35 years of N=4 Yang-Mills theory
Luboš Motl, March 31, 2012
George Musser who writes for Scientific American has authored an unusually (for SciAm) scientifically oriented blog entry
The Emperor, Darth Vader, and the Ultimate Ultimate Theory of Physics
about the 35th birthday of the maximally supersymmetric i.e. \( \mathcal{N}=4\) Yang-Mills theory in \(d=4\). It’s not just the anniversary. A conference is taking place at Caltech between Thursday and Saturday (it ends today). The list of big shots who are participating is spectacular.
This quantum field theory has become one of the modern era’s «quantum harmonic oscillators». It involves much more nontrivial structures and relationships than the quantum harmonic oscillator but it has a comparably far-reaching and intellectual diverse impact on a large fraction of the landscape of the modern theoretical physics research.
One of the most efficient ways to determine that someone who talks about high-energy physics is a hopeless and worthless deluded crank is to notice that he either dismisses the importance of the \(\mathcal{N}=4\) gauge theory or he doesn’t even know what it is.
The maximally supersymmetric gauge theory was born in a 1977 paper in Nuclear Physics B,
Lars Brink, John Schwarz and Joel Scherk (the latter sadly committed suicide while the others were working at Caltech) decided to find all spacetime dimensions in which the simple \(\mathcal{N}=1\) gauge theories can exist. They ended up with the list
\[d\in \{2,4,6,10\}\]
as the abstract shows. Well, I think that this was a bit imperfect. A more updated claim would probably say
\[d\in \{3,4,6,10\}\]
Note the difference between two and three. The 3-dimensional gauge theory may be dimensionally reduced to two spacetime dimensions; the three-dimensional minimal gauge theory is more fundamental.
The authors studied the dimensional reductions of those theories to \(d=2\) and \(d=4\) and found the maximally supersymmetric theory in \(d=4\) as one of the very important examples. It’s more important than others because it’s fully consistent as a quantum theory.
Minimal gauge theories
Photons and gluons – and other quanta of gauge fields – carry the spin \(j=1\). If we have the minimal supersymmetry only, it may transform the \(j=1\) multiplet of physical states either to \(j=3/2\) or \(j=1/2\). However, the particles with \(j=3/2\) may be immediately eliminated. The corresponding fields would carry negative-norm polarizations (particles leading to negative probabilities). To decouple these bad ghosts, one would need a local symmetry whose conserved charges transform as spinors.
However, those would have to be supercharges and they would anticommute to the energy-momentum vector (or stress-energy tensor, locally) in any consistent interacting theory, by the extended Coleman-Mandula logic. If you think about these statements, it follows that \(j=3/2\) light or massless fields may only exist in supergravity theories and they must have superpartners with \(j=2\), the gravitons. The fields with \(j=3/2\) have to be gravitinos.
I just wanted to write a paragraph about this «distraction». Now we may jump to the right form of the theories. Minimally, \(\mathcal{N}=1\) supersymmetric gauge theories inevitably contain gauge fields and fermionic spinors with \(j=1/2\). If we want the gauge theory to be «pure», it can’t contain any scalars with \(j=0\). If there were such a scalar, this scalar and its \(j=1/2\) superpartners would form a separate multiplet from the gauge supermultiplet.
Even if the justification isn’t quite complete, the conclusion is true: the minimal supersymmetric gauge theories only contain fields with spin \(j=1\) and \(j=1/2\). Because the theory is supersymmetric, it must have the same number of bosonic and fermionic polarizations.
What’s the number of bosonic polarizations? Well, a photon has \(d-2=2\) independent transverse linear polarizations (or, equivalently, two independent transverse circular physical polarizations). Two of the polarizations, the time-like one and the longitudinal one, are made unphysical by the gauge symmetry (an equivalence) and the Gauss’s law.
