Hassaan Saleem

This brief post answers the question, “Have we found all string theories?”. It will require a basic understanding of QFT and symmetry algebras. I must mention that I will spend some time talking about the background material before coming to answer the question mentioned at the start.

We recall that when strings move in the target space, they sweep a surface called the worldsheet. This sheet is two-dimensional. A way to study string theory is to study it as a field theory on this worksheet. This field theory has symmetries (referred to as worldsheet symmetries) and those symmetries have symmetry algebras.

Mostly when we talk about symmetries algebras, we mean the set of commutators of the generators of symmetries involved. Now, if the symmetry involved is conformal symmetry, then it turns out that two-dimensional conformal symmetry has infinite generators.

These generators can be thought of as Laurent modes of a field. This field turns out to be the stress-energy tensor (denoted as T(z) where z is the location of the field). Every field has a property called the conformal weight (which is just a number and is denoted by h) which dictates the way that field transforms under conformal transformations. It turns out that for T(z), h=2.

Since the generators of conformal transformations in two dimensions (called the Virasoro modes) are modes of the T(z), the commutator of Virasoro modes (called the Virasoro algebra) contains information about something called the operator product expansion (OPE) of T(z),

OPE of a field T at a point z (denoted as T(z)) with another field X at point w (denoted as X(w)) denotes the product of T(z) and X(w) as a sum of fields and their derivatives. This sum can have a singular part (because z and w can come very close and in that case, the product T(z)X(w) blows up.

OPE of T(z) means OPE of T(z) with itself i.e. T(z)T(w). Now, it turns out that the singular part of T(z)T(w) contains the information in the Virasoro algebra. We refer to T(z)T(w) as the “current algebra” or the “operator algebra” corresponding to the Virasoro algebra. T(z) is correspondingly referred to as a “current” here.

Now, if we try to make a string theory, we have two “sectors” of the theory called the left and right sectors. So, we have to specify the symmetries (and hence, currents) of both of these sectors. If the stress tensor is the only current in both of these sectors, we get the bosonic string theory.

As an aside, conformal symmetry is not something that we put in by hand in bosonic string theory. Instead, it comes out as an inevitable condition to be fulfilled. The classification scheme that we are pursuing here paints the picture as if we put in conformal symmetry by hand. This is not the case.

Now, we can generalize this idea (with some assumptions that I will talk about in the following). We can think of more symmetries that have some other currents and write down their current algebras. Can we make new string theories in this way? One obvious symmetry to include is supersymmetry. Supersymmetry has a current called supercurrent. For this current, h=3/2. If we add stress tensor and supercurrent in both sectors of a string theory, we get type II string theories. Both Type II A and Type II B are in this category. They are different in some other regards.

Let’s develop some convenient notation here. We can label these theories by two numbers (N, N’). N and N’ tell the number of supercurrents that we have in the left and right sectors respectively. Therefore, bosonic string theory is (0,0) and Type II theories are (1,1). What about other (N, N’)?

Now, the critical dimension of a string theory is the maximum spacetime dimension of that theory (it is well known to be 26 for bosonic strings and 10 for superstrings). It turns out that if N or N’ is larger than 2, then the critical dimension is negative. So, these theories don’t make physical sense. So, N and N’ can be 0, 1, and 2.

We will list the cases apart from the cases that we have discussed above. These cases are (0,1), (0,2), (1,2), and (2,2). We have not included the cases (1,0), (2,0), and (2,1) as they are equivalent to (0,1), (0,2), and (1,2) respectively.

(0,1) is nothing but the heterotic string theory. The two well-known heterotic theories i.e. SO(32) and E8 X E8 are in this category. They are different in other regards. The theories corresponding to (0,2) and (1,2) are interesting but mostly of mathematical interest. (0,2) is used in some other places e.g. for Landau Ginzburg models e.g. see this paper.

The (2,2) theory is quite interesting because the critical dimension of this theory is 4 which is the dimension of our macroscopic spacetime!!! However, these 4 dimensions should either be all space dimensions, all time dimensions or 2 space and 2-time dimensions (damn it, so close!).

Until now, there were two unspecified assumptions that we were making. First of all, we were assuming that there are no currents in our theory that have a conformal weight higher than 2. Why this assumption? Well, we know about current algebras and their corresponding commutators that have h>2. These algebras are called W algebras.

W algebras are hard to study because they contain nonlinear terms. These nonlinear terms impose a lot of problems if we want to make a string theory out of W algebras. For readers that want more details, I can say that constructing the BRST current is a nightmare with these algebras.

The simplest W algebra is called W3 algebra and the corresponding strings are called W3 strings. An interested reader can read Chris Pope’s manuscript.

Talking about the second assumption requires a bit of background. When we have a current (let’s call it j(z)) then the first term in the OPE j(z)j(w) has to be proportional to (z-w)^(-2h) (this has to be true because if it isn’t true, then the product j(z)j(w) and the OPE don’t transform in the same way under conformal transformations). We can show that this first term can’t vanish. Now, if h is not an integer or a half-integer, it is easy to see that (z-w)^(-2h) won’t be single-valued when w circles around z.

This multivaluedness can pose some problems. To avoid these problems, we assume that all the currents in our theory have integer or half-integer conformal weights. But what will happen if we drop this assumption? We get something called fractional strings. An interested reader can read Tye’s review for details.

It should be mentioned that there are some string theories that can’t be put in this classification scheme e.g. Green Schwarz string. Moreover, there are some other questions regarding string theories on nongeometric backgrounds that still aren’t understood very well. People occasionally come up with suggestions for new string theories while studying strings in different settings e.g. see this paper by Savdeep Sethi (who is also the advisor of my advisor).

So, if someone asks me the question, “Have all string theories been found?”, I would say that “most probably no, but a significant number of them have been found”. However, you never know what the future has for us. Peace.