Conformal Transformations on Operators

Action on Primary Operators

In the previous lesson, we saw the action of the generators on a primary operator placed at the origin, $x=0$. The real utility, however, comes from knowing how they act at an arbitrary point $x$. The full expressions can be derived from the definitions at the origin combined with the conformal algebra, but for our purposes, we can take the results directly as a starting point for our computations.

Let's summarize the key results from Eq. (3.17) for a primary operator $\mathcal{O}(x)$ with scaling dimension $\Delta$ and spin representation matrices ${\Sigma }_{\mu \nu }$:

• Translations ($P_{\mu }$):
  $[P_{\mu },\mathcal{O}(x)]=-i{\partial }_{\mu }\mathcal{O}(x)$
  This is the standard expression from QFT, showing that $P_{\mu }$ generates translations.

• Rotations & Boosts ($M_{\mu \nu }$):
  $[M_{\mu \nu },\mathcal{O}(x)]=-i(x_{\mu }{\partial }_{\nu }-x_{\nu }{\partial }_{\mu })\mathcal{O}(x)-i{\Sigma }_{\mu \nu }\mathcal{O}(x)$
  The first part is the orbital angular momentum, describing how the operator's position is rotated. The second part, involving ${\Sigma }_{\mu \nu }$, is the intrinsic spin contribution.

• Dilatations ($D$):
  $[D,\mathcal{O}(x)]=-i(x^{\mu }{\partial }_{\mu }+\Delta )\mathcal{O}(x)$
  This shows that the dilatation generator acts with a combination of scaling the coordinate ($x^{\mu }{\partial }_{\mu }$) and scaling the operator itself by its dimension $\Delta$.

• Special Conformal Transformations ($K_{\mu }$):
  $[K_{\mu },\mathcal{O}(x)]=-i(2x_{\mu }(x\cdot \partial )-x^2{\partial }_{\mu })\mathcal{O}(x)-i(2\Delta x_{\mu }+2x^{\lambda }{\Sigma }_{\lambda \mu })\mathcal{O}(x)$
  This is the most complex one, but you can see how it combines pieces that look like translations (${\partial }_{\mu }$), dilatations ($\Delta$), and rotations (${\Sigma }_{\lambda \mu }$), all with position-dependent parameters. Notice how at $x=0$, this entire expression vanishes, recovering the definition of a primary operator.

Action on Descendant Operators

Now that we have the rules for primaries, how do we handle descendants? A descendant is simply a derivative of a primary, like $\mathcal{D}(x)={\partial }_{\nu }\mathcal{O}(x)$. We can compute the action of a generator $G$ on $\mathcal{D}(x)$ by using the Leibniz rule for commutators:
$[G,{\partial }_{\nu }\mathcal{O}(x)]={\partial }_{\nu }[G,\mathcal{O}(x)]+[G,{\partial }_{\nu }]\mathcal{O}(x)$
The first term involves the action on the primary (which we know), and the second is a commutator of two differential operators. Let's work through a concrete example.

Example: Action of Dilatation on a Descendant

Let's compute $[D,{\partial }_{\nu }\mathcal{O}(x)]$ for a scalar primary $\mathcal{O}$ with dimension $\Delta$.

1. First term: ${\partial }_{\nu }[D,\mathcal{O}(x)]$
   Using the formula above, this is ${\partial }_{\nu }[-i(x\cdot \partial +\Delta )\mathcal{O}(x)]$.

