string theory - Same partition functions, different theories

I am reading the book "Basic Concepts of String Theory" by Blumenhagen, Lust and Theisen and in page 290 they say:

"It follows that the $E8\times E8$ and the $SO(32)$ heterotic string theories have the same number of states at every mass level which are however differently organized under the internal gauge symmetries. So, even though the partition functions are identical, the theories are nevertheless different. The differences show up in correlation functions."

I am trying to understand how these differences would show up in correlation functions, ie what would be an example of two correlation functions that are different in these two theories. Any references where this is done/explained would be greatly appreciated!

Answer

The differences will show up in the correlation functions because the correlation functions "know" about the group under which the states transform.

For example, the first excited level of both CFTs, one with $E_8\times E_8$ (HE) and one with $SO(32)$ (HO), contains $248+248=32\times 31/(2\times 1)=496$ states (and therefore the corresponding operators $K_i$, $i=1,2,\dots, 496$, assigned by the state-operator correspondence) that transform as the adjoint of the gauge group.

These $K_i(z)$ operators may become factors in the vertex operators in string theory (the operators encoding the external gauge bosons and their superpartners). The scattering amplitudes involving two gauge bosons (and something else) are integrals of integrands that are proportional to the correlators

$$\langle K_i(z_1) K_j(z_2) \cdot \cdots \rangle$$ perhaps with some additional operators. But these correlators may be computed if you replace the $KK$ according to the OPEs (operator product expansions). These OPEs will also contain terms proportional to $K_k$ operators themselves: $$K_i(z_1) K_j(z_2) \sim \frac 1{z_1-z_2} f_{ij}{}^k K_k(z_2)$$ where $f_{ij}{}^k$ are the structure constants of the Lie group. The structure constants of $E_8\times E_8$ and $SO(32)$ groups are different from each other – regardless of the basis of the Lie algebra you choose (for example, it's because the $E_8\times E_8$ algebra splits into two decoupled pieces while the $SO(32)$ algebra does not) so if you study the OPEs of the 496 operators $K_i$, you will be able to extract the structure constants $f_{ij}{}^k$ and therefore determine which of the two gauge groups is involved, too.

The structure constants may be extracted not only from the OPEs but also from the correlators of three vertex operators such as

$$\langle K_i(z_1) K_j(z_2)K_k(z_3) \rangle \sim f_{ij}{}^k$$ so these correlators "know" about the representations under which the operators $K_i(z)$ transform. Although these operators may look like a collection of 496 operators with the same dimension in both cases, when you switch to the interacting theory, the difference between the HE and HO theories appears through the correlators (and structure constants in them).

The formulae above were just examples for the basic operators in the adjoint. The formulae for more complicated correlators possibly involving more than 3 operators and perhaps some operators that are more excited (higher dimensions) are more involved but they display differences between HE and HO, too. At any rate, the correlators of the simplest states above (the gauge bosons) are enough to show that the theories are different at the interacting, perturbative level.

One should emphasize that after one bosonic dimension of the heterotic string is compactified on a circle, the theories actually become equivalent – T-dual to each other – but the Wilson lines and other moduli must be carefully adjusted on both sides for the theories to match. It's because the even self-dual lattices of signature 17+1 are all isometric to each other. In particular

$$\Gamma^{16}_{Spin(32)/Z_2} +\Gamma^{1,1} \equiv \Gamma^8_{E_8} + \Gamma^8_{E_8} + \Gamma^{1,1}$$ after an appropriate $SO(17,1)$ "Lorentz" transformation of the lattices on both sides.