I've known Jonathan Heckman as a brilliant Harvard student – I think that he was still an undergrad when he was greatly contributing to papers with Vafa and others. And it's great to see this UPenn assistant professor (the same state where Ashtekar works at PSU – my point is that the string theorists could potentially face some extra friendly hostility in Pennsylvania) as a senior co-author of a provoking paper:
It's a very novel scenario to explain the small cosmological constant – or dark energy or, even more generally, the accelerated expansion of the Universe. You must have seen many papers written by authors with great egos. Their lists of references – especially if they're full-blown crackpots – often look like this: [1] I, [2] I, [3] I, I am so great, and so on. ;-)
Jonathan is different. So their paper has 227 references and the most important one, one by the same set of authors minus Sakstein from November 2018, F-theory And Dark Energy, is the reference number 160 in the new paper. :-) That's what I call a quantification of humility.
OK, Heckman and co-authors ignore the "anthropic" landscape of flux vacua but also Vafa's swampland-justified quintessence scenario, as well as K-essence and infrared modifications of gravity. Instead, they recycle some old ideas, including those by Witten, and add some cute and cutting-edge F-theory twists.
Recall that in the Planck units, the cosmological constant (the apparent vacuum energy density with the negative pressure \(p=-\rho\) that makes the expansion of the Universe accelerate) is tiny. In the Planck units, it may be estimated as\[
\Lambda \approx 10^{-120}.
\] Well, the exponent is around \(122-123\) but let's not be picky and use their units in which \(120=122\). :-D This tiny cosmological constant is not only unnatural but within quantum field theory, it's also unstable with respect to the quantum corrections. The second problem is solved by Heckman et al. by deriving the cosmological constant as\[
\Lambda=\frac{1}{N}, \quad N\approx 10^{120}
\] the inverse of an integer! The integer \(N\) is referred to as the number of pixels. Because \(N\) is integer-valued and discrete, \(\Lambda\) is discrete as well. So it can't continuously drift anywhere. Great. Why would the cosmological constant be the inverse of an integer?
I don't claim to understand their derivations so far which means that what I am writing isn't really a guaranteed result of my rediscovery or verification. Instead, I am mostly parroting pieces of information in their papers that caught my attention.
The number of pixels \(N\) is integer because the pixels are really fivebrane instantons wrapped on meta-stable six-cycles in a string compactification. Geometrically, the pixels are literally points in \(S^3\), the spatial part of the geometry of the Universe, and you may see Figure 2 with a sphere \(S^2\) which is caricature of an \(S^3\) and is equipped with lots of pictures.
Because they treat the \(S^3\) spatial part of the Universe separately from time, they break the Lorentz symmetry. If that breaking were of order 100%, it would be lethal, and I would throw away the paper immediately. So I sincerely hope that while they admit that the Lorentz symmetry is broken, they have some reason to believe that the breaking is small if \(N\) is large – similarly small as the cosmological constant itself.
Now, their paper also claims to be a realization of the old well-known numerical coincidence that has also been discussed in some TRF blog posts,\[
\Delta m \sim \sqrt{M_{IR} M_{UV}}.
\] There is some IR cutoff given by the cosmological constant, \(\rho_{vac}\sim M_{IR}^4\) (the mass scale is roughly the inverse millimeter), and the UV cutoff, \(M_{UV}\sim M_{\rm Planck,4}\). If you take the geometric average of the two, of \(10^{19}\GeV\) and \(10^{-13}\GeV\), you will get close to \(10^{3}\GeV\), close to the energies probed by the LHC or FCC, which are their plausible superpartner masses.
It has never been clear why the superpartner mass differences should depend on the vacuum energy in this way – the normal default expectation of quantum field theory is a much larger, parameterically larger, vacuum energy density as a function of the superpartner masses – and it is not even clear to me whether they really claim to derive this miraculously numerological (parameteric) relationship from their string theory picture. If they did, it would be great.
What is their stringy compactification?