Fine, so we have \(d-2\) polarizations of the gauge field (per generator of the gauge group) and we must have the same number of fermionic components. But the fermionic components are organized as a spinor. Its dimension is a power of two. We must therefore have
\[d-2 = 2^K\] where \(K\) is approximately equal to \(d/2\) plus minus one or so. So we’re interested in values of \(d\) such that \(d-2\) is a power of two. Such values of \(d\) can be easily found: take powers of two, namely \(1,2,4,8,16,\dots\), and add two to get
\[d = 3,4,6,10,18,\dots\]
The last entry in my list is \(d=18\) but it’s already too high because the smallest spinor in \(d=18\) has something like \(2^{18/2}/2 = 256 \) real components (even if I add the factor of one-half for the reduced chirality) which is much more than \(d-2=16\). So the minimal supersymmetry simply can’t match the bosonic and fermionic degrees of freedom. Needless to say, this discrepancy becomes even worse for even higher powers of two.
We have just derived that we only need to consider dimensions
\[d=3,4,6,10\]
Interestingly enough, each of these spacetime dimensions offers us a nice minimally supersymmetric Yang-Mills theory. How is it possible that the equation
\[d-2 = 2^{d/2\pm 1}\]
has four solutions instead of one? Well, it’s because we may exploit the \(\pm 1\) uncertainty of the exponent. In particular, the \(j=1/2\) fermionic degrees of freedom may be either the full Dirac spinors, Weyl spinors, Majorana spinors, or Majorana-Weyl spinors. The adjective «Dirac» means that the spinor has \(2^{d/2}\) components that are unrestricted and complex; «Majorana» means that there is a reality condition for the spinor (it has to be related to its complex conjugate); «Weyl» means that the a priori Dirac spinor has to be chiral (i.e. left-handed); and «Majorana-Weyl» means that we impose both previous conditions simultaneously.
When we carefully correct various factors of two in the counting etc., we find out that in \(d=10\), the minimum spinor representations are 16-dimensional, chiral, and real. That’s great because we take a Majorana-Weyl (chiral, real) spinor with 16 components and interpret these components as 8 partners of the \(d-2\) gauge boson polarizations and their 8 antiparticles which is how the right matching should actually be. The \(d=10\) case of the minimum supersymmetric gauge theory is the «highest-dimensional» one which makes it very important.
However, the Weyl spinor in \(d=6\) is 4-complex-dimensional. Due to the necessary complexity of the representation, we can’t impose the Majorana condition in this case, only the chiral one. Again, we get the equivalent of 8 real components which is good to match \(d-2=4\) gauge boson polarizations.
In \(d=4\) which is the dimension many of us consider our home, the Majorana and Weyl conditions give pretty much the same constraint on the Dirac spinor. So we may say it’s only the Majorana condition we’re imposing on the \(j=1/2\) gauginos here, to make it different from \(d=6\). The Majorana spinor has 4 real components in our spacetime dimension, the right number to match \(d-2=2\) polarizations of a gauge boson.
Finally, in \(d=3\), the spinor is real 2-dimensional. It’s automatically real but there’s no chirality in \(d=3\) because it’s an odd number. Again, these two real polarizations of the spinor match the \(d-2=1\) polarization of the gauge field in this dimension. Alternatively, one may understand the \(d=3\) real 2-dimensional spinor as the complex 1-dimensional Dirac spinor in \(d=2\) which is what the authors originally did.
Returning to ten dimensions
You have noticed that \(d=10\) is a kind of a master case. It’s the highest dimension in which we found a natural simple supersymmetric theory. It just happens to be the spacetime dimension of superstring theory, too. However, in the superstring case, the \(d=10\) solution is unique; we don’t really get four equally good dimensions in superstring theory.
Still, the \(d=10\) superstring theory has some rather direct relations to the \(d=10\) Yang-Mills theory. In particular, the Yang-Mills theory appears in the low-energy limit of type I string theory (the gauge theory lives on the D9-branes that fill the space of type IIB string theory, together with orientifold O9-planes, to produce type I) and heterotic string theories in \(d=10\). In the heterotic case, the Yang-Mills fields arise from the 16 excessive left-moving excitations of the hybrid 26/10-dimensional string and they appear at the same order as gravitons.
