Explain these concepts of String Theory in easy words for mathematicians, from a popular point of view

Question

In the Wikipedia's article for the number 496 in the section Physics is related the condition found by Green and Schwarz about this perfect number in their string theory.

Question. Can you tell us from a popularizing point of view what's means the dimension of the gauge group of a type of string theory? I say an explanation with words/plots and/or some formulas but that it is understandable for mathematicians about what is the idea beyond this formalism (additionally if you know explain us why one hope a whole number, an integer, in those equations where was $496$, you are welcome if it is understandable in words). Many thanks.

Is not required (currently I don't need it, and I don't understand such theories) an explanation about 496 and string theory, only is required the background with the purpose to understand from a popular point of view the words in previous article of Wikipedia. I believe that it is possible in several paragraphs.

That is, imagine that I want provide a better/expanded explanation of such fact referenced by Wikipedia to a mathematician friend, from a popularizing point of view, what can I tell to my friend with the purpose that understand such claim about $496$ concerning a condition about the dimension of the gauge group of a type of string theory? With easy words, draws, analogies with equations or definitions well known for a mathematician.

Answer 1

So let me start by citing the sentence from Wikipedia:

The number $496$ is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be $496$. The group is therefore $\mathrm{SO}(32)$. Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely $E_8 \times E_8$.

OK, so let me start with gauge theory. To this, let's have a look at nonrelativistic quantum mechanics. As you might have heard, the state of a system in quantum mechanics is described by the so-called wave function. The wave function is a complex-valued function $\psi$ that depends on the coordinates and the time. If you know the wave function of the system, you can calculate the possible measurement results for any observable quantity.

Now it turns out that the observable physics does not change the slightest if you multiply the wave function by a constant "phase factor" $\mathrm e^{\mathrm i\phi}$, $\phi\in\mathbb R$. In other words, the wave function $\psi(x,t)$ and the wave function $\mathrm e^{\mathrm i\phi}\psi(x,t)$ describe the exact same physical state.

Now one may ask what happens if one multiplies a phase factor that is not constant, but instead depends on space and time, that is, $$\tilde\psi(x,t) = \mathrm e^{\mathrm i\phi(x,t)}\psi(x,t)$$ Well, that does change something, but it turns out one can "fix" those changes. In particular, if you look at the derivatives, you find e.g. $$\frac{\partial\tilde\psi}{\partial t} = \frac{\partial}{\partial t}\left(\mathrm e^{\mathrm i\phi(x,t)}\right) = \mathrm e^{\mathrm i\phi(x,t)}\frac{\partial\psi}{\partial t} + \mathrm i\frac{\partial\phi}{\partial t}e^{\mathrm i\phi(x,t)}\psi(x,t)$$ Now those terms turn out to have exactly the form of the transformations of electromagnetic potentials that do not change the electromagnetic field. In other words, you can indeed do such local changes of the wave function, as long as your equations contain an electromagnetic field that can "absorb" those extra terms.

Such transformations that do not change the physics are called gauge transformations. In particular, the factors $\mathrm e^{\mathrm i\phi}$ form a group under multiplication, namely the unitary group $U(1)$. Since the extra terms caused by those $U(1)$ factors effectively determine the interactions of matter with the electromagnetic fields (by determining what the products of the wave function with the potentials have to be, as they have to have the right form in order to "absorb" the extra terms), the gauge group effectively determines the field.

Now it turns out that the simple complex-valued function is not sufficient to describe physics; rather you need a full vector of such complex values. Of course, with such a vector, you can have many more symmetries, and those other symmetries correspond to other fields. Of course it also means that you need more complicated symmetry groups. For example, the symmetry group of the standard model is $SU(3)\times SU(2)\times U(1)$.

Now a Lie group has a dimension. For example, the group $U(1)$ is one-dimensional (the phase is the only dimension), as is the group of rotations in the plane, $SO(2)$ (the two are actually isomorphic, as seen by the fact that multiplication with a phase factor amounts to a rotation in the complex plane). The group of rotations in space, $SO(3)$ has 3 dimensions (you have 3 independent rotations), and so on.

All modern quantum field theories are gauge theories, and so is string theory. And for reasons I don't know (I'm no string theorist), a certain type of string theory (type I) only works if the gauge group has dimension $496$.

Now $496 = \frac{31\cdot 32}{2}$, a triangular number. Now the dimension of the orthogonal group $SO(n)$, that is, the rotations in an $n$-dimensional Euclidean space, is $\frac{n(n-1)}{2}$, therefore $SO(32)$ has the right dimension.

$E_8$ is an exceptional lie group which has dimension 248. Therefore $E_8\times E_8$ also has dimension $248 + 248=496$.

Answer 2

Here's a quick and imprecise description (that would be better suited for a comment except for its length). A gauge symmetry is an inherent symmetry of an interaction. It doesn't mean that the system is precisely invariant under it, but that it transforms under a certain precise way. Consider, for example, the behavior of a Newtonian system under translating or rotation space; given the behavior in the untransformed system, we can obtain the behavior in the transformed systems, and vice versa. These symmetries are basically coordinate transformations, for liberal definitions of "coordinate." Similar things happen in special relativity with more general inertial frames, or with classical electromagnetism in taking $(A, V) \to (A + \nabla f, V - \partial f/\partial t)$. A choice of a particular $f$ is often referred to as a gauge.

Now consider quantum field theory. Here the interaction is described in terms of a Lagrangian $L$, analogous to the classical Lagrangian. (It's a bit more complicated for reasons I won't get into here.) We combine these symmetries in a gauge group $G$, and we usually care about infinitesimal symmetries. Thus we can think about them as a Lie algebra or as a Lie group. In the former case, we have a vector space of a certain dimension; in the latter case, we have a manifold of a certain (same) dimension. Unlike the classical case, the point of quantum field theory is often to figure out the appropriate Lagrangian rather than to consider the equations of motion for one that's conveniently given to you, so gauge symmetries impose various conditions on $L$.

I'm not very familiar with string theory, but the basic setup involves a vaguely similar Lagrangian, and the gauge group presumably corresponds to its group of symmetries.