LOG#211. Understanding gravity (I).

Hi, everywhere! Have you missed me?

A little short thread begins. This thread is based on some Nima’s talk on the understanding on gravity. Do we understand gravity? Classically, yes. Quantumly, it is not so easy. String theory is the most successful approach, but canonical quantum gravity or loop quantum gravity is interesting as well, despite some hatred and pessimist reactions to this conservative alternative to strings. Even more recently, two papers appear suggestion a deeper relation between these two approaches, sometimes considered far away from each other…

Nima Arkani-Hamed is a powerful theoretical physicist. In the last years, he has fueled some insights on string theory and quantum field theory pondering about the nature of the ultimate physical theory, the theory of everything, and quantum gravity. What is he working out? A summary:

  • General relativity (GR) and Yang-Mills theory are inevitable and, in the end, they must “merge” or “emerge” somehow from the microworld. Frank Wilczek told me once on twitter, that GR is strikingly similar to some non-linear sigma models known in quantum field theory but not just “the same thing”.
  • The cosmological constant problem and the epic fail of quantum field theory to guess a reasonable value of it (compared with the observed value), and the most certain fail of modified IR gravity theories (till now), as solution. Even MOND (MOdified Newtonian Dynamics is unclear on this stuff) is not accepted as plain solution despite the absence of positive finding of dark matter, so its surprisingly precise fit to galactic scales (not so good at bigger scales) remains as a mystery and issue.
  • Massive gravity theories and related issues with higher derivative theories (Fierz-Pauli, DGP, galileon theories, TeVeS, ghost-free higher derivative theories,…). Experimentally, they are not supported and yet they have been evolving in this experimental era. Likely, the rediscovery of certain ghost-free higher derivative theories, and new BH solutions with scalar hair have rebooted the interest in these theories, again, despite the lack of experimental support, what is not bad since we are limiting the space of available theories to fit the data.
  • Horizon thermodynamics and violent crashes with non-locality. The black hole information problem remains after more than 40 years of the discovery of Hawking radiation (R.I.P., in memoriam). Are black hole horizons apparent as recent ideas claim or are they real? Reality is a powerful but likely relative word.
  • Inflation expectations from the CMB observations. In the next years, the hope to observe the B-mode of the inflationary phase of the Universe in the Cosmic Microwave Background has grown. Living with eternal inflation would push us to adopt the Multiverse scenario, something many people seem to see as cumbersome, but others see it as natural as the solution to the Quantum Mechanics interpretation problem.
  • The end of space-time. Particle physicists are familiar with the idea of unstable or metastable particles. Even the proton would be unstable in many Grand Unified Theories (GUT) and the TOE. Hawking radiation is important because it also would imply that not only black holes, the space-time itself could end or evaporate into “something” or into nothing! Of course, this seems crazy. Accepted known Quantum Mechanics only allows for unitary evolution, allowing something to disappear, would introduce the idea that space-time itself will decay or disappear in the far future. We want to know in such a case why or not the spacetime disappear, and what is the final destiny of the Universe and its spacetime. We don’t know that, as we do not know what happened in the Planck era or before. That is important,…Not for us, but the destiny of our species depends on it. It is a thing that will be important for our descendants unless you think we will not survive the forthcoming crisis.
  • Physics with no space-time. Can we formulate physical theories without any reference to space-time at all. How could such theories look like?
  • Surprising new mathematical structures arising in quantum field theories and general relativity, specially those arising from algebraic geometry and/or combinatorial geometry. Physics has always been boosted by advances in mathematics. It is very likely the next big revolution in physics will happen when new mathematical structures and methods emerge from some sides. In QFT, the polyzeta and polylogarithms have appeared as wonderful recipes to some closed formulae problems. In general relativity and supergravity or superstrings, the duality and the brane revolution are pushing towards a categorical and algebraic differential (likely differentigral in near future) structures yet to be fully understood, and likely appreciated, by the academical community.