Fine, what is the Lagrangian for the \(d=10\) \(\mathcal{N}=1\) Yang-Mills theory? It’s actually as straightforward as you might guess.
\[ \mathcal{L} ={\rm Tr}\left[ -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i\bar\Psi D^\mu \gamma_\mu \Psi\right]\]
No kidding. It’s one line, even when you use the very short and constraining TRF lines. The kinetic term for the gauge field is the same as in QCD; you just need to go to \(d=10\) and use the right gauge group. The term for the Majorana-Weyl spinorial fermionic field is just the usual Dirac-style Lagrangian with derivatives. The gaugino masses have to be zero because they’re equal to the gauge field masses by supersymmetry and the gauge bosons have to be massless because there are no scalar fields that would offer you the God particles needed to break the gauge symmetry.
The trace is one over the adjoint indices; both the gauge field and the gaugino fermionic field transform in the adjoint representation of the gauge group.
This theory is interacting because the gauge-field is non-Abelian and includes the \([A_\mu,A_\nu]\) terms. Both terms in the Lagrangian above contain higher-order terms, proportional to the structure constants of the Lie group that we used as the gauge group. Note that the second term has a covariant derivative which induces a cubic coupling between the fermion (gaugino) and the gauge field.
You might think that we just added the gauge field terms and some interacting Dirac-like terms and the supersymmetry will be broken by the interactions that look different. But remarkably enough, the theory defined by the Lagrangian above is supersymmetric even if you compute all the interaction terms exactly. That’s true despite the fact that there’s no known superspace that would make the supersymmetry of the theory immediately self-evident.
In \(d=10\) which is very high, the field theory is non-renormalizable. If you want to know how physics behaves at energy scales comparable to or higher than the typical energy scale associated with the dimensionful gauge coupling (which I have set equal to one for simplicity), you need to study the whole string theory in which the Yang-Mills theory is embedded. You will find out that consistency requires you to include the rest of the string theory as well, including gravity and black holes.
Compactification to \(d=4\)
However, if you compactify the \(d=10\) gauge theory to \(d=4\), the spacetime dimensions many of us still know and love, you obtain a gauge theory that is renormalizable even as a quantum theory. It’s because the Lagrangian above contains at most quartic (fourth order) terms and that’s exactly what you need for renormalizability. (Note that because some gauge field components were transformed to scalars, you also produce quartic terms in the potential and Yukawa couplings, among other interactions.)
In fact, the theory has a much better UV behavior than you could a priori assume. Supersymmetry guarantees that.
What does it mean to «dimensionally reduce» the theory to \(d=4\)? It means that you suppress or forbid the dependence of all the fields on 6 «unwanted» spatial coordinates and you rename the corresponding components of the gauge field \(A_{4}\dots A_9\) as scalars. As you can see, it’s six scalars. They still transform in the adjoint of the gauge group, much like the gauge field and the gaugino.
Meanwhile, the 10-dimensional spinor with 16 real components splits into 4 four-dimensional Majorana spinors, each of which contains the equivalent of 4 real components. Nevertheless, there are 16 real fermionic components. As always, we divide the number by two (imagine that it’s because the canonical momenta for fermions aren’t the time derivatives but some other components of the fermions). So these 16 real fermions must be matched to 8 real physical bosonic polarizations. And indeed, we have 2 polarizations from the \(d=4\) gauge field and 6 polarizations from the scalars. The counting works; there is an \(SO(6)\sim SU(4)\) symmetry, the R-symmetry, rotating the six scalars as well as the four gauginos. The supersymmetry algebra works even at the interacting level. It’s actually enhanced to a superconformal algebra I will mention later.
Finiteness
So it’s a nice, interacting, but heavily symmetric (supersymmetric) theory. The allowed interactions are essentially uniquely dictated by supersymmetry. You can’t change them in any way. You may only change the gauge coupling \(g\) which is dimensionless in \(d=4\). Well, there is another parameter, the angle \(\theta\) in front of the \(F\wedge F\) term. The parameters \(g\) and \(\theta\) may be combined to a complex coupling I will call \(\tau\), something like
\[\tau = \frac{i}{g^2}+\frac{\theta}{2\pi}\]
I guess that I have deviated from some conventions. In QCD, you know that the gauge coupling runs so it’s not strictly dimensionless at the quantum level (dimensional transmutation). However, the cancellations in the \(\mathcal{N}=4\) gauge theory are so powerful that they guarantee that the beta-function vanishes – at all orders, actually – so this gauge coupling is a genuine dimensionless parameter that doesn’t run. The theory is classically scale-invariant and the vanishing beta-function (and vanishing analogous quantities) guarantee(s) the scale invariance at the quantum level, too.
