Supergravity

Prerequisites: Supersymmetry, General Relativity. Full path (33 pages)

Intuitive
Concrete
Abstract
Why is it interesting?
Criticism

Intuitive

The basic idea of supergravity is that supersymmetry is a gauge symmetry.

When we combine these two ideas we end up automatically with a particle that looks like the particle that we expect to mediate gravity - the graviton.

Concrete

The basic idea of supergravity is to gauge supersymmetry. In the process of doing this at least one spin $2$ particle emerges, which is then interpreted as the graviton that mediates gravity.

Abstract

Rigid $\mathcal N = 1$ supersymmetry in four dimensions extends the Poincaré algebra by Majorana spinor charges $Q$ with, in two-component notation,

$\{Q_\alpha, \bar Q_{\dot\beta}\} = 2\sigma^\mu_{\alpha\dot\beta} P_\mu .$

Making the spinor parameter $\epsilon$ depend on spacetime forces gravity into the theory. The anticommutator shows that the commutator of two supersymmetry transformations with parameters $\epsilon_1(x)$ , $\epsilon_2(x)$ is a translation by $\xi^\mu(x) \propto \bar\epsilon_2 \gamma^\mu \epsilon_1$ , which now varies from point to point: a general coordinate transformation. A theory with local supersymmetry is therefore invariant under diffeomorphisms and contains a metric, or a vielbein $e_\mu{}^a$ . The gauge field of local supersymmetry is a vector-spinor $\psi_\mu$ , the gravitino, with helicities $\pm 3/2$ .

Pure $\mathcal N=1$ , $D=4$ supergravity. The fields are the vielbein and a Majorana gravitino. In one common convention (mostly plus signature, $\gamma^{\mu\nu\rho}$ the antisymmetrised product of three gamma matrices, $\kappa^2 = 8\pi G$ ) the action is

$S = \frac{1}{2\kappa^2}\int d^4x\, e \left[ R(e,\omega) - \bar\psi_\mu \gamma^{\mu\nu\rho} D_\nu \psi_\rho \right],$

with $e = \det e_\mu{}^a$ and $D_\nu \psi_\rho = \partial_\nu\psi_\rho + \frac{1}{4}\omega_\nu{}^{ab}\gamma_{ab}\psi_\rho$ . It is invariant under

$\delta e_\mu{}^a = \frac{1}{2}\bar\epsilon\gamma^a\psi_\mu, \qquad \delta \psi_\mu = D_\mu \epsilon ,$

provided the spin connection $\omega$ contains the torsion built from gravitino bilinears (second order formalism), or is an independent field whose own field equation sets it to that value (first order formalism). The gravitino transformation $\delta\psi_\mu = D_\mu \epsilon$ has the form of a gauge transformation, like $\delta A_\mu = D_\mu \lambda$ for a Yang-Mills potential. Without auxiliary fields the algebra of transformations closes only on shell. Off-shell formulations add auxiliary fields; in "old minimal" supergravity these are a complex scalar and an axial vector, giving 12 bosonic and 12 fermionic off-shell degrees of freedom.

Extended and higher dimensional supergravity. With $\mathcal N$ supercharges, a massless multiplet whose helicities do not exceed 2 requires $\mathcal N \le 8$ in four dimensions. The $\mathcal N = 8$ theory has a single multiplet with 128 bosonic and 128 fermionic states: one graviton, 8 gravitini, 28 vectors, 56 spin- $\frac{1}{2}$ fermions and 70 scalars. Counting real supercharges, 32 is the maximum, and the highest spacetime dimension that admits a supergravity theory with this many supercharges and no fields of helicity above 2 is $D = 11$ . Eleven dimensional supergravity (Cremmer, Julia, Scherk) contains the metric, a Majorana gravitino and a three-form gauge field. Compactifications such as $AdS_4 \times S^7$ are the seven-sphere solutions discussed in the criticism below. The ten dimensional type IIA, type IIB and type I supergravities are the low energy limits of the corresponding superstring theories.

Matter couplings and breaking. $\mathcal N = 1$ supergravity coupled to chiral multiplets is specified by a Kähler potential $K$ and a holomorphic superpotential $W$ . The scalar potential is

$V = e^{K/M_P^2}\left( K^{i\bar j} D_i W\, \overline{D_j W} - \frac{3}{M_P^2}|W|^2 \right), \qquad D_i W = \partial_i W + \frac{1}{M_P^2}(\partial_i K) W ,$

with $M_P$ the reduced Planck mass. The negative term allows supersymmetry to be broken with vanishing vacuum energy. When supersymmetry is broken spontaneously, the gravitino absorbs the goldstino and becomes massive (the super-Higgs mechanism).

As a quantum theory, four dimensional supergravity is non-renormalisable by power counting. Supersymmetry cancels many divergences, and whether $\mathcal N = 8$ supergravity is finite at all loop orders is an open question.

Why is it interesting?

Supergravity is a proposed theory of quantum gravity, i.e. a theory that unifies general relativity with quantum theory.