Where are we now? It is hard to say. The Heisenberg uncertainty principle, as you know, surely implies that there is something like an “ultimate microscope”. \Delta E\sim 1/\Delta t means that, eventually, as you put more and more energy into a single “point”, you would create a black hole. No distance or time, based on usual or commonly accepted riemannian (and likely differential!), can be have any sense when you reach the fantastic ultrashort size of 10^{-35}m, or the time scale of 10^{-43}s. No operational definition of space or time is available, in terms of conventional geometries, at those scales of distance and time. Volovich suggested, almost 30 years ago, in 1988, that the ultimate physical theory would be number theory. However, this statement has not been developed further, beyond some exotic researches on p-adic and adelic geometries that, however, are being growing in the last years. The point is, of course, where space-time notions end. There are two main places where physicists are for sure convinced they need something else to current theories: the Big Bang and the black hole singularities (and maybe, tangentially, at the black hole horizon and or those dark matter and dark energy stuff nobody understands, due to the fact we do not understand GRAVITY?).

Let me be more technical here. Quantum Field Theory (QFT for short from now), says that vacuum is not static, is also dynamical. It polarizes. So, every old grade course of electromagnetism is not completly fair when telling you vacuum does not polarize. Classically not! Quantumly, YES! Mathematically, QFT provides a recipe to calculate the effect of vacuum polarization through loop integrals in Feynman graphs that are mathematically evaluated into logarithmic integrals, WHENEVER you plug a regulator. That is, a high energy scale \Lambda_{UV} where the usually divergent integral gets regularized to provide a finite value. For one loop, momentum P, and supposing M_P^2 finite, you get something like this:

(1)   \begin{equation*}\dfrac{P^4}{M_P^2}\log \left(\dfrac{\Lambda_{UV}^2}{P^2}\right)\end{equation*}

The Planck mass squared comes from the two vertices from the loop, the 4th power comes from the edges and the logarithmic regulator of the squared cut-off appears due to finite expectations on general grounds. Actually, Higgs physics is important because Higgs mass is sensitive for such logarithmic corrections as there is NOTHING, absolutely nothing, in the Standard Model allowing the Higgs remains so light (125 GeV/c²) as we measure it! Note the similarity between this Higgs physics and the formula above. Any reasonable force, indeed, can be expanded in terms of energy scales in QFT. You would get:

(2)   \begin{equation*}f=F_0+\dfrac{1}{p^2}+\dfrac{1}{M_P^2}\log P^2+\dfrac{1}{M_P^2}\delta^3(r)+\ldots\end{equation*}

and where the divergent parts are usually neglected hoping that some further theoretical approach will teach us why they are certainly negligible under the floor…

Now, enter into the gravity realm. Some times is said that quantum gravity is “hard” or impossible. That is not exactly true or accurate. There are in fact some quantum gravity calculations available. For instance, the leading quantum correction to the newtonian force is provided by the following formula (up to some conventions with the numerical prefactors):

    \[F_q=\dfrac{GM_1M_2}{r^2}\left(1-\dfrac{27}{2\pi^2}\dfrac{G\hbar}{r^2c^2}+\cdots\right)\]

and you can see the classical newtonian force plus a leading order correction. I recommend you the papers by Bjerrum-Bohr on this subject. Then, you may ask, what is the problem? Well, the problem is…You can not do the above calculation for any LOOP order. Only certain theories, like maximal supergravity (due to its hidden and exotic Chern-Simons terms), and superstring theory (I am not sure with que Loop Quantum Gravity approach here), can, in principle provide a recipe to calculate a finite quantum gravity interaction between gravitons and gravitons and matter at any loop order. Why? The increasing number of Feynman diagrams and the mathematically complexity of non-linearity of GR makes the problem formidable. Likely, only a supercomputer or trained AI could manage to add all these diagrams in the most advanced theories and tell us if they are correct when contrasted with experimental data. But that is a future today.

