Finding irreducible representations

[llorgos]

Hello!

Can someone explain to me, as clearly as possible, how one can find irreducible representations of Lie groups (and especially in the context of finding the spectrum of e.g. the bosonic string theory)?

I am following BB&S and Polchinski but I cannot really understand how they construct the spectrum of the theories.

Thank you very much!

 

[fzero]

I can sketch a bit of how it works. In the bosonic string, we have the worldsheet bosons $X^M(z,\bar{z})$. In terms of the worldsheet CFT, the operators $\partial_a X^M$ can be used to construct the primary operators, which can be expressed in terms of multilinear combinations

$$O^{(n)} = C^{a_1\cdots a_n}_{M_1 \cdots M_n} : \partial_{ a_1} X^{M_1} \cdots \partial_{ a_n} X^{M_n} e^{ik\cdot X}: .$$

By primary operators, we mean that these operators have a definite conformal weight (which in fact is $n$) and have a simple OPE with the stress tensor. It turns out that these are also highest weight states of the conformal algebra: we can act on these operators with the conformal generators to obtain the other operators in the CFT.

Now the tensors $C^{a_1\cdots a_n}_{M_1 \cdots M_n}$ haven't been given any particular structure yet. However, it is quite natural to decompose these into the irreducible generators of the worldsheet and target space symmetries (acting on the indices $a$ and $M$). For instance, if the target space was $D+1$-dimensional Minkowski space, we would decompose into irreducible representations of the Lorentz group $SO(1,D)$. We would then find corresponding states in the theory corresponding to spacetime spin 0, 1, 2, etc. states. We can use the CFT to determine the masses of these states.

For instance, $: e^{ik\cdot X}:$ corresponds to the tachyon. The states generated by $: \partial^a X^M \partial_a X^N e^{ik\cdot X}:$ are massless. The trace over $MN$ corresponds to the dilaton, the symmetric part is the graviton, while the antisymmetric part is the antisymmetric tensor $B_{MN}$. The higher order states are all massive, including higher-spin fields.

So far we've been able to generator representations of the Lorentz group. If we want gauge symmetries, we could compactify some of the target space directions, then components of the graviton and tensor fields with one compact index correspond to abelian gauge fields. We could also consider open strings, where we would be allowed to have odd numbers of $\partial X$ appearing as allowed by boundary conditions.

For a more interesting closed-string spectrum, we would need to add fermions, while to get an interesting gauge structure in the perturbative string, we really want to consider the heterotic string. In the heterotic string, we have 10 bosons $X^\mu (z,\bar{z})$, 10 right-moving fermions $\tilde{\psi}^\mu(\bar{z})$ and 32 left-moving fermions $\lambda^A(z)$. Vertex operators can be constructed in an analogous manner as in the bosonic string, however, here we have additional structure from the fermions. This includes the GSO projection, which leads to spacetime supersymmetry and the removal of the bosonic string tachyon. The details can be found in Polchinski, but the upshot is that we have vertex operators

$$ \lambda^A \lambda^B \tilde{\psi}^\mu e^{ik\cdot X}.$$

Fermionic statistics means that this is antisymmetric in $AB$. With the correct choice of projections the corresponding state is a gauge boson in the adjoint representation of $SO(32)$.

Again, this is all a rough sketch. You really need to consult the texts to understand how one determines the mass of the corresponding states, etc.