Global Conformal Generators and Descendant States

The previous lesson established the Virasoro algebra and the Schwarzian transformation law of the stress tensor. We now zoom in on the part of that structure that is already visible before using the full infinite-dimensional algebra: the global conformal generators. On the Riemann sphere these are $L_{-1}$, $L_0$, and $L_1$, corresponding to translations, dilatations/rotations, and special conformal transformations.

This page has two intertwined goals. The first is to derive the global Ward identities that follow from the invariance of the vacuum. These identities are the quickest way to understand why two- and three-point functions in a two-dimensional CFT have their familiar fixed forms. The second is to translate the same contour logic into the state language of radial quantization: primary states sit at the top of representation towers, and negative Virasoro modes generate descendants.

A final section rewrites the same idea in the more general language of conserved currents. This is not a detour. It explains why the formula $\langle 0|[Q,\mathcal O]|0\rangle=0$ is harmless when the symmetry is unbroken, but becomes a massless-particle statement when the vacuum is not invariant.

The three global generators

Holomorphic conventions

We work in the holomorphic sector, with

$$L_n={1\over 2\pi i}\oint dz\,z^{n+1}T(z), \qquad T(z)=\sum_{n\in\mathbb Z}L_n z^{-n-2},$$

and

$$[L_n,L_m]=(n-m)L_{n+m}+{c\over 12}n(n^2-1)\delta_{n+m,0}.$$

For a holomorphic primary field $\phi(z)$ of weight $h$,

$$T(z)\phi(w)\sim {h\phi(w)\over (z-w)^2}+{\partial\phi(w)\over z-w}.$$

The antiholomorphic sector is obtained by adding bars everywhere.

The infinitesimal conformal transformation generated by a holomorphic vector field $\epsilon(z)\partial_z$ acts on a primary field by

$$\delta_\epsilon\phi(z) = {1\over 2\pi i}\oint_z dw\,\epsilon(w)T(w)\phi(z) = \epsilon(z)\partial\phi(z)+h\partial\epsilon(z)\phi(z).$$

Choosing $\epsilon(z)=z^{n+1}$ gives the mode action

$$\boxed{ [L_n,\phi(z)] = \left(z^{n+1}\partial_z+(n+1)h z^n\right)\phi(z). }$$

The modes $L_{-1}$, $L_0$, and $L_1$ correspond to

$$\begin{aligned} [L_{-1},\phi(z)]&=\partial\phi(z),\\ [L_0,\phi(z)]&=(z\partial+h)\phi(z),\\ [L_1,\phi(z)]&=(z^2\partial+2hz)\phi(z). \end{aligned}$$

They close among themselves:

$$[L_0,L_{-1}]=L_{-1}, \qquad [L_0,L_1]=-L_1, \qquad [L_1,L_{-1}]=2L_0.$$

This is the global conformal algebra $sl_2$. The central term vanishes on this subalgebra because $n(n^2-1)=0$ for $n=-1,0,1$. That is why the global conformal group acts without the Schwarzian anomaly discussed on the previous page.

The globally defined holomorphic vector fields on the Riemann sphere are generated by $\epsilon(z)=1,z,z^2$. On a primary field they act as $\partial_z$, $z\partial_z+h$, and $z^2\partial_z+2hz$.

The word “global” is doing real work. A general holomorphic function $\epsilon(z)$ is locally allowed in two dimensions, but on the compact Riemann sphere a globally nonsingular holomorphic vector field can have at most a quadratic polynomial coefficient. The infinite Virasoro algebra is local; the $sl_2$ algebra is globally well-defined on the sphere.

Ward identities from vacuum invariance

In radial quantization the vacuum is invariant under the global conformal group:

$$L_{-1}|0\rangle=L_0|0\rangle=L_1|0\rangle=0,$$

and similarly for the bra vacuum. Therefore, for a product of primary fields,

$$0= \langle 0|[L_n,\phi_1(z_1)\cdots\phi_N(z_N)]|0\rangle, \qquad n=-1,0,1.$$

Using the commutator as a derivation gives the global conformal Ward identities

$$\boxed{ \sum_{i=1}^N \left(z_i^{n+1}\partial_{z_i}+(n+1)h_i z_i^n\right) \langle \phi_1(z_1)\cdots\phi_N(z_N)\rangle=0, \qquad n=-1,0,1. }$$

Equivalently, insert the charge

$$Q_\epsilon={1\over 2\pi i}\oint dz\,\epsilon(z)T(z)$$

on a contour surrounding all insertions. Because the charge annihilates the vacuum, the large contour gives zero. Deforming it inward to small contours around the insertions produces the sum of residues above.

A global conformal generator annihilates the vacuum, so a contour surrounding all insertions gives zero. Deforming the contour to small circles around the insertions converts the statement into a sum of local transformations.

The case $n=-1$ is translation invariance:

$$\sum_i\partial_{z_i}G_N=0.$$

The case $n=0$ is scale and rotation covariance:

$$\sum_i(z_i\partial_{z_i}+h_i)G_N=0.$$

The case $n=1$ is special conformal covariance:

$$\sum_i(z_i^2\partial_{z_i}+2h_i z_i)G_N=0.$$

These three equations already fix the holomorphic dependence of two- and three-point functions. For example, translation invariance implies that a two-point function depends only on $z_{12}=z_1-z_2$. Scaling then gives

$$G_{12}(z_1,z_2)={C_{12}\over z_{12}^{h_1+h_2}},$$

and the special conformal Ward identity forces $h_1=h_2$ unless $C_{12}=0$. Thus, more precisely,

$$\boxed{ \langle \phi_1(z_1)\phi_2(z_2)\rangle ={C_{12}\over (z_1-z_2)^{2h_1}}, \qquad C_{12}=0\ \text{unless}\ h_1=h_2. }$$

Equal weights are necessary, not sufficient: internal quantum numbers and the choice of operator basis determine the matrix $C_{12}$.