Helicities of particles like gravitons or photons enter into the difficult calculations of quantum gravity or QFT. Locality, imposed by our preconceptions on space, time and field formulations due to Lorentz symmetry and causality, are solid in local QFT based on relativity and Quantum Mechanics (QM). For instance, a photon field can be coded into a vector field with polarization in form of plane waves:

    \[A_\mu=\varepsilon_\mu e^{ipx}\]

Transversality of the photon field, i.e.,

    \[\varepsilon_\mu\cdot p^\mu=0\]

plus gauge redundancy

    \[\varepsilon_ \mu\sim\varepsilon_\mu+\alpha p_\mu\]

    \[A_\mu\sim A_\mu+\partial_\mu\Lambda\]

implies a constraint

    \[\sum_i k_iP_i^\mu=0\]

for all equal k_i. That is, the equivalence principle somehow is telling us that, the whole structure of interactions, based on QM and special relativity, leads long distance interactions for spin 1 or spin 2 forces (electromagnetism and gravity!). Thus, it yields that, whatever the TOE is, relativity (special and general somehow) and QM (the SM somehow as well), they must emerge from it. In fact, the general structure of any Yang-Mills (YM) theory is pretty simple, it has an Y diagram form and three labels for any spin, s=0,1/2,1,3/2,2. We have found fundamental particles of spin 0, 1/2, 1 and 2 (0 with the Higgs, 2 if you count gravitational waves as gravitons). The only lacking fundamental particle is that with spin 3/2, a general prediction of supersymmetry (SUSY). Gravity is unique, at minimal sense, thanks to Einstein discovery of gravity as curved spacetime geometry. Of course, you could extend GR to include extra fields or gravitational degrees of freedom (massive gravitons, dilatons, graviphotons,…), but all of these have failed till now. SUSY has not been found, but it will be found in the future for sure. Black holes, indeed, in GR have some exotic supersymmetries, even in the simple Kerr case. That is not very known but it is true.

Moreover, move to the cosmological constant problem. Vacuum energy density problem if you prefer the alternative name for it. The fact that

    \[\Lambda_{observed}\sim 10^{-122}\Lambda_{theory}\]

have been known since the times where Einstein introduced the cosmological term. Worst, now we do know it is not zero, it makes the situation more unconfortable for many. Before 1998 you could simply argue that some unknown symmetry would erase the cosmological term of your gravitational theories. Now, you can not do that. Evidence is conclusive in which the cosmological term seems to be positive and non-null. Just having a theory, that allows for fantastically tiny cancellations, is just weird. Weirder if that cancellation is precise in 122 or lets say the falf 61 orders of magnitude. Such a fine tuning is disturbing. Ludicrous! Ridiculous. Anyway, this fact has not stopped theorists to make some guesses of how to life with that. One idea arised after the second string revolution, circa 1995. Brane worlds. Plug some damping curvature into the bulk of spacetime. The tension on a brane could just explain why gravity is weak, and maybe, explain why the cosmological constant is tiny. The problem is, that the mechanism is much more consistent with NEGATIVE cosmological constant. The DGP model gets 4D GR plus massless matter in an AdS (Anti-deSitter, negative cosmological constant Universe). In technical words, a negative tension provides a modified propagator in QFT solving the cosmological constant problem if this is negative (otherwise this blows up!). Another problem with brane worlds is that no causal modification can work on fields of the main brane Universe. Massive gravities, both in brane gravity and independent models, imply, on very general grounds, that long distance interactions should include SCALAR new degrees of freedom. Remember the Higgs mass problem I mentioned before? Well, you loose the control of scalars in theories without symmetries. In fact, the good and the evil of scalar degrees of freedom: at some point they introduce modifications or violations of either the Einstein equivalence problem (or some soft/hard variation), or the Lorentz symmetry behind it. Exciting news for experimentalists: you can seek these violations in designed experiments. Bad news for conservative theorists: nonlinear interactions can introduce conflicts with the usual conceptions and features of locality (even causality), thermodynamics or Lorentz symmetry expected from current well-tested theories. To save locality, the Einstein equivalence principle or usual properties of QM seems to be inevitable at some point (something also triggered by the yet unsolved black hole information paradox). Unless a loophole is found, it seems the combination of the current theories will imply that we must abandon some yet untouched principle. A toy model in DGP massive gravity, the so called galileon gravity lagrangian:

    \[L_s=\partial^2\phi+\dfrac{\partial^2\phi}{\Lambda^3}\square\phi+\cdots\]

