A question about Virasoro algebra (skip disclaimer)
I have a question in Polchinski's string theory book volume 1 p54, related to the Virasoro algebra.
Introducing complex coordinates
$$w=\sigma^1 + i \sigma^2 $$ 
$$z=\exp (-i \omega) =\exp( -i \sigma^1 + \sigma^2) $$
and $$T_m=L_m - \delta_{m0} \frac{c}{24}$$
$$ \tilde{T}_m=\tilde{L}_m - \delta_{m0} \frac{\tilde{c}}{24}$$
It is said 

The Hamiltonian H of time translation in the $w=\sigma^1+i\sigma^2$ frame is
  $$H=\int_0^{2 \pi} \frac{ d \sigma^1}{2 \pi} T_{22} = L_0 + \tilde{L}_0 - \frac{ c +\tilde{c}}{24} (2.6.10) $$

My questions are  (sorry ask two questions in a thread, since they are closely related)
(i) Does $T_{22}$ mean $T_{zz}$ or $T_{\sigma^2 \sigma^2}$?
(ii) How the  anti holomorphic operator $\tilde{L}_0$ are obtained in Eq.  (2.6.10)?
 A: (1) The symbol $T_{22}$ obviously means what you call – unusually – $T_{\sigma_2\sigma_2}$. The letter $z$ has no relationship with the number $2$, except that they look similar so one could make a typo if his handwriting were bad – and the expression where $T_{22}$ appears uses $\sigma_1$ so it is obvious that these are the coordinates used in the expression.
(2) Pretty much all equations in section 2.6 are written twice so everything that is done for $L_n$ is also done for $\tilde L_n$. $\tilde L_0$ itself is introduced by nothing else that eqn (2.6.10) or, more generally for all generators $\tilde L_n$, by (2.6.5). The only equation that isn't doubled is (2.6.6); the formula for $\tilde L_n$ just has tildes and bars at the obvious places and changed sign of $i$ (or, equivalently, the direction of the contour integration).
A: For (ii) : 
Using  $(2.1.3)$,$(2.3.15b)$ , you get : 
$$T_{22} = -(T_{ww} + T_{\bar w \bar w} )$$
Now, looking at the expansion $(2.6.7a)$ and $(2.6.7b)$ ,we see that if we integrate $T_{ww}$ or $T_{\bar w \bar w}$ with $\sigma_1$ going from $0$ to $2\pi$, the only non-null terms are for $m=0$, so, finally : 
$H = \frac{1}{2\pi} \int d\sigma^1 T_{22} = -\frac{1}{2\pi}\int d\sigma^1(T_{ww} + T_{\bar w \bar w} )=(T_0+\tilde T_0)$  
By using $2.6.8$, which itself comes from $(2.6.9)$(that is $(2.4.26)$ with $z=e^{-i\omega}$), and the comparison of the expansions $2.6.5$ and $2.6.7 a b$, we have finally:
$$H = L_0 + \tilde L_0 -\frac{c+\tilde c}{24}$$