Not only the beta-function cancels. All divergences cancel as well. So you don’t really need to renormalize anything. Individual Feynman diagrams may have short-distance divergences but if you write them consistently, they always cancel in the sum. I mean if you try to compute scattering amplitudes for gauge bosons or gauginos or the scalars. The finiteness was proved rather quickly after the theory was presented.
S-duality
Even if you just use the ordinary perturbative methods for quantum field theory that you learned in your undergraduate or high school courses on quantum field theory, you will quickly find out that the theory has remarkable properties: symmetries, cancellations, finiteness, uniqueness for a given gauge group (up to the coupling constant \(\tau\).
But that’s nothing compared to what you experience if you study the behavior of the theory beyond the perturbative regime and/or in various scaling limits in which you change the number of colors \(N_c\) from the gauge group \(SU(N_c)\), the simplest interesting choice for the gauge group.
For example, what happens when you just try to send the dimensionless coupling \(g\to\infty\)? The strong coupling of any theory used to be an enigma, a sea filled with seadragons and witches. However, the perturbative beauty of this particular theory suggested that the theory should be consistent at strong coupling. In fact, it’s not only consistent: it’s equivalent to itself at the weak coupling.
This property is known as the Montonen-Olive duality but people usually talk about S-duality these days even though the latter concept is more general and applies to other theories as well. This property of the theory is a surprisingly exact and nontrivial generalization of the electromagnetic duality of Maxwell’s equations.
When you learned about Maxwell’s equations while you were in the kindergarten (or in the elementary school, assuming that you were a retarded baby), you must have noticed the symmetry
\[\vec E \leftrightarrow \vec B\] or, equivalently, \[F_{\mu\nu} \leftrightarrow *F_{\mu\nu}\]
In particular, the vacuum (sourceless) Maxwell’s equations have a complete symmetry between electric and magnetic fields. The non-vacuum equations don’t have the symmetry because electric charges (monopoles) exist while the magnetic sources (monopoles) don’t, at least not in the everyday environments. But if you add the magnetic monopole densities and fluxes to the equations, Maxwell’s equations become totally «electro-magnetically» symmetric once again (even though the physical properties such as masses of the electrically charged and magnetically charged particles are obviously different in the real world).
The supersymmetric Yang-Mills theory has equations of motion that look like Maxwell’s equations with sources; the sources are made out of other fields of the theory, too. Montonen and Olive conjectured and it’s been later established – although the certainty of the claim is indisputable only if you accept physicists’ definition of rigor – that for every electric excitation, there is an object similar to a magnetic monopole in the theory. And they also transform under supersymmetry and if you exchange the electric and magnetic sources, everything remains unchanged up to a different value of the coupling constant.
In fact, the complex coupling constant \(\tau\) I have mentioned may be transformed as
\[\tau\to \tau’ = \frac{a\tau +b}{c\tau +d}\] where the numbers \((a,b,c,d)\) define an element of a discrete group,\[\pmatrix{ a&b;\\c&d; }\in SL(2,\mathbb{Z})\]
The group contains all matrices with integral entries whose determinant \(ad-bc\) is equal to one. The Montonen-Olive duality means that the theories with values of \(\tau\) that are related by such a transformation, i.e. those with \(\tau\) and any \(\tau’\), are actually physically equivalent. Not all values of \(\tau\) lead to inequivalent theories. You may use the symmetry to bring \(\tau\) to the so-called fundamental domain obeying
\[-\frac 12\lt \tau_1\lt +\frac 12,\qquad |\tau|\gt 1\]
It’s a semi-infinite strip in the complex plane (whose width equals one) with an arc-shaped boundary. This \(\tau\)-theme appears in many contexts of modern theoretical physics.