It has a shift symmetry (galilean symmetry) given by transformations

    \[\partial_\mu  \phi\rightarrow \partial_\mu\phi+V_\mu\]

In the simplest Higgs phase of gravity, you would have a zero expectation value for the scalar field. However, radiative corrections to this vacuum are expected to arise. And we do NOT know how to handle with it.

20th century physics is made by the triumph of the current two pilar theories: relativity (special and general) and quantum mechanics. The apparent difficulties to get gravity into the quantum game is much a deep question, but it could allow us to explain the Big Bang, the Universe and the future of it. These difficulties, are also triggering doubts about the role and formulation of QM (annoying for many to accept, from philosophical reasons, more than experimental precision). Surely, quantum mechanics could be modified by a further TOE or GUT theory. However, it will remain true just as the Bohr toy model or newtonian mechanics. Why is our Universe big with big things in it? The reason is QM, or more precisely QFT. QFT=SR+QM. And it is true. Particles are classified as entities with mass (energy) and spin (times \hbar). Experimentally, with the exception of spin 3/2, we have found every particle (GW counted as graviton wavepackets) till spin 2 (spin zero is the Higgs particle). Why not spin 5/2 or 3? Well, there are higher spin theories. They have other issues, with locality or their definition as interacting field theories (excepting some special theories as those by Vasiliev). The simple Y form of Feynman graphs in known theories is particularly striking and simple (beyond some technicalities due to the well-defined processes of regularization and renormalization of physical quantities, that some one should study better at these crisis times?). However, the structure of the Y shape interaction of the Higgs field CANNOT be investigated in the LHC very well. It is a nasty hadron collider. We will need a linear or muon collider. Or a circular collider adjusted to the resonance of the Higgs particle to study his self-interactions (in the standar model, the Higgs potential is simply a cubic plus quartic potential). Otherwise, a 100TeV collider would be better for this as well. A 100 TeV collider would probe vacuum fluctuations of the Higgs field itself (or the muon collider or any other special collider tuned to the Higgs).

There are some critics towards the waste of money those machines would be. Or course, there is no guarantee that we will find something new. But the Higgs particle interactions MUST be studied precisely. The universe is surprisingly very close to Higgs field metastability. It is something that well deserve the money, perhaps, I would only complain about not doing this crazily. We need to plan the Higgs potential study further. However, note that the LHC is about 10TeV, the future colliders will be clean lepton (or photon!) colliders tuned to the Higgs resonance or 100TeV/1000TeV (the latter in my lifetime I wish to see it) and those energies are yet much smaller than Planck energy, 10^{16} TeV. Neutrino physics, gamma rays, likely X-rays and radio bursts or gravitational wave astronomy can probe surely strong gravity and extreme processes much better. For free. Of course, you have to be good enough to catch those phenomena and that, again, cost time and money.

String theory news…But firstly, a little bit history. Strings were discovered in the context of S-matrix theory and strong interactions. The Veneziano amplitude was the key to find out that string theory has something to do with the nuclear realm (despite this is surely ignored by quantum stringers right now, or not so appreciated as 30 years ago!). String amplitudes of four point particles has a simple structure:

