# Has hep-th/0312070 forgotten to fix $s_{0} = 1/2$ for the fermionic states in the second table on page 52?

Link to the original paper: The Gauge/String Correspondence Towards Realistic Gauge Theories (arXiv paper)

On page 52 we see that, for a theory of Dp-branes placed at an orbifold (orbifold = $C_{2}$/$Z2$) fixed point, we have a certain spectrum of open string NS and R states. The first thing to note is that the orbifold action has the effect of making the spinor labels behave as follows: $s_{3} = s_{4}$ or $s_{3} = -s_{4}$. This means the 3rd and 4th components are no longer independent. Since $s_{i}$ equals $1/2$ or $-1/2$, this means there are now 16 different combinations of the state $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ corresponding to the different 16 permutations of the values of $s_{0}, s_{1}, s_{2},s_{3} = -1/2,+1/2$. We would have had 32 different permutations corresponding to $s_{0}, s_{1}, s_{2},s_{3}, s_{4} = -1/2,+1/2$ if $s_{4}$ wasn't fixed by the value of $s_{3}$. However, this isn't the end of it. We must also take a GSO projection which projects out half the states. This leaves 8 states (8 permutations) for each $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. So far so good; we now have 2 different types of Chan-Paton factor in front of our $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. Thus, we must consider two types of $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. This is why each row of the second table on page 52 has 2 states with different Pauli matrices in front. Each of these 2 states contributes 8 permutations so we have 16 fermions for each row of the table.

But this is where I threw a spanner into the problem. For the physical Ramond sector states, don't we normally fix $s_{0}$ to equal 1/2. This is the case for flat spacetime and is explained on page 23 of Notes on D-branes. So now we have $s_{0}$ and $s_{4}$ fixed leaving permutations of $s_{1}, s_{2}, s_{3} = 1/2, -1/2$. This would give us 8 states per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ and after GSO projecting half the states out we'd be left with 4 per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. The result: I'd expect 8 fermions per row of the 2nd table on page 52 of our original source.

My interpretation seems to be supported by page 30 of UV/IR Mixing in Noncommutative Field Theories and Open Closed String Duality which says we have 4 copies of each spinor. Note: It ignores $s_{0}$. I'm guessing they do this as it is fixed by the physicality condition.

Sorry for the length of my description and thanks in advance.

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I think there is another piece of evidence that may help answer this question, for those (if any!) that are interested. The theory of a D3-brane at the orbifold singularity produces the 4d field content of 2 N=2 vector multiplets and 2 hypermultiplets. The bosonic spectrum thus consists of 2 gauge fields, 4 scalars corresponding to Higgs fields and a further 8 scalars. As a result we get 16 bosonic degrees of freedom. We also get 16 fermionic degrees of freedom as the two need to match. Fixing $s_{0} = 1/2$ would halve the 32 fermionic d.o.f of hep-th/0312070, pg 52 to give this. –  Siraj R Khan Mar 6 '13 at 13:08