It is some F-theory – well, Heckman has worked on F-theory since his kindergarten years. Recall that "F" in F-theory stands for "father" or "Vafa" (more precisely "Fafa"), the surname of the father of F-theory. ;-) F-theory is naively a 12-dimensional theory within the general string theory landscape, just like M-theory ("M" stands for "mother" or upside-down "Witten") is an 11-dimensional vacuum.
However, unlike M-theory, F-theory doesn't want to keep its 12 dimensions decompactified, so it normally loves to compactify two of its dimensions on a formally infinitesimal 2-torus of an unclear but uniform signature (which is not unexpected, given it's infinitesimal). For this reason, what is left are 10 dimensions, or 9+1 dimensions, of type IIB string theory and the shape (complex structure) of the 2-toroidal fiber encodes the dilaton and RR-axion at the given point of the 10-dimensional base.
F-theory is therefore a fancy way to describe type IIB string theory vacua where the dilaton-axion may be spacetime-dependent and may even produce special points in 10 dimensions with monodromies. However, the main point of this exercise is that you still want to imagine that the whole spacetime is 12-dimensional except that the 12-dimensional geometry is a fiber bundle with \(T^2\) fibers.
Such 12-dimensional geometry may be compactified on some manifolds to produce semi-realistic four-dimensional vacua. To compare, heterotic string theory has 16 supercharges in \(D=10\) and you reduce the number to \(1/4\) i.e. 4 supercharges if you compactify the heterotic string theory on a manifold of the \(SU(3)\) holonomy, the Calabi-Yau manifold. The generic holonomy of six-real-dimensional manifolds is \(SU(4)\sim SO(6)\) and \(SU(3)\) is a subgroup of \(SU(4)\) that allows 3 of the 4 spinor components to have coitus with each other (rotate and twist into each other) while 1 component is protected – and that's the unbroken supercharge.
Similarly, the 32 supercharges of the \(D=11\) M-theory may be reduced to semi-realistic four supercharges by compactifying M-theory on a seven-real-dimensional manifold of \(G_2\) holonomy, note that \(11-7=4\). \(G_2\) is a subgroup of \(Spin(7)\) that preserves one component of the single 8-dimensional spinor \(s\) of \(Spin(7)\) or, equivalently, that preserves the 3-form \(f_{abc}=\bar s \gamma_{abc} s\) encoding the multiplication table for seven imaginary octonions.
(A reader who has a psychological problem with the existence of any supersymmetry in Nature is encouraged to stop reading and splash herself into the toilet.)
Now, F-theory also has 32 supercharges in 12 dimensions and you need to reduce it to 4 supercharges in \(D=4\). The normal holonomy of the manifold that achieves it is \(SU(4)\). Manifolds of \(SU(4)\) holonomy are also "Calabi-Yau manifolds of some type", namely four-folds. And manifolds of this holonomy which directly generalize the \(SU(3)\) holonomy manifold are the conventional framework that is being assumed in various types of F-theory phenomenology, including KKLT "anthropic" flux vacua as well as some Vafa-Heckman "non-anthropic" F-theory phenomenology.
But real-eight-dimensional manifolds may also have another cool holonomy group, namely \(Spin(7)\). That denotes manifolds on which the parallel transports around any curves induce a rotation on the tangent space that are a priori in \(Spin(8)\) but due to the special character of the manifold, they happen to be confined in \(Spin(7)\) – rotating just 7 out of 8 components of the 8-dimensional spinor of the tangential \(Spin(8)\) group!
Because there is a natural embedding \(SU(4)\subseteq Spin(7)\), the manifolds of the \(SU(4)\) holonomy may be considered a special subclass of those with the \(Spin(7)\) holonomy. By pure counting, the \(Spin(7)\) holonomy manifold only preserve 1/2 of the supercharges that the \(SU(4)\) holonomy manifolds do. They preserve 1 component of 8 in a chiral 8-dimensional spinor – but preserve nothing from the spinor of the opposite chirality.