(3)   \begin{equation*}A_s=\dfrac{<12>^4\left[34\right]^4}{stu}\mathcal{C}\end{equation*}

and the general amplitude takes the form

(4)   \begin{equation*}\mathcal{A}=\dfrac{<12>^4\left[34\right]^4}{stu}\Pi_{i=1}^\infty\dfrac{(s+i)(t+i)(u+i)}{(s-i)(t-i)(u-i)}=\dfrac{<12>^4\left[34\right]^4}{stu}\dfrac{\Gamma(-s)\Gamma(-t)\Gamma(-u)}{\Gamma(1+s)\Gamma(1+t)\Gamma(1+u)}\end{equation*}

Here, the s, t, u are variables encoding the energies of the colliding strings in certain frame, \Gamma is the gamma function of Euler, a generalization of the factorial function for real AND complex values. The numbers between brackets and powers are certain spinor/vector quantities coding helicities. Well, take weak coupled string theory amplitudes, and 4 point interactions at tree level (no loops in the classical sense), independently of compactification you can get a wonderful universal formulae for gravity and YM amplitudes:

1st. For gravity, you get:

(5)   \begin{equation*}\mathcal{A}_G=\dfrac{<12>^4\left[34\right]^4}{stu}\dfrac{\Gamma(-s)\Gamma(-t)\Gamma(-u)}{\Gamma(1+s)\Gamma(1+t)\Gamma(1+u)}\times \mathcal{K}\end{equation*}

where\mathcal{K} is

    \[\mathcal{K}=-\dfrac{1}{stu}\]

for Type (II) strings at the level of three gravitons interaction, and

    \[\mathcal{K}=-\dfrac{1}{stu}+\dfrac{1}{s(1+s)}\]

for heterotic strings at the level of 2 graviton-scalar interaction, and

    \[\mathcal{K}=-\dfrac{1}{stu}+\dfrac{8}{(1+s)s}-\dfrac{tu}{s(1+s)^2}\]

for bosonic strings at the level of 2 graviton-scalar interaction. Note the universal pole structure of the formulae above.

    \[\mathcal{C}=\dfrac{\Gamma(-s)\Gamma(-t)\Gamma(-u)}{\Gamma(1+s)\Gamma(1+t)\Gamma(1+u)}\]

2nd. YM theory in string theories have the following 4 point tree level amplitudes:

(6)   \begin{equation*}\mathcal{A}_G=<12>^2\left[34\right]^2\dfrac{\Gamma(1-s)\Gamma(1-t)}{\Gamma(1+s)\Gamma(1-s-t)}\times \mathcal{K}'\end{equation*}

where\mathcal{K}' is

    \[\mathcal{K}'=-\dfrac{1}{st}\]

for Type (I) strings at the level of three gravitons interaction, and

    \[\mathcal{K}=-\dfrac{1}{st}-\dfrac{u}{s(1+s)}\]

for bosonic strings at the level of 2 graviton-scalar interaction, and

    \[\mathcal{K}=-\dfrac{1}{stu}+\dfrac{8}{(1+s)s}-\dfrac{tu}{s(1+s)^2}\]

for bosonic strings at the level of 2 graviton-scalar interaction. Heterotic theories would have massive pole structures in the similar expression. The pole structure (beyond the massive corrections in the heteorica case) arises from

    \[\mathcal{C}'=\dfrac{\Gamma(-s)\Gamma(-t)}{\Gamma(1-s-t)}\]

These gamma functions are related to the Euler beta function and can be seen as generalizations of the famous Veneziano amplitude who gave birth to string theory in the 70s or the 20th century. In fact, the S-matrix programme can be approached for any mass and spin. The string theory “magic” procedures with only massless states (external) and playing with another interactions or deformations of the above formula give rise to some open problems (some of them, known from the old string theory times).

