# Central Charge of large $N$ Gauge theory in 't Hooft limit

It is well known that large N gauge theory in t'Hooft limit has central charge ~ $$N^2$$
I want to convince myself in this by considering simple example of:

1 flavor(meaning that we have only one generation of quarks)
strongly coupled ($$g>>m_q$$)
QCD in 1+1 dim
in t'Hooft limit ($$N_c \to \infty, g^2N_c =const$$)
with action given by: $$S=\int d^2x[\sum_{a=1}^{N_c} \overline\psi^{a}(\gamma_{\mu}D^{\mu}+m_q)\psi_a^{\mu} - \sum_{a,b=1}^{N_c}\frac{1}{2}F^{a}_{\mu\nu b} F^{\mu\nu b}_{a}]$$ where
$$F^{a}_{\mu \nu b} = \partial_{\mu} A^{a}_{b \nu} - \partial_{\nu} A^{b}_{a \mu} + i g[A_{\mu},A_{\nu}]^a_{b}$$ $$D_\mu \psi^{ a} = \partial_{\mu} \psi^{ a}+gA^{a}_{b \mu} \psi^{b}$$ (I believe that)In strongly coupled limit $$m_q \to 0$$

One way to obtain central charge would be to evaluate OPE of T with itself and look at term proportional to $$\frac{1}{(z-w)^{4}}$$. Yet so far this approach looks to me too complicated. Is there any better one? Or should I brace myself and go through calculations?

• may be it is possible to get the 2pt function for T just by appealing to conformal symmetry constraints. Jul 31 '17 at 18:57