For this reason, these \(Spin(7)\) compactifications may be creatively called \(\NNN=1/2\) vacua - with one-half of the supersymmetry – and that's exactly the meme that Heckman et al. have promoted since their November 2018 paper, too. What can it mean when the number of "full spinors of supercharges" is one-half of the minimal allowed positive package? ;-)
It is a strange situation and needs to be understood properly. Because I know how confusing this thing is for me, I want to believe that Jonathan and pals understand what's going on much more than I do. At any rate, the vacuum of the \(\NNN=1/2\) vacua behaves like \(\NNN=1\) but all the excited states behave like \(\NNN=0\). In some sense that may be made more rigorous, I hope, the "halved" number of supercharges means that \(\NNN=1/2\) is the geometric average between the number of supersymmetries in the vacuum and outside the vacuum. The vacuum respects the supersymmetry but the excited states don't.
The answer to the question "is there minimal supersymmetry in Nature?" could therefore be "one-half of it". ;-)
Great. They surely use these surprising statements in their construction so if you want to follow their paper, one of your homework exercises is to understand how this stuff may possibly work. I think that they also claim to bring a specific realization of some papers by Witten from the mid 1990s – in which the seemingly non-supersymmetric \(D=4\) vacuum is a manifestation of some underlying supersymmetric \(D=3\) vacuum.
On top of that, Heckman et al. use the pixels. Type IIB string theory has 6 compactified dimensions. Add the 2 dimensions of the toroidal fiber on top of that. It's enough dimensions to consider wrapped fivebranes. They wrap fivebranes on metastable six-dimensional cycles. At least if I understood some sentences in the November paper correctly, they don't mean "static fivebranes". Instead, they mean some fivebrane instantons (localized in the Euclidean time, like any instantons) that are wrapped on some six-cycles in the geometry.
Great. To verify whether this picture may work, you need to understand the bizarre SUSY properties of the \(Spin(7)\) holonomy manifolds, fivebrane instantons wrapped on six-dimensional cycles, the reason why they should behave as pixels and why they should count the inverse of the cosmological constant, and add some "stiff fluid" with \(p=+\rho\) (the equation of state for a black hole gas) on top of the cosmological constant with \(p=-\rho\).
When you understand these things and their particular union used by Heckman et al., you may be competent in judging whether this picture may work. There are too many things whose details are not comprehensible to me at this moment. I don't understand whether the metastable six-cycles are generically there in the compactification, whether the wrapped fivebranes are something you can't avoid (and there's some proof of it) or something that they artificially hypothesized. I don't understand how the number of pixels is constant or non-constant (because \(N\) just counts instantons, there is no reason for \(N\) to be conserved or even well-defined at one moment of time, right?), why they're related to the dark energy at all, and other things.
This conglomerate of cool ideas is obviously more esoteric and surprising than the KKLT construction, for example. The KKLT construction was "trivial to understand" qualitatively. Fluxes create the discretuum of AdS vacua and some wrapped D3-branes are believed to turn them to dS vacua. All these steps have a straightforward interpretation in terms of the 4-dimensional effective field theory. These KKLT steps are so "isolated" that the construction looks "man-made" and therefore unnatural.
Heckman et al. are proposing a more surrealist or dadaist construction that is also composed of many ideas but they're harder to understand. But maybe if you understand them as well as they do, you will see that they're more unavoidable and natural than e.g. the KKLT construction. I am not that far at this point, however. Nevertheless, they claim to combine many pieces that many of us have considered "cute observations" that seem to be "relevant for the cosmological constant problem". Whether the conglomerate is coherent remains to be seen – at least by non-authors of these Pennsylvanian papers.
If Heckman et al. have merged them to create a picture that really makes sense and may be considered a competitor of the KKLT or stringy quintessence models, that would be great, indeed. Even if that papers passed some tests, it would still look surprising why exactly this collection of ideas should be crucial for the understanding of the cosmological constant problem. Fivebrane instantons as pixels with some stiff fluid. Great. Why not some other combination of \(p\)-brane instantons or non-instantons and some other condensed matter-like phenomenon? If the construction were really the "right solution", we should ultimately have some derivation of the picture from the first principle that makes the steps look unavoidable.