For external interactions of 3 point interactions, at tree level, you get an Y-amplitude

    \[g\varepsilon^{\mu_1\mu_2\cdots \mu_N}(p_1-p_2)_{\mu_1}\cdots (p_1-p_2)_{\mu_N}\]

For 4 point amplitudes Y+Y at tree level, you get

    \[\dfrac{gg'}{s-M^2}G_N^{(d)}\left[\cos\theta\right]\]

where G_N^{(d)}(\cos\theta) are Gegenbauer polynomials, G_0=1, G_1=x, G_2=dx^2-1, that arise in the expansion of the fraction (newtonian like force) in d-dimensions:

    \[\dfrac{1}{\vert z-r\vert^{d-2}}=\sum_N r^NG_N^{(d)})\left[\cos(\theta)\right]\]

These remarkable formulae link with the Veneziano amplitude:

(7)   \begin{equation*}A_V=\dfrac{\Gamma(-1-s)\Gamma(-1-t)}{\Gamma(-2-s-t)}\end{equation*}

It has a striking pole structure, with residua at s=-1, s=0, s=1,…, s=N; or equivalently at 1, t+2, (t+2)(t+3),...,(t+2)\cdots(t+N+2). It yields the residue at s=1 provides

    \[\mbox{Res}(t+2)(t+3)=\dfrac{25}{4}\left(\cos^2\theta-\dfrac{1}{25}\right)=\dfrac{25}{4}\left(\cos^2-\dfrac{1}{d}\right)+\dfrac{25}{4}\left(\dfrac{1}{d}-\dfrac{1}{25}\right)\]

This expresion is POSITIVE as you keep d\leq 25, a fact known from bosonic string theory (living in 25+1 spacetime). Similarly, the open superstring amplitude

    \[A(1^-2^-3^+4^+)=<12>^2<34>^2\dfrac{\Gamma(-s)\Gamma(-t)}{\Gamma(1-s-t)}\]

has an analogue residue at s=3 (levels 1,t+1,(t+1)(t+2),...,(1+t)\cdots(N-1+t) corresponding to s=1,..,s=N) and you get

    \[\mbox{Res}_{s=3}(t+1)(t+2)=\dfrac{9}{4}\left(\cos^2\theta-\dfrac{1}{9}\right)=\dfrac{9}{4}\left(\cos^2-\dfrac{1}{d}\right)+\dfrac{9}{4}\left(\dfrac{1}{d}-\dfrac{1}{9}\right)\]

that is OK iff d\leq 9. You can enforce positivity to every level as well! But a price is to ensure it. You have to pay a d=2 conformal setting (only known theory to do that is string theory!). If you want a ghost-free theory, positive amplitude theory, such as

    \[a=\sum_k c_k\cos (k\theta)\]

remains positive with c_k>0 you have to live in d=2 dimensions (string theory magic do that, in the abstract worldsheet!). However, this is hard statement as c_k become increasingly exponentially small, but adding a \cos \theta-1 factor makes it false! The positivity of these amplitudes and the analysis of the hidden symmetry structure of the string diagrams have revealed a “jewel” or hidden geometric object in quantum mechanics/string theory/quantum field theory. How? The residue structure of gravity amplitudes are correlated to open superstrings:

    \[ \mbox{Res}_N^{Gravity}(\cos\theta)=\left(\mbox{Res}_N^{OpenS}(\cos\theta)\right)^2\]

This is a new hint of the Gravity=YM^2 mantra of these times, but there is more. Departure of positivity seems to indicate non consistent theories. Positivity magic struggles with massive higher spin states, where problems really live too. What higher spin amplitudes should we include and what to exclude? Not easy task. The idea is that we should search for a way to understand higher spin amplitudes without the worldsheet picture as primary entity (that d=2 restriction is hard for practical purposes in the real world). There is another reason, and that is gravity. Gravity is, from certain viewpoint, more positive than open superstrings (note the power in the amplitude coming from the spinors/polarizations). Here it comes, the amplituhedron. What is the amplituhedron? Well, it is a new object encoding the positivity of amplitudes in QFT and string theory. A formal definition is something like this:

    \[\boxed{\mathcal{M}_{n,k,L}\left[Z_a\right]=\mbox{Vol}\left[\mathcal{A}_{n,k,L}\left[Z_a\right]\right]}\]

What is this? Well, roughly speaking, the amplituhedron is certain generalized polytope (higher dimensional polyhedron) in projective geometry (technically, a generalization of the positive grassmannian) such as its volume is the all-loop scattering amplitude of particle physics processes. There, n is the number of vertices (or particles) interaction, k is the plane dimensionally specifying the helicity structure of the particles, and L is the loop order. Certainly, that amplitudes are lower dimensionally projected shadows of higher dimensional, maybe discrete, structures is a powerful language. I will talk about the amplituhedra and related stuff in the next posts.