2. Second term: $[D,{\partial }_{\nu }]\mathcal{O}(x)$
   The generators are differential operators: $D=-i{x}^{\rho }{\partial }_{\rho }$ and $P_{\nu }=-i{\partial }_{\nu }$. We are interested in the action of $D$ and ${\partial }_{\nu }$, so we can ignore the factors of $-i$ for the commutator of operators.
   $[x^{\rho }{\partial }_{\rho },{\partial }_{\nu }]=x^{\rho }[{\partial }_{\rho },{\partial }_{\nu }]+[x^{\rho },{\partial }_{\nu }]{\partial }_{\rho }=0+(-{\delta }_{\nu }^{\rho }){\partial }_{\rho }=-{\partial }_{\nu }$
   So, $[D,{\partial }_{\nu }]=[-i{x}^{\rho }{\partial }_{\rho },-i{\partial }_{\nu }]=-[x^{\rho }{\partial }_{\rho },{\partial }_{\nu }]=-(-{\partial }_{\nu })={\partial }_{\nu }$.
   Wait, let's be careful with the factors of $-i$.
   The action of the generator is $G\cdot \varphi =[G,\varphi ]$. The differential operator representing $D$ is $-i(x\cdot \partial +\Delta )$. So we want to compute the commutator of the operator expressions for the action. Let's stick to the generators themselves. $[D,{\partial }_{\nu }]$ as operators on the fields is what we need. Since ${\partial }_{\nu }=i{P}_{\nu }$, we can use the algebra:
   $[D,P_{\nu }]=-i{P}_{\nu }$.
   So, $[D,{\partial }_{\nu }\mathcal{O}]=[D,i[P_{\nu },\mathcal{O}]]$. This gets complicated.

   Let's return to the Leibniz rule, which is more direct. We need the commutator of the differential operators that represent the action.
   Action of D: ${\delta }_D\mathcal{O}=-(x\cdot \partial +\Delta )\mathcal{O}$
   Action of ${\partial }_{\nu }$: ${\delta }_{{\partial }_{\nu }}\mathcal{O}={\partial }_{\nu }\mathcal{O}$
   The commutator of these actions is:
   $[{\delta }_D,{\partial }_{\nu }]\mathcal{O}=-[(x\cdot \partial +\Delta ),{\partial }_{\nu }]\mathcal{O}=-[x\cdot \partial ,{\partial }_{\nu }]\mathcal{O}=-(-{\partial }_{\nu })\mathcal{O}={\partial }_{\nu }\mathcal{O}$
   Combining the two terms:
   \begin{align*} [D, \partial_\nu \mathcal{O}] &= \partial_\nu[D, \mathcal{O}] - i(\partial_\nu \mathcal{O}) \ &= \partial_\nu[-i(x\cdot\partial + \Delta)\mathcal{O}] - i\partial_\nu \mathcal{O} \ &= -i[\partial_\nu(x\cdot\partial)\mathcal{O} + \Delta\partial_\nu\mathcal{O}] - i\partial_\nu\mathcal{O} \ &= -i[(x\cdot\partial)\partial_\nu\mathcal{O} + \partial_\nu\mathcal{O} + \Delta\partial_\nu\mathcal{O}] - i\partial_\nu\mathcal{O} \ &= -i[(x\cdot\partial)\partial_\nu\mathcal{O} + (\Delta+1)\partial_\nu\mathcal{O}] \ &= -ix\cdot\partial + (\Delta+1) \end{align*}
   This is exactly the correct transformation law for an operator $\mathcal{D}={\partial }_{\nu }\mathcal{O}$ with scaling dimension $\Delta +1$. The method works!

The 2D Perspective: Action of Virasoro Generators

For 2D CFTs, the infinite-dimensional conformal algebra allows for an even more elegant and powerful method. The conformal generators are the Laurent modes of the stress-energy tensor, $L_n$ and ${\tilde{L}}_n$. Their action on any operator $\mathcal{O}(w)$ can be computed via a contour integral of the Operator Product Expansion (OPE).

The action of $L_n$ on $\mathcal{O}(w)$ is defined by the commutator, which can be expressed as:
$[L_n,\mathcal{O}(w)]={\oint }_w\frac{dz}{2\pi i}(z-w{)}^{n+1}T(z)\mathcal{O}(w)$
where the contour encloses the point $w$. Your familiarity with complex analysis makes this a potent computational tool.

