y order in a semi-classical expansion. We will show that the knowledge of the poles in these OPEs is enough to fix all the subleading terms. The idea driving the bootstrap is to ask for the compatibility of the elementary OPEs with both current conservation and the Maurer-Cartan equation.

Current-current OPEs {#current-current-opes .unnumbered}
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First let us consider the current-current OPEs. Current conservation gives the first constraints: j\^a\_[L,z]{}(z) = 0 j\^a\_[L,|z]{}(z) = 0. The first line implies a one-to-one correspondence between the terms in the $j^a_{L,z} j^b_{L,z}$ and $j^a_{L,z} j^b_{L,\bar z}$ OPEs. The second line then links the $j^a_{L,\bar z}
j^b_{L,\bar z}$ and the $j^a_{L,z} j^b_{L,\bar z}$ OPEs. These OPEs are expected to vanish up to contact terms. Indeed the same OPEs code the Ward identity for the global symmetry $G_L$. It follows that the contact terms in these OPEs are given by the transformation properties of the left current under the left action of the group on itself [^4].

The second constraint comes from the Maurer-Cartan equation : j\^a\_[L,z]{}(z) = 0. Contact terms in this OPE should vanish. Using current conservation and the fact that $c_++c_-=-f^{-2}$ we rewrite this constraint as : \[jmodMC\] j\^a\_[L,z]{}(z) |j\^b\_[L,z]{}(w) = f\^2 j\^a\_[L,z]{}(z) i [f\^b]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w). Thanks to the factor of $f^2$ on the right-hand side of the previous equation, it becomes manifest that the knowledge of the current algebra at a given order in $f^2$ will also determine the current algebra at the next order. The discussion of appendix \[XXOPEs\] shows that the terms appearing in the current-current OPEs at order $f^{2n}$ are composites of at most $n+1$ currents. This allows us to make an ansatz for the current-current OPE at higher-order. Then equation fixes the coefficients in this ansatz. This method is illustrated in appendix \[jMCOPE\] where we compute the current algebra up to order $f^2$.

Current-primary OPEs {#current-primary-o