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I will give yet a different, simpler, answer. All of these discussions are operator equations. Now as an operator equation $\bar{\partial}\partial X^\mu = \text{contact terms}$, see the discussion on p 35. But contact terms would arise from any operators inside the contour. The only such operator is the unit operator, and so there are no contact terms and $\... 4 A descendant is not something that has derivatives in it. It is something that is a total derivative of something else$\mathcal{O}' = \partial_\mu \mathcal{O}$. More precisely, the definition of a primary operator is $$K_\mu \mathcal{O}(x) = 0\,,$$ where$K_\mu$is the generator of special conformal transformations. You can prove that you operator is a ... 0 The S matrix of affine Lie algebra can be calculated by the Kac-Peterson formula, which can also be found in Theorem 13.8 in Kac's book. 0 I believe you are both missing something. As$z$is related to$w$by a conformal transformation you need to use (2.5.17) to relate$j_w(w)$and$j_z(z). This is a my derivation: \begin{align} N^\mathrm{g} = \frac{1}{2\pi i } \int_0^{2\pi}dw j(w) \end{align} From (2.5.17) we have \begin{align} (\partial_w z) j_z(z) = j_w(w) + \frac{2\lambda-1}{2} \frac{\... 3 The character of a representation of the Virasoro algebra depends on the whole structure of the representation, not just of the conformal dimensionh$of the highest-weight state. For a Verma module, the character is simply $$\chi_{c,h}^\text{Verma}(\tau) = \frac{q^{h+\frac{1-c}{24}}}{\eta(\tau)}$$ as you wrote. However, for generic$c$there are two ... 1 Indeed there is a version of the state operator correspondence that holds in 1d CFT/Quantum mechanics. Note that operator -> state map is trivially true in any general QFT in arbitrary dimensions. However, state -> operator map is the non-trivial bit that holds only in a CFT, and follows from scale invariance as depicted by the diagram in your question. In a ... 2 Your claim that (1.7) only considers infinitesimal conformal transformations with singularity at zero is simply wrong: The most general generator that you get from (1.7) is a linear combination of all the basic generators there, i.e. a Laurent series$\sum a_n z^n$. While the center of a Laurent series may be its only singularity, it need not be - it has an ... 4 The constant term is needed because on a compact manifold with periodic boundary conditions there is a zero mode in the spectrum of the laplacian. This is easier to see on the circle, where$\frac{d^2}{dt^2}f = \lambda f$has the periodic constant solution$f(t) =c$. This makes the operator not-inveritble. One should instead work in the space orthogonal to ... 7 When both "space" ($x$) and "time" ($y$) directions are periodic, the Laplacian on torus with coodinate$z=x+iy$has a normalized zero mode $$\varphi_0(z) = \frac 1 {\sqrt{{\rm Im}(\tau)}}$$ (Here$\tau$is the modular parameter defining the torus.) As $$-\nabla^2 \varphi_0=0.$$ the zero mode means that the Laplace operator is not 1-1 and so prevents ... 2 They have put all the anti-holomorphic dependence into$\bar{v}(\bar{z})$. So for holomorphic modes of stress energy tensor$[L_n, \bar{v}(\bar{z})]=0$. For anti-holomorphic generators$[\bar{L}_n, \bar{v}(\bar{z})]\neq0$. Field$X(z, \bar{z})$have holomorphic and anti-holomorphic parts in Laurent expansion (2.89). Then from equation (2.40) with$h=\bar{h}=...