Criticism

The particular virtue of supergravity, so it seemed a few years ago, is that the infinities introduced by the gravitinos cancel out the infinities due to the graviton, leaving a renormalizable quantum theory of gravity. To be precise, the quantum theory of supergravity was better than renormalizable: the countervailing actions of the gravitons and the gravitinos suggested that no unpleasant infinities of any sort would arise in the first place, so there would never be any need to resort to the somewhat under handed subtraction that was needed in quantum electrodynamics.

Better yet was the perception that supergravity led to an almost unique choice for the unified theory encompassing gravity and the other interactions. Supergravity is not one theory but a family of them, characterized by the number of distinct species of gravitino they contain. There was N = 1 supergravity, which contained but a single gravitino type, up to N = 8, which contained eight.(There are theories with more than eight gravitinos, as it happens,but they also contain new kinds of particles that are not wanted at all.) N = 8 supergravity contains not just the graviton and eight gravitino (which have quantum spins of 2 and 3/2 respectively)but a stack of other particles: twenty-eight with one unit of spin. Better yet was the perception that supergravity led to an almost unique choice for the unified theory encompassing gravity and the other interactions. Supergravity is not one theory but a family of them, characterized by the number of distinct species of gravitino they contain. There was N = 1 supergravity, which contained but a single gravitino type, up to N = 8, which contained eight.(There are theories with more than eight gravitinos, as it happens,but they also contain new kinds of particles that are not wanted at all.) N = 8 supergravity contains not just the graviton and eight gravitino (which have quantum spins of 2 and 2/3 respectively)but a stack of other particles: twenty-eight with one unit of spin.

this is the case, then each point of our conventional four-dimensional spacetime has to be regarded no longer asa point but as a little seven-sphere (as it is called), requiring an extra seven numbers (the equivalent of a set of latitudes and longitudes on a seven-dimensional globe) to say exactly where each point is. These seven numbers then say, in effect, what kind of particle is present: one combination of latitudes and longitudes is the specification of a muon, another combination specifies a quark, and a certain geometrical transformation—a particular rotation of the seven-sphere—turns the muon into the quark.This is the way in which supersymmetry makes all particles cousins by means of higher geometry.

One immediate worry, though, is that the seven-sphere is by no means the only possible arrangement of the extra dimensions.The radius of some of the extra dimensions can be made different,so that the seven-sphere turns into a kind of seven-dimensional football. Or the circles can be interlinked to make a seven dimensional doughnut instead of a seven-dimensional sphere or foot ball.

While the general idea of an eleven-dimensional world did not altogether dismay the physicists, the fact that there was a large choice in the geometrical disposition of these extra dimensions meant that supergravity was not unique. Nothing in supergravity itself suggested that one way of rolling up the extra dimensions would be better than the others. So the choice had to be made bi physicists and not by nature. If no one way of rolling up extra dimensions, the hope that supergravity could be a unique ""theory of everything'' had to be put aside. Either the physicists have to make the dimensions roll up in whatever way they find most convenient, or else they must live in hope of finding some higher theory, beyond super gravity, that dictates how supergravity is turned into an accept able account of our particular universe, with its four dimension

In the end, there were also technical reasons for supergravity failure to live up to Hawking's billing of it as the end of theoretical physics. The surpassing virtue of the theory was its apparent ability to do away with the infinities plaguing all attempts at realistic calculation in quantum-mechanical versions of gravity, but in fact this freedom from infinities had never been incontrovertibly established. What had happened originally was that while calcula tions involving a single graviton had thrown up the usual intractable sort of infinity, the addition of the gravitino produced,by virtue of the intrinsic symmetries of supergravity, equivalent infinities with the opposite sign: everything canceled out, and the theory produced sensible answers entirely of its own accord, with no need for the subterfuge of renormalization. But this was demonstrably true only for the one-graviton calculation, and there was no proof that the same sort of cancellation would so obligingly occur in calculations involving several gravitons and gravitino. Supergravity contains so many particles and so many interconnections that performing calculations of interactions involving more than one graviton or gravitino required hours of effort and pages of scientific journals. Some examples were found of two-graviton calculations that appeared to be finite, but as the calculations became more complicated, infinities began to blossom in number and variety, like weeds in a well-kept garden. It looked increasingly less likely that the theory would actually work as intended, producing sensible answers free of infinities to all levels of calculation. In the end, supergravity died, choked by its weeds.

It is hard to find an obituary for supergravity in any scientific journal. Many physicist-hours were spent on supergravity, efforts either to prove by direct mathematical demonstration that it was free of infinities at all levels, or to find examples of infinities that did not disappear quietly. The latter was the eventual outcome. No doubt some physicists paused briefly to reflect on the passing of what had seemed a wonderful idea, but no one took the time to eulogize. Scientists are not inclined to pause and reflect at the demise of a once-bright idea; if they experience any grief, they find release from it in action, not in mourning. Their Idea of recovery is to move as quickly as possible to the next bright idea. And, as it happens, a new idea—another super idea—was waiting for them. The new object of their affections was the superstring.

From The end of physics by Lindley