process is understood). in the gravitation of Einstein).
In string theory we talk about cosmology in the context of a supergravity (SUGRA), a theory of gravitation with one or several supersymmetries, and extra dimensions of space, for example in a SUGRA in 9 + 1 dimensions for the five theories of superstrings in 10D, or in a SUGRA in 10 + 1 dimensions for theory M in 11D. There is also talk of cuerdista cosmology when phenomenological models with extra dimensions (such as the ecpirotic universe) are used, but these ideas are not relevant in the discussions about the problem of the cosmological constant in string theory.
The landscape is defined as the set of effective theories at low energy in 3 + 1 dimensions, also called empty, minimum or solutions in string theory, obtained by an adequate compactification of the extra dimensions of space -an element of the landscape corresponds to a certain set of quantum fields of gauge type in a space-time of constant curvature. It is often said that the landscape groups effective theories at low energy (infrared limit) that are compatible with a quantum gravitation at high energy (ultraviolet limit). I do not like this definition because implicitly many things are taken for granted; for example, that quantum gravitation is understood as string theory, that a supergravity is selected as a classical vacuum of string theory, that the extra dimensions of space have been compactified, etc. Moreover, an ultraviolet limit with supersymmetry is assumed, which must be broken before reaching the infrared limit, which is not supersymmetric (at least up to the scale of energy explored up to now).
The fundamental cosmological hypothesis in string theory is that our low-energy universe (described by the standard model of particle physics and the consensus cosmological model based on Einstein's gravitation) corresponds to a solution that is in the landscape. The usual thing is to support the Weinberg in the anthropic principle; With the eye of a good cubero the number of possible models for the universe in the landscape is estimated and a great value is obtained (a popular value is 10500) that it is difficult to conceive that a model of our universe will not be hidden among all of them .
However, there is no mathematical proof that the landscape exists, since the equation of string theory that these solutions solve is ignored; neither is a precise mathematical characterization of the landscape known. In the last 20 years only many reasonable conjectures have been proposed for the generic properties of the solutions of string theory that are in the landscape; in fact, there is not even an agreement among the theoretical physicists experts in string theory about these conjectures, hence the bitter debates that are generated in many scientific congresses. Unfortunately, for want of something else, these conjectures are the only guide for those who try to unveil the generic predictions of string theory. Fortunately, they are very productive, having generated a great stir after unveiling that seem to indicate that in the landscape there are no solutions with positive cosmological constant.
Vafa introduced the concept of swampland in 2005 (arXiv: hep-th / 0509212). It is often said that it groups effective low-energy theories that are incompatible with high-energy quantum gravitation. I do not like this definition because it suggests that it groups the quantum theories of fields that can not be described by string theory. Actually Vafa had in mind the effective infrared theories that can be described using the mathematical language of string theory (swampland only makes sense in the context of string theory), but they do not have a corresponding ultraviolet limit. to a supergravity (in 10D or in 11D). Its objective was to start the search for properties (conjectures) related to these effective theories that allow to determine if they belong to the landscape or to the swampland.
Before continuing, I would like to remind you that within the framework of string theory there are quantum gravitation theories, which are mathematically coherent, but which are not in the landscape. For example, the heterotic string theory SO (16) × SO (16), which is a chiral theory in 10D free of anomalies (doi: 10.1016 / 0370-2693 (86) 91524-8); the 3 + 1 compactifications of this theory are in the swampland, outside the landscape, because this theory does not have supersymmetry (it does not need it to eliminate the problem of tachyons), then it does not contain a supergravity in 10D; By the way, this theory predicts a positive cosmological constant (doi: 10.1016 / S0370-2693 (97) 01172-6). There are also supercritical string theories in more than ten dimensions whose compactifications are outside the landscape and which predict a positive cosmological constant (doi: 10.1007 / 978-94-010-0211-0_32). Many examples could be given, but all are little studied; One of the objectives of Vafa is to provoke the youngest to study these alternative cuerdistas al landscape.
During the first 30 years of string theory the objective was to develop models without a cosmological constant (with a Minkowski-type spacetime in the infrared). The discovery in 1998 of the acceleration of the cosmic expansion caused by the so-called dark energy, which is compatible with a positive cosmological constant, caught the string theorists by surprise. Not only because its value is extremely small, Λ ~ 3 × 10-122 in Planck units, which requires a huge fine adjustment, but because its sign was positive, Λ> 0. As the most powerful string idea after the second revolution, the Adda / CFT correspondence of Maldacena, in natural string theory, Λ <0 (a AdS type spacetime). What seemed like a great advantage of string theory (in any other theory you can blithely change the sign of the cosmological constant without anything substantial happening) is becoming a nightmare.
The simplest solution for dark energy in string theory is the quintessence, that is, a dynamic model based on a scalar field that stabilizes at a certain certain minimum where its value equals the cosmological constant (V ~ 10-122 ), but a fine adjustment is required. Moreover, said scalar field will be coupled to the rest of the fields, with which it will cause observable effects at astrophysical distances, and even small variations of the fine structure constant and other couplings between fields. To avoid these unobserved effects, we must introduce a second fine adjustment in the scalar potential, requiring that its spatial gradient be | ∇V | ≤ 10-122 (in Planck units). Only this double fine-tuning is avoided if a model in string theory is used that naturally predicts that | ∇V | ~ V. Vafa is one of the leaders in this approach, but in the last 20 years it was considered unnecessary since there was an alternative, the KKLT model.