Can we understand where the remarkable equivalence between different values of \(\tau\) comes from? Perturbative methods aren’t quite sufficient but we may understand the S-duality by many constructions based on string theory. For example, the maximally supersymmetric gauge theory appears as the low-energy limit of the dynamics on D3-branes, stringy solitons (branes when it comes to their shape) with 3+1 dimensions. D3-branes are similar (T-dual) to D9-branes which produce the 10-dimensional gauge theory in type I string theory, as I have already discussed in the \(d=10\) section. D3-branes exist in type IIB string theory which has the same \(SL(2, \mathbb{Z})\) symmetry.
Why does type IIB string theory have this symmetry? It’s because type IIB string theory may be visualized as Vafa’s F-theory in \(d=12\) which is compactified on an infinitesimal two-torus \(T^2\). The complex coordinate \(z\) describes the dimensions along the two-torus. The shape of the torus is given by identifications encoded in a lattice
\[z\equiv z+1,\qquad z\equiv z+\tau\]
but you may redefine the basis of such a lattice by the same matrix based on the numbers \((a,b,c,d)\) as we mentioned above. You simply get a manifestly the same lattice and the same geometry if the shape of the torus is given by values of \(\tau\) related by an \(SL(2,\mathbb{Z})\) transformation.
(The point of F-theory is that it allows to study «difficult» or «strongly coupled» type IIB spacetimes in which \(\tau\) is a function of the remaining 9+1 spacetime coordinates. In fact, \(\tau\) doesn’t have to be single-valued; it may become any of the \(\tau’\) related to \(\tau\) by the \(SL(2,\mathbb{Z})\) symmetry if you make a round trip around a co-dimension 2 singularity, a «7-brane». Realistic model building based on F-theory has been one of the fascinating themes in the recent 5 years or so.)
In fact, this trick involving the secret hidden two-torus may be done without the glory of F-theory with all of its type IIB string theoretical gravitational physics and all the other mess. How? We may describe the \(\mathcal{N}=4\) gauge theory as the limit of a \(d=6\) theory known as the (2,0) superconformal theory which is called the Emperor theory by Nima Arkani-Hamed (and therefore George Musser).
This theory is also superconformal and arises from the low-energy dynamics of M5-branes in M-theory (among other stringy ways to get it). It doesn’t have a simple Lagrangian description we may write down (in fact, it has no dimensionless coupling that could be made small, i.e. it has no classical limit as we know from weakly coupled field theories! An inherently quantum theory) but we may still prove by stringy methods that the theory exists, is local, superconformal, and has many other properties that have been proven.
If you compactify the (2,0) theory on a tiny 2-torus, you get the \(d=4\) \(\mathcal{N}=4\) gauge theory as the low-energy limit describing dynamics of the remaining 4-dimensional spacetime. The proof of the \(SL(2,\mathbb{Z})\) symmetry is then unchanged.
There are also other ways how the maximally supersymmetric theory is embedded in string theory but let me stop with the flat-space discussion. In 1997, Juan Maldacena found something about this gauge theory that was much more surprising. For the first time, he found a relationship of the gauge theory to string theory which critically included a curved spacetime; and a relationship in which the gauge theory wasn’t just a long-distance limit of string theory but all of string theory.
The AdS/CFT correspondence
In 1997, Juan proposed a much more explicit version of the holographic principle of quantum gravity, something discussed at a much more heuristic level by Gerard ‘t Hooft and Leonard Susskind about 5 years earlier. He said that stringy/M quantum gravitational theories describing physics on anti de Sitter (AdS) spacetimes are exactly equivalent to non-gravitational «conformal» or «superconformal» (supersymmetric and conformal) quantum field theories defined on the causal boundaries of these AdS spacetimes.
The \(\mathcal{N}=4\) gauge theory in \(d=4\) is by far the most carefully analyzed example of the AdS/CFT correspondence. A large portion of the 5,000-10,000 papers written on this topic contain some evidence that Maldacena’s equivalence works. In various subcontexts, one may say that the equivalence has been exactly proven, and so on.
But what is the other side of the equivalence for the gauge theory discussed in this blog entry?