Pixelated Dark Energy.They did very well in the speed contest. By several milliseconds, they have beaten two competitors who also submitted the paper at 19:00:03 UTC :-) so their paper appeared at the top of the hep-th listings today.
It's a very novel scenario to explain the small cosmological constant – or dark energy or, even more generally, the accelerated expansion of the Universe. You must have seen many papers written by authors with great egos. Their lists of references – especially if they're full-blown crackpots – often look like this: [1] I, [2] I, [3] I, I am so great, and so on. ;-)
Jonathan is different. So their paper has 227 references and the most important one, one by the same set of authors minus Sakstein from November 2018, F-theory And Dark Energy, is the reference number 160 in the new paper. :-) That's what I call a quantification of humility.
OK, Heckman and co-authors ignore the "anthropic" landscape of flux vacua but also Vafa's swampland-justified quintessence scenario, as well as K-essence and infrared modifications of gravity. Instead, they recycle some old ideas, including those by Witten, and add some cute and cutting-edge F-theory twists.
Recall that in the Planck units, the cosmological constant (the apparent vacuum energy density with the negative pressure \(p=-\rho\) that makes the expansion of the Universe accelerate) is tiny. In the Planck units, it may be estimated as\[
\Lambda \approx 10^{-120}.
\] Well, the exponent is around \(122-123\) but let's not be picky and use their units in which \(120=122\). :-D This tiny cosmological constant is not only unnatural but within quantum field theory, it's also unstable with respect to the quantum corrections. The second problem is solved by Heckman et al. by deriving the cosmological constant as\[
\Lambda=\frac{1}{N}, \quad N\approx 10^{120}
\] the inverse of an integer! The integer \(N\) is referred to as the number of pixels. Because \(N\) is integer-valued and discrete, \(\Lambda\) is discrete as well. So it can't continuously drift anywhere. Great. Why would the cosmological constant be the inverse of an integer?
I don't claim to understand their derivations so far which means that what I am writing isn't really a guaranteed result of my rediscovery or verification. Instead, I am mostly parroting pieces of information in their papers that caught my attention.
The number of pixels \(N\) is integer because the pixels are really fivebrane instantons wrapped on meta-stable six-cycles in a string compactification. Geometrically, the pixels are literally points in \(S^3\), the spatial part of the geometry of the Universe, and you may see Figure 2 with a sphere \(S^2\) which is caricature of an \(S^3\) and is equipped with lots of pictures.
Because they treat the \(S^3\) spatial part of the Universe separately from time, they break the Lorentz symmetry. If that breaking were of order 100%, it would be lethal, and I would throw away the paper immediately. So I sincerely hope that while they admit that the Lorentz symmetry is broken, they have some reason to believe that the breaking is small if \(N\) is large – similarly small as the cosmological constant itself.
Now, their paper also claims to be a realization of the old well-known numerical coincidence that has also been discussed in some TRF blog posts,\[
\Delta m \sim \sqrt{M_{IR} M_{UV}}.
\] There is some IR cutoff given by the cosmological constant, \(\rho_{vac}\sim M_{IR}^4\) (the mass scale is roughly the inverse millimeter), and the UV cutoff, \(M_{UV}\sim M_{\rm Planck,4}\). If you take the geometric average of the two, of \(10^{19}\GeV\) and \(10^{-13}\GeV\), you will get close to \(10^{3}\GeV\), close to the energies probed by the LHC or FCC, which are their plausible superpartner masses.
It has never been clear why the superpartner mass differences should depend on the vacuum energy in this way – the normal default expectation of quantum field theory is a much larger, parameterically larger, vacuum energy density as a function of the superpartner masses – and it is not even clear to me whether they really claim to derive this miraculously numerological (parameteric) relationship from their string theory picture. If they did, it would be great.
What is their stringy compactification?
It is some F-theory – well, Heckman has worked on F-theory since his kindergarten years. Recall that "F" in F-theory stands for "father" or "Vafa" (more precisely "Fafa"), the surname of the father of F-theory. ;-) F-theory is naively a 12-dimensional theory within the general string theory landscape, just like M-theory ("M" stands for "mother" or upside-down "Witten") is an 11-dimensional vacuum.
