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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 $\...


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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 ...


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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.


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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{\...


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The character of a representation of the Virasoro algebra depends on the whole structure of the representation, not just of the conformal dimension $h$ 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 ...


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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 ...


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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 ...


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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 ...


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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 ...


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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}=...


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