Finally, to end this long boring post, let me review some of the magic. The amplitude with no negative probability (positive residue!) for gravity and massive particles reads off:

(8)   \begin{equation*}\mathcal{A}=G_N\dfrac{<12>^4<34>^4}{stu}\dfrac{\Gamma(1-\frac{s}{M^2})\Gamma(1-\frac{t}{M^2})\Gamma(1-\frac{u}{M^2})}{\Gamma(1+\frac{s}{M^2})\Gamma(1+\frac{t}{M^2})\Gamma(1+\frac{u}{M^2})}\end{equation*}

However, emphasis is usally done in the spacetime picture of strings, instead of the amplitude structure inherited from S-matrices! The equation of a superstring is usally given by

    \[\partial_\tau^2 X^\mu(\sigma,\tau)-\partial_\sigma^2 X^\mu(\sigma,\tau)\]

The so-called Green-Schwarz theory contains a Super-Yang-Mills (supersymmetric extension of YM) theory with action

(9)   \begin{equation*}S=\int\left(-\dfrac{1}{4}Tr F^2+i\overline{\Psi} \Gamma \cdot D\Psi\right) dvol\end{equation*}

String theory has a problem. It yields to too many consistent vacua for the Universe. Our SM+GR world is only one betwen 10^{500} in general superstring models, or 10^{272000} F-theory (10+2) different universes. These are Universes very similar to ours (or different), differing in coupling constants and vacuum expectactions values no hint of how to selec. This is the string theory nasty trick. There is no adjustable parameter, but you are driven to accept there are plenty of Universes. To my knowledge, no one has even proved that our constants and field theory expectation values can be derived from one of those Universes in clear plain way. However, there are no too many consistent quantum theories of gravity in the market…

For instance, the (quantum) supermembrane allows you get a bosonic equation (free) of motion given by

(10)   \begin{equation*}\partial_i\left(\sqrt{-g}g^{ij}E^\mu_j\right)=0\end{equation*}

and where

    \[g_{ij}(X)=\eta_{\mu\nu}\partial_iX^\mu\partial_jX^\nu=E^\mu_iE^\nu_j\eta_{\mu\nu}\]

Going to superspace Z=(X,\theta) membranes, the supermembrane equation gets modified

(11)   \begin{equation*}\partial_i\left(\sqrt{-g}g^{ij}E^\mu_j\right)=\varepsilon^{ijk}E_i^\nu\partial_j\overline{\theta}\Gamma^\mu_\nu\partial_k\theta\end{equation*}

SUSY and coherence of the theory in minkovskian(lorentzian) signature force you to match bosonic and fermionic degrees of freedom for branes, i.e., N_B=N_F, such as for a p-brane in D-spacetime (generally lorentzian, giving up this allows you to go beyond eleven dimensions), you have:

    \[N_B=D-p-1\]

    \[N_F=\dfrac{MN}{4}\]

where M is the number of fermionic degrees of freedom, and N is the number of supersymmetries on the superspace target. From this simple equation, you can easily derive that

    \[D-p-1=\dfrac{MN}{4}\]

Take any 1-brane, so you get D-2=\dfrac{MN}{4}. If you impose N=1 SUSY, you get superstring theory with M=4 generators (fermionic d.o.f.) in D=3, you get superstring theory with M=8 generators in D=4, M=16 generators in D=6 and M=32 generators in D=10. You can play with N=2, N=4 and N=8 supersymmetries in these dimensions too.

See you in the next amplituhedron post!

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