It’s type IIB string theory on a particular 10-dimensional background, and be sure that proper backgrounds of type IIB string theory have to be 10-dimensional. The background is
\[AdS_5 \times S^5\]
where the curvature radii of both factors match. The curvature radius in type IIB string units is proportional to a power of the ‘t Hooft coupling of the gauge theory, \(\lambda = g^2 N_c\) where \(N_c\) is the number of colors. It’s really ‘t Hooft’s coupling, and not the simple Yang-Mills coupling \(g\), that determines whether the sum over multi-loop diagrams in the gauge theory tends to converge; Gerard ‘t Hooft has understood that the large-\(\lambda\) limit of gauge theories have to be «some kinds of string theory» (with the nontrivial topology of stringy world sheet emerging from a \(1/N_c\) expansion) already in 1974 but he didn’t know much how to define these string theories and where (in which spacetimes) the strings lived.
The type IIB string coupling is the square of the Yang-Mills coupling, up to a normalization factor. The \(SL(2,\mathbb{Z})\) S-dualities act equally on both sides.
Note that the background above isn’t Ricci-flat so it doesn’t solve the vacuum Einstein’s equations. But that’s not a problem because there should be sources. The relevant sources (stress-energy tensor) is induced by a 5-form generalized electromagnetic flux with five indices inside \(AdS_5\) or five indices inside \(S^5\).
At long distances, assuming a large \(\lambda\), the type IIB string theory may be approximated simply by supergravity, and typically the classical one. That’s the limit on which the early papers focused.
BMN and the Penrose’s pp-wave limits
However, it didn’t take too much time and people began to master the beyond-the-supergravity part of the duality as well; you need to calculate what happens in the theory for «not so huge» values of \(\lambda\), too. Witten and others could have found wrapped branes in the gauge theory (represented by creation operators for baryons of some kind). The most explicit proof that the gauge theory is equivalent to the full string theory was discovered by Berenstein, Maldacena, and Nastase (BMN) a decade ago.
They decided to solve a particular limit of the anti de Sitter space geometry, the Penrose limit, which produces something that isn’t flat yet but it isn’t the full original space, either: it’s a «pp-wave». One may show that there’s a corresponding limit of the gauge theory as well: however, one must consider operators with a large value of \(J\), some charge under the \(SO(6)\) symmetry rotating the six scalars I have mentioned earlier – or rotating the five-sphere which is the same rotation.
BMN realized that strings moving on the pp-wave geometry (the bulk, gravitational side of the duality) may be exactly identified with operators of the type
\[{\rm Tr} ( ZZZZAZZZZZZZBZZZZZ )\]
where I chose names \(A,B,Z\) for the three complex scalars (mixtures of the six Hermitian scalars) in the gauge theory. The pieces in the trace are connected by contracted indices just like bits of a string. The string is a closed one because it’s a trace. The trace of a long product of matrices looks like a string – and it doesn’t just look like a string, it is a string. Most of the string is made out of the \(Z\) bits; the operators \(A,B\) are some impurities or transverse excitations that may move along the string. Of course, to get the energy eigenstates, you need to combine the operators into «Bloch waves» with some phases multiplying the traces with different locations of the impurities – i.e. into the so-called BMN operators.
Other people, including your humble correspondent, have proven that the interactions of these «strings made out of traces» match the interactions of strings on the pp-wave curved background, too. Every test that has been done has always agreed with the stringy and Juan’s predictions and confirmed the duality proposed by Maldacena. It’s been established beyond any reasonable doubt – and even beyond any unreasonable doubt that isn’t excessively unreasonable – that Maldacena’s equivalence works. For a short time, it was called «Maldacena’s conjecture» but if you hear someone using this term today instead of «Maldacena’s correspondence», you may be pretty sure that he or she has missed the last 15 years in theoretical physics.
Integrability
The theory has some wonderful symmetries. Among them one hasn’t been discussed too explicitly: the superconformal symmetry. The theory isn’t just symmetric under the large super-Poincaré algebra, involving Lorentz transformations and four packages of supersymmetry generators. It’s invariant under all coordinate transformations that preserve the angle, i.e. all conformal transformations, as well as some supersymmetric fermionic counterparts of them.
The vanishing of the beta-function I discussed at the beginning actually means that the theory is exactly scale-invariant which is a necessary condition (and, under some reasonable conditions, also sufficient condition) for the Lorentz-invariant theory to be conformally invariant as well.