Let's compute the action of the first few Virasoro generators on a primary operator $\mathcal{O}(w)$ with weight $(h,\tilde{h})$. Its OPE with the stress tensor is:
$T(z)\mathcal{O}(w,\bar{w})=\frac{h\mathcal{O}(w,\bar{w})}{(z-w{)}^2}+\frac{{\partial }_w\mathcal{O}(w,\bar{w})}{z-w}+\text{non-singular terms}$

• $L_{-1}$ (Translation):
  $[L_{-1},\mathcal{O}(w)]={\oint }_w\frac{dz}{2\pi i}T(z)\mathcal{O}(w)={\text{Res}}_{z=w}\left(\frac{h\mathcal{O}}{(z-w{)}^2}+\frac{{\partial }_w\mathcal{O}}{z-w}\right)={\partial }_w\mathcal{O}(w)$

• $L_0$ (Dilatation + Rotation):
  $[L_0,\mathcal{O}(w)]={\oint }_w\frac{dz}{2\pi i}(z-w)T(z)\mathcal{O}(w)={\text{Res}}_{z=w}\left(\frac{h\mathcal{O}}{z-w}+\dots \right)=h\mathcal{O}(w)$

• $L_{n\ge 1}$:
  $[L_n,\mathcal{O}(w)]={\oint }_w\frac{dz}{2\pi i}(z-w{)}^{n+1}T(z)\mathcal{O}(w)=0$
  The integral is zero because for $n\ge 1$, the term $(z-w{)}^{n+1}$ cancels all the poles in the OPE, leaving a holomorphic integrand, which gives zero by Cauchy's theorem. This is the 2D equivalent of $[K_{\mu },\mathcal{O}(0)]=0$.

Action on 2D Descendants
The true power of this method is revealed when acting on descendants. A descendant state is created by acting with raising operators, e.g., $L_{-k}|h⟩$ for $k>0$. An $n$-th level descendant operator corresponds to acting $n$ times, e.g., $\mathcal{D}(w)=L_{-k_1}\dots L_{-k_n}\mathcal{O}(w)$.

To compute the action of $L_m$ on such a descendant, we simply use the Virasoro algebra:
$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m){\delta }_{m+n,0}$
Example: Compute the action of $L_1$ on the first descendant $L_{-1}\mathcal{O}(w)$.
\begin{align*}
[L_1, L_{-1}\mathcal{O}(w)] &= [L_1, L_{-1}]\mathcal{O}(w) + L_{-1}[L_1, \mathcal{O}(w)] \
&= ((1 - (-1))L_{1-1})\mathcal{O}(w) + L_{-1}(0) \quad (\text{since }\mathcal{O} \text{ is primary}) \
&= 2L_0 \mathcal{O}(w) \
&= 2h \mathcal{O}(w)
\end{align*}
This is a remarkably efficient and purely algebraic way to carry out the computation.

Conclusion

Today we have built a practical toolkit for calculating the effect of conformal transformations on the operators of a CFT. This is the engine room of the theory, allowing us to manipulate operators and, eventually, correlation functions.

Key Takeaways:

• In any dimension, the action of a conformal generator on a primary operator at a point $x$ is given by a specific differential operator, as listed in Rychkov's notes Eq. (3.17).
• The action on a descendant operator can be computed systematically by applying the Leibniz rule for commutators, reducing the problem to the known action on primaries and the algebra of generators.
• In 2D, this process is streamlined. The action of Virasoro generators $L_n$ on any operator is found by computing a contour integral of its OPE with the stress tensor.
• For 2D descendants, the action can be computed purely algebraically by commuting the Virasoro generators through each other until they act on the primary operator.

In this lesson and the last, we have treated states and operators as related but distinct concepts. Our next lesson will forge the final, crucial link between them by introducing the state-operator correspondence, a profound isomorphism that lies at the heart of CFT.