However, unlike M-theory, F-theory doesn't want to keep its 12 dimensions decompactified, so it normally loves to compactify two of its dimensions on a formally infinitesimal 2-torus of an unclear but uniform signature (which is not unexpected, given it's infinitesimal). For this reason, what is left are 10 dimensions, or 9+1 dimensions, of type IIB string theory and the shape (complex structure) of the 2-toroidal fiber encodes the dilaton and RR-axion at the given point of the 10-dimensional base.
F-theory is therefore a fancy way to describe type IIB string theory vacua where the dilaton-axion may be spacetime-dependent and may even produce special points in 10 dimensions with monodromies. However, the main point of this exercise is that you still want to imagine that the whole spacetime is 12-dimensional except that the 12-dimensional geometry is a fiber bundle with \(T^2\) fibers.
Such 12-dimensional geometry may be compactified on some manifolds to produce semi-realistic four-dimensional vacua. To compare, heterotic string theory has 16 supercharges in \(D=10\) and you reduce the number to \(1/4\) i.e. 4 supercharges if you compactify the heterotic string theory on a manifold of the \(SU(3)\) holonomy, the Calabi-Yau manifold. The generic holonomy of six-real-dimensional manifolds is \(SU(4)\sim SO(6)\) and \(SU(3)\) is a subgroup of \(SU(4)\) that allows 3 of the 4 spinor components to have coitus with each other (rotate and twist into each other) while 1 component is protected – and that's the unbroken supercharge.
Similarly, the 32 supercharges of the \(D=11\) M-theory may be reduced to semi-realistic four supercharges by compactifying M-theory on a seven-real-dimensional manifold of \(G_2\) holonomy, note that \(11-7=4\). \(G_2\) is a subgroup of \(Spin(7)\) that preserves one component of the single 8-dimensional spinor \(s\) of \(Spin(7)\) or, equivalently, that preserves the 3-form \(f_{abc}=\bar s \gamma_{abc} s\) encoding the multiplication table for seven imaginary octonions.
(A reader who has a psychological problem with the existence of any supersymmetry in Nature is encouraged to stop reading and splash herself into the toilet.)
Now, F-theory also has 32 supercharges in 12 dimensions and you need to reduce it to 4 supercharges in \(D=4\). The normal holonomy of the manifold that achieves it is \(SU(4)\). Manifolds of \(SU(4)\) holonomy are also "Calabi-Yau manifolds of some type", namely four-folds. And manifolds of this holonomy which directly generalize the \(SU(3)\) holonomy manifold are the conventional framework that is being assumed in various types of F-theory phenomenology, including KKLT "anthropic" flux vacua as well as some Vafa-Heckman "non-anthropic" F-theory phenomenology.
But real-eight-dimensional manifolds may also have another cool holonomy group, namely \(Spin(7)\). That denotes manifolds on which the parallel transports around any curves induce a rotation on the tangent space that are a priori in \(Spin(8)\) but due to the special character of the manifold, they happen to be confined in \(Spin(7)\) – rotating just 7 out of 8 components of the 8-dimensional spinor of the tangential \(Spin(8)\) group!
Because there is a natural embedding \(SU(4)\subseteq Spin(7)\), the manifolds of the \(SU(4)\) holonomy may be considered a special subclass of those with the \(Spin(7)\) holonomy. By pure counting, the \(Spin(7)\) holonomy manifold only preserve 1/2 of the supercharges that the \(SU(4)\) holonomy manifolds do. They preserve 1 component of 8 in a chiral 8-dimensional spinor – but preserve nothing from the spinor of the opposite chirality.
