# Existence of ground states in $bc$ CFT

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.

• 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 Sep 27 at 11:28