The KKLT solution (by Shamit Kachru, Renata Kallosh, Andrei Linde and Sandip P. Trivedi) was proposed in 2003 (10.1103 / PhysRevD.68.046005) to incorporate a metastable dS spacetime in type IIB string theory. Simplifying a lot, we start from a spacetime AdS4 stabilized adequately at a minimum of the potential with V <0; the idea is to "lift" the potential by adding an anti-D3-brane so that the potential happens to have a minimum V> 0, with V ≈0; thus the space AdS4 has been transformed into a space dS4; but the result is a metastable Sitter with broken supersymmetry because of the anti-D3-brane; adjusting everything so that the anti-D3-brane is also metastable, it manages to decay spontaneously in a vacuum at low energy with supersymmetry recovered. Thus at low energy we obtain a positive cosmological constant, very small, with an almost constant value. This solution was accepted by most string theorists until last year, when critical voices began to flourish.
On the one hand, as Thomas Van Riet (Univ. Catholic of Louvain (KU Leuven), Bégica) and Ulf H. Danielsson (Univ. Of Uppsala, Sweden) tell us, the KKLT solution is the product of a circular argument. We observe a Sitter-type universe, we want string theory to be the theory that describes it, so we conclude that string theory must contain Sitter-type solutions. So we look for a solution fitted with pins that conceals under the carpet the problem that there is no known natural solution that presents a vacuum in 4D that is a type of Sitter. According to Van Riet we must accept a paradigm shift in how to deal with dark energy in string theory; You have to explore more exotic string theories, even those that were discarded when you detested that Λ ≠ 0 and the goal was to describe a universe with Λ = 0. Rethinking the past with new ideas seems a fruitful way.
And on the other hand, as Vafa and his colleagues tell us there are conjectures to differentiate the landscape of the swampland that imply that the landscape does not contain any dS type solution. His argument is based on generalizing the theorem no go of Maldacena and Nuñez in 2001 (10.1142 / S0217751X01003937) to conclude that the supersymmetry at low energy requires that every potential field V (φ) ≤ 0 at its minimum, leading to AdS spaces. Thus it is conjectured that, in the landscape, all scalar potential satisfies that | ∇V | ≥ c · V, with c a constant of the order of the unit. This condition is trivial for V <0 (for AdS solutions) and is fulfilled by all known supersymmetric solutions with Λ = 0; but this condition radically rules out the dS space in the landscape. In fact, Vafa has challenged the string community to find a counterexample.
The work of Vafa and his colleagues on this new conjecture goes further, obtaining an estimate of the constant c in some cases. For compacting AdS4 × S7, we have | ∇V | / V = c = 6 / √14 ~ 1.6; generalizing this dimension, it is conjectured that for all compactification of the theory M that reduces the 11 dimensions to only d dimensions, it will be fulfilled that | ∇V | / V ≥ c = 6 / √ (d-2) (11-d), which for d = 4 offers c = 6 / √14 ~ 1.6. Note that it is also true in certain regions of the swampland outside the landscape; for example, for the heterotic theory SO (16) × SO (16) we obtain | ∇V | / V = c = 5 / √2 ~ 3.5.
The new conjecture in the swampland is added to many others (as there are no stable AdS vacuums without supersymmetry in the landscape, or that there are no gaps with global symmetries, only with gauge symmetries, or that the conjecture of the weak gravity is always fulfilled , etc.) that are helping to explore the landscape from new approaches. Let me highlight the optimistic path that Luis E. Ibáñez (Univ. Autónoma de Madrid, Spain) is leading, which together with several colleagues is using these conjectures to make predictions of string theory for the particles of the standard model. For example, if the dS type space is obtained by "elevation" of a stable AdS space, then the neutrinos can not be Majorana particles and have to be Dirac particles; Moreover, if the mass hierarchy of neutrinos is normal, the lightest neutrino has a mass between 6.7 meV <m (ν1) <7.7 meV, and if it is inverted, it has a mass between 2.1 meV <m (ν3) < 2.56 meV. It seems incredible that such precise estimates can be made using only reasonable conjectures in string theory (and with a simple process of counting and adjusting degrees of freedom). Estimates can also be made beyond the standard model, for example, for the mass of sterile neutrinos and the axions, although the results are not as accurate.
In short, we are in an exciting moment in string theory. Maybe on the threshold of the third revolution. It seems clear that there are no gaps (or solutions) with a Sitter-type spacetime in the landscape. It is not a problem but an opportunity. Guided by Vafa, the new guru of string theory, we must begin to reject certain prejudices that bias the exploration of the landscape. We must go further, with an open mind to string theories that were discarded decades ago. You have to remove the earmuffs that force us to see only the path that crosses the landscape. We can enjoy much more, because the mathematical formalism of string theory is very powerful and offers many alternatives. The solution to the problem of dark energy in string theory requires new approaches. And they will surely appear in the coming years. Theoretical physics is still fascinating!