For this reason, these \(Spin(7)\) compactifications may be creatively called \(\NNN=1/2\) vacua - with one-half of the supersymmetry – and that's exactly the meme that Heckman et al. have promoted since their November 2018 paper, too. What can it mean when the number of "full spinors of supercharges" is one-half of the minimal allowed positive package? ;-)
It is a strange situation and needs to be understood properly. Because I know how confusing this thing is for me, I want to believe that Jonathan and pals understand what's going on much more than I do. At any rate, the vacuum of the \(\NNN=1/2\) vacua behaves like \(\NNN=1\) but all the excited states behave like \(\NNN=0\). In some sense that may be made more rigorous, I hope, the "halved" number of supercharges means that \(\NNN=1/2\) is the geometric average between the number of supersymmetries in the vacuum and outside the vacuum. The vacuum respects the supersymmetry but the excited states don't.
The answer to the question "is there minimal supersymmetry in Nature?" could therefore be "one-half of it". ;-)
Great. They surely use these surprising statements in their construction so if you want to follow their paper, one of your homework exercises is to understand how this stuff may possibly work. I think that they also claim to bring a specific realization of some papers by Witten from the mid 1990s – in which the seemingly non-supersymmetric \(D=4\) vacuum is a manifestation of some underlying supersymmetric \(D=3\) vacuum.
On top of that, Heckman et al. use the pixels. Type IIB string theory has 6 compactified dimensions. Add the 2 dimensions of the toroidal fiber on top of that. It's enough dimensions to consider wrapped fivebranes. They wrap fivebranes on metastable six-dimensional cycles. At least if I understood some sentences in the November paper correctly, they don't mean "static fivebranes". Instead, they mean some fivebrane instantons (localized in the Euclidean time, like any instantons) that are wrapped on some six-cycles in the geometry.
Great. To verify whether this picture may work, you need to understand the bizarre SUSY properties of the \(Spin(7)\) holonomy manifolds, fivebrane instantons wrapped on six-dimensional cycles, the reason why they should behave as pixels and why they should count the inverse of the cosmological constant, and add some "stiff fluid" with \(p=+\rho\) (the equation of state for a black hole gas) on top of the cosmological constant with \(p=-\rho\).
When you understand these things and their particular union used by Heckman et al., you may be competent in judging whether this picture may work. There are too many things whose details are not comprehensible to me at this moment. I don't understand whether the metastable six-cycles are generically there in the compactification, whether the wrapped fivebranes are something you can't avoid (and there's some proof of it) or something that they artificially hypothesized. I don't understand how the number of pixels is constant or non-constant (because \(N\) just counts instantons, there is no reason for \(N\) to be conserved or even well-defined at one moment of time, right?), why they're related to the dark energy at all, and other things.
This conglomerate of cool ideas is obviously more esoteric and surprising than the KKLT construction, for example. The KKLT construction was "trivial to understand" qualitatively. Fluxes create the discretuum of AdS vacua and some wrapped D3-branes are believed to turn them to dS vacua. All these steps have a straightforward interpretation in terms of the 4-dimensional effective field theory. These KKLT steps are so "isolated" that the construction looks "man-made" and therefore unnatural.
Heckman et al. are proposing a more surrealist or dadaist construction that is also composed of many ideas but they're harder to understand. But maybe if you understand them as well as they do, you will see that they're more unavoidable and natural than e.g. the KKLT construction. I am not that far at this point, however. Nevertheless, they claim to combine many pieces that many of us have considered "cute observations" that seem to be "relevant for the cosmological constant problem". Whether the conglomerate is coherent remains to be seen – at least by non-authors of these Pennsylvanian papers.
If Heckman et al. have merged them to create a picture that really makes sense and may be considered a competitor of the KKLT or stringy quintessence models, that would be great, indeed. Even if that papers passed some tests, it would still look surprising why exactly this collection of ideas should be crucial for the understanding of the cosmological constant problem. Fivebrane instantons as pixels with some stiff fluid. Great. Why not some other combination of \(p\)-brane instantons or non-instantons and some other condensed matter-like phenomenon? If the construction were really the "right solution", we should ultimately have some derivation of the picture from the first principle that makes the steps look unavoidable.
Cosmological constant from pixels
![Cosmological constant from pixels]()
Reviewed by MCH
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January 31, 2019
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