Because the theory is so well-behaved, you might think that it must be «really simple» and «solvable» in some way. And indeed, people have believed such a thing for decades, too. In particular, they believed that it should be easy to solve the «planar limit» which is a limit ignoring \(1/N_c\) corrections for large \(N_c\) or, equivalently in the stringy language, the tree-level limit in the holographically dual stringy description.
A reason why this «integrability» or «solvability» is a sensible assumption is that there is actually a whole infinite-dimensional symmetry algebra that the theory respects at the stringy tree level i.e. in the so-called planar limit. This big symmetry is known as the «Yangian» and it may be generated by the superconformal generators together with the «dual superconformal symmetry generators» which may be interpreted as generators of a symmetry on some kind of T-dual background obtained by string theory methods.
Twistor magic
Finally, there is an aspect of this maximally supersymmetric gauge theory that may deserve a much longer introduction than the text above, and it is related to Roger Penrose’s twistors. Using various twistor variables, some properties of the amplitudes of this gauge theory – especially the scattering on-shell amplitudes – acquire a very transparent, geometric explanation.
Nima Arkani-Hamed believes that there is a whole third «big animal» aside from AdS and CFT, a twistor-based description that will ultimately clarify not only some properties of «nearly maximally helicity violating» scattering amplitudes but that may also sketch a much more transparent and deeper connection between the AdS description and the CFT description because this twistor description may be a sort of an interpolation between the AdS and CFT spaces.
See e.g. Nima Arkani-Hamed on the twistor uprising for some discussion of this picture.
Much of these remarkable properties of the twistor description have already been established by fascinatingly difficult and beautiful geometric and algebraic methods involving twistor spaces; some of the comments remain a wishful thinking.
But it’s true that at least 50% of my objections against «taking this program too seriously» have been shown unsubstantiated. There were two complaints I have always had. One of them is that the CFT used in AdS/CFT has to be an off-shell gauge theory; the off-shell physics is totally essential to see the «bulk» of the AdS space, including the new holographic dimension. I am still afraid that this objection is legitimate and the twistor methods could never really transcend the on-shell games with scattering amplitudes.
The other problem I had was that scattering amplitudes in the \(\mathcal{N}=4\) gauge theory are really ill-defined due to infrared (long-distance) divergences. But this problem was really unsubstantiated. Even though the forces in this gauge theory are long-range forces (infinite range) due to the conformal symmetry, the (IR-divergent) scattering amplitudes may be rewritten as some appropriate integrals and the origin of the infrared divergences may be blamed at a particular region in the integration domain.
But even the integrand itself is meaningful and has been re-expressed in new ways.
This was a sketch of some of the basic mathematical facts about the theory that celebrates its 35th birthday these days. The mathematical background has lots of ramifications; for example, because of variations of the AdS/CFT, the gauge theory has been used for some «really tangible» physics questions such as the viscosity of the quark-gluon plasma or some superconductors and other things (explained largely by black holes in a five-dimensional AdS space).
The quantum harmonic oscillator is something that a modern theoretical physicist obviously has to know; it’s like the multiplication table for numbers up to 10. But there are also other mathematical structures whose importance across physics is huge and whose mathematical properties make it unique. Even if you haven’t understood everything in this blog entry, you may believe me that \(\mathcal{N}=4\) gauge theory is one of those gems. In some proper sense, it’s simpler and more fundamental than any gauge theory without (or with less extended) supersymmetry. Because of direct and exact links between this theory and string theory plus its links to «truly experimental» problems in physics, this gauge theory is also one of numerous ways to prove that it’s absurd to expect that string theory may be cut off of empirically rooted physics sometime in the future. The link has already been established; string theory is irrevocably needed for a proper understanding of the laws underlying the Universe – and I mean our Universe.
But I have to stop at some moment and I decided that the moment is 35 seconds from now. Happy 35th birthday, \(\mathcal{N}=4\) gauge theory. I think it’s likely that Darth Vader will show us some of her new colors in the future, too – whether we talk about her in the limit of a large number of colors or outside this limit.
