# How to find explicit QFT particles from a heterotic String theorie partition function

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Starting with a SUSY $E_8$ x $E_8$ heterotic partition function:

$Z = - \frac{1}{8} \sum Z^8_x Z_4[^s_{s'}] Z_8[^t_{t'}] Z_8[^u_{u'}]$

where the sum is over all spin structures $s,t,u,s',t',u' = 0, 1$.

I can go to a lattice formulation of this partition function:

$$Z = Z^8_x (V_4 + S_4) \times (R_8 + S_8) \times (R_8 + S_8)$$

Where $V,S,R$ are the partition sums of the corresponding lattices.

1. How does the last part (gauge representation) define which of the fermions/bosons I get in the $(R_8 \otimes S_8)$ representation?

2. How can I get from the partition function to a particle of the QFT using the massless part of the partition function? If I choose corresponding vectors in the three lattices how can I see the properties of the particles?

This post imported from StackExchange Physics at 2015-12-08 22:39 (UTC), posted by SE-user LOQ
This could use more context/explanation, especially notation-wise (you cannot assume everyone uses the same notation as you do). What are all those $Z_i$ in your first equation (i.e. how are they defined)? Why are you writing a partition function (which is usually a number) as a tensor and direct sum of other partition sums as if they were representation spaces?
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