I am reading Polchinski's Vol. 1 on String Theory, and I have some basic doubts on how he introduces the $bc$ conformal field theory (see section 2.7, page 61).

He basically starts from the anticommutators of the $b_n$ and $c_n$ operators, $\{b_m,c_n\}=\delta_{m,-n}$; he then considers first "the states that are annihilated by all of the $n>0$ operators. The $b_0$, $c_0$ oscillator algebra generates two such ground states $\left| \uparrow \right>$ and $\left| \downarrow \right>$"

Do those states necessarily exist? What is special about $n>0$? If he is using only the anticommutation relations as assumptions, probably I am missing some steps.

  • $\begingroup$ 1. These states exist definitionally, 2. The anticommutativity of the modes is because the ghost fields must anticommute for the Faddeev-Popov procedure to work, 3. The states annihilated by the operators with $n>0$ are important because they are used to construct a unique representative for each ghost cohomology class $\endgroup$ Sep 27 at 11:28

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