The enhanced gauge groups look fascinating, indeed.

The enhanced continuous gauge symmetry arise because in many cases, one may prove that for particular values of the radii and Wilson lines in various compactifications etc., there exist additional massless spin-one states in the single-string spectrum.

Because string theory produces consistent physics in spacetime for any value of the radii and Wilson lines, and because Yang-Mills theory is the only consistent way to make spin-one fields interact, it follows that spacetime dynamics has to include the enhanced Yang-Mills symmetry.

Moving away from the enhanced-symmetry point (in the space of the radii and Wilson lines) is equivalent to the Higgs mechanism. The scalar fields that measure the deviation (of the radii and Wilson lines) from the enhanced-symmetry points may be interpreted as Higgs fields that spontaneously break the enhanced symmetry.

The emergence of Yang-Mills symmetry was proved from the consistency of string theory which may look like too powerful and abstract a method. However, the emergence may also be showed very explicitly.

The new massless gauge-bosons - analogous to W-bosons that become massless at some point, enhancing $U(1)$ to $SU(2)$, if I mention the simplest example - are obtained by a "purely stringy effect", namely from strings that carry a nonzero winding number. That's why these states can't be seen in the "effective higher-dimensional field theory": the latter has no wound strings.

If those new string states are spin-one states and they are charged under the $U(1)$ and similar "generic" groups, it follows that the physics has to be Yang-Mills theory. But it may be useful to see why there's at least the "global enhanced symmetry group" such as $SU(2)$ at the enhanced point.

The appearance of the global symmetry may be seen by the method of "current algebras". Instead of states of a closed string, it's very useful to switch, by the state-operator correspondence, to the local operators on the world sheet. Much like the total $U(1)$ generator may be written as an integral of a charge density over the string, the remaining $SU(2)$ or other generators may also be written as integrals of a current (its temporal component) over the world sheet.

And indeed, it may be proved that at the enhanced point, the currents have the right OPEs (operator product expansions)
$$ j^a(z) j^b(0) \sim \frac{ k^{ab}}{z^2} +\frac{ic^{ab}_{\,\,\,\,\,c}}{z}j^c(0) $$
Because of the general CFT calculus, these OPEs know all about the commutators of the resulting charges, so one may prove that the charges obtained as integrals of those currents have the right commutation relations. One also has to show that they commute with the Hamiltonian - they're conserved i.e. they're symmetries.

The simplest and most important triplet of the currents that one should understand are those that generate the $SU(2)$ algebra in a circular compactification on the self-dual radius:
$$\partial X, \quad :\exp(iX):, \quad :\exp(-iX):$$
I omit constants, both prefactors and those in the exponent. The $\partial X$ current for a compactified dimension $X$ has the OPEs with others where one simply differentiate with respect to $X$. So it shows that the first current is that of $J_z$ and the other two are $J^\pm$. The most nontrivial commutator or OPE that has to come out right is the OPE of the last two exponentials, but it indeed produces the right result when it should. (Quantum subtleties are critical to calculate the OPE: the commutator is not just the Poisson bracket in this case!)

It is typical that the enhanced symmetries separately arise from left-movers (holomorphic $z$ dependence) and right-movers (antiholomorphic $\bar z$ dependence) - two CFTs that just happen to be combined.

From a more overall perspective, one can't say that the enhanced symmetry is an "unnatural" point and the generic one is "natural". Any point of the moduli space is equally consistent as any other. It sometimes doesn't even make sense to ask what's the "maximum gauge group" that may be broken by the Higgs mechanism.

This has fascinating consequences.

A simple example. Take the $SO(32)$ heterotic string: the rank of the gauge group is 16 and the dimension is 496. It looks pretty big. Compactify it on a circle. Turn on generic Wilson lines. It will break the gauge group to $U(1)^{16}$, aside from the additional $U(1)$ from the circle itself (one $U(1)$ from the Kaluza-Klein $U(1)$, another $U(1)$ from a reduction of the B-field along the circle).

At an enhanced point, i.e. for proper values of the radius and the Wilson lines, the gauge group becomes $E_8 \times E_8$ - times the same $U(1)^2$. You may decompactify the dimensions differently: if you allow compactifications on a circle, one may interpolate between the $SO(32)$ and $E_8\times E_8$ heterotic string theories by totally consistent theories! Now, notice that $E_8\times E_8$ has rank 16 and dimension 496, much like $SO(32)$.

In field theory, you would think that there's always the "biggest gauge group" that may perhaps be broken by the Higgs mechanism. In string theory, however, there may be two totally equally big (both rank and dimension), totally different "initial" gauge groups that may be broken to a more generic gauge group. It makes absolutely no sense to ask which of the two heterotic string vacua is more unbroken, more symmetric, or more fundamental. They're clearly equally unbroken, equally symmetric, equally fundamental.

For more general cases of enhanced symmetry groups, one has to learn the Dynkin diagrams, weights, roots, Cartan subalgebras etc., but the derivations are de facto just straightforward generalizations of the simplest case of the $SU(2)$ enhanced symmetry. And it is very typical for the CFTs - that are used as world sheet CFTs here in string theory - have several totally equivalent descriptions that use totally different degrees of freedom. For example, 1 boson is equivalent to 2 fermions when the allowed projections and boundary conditions of both kinds of fields are properly adjusted.

These were gauge symmetries coming from bulk closed strings. Non-Abelian gauge symmetries in string theory also arise on stacks of D-branes - from open strings - and on singularities - the ADE singularities in M-theory (and their F-theory generalizations) are the key example. In M-theory (and type IIA), the enhanced non-Abelian gauge symmetries on the singularities arise from M2-branes (or D2-branes) wrapped on 2-cycles that shrink to zero size. The gauge bosons are literally tiny membranes, much like they're wound strings or open strings in the other pictures. All these pictures how non-Abelian gauge symmetries may emerge in string theory may be related by various dualities.

Of course, I skipped one more method how a non-Abelian group emerges - the $E_8$ group (and the vector supermultiplet) that has to live on all the boundaries of the 11-dimensional spacetime of M-theory.

This post imported from StackExchange Physics at 2014-04-01 16:34 (UCT), posted by SE-user Luboš Motl