Similarly, three primary fields obey

$$\boxed{ \langle \phi_1(z_1)\phi_2(z_2)\phi_3(z_3)\rangle = {C_{123}\over z_{12}^{h_1+h_2-h_3} z_{13}^{h_1+h_3-h_2} z_{23}^{h_2+h_3-h_1}} }$$

in the holomorphic sector. The constants $C_{123}$ are dynamical CFT data. Symmetry fixes the coordinate dependence, but it does not determine which fields exist or what their OPE coefficients are.

Primary states and descendants

The state–operator correspondence assigns to a local operator the state

$$|\phi\rangle=\lim_{z\to0}\phi(z)|0\rangle.$$

If $\phi$ is primary of holomorphic weight $h$, the commutator formula immediately implies

$$L_n|\phi\rangle=0\quad(n>0), \qquad L_0|\phi\rangle=h|\phi\rangle.$$

The primary state is therefore a highest-weight state for the Virasoro algebra. Negative modes generate descendant states:

$$L_{-n_1}L_{-n_2}\cdots L_{-n_k}|\phi\rangle, \qquad n_i>0.$$

The level of this descendant is

$$N=n_1+n_2+\cdots+n_k,$$

and its $L_0$ eigenvalue is $h+N$, because

$$[L_0,L_{-n}]=nL_{-n}.$$

The simplest descendant is the translation descendant:

$$L_{-1}|\phi\rangle = \partial\phi(0)|0\rangle.$$

This is the state-language version of the local commutator $[L_{-1},\phi(z)]=\partial\phi(z)$.

A primary state $|h\rangle$ is annihilated by all $L_n$ with $n>0$. Negative modes generate a descendant tower. In the identity module the global descendants vanish; the first state not forced to vanish kinematically is $L_{-2}|0\rangle$, represented by the stress tensor.

At a fixed level, there can be several descendants. At level $2$, for example, a generic highest-weight module contains

$$L_{-2}|h\rangle, \qquad L_{-1}^2|h\rangle.$$

They have the same scaling dimension $h+2$, so they can mix under changes of basis. They need not both be physical independent states; null vectors may appear at special values of $h$ and $c$. That is the entry point to BPZ equations and minimal models on the next pages.

The identity module and the stress tensor

The vacuum corresponds to the identity operator. Since the identity has $h=0$ and is constant,

$$L_{-1}|0\rangle=0.$$

Global conformal invariance also gives

$$L_0|0\rangle=L_1|0\rangle=0.$$

More generally, regularity of the stress tensor at the origin when acting on the vacuum implies

$$L_n|0\rangle=0, \qquad n\ge -1.$$

The first Virasoro descendant not forced to vanish by global invariance is therefore

$$L_{-2}|0\rangle.$$

This state is the stress tensor state. Indeed, from the mode expansion,

$$T(z)|0\rangle = \sum_n L_n z^{-n-2}|0\rangle =L_{-2}|0\rangle+zL_{-3}|0\rangle+z^2L_{-4}|0\rangle+\cdots,$$

so

$$T(0)|0\rangle=L_{-2}|0\rangle.$$

Its norm is controlled by the central charge:

$$\langle 0|L_2L_{-2}|0\rangle = \langle 0|[L_2,L_{-2}]|0\rangle = {c\over2}.$$

This is the state version of the two-point function

$$\langle T(z)T(w)\rangle={c/2\over (z-w)^4}.$$

Thus $c$ fixes the normalization of the stress-tensor excitation above the vacuum; it should not in general be read as a literal count of fields. Reflection positivity in a unitary CFT implies $c\ge0$, and $c=0$ forces the stress-tensor state to be null.

Lorentzian current language

The same algebraic statement can be written in Lorentzian coordinates. Let

$$x^\pm=x^0\pm x^1.$$

In a two-dimensional CFT, tracelessness and conservation imply that the stress tensor splits into chiral components,

$$\partial_-T_{++}=0, \qquad \partial_+T_{--}=0,$$

away from operator insertions. The right-moving conformal charges are schematically

$$Q_\epsilon(x^-)=\int dx^+\,\epsilon(x^+)T_{++}(x^+,x^-).$$

The conservation equation says that $Q_\epsilon$ is independent of $x^-$ when no insertion is crossed. When the contour crosses an operator, the conservation equation acquires contact terms. These contact terms are exactly the Lorentzian version of the OPE residues:

$$[Q_\epsilon,\phi(x^+)] =\epsilon(x^+)\partial_+\phi(x^+)+h\partial_+\epsilon(x^+)\phi(x^+).$$

For $\epsilon=1$, this says that $Q_1$ is the momentum generator along $x^+$; for $\epsilon=x^+$, it is a scale/boost generator; for $\epsilon=(x^+)^2$, it is a special conformal generator. The holomorphic contour formalism is therefore not a trick detached from real time. It is an efficient Euclidean encoding of current conservation plus equal-time commutators.

Momentum-space Ward identities and soft limits

For an ordinary internal symmetry with conserved current $J^\mu$, write $D$ for spacetime dimension. In the convention

$$\delta\mathcal O_k=i e_k\mathcal O_k,$$

the Euclidean local Ward identity has the form

$$\partial_\mu\langle J^\mu(x)\mathcal O_1(x_1)\cdots\mathcal O_N(x_N)\rangle = -i\sum_{k=1}^N e_k\delta^{(D)}(x-x_k) \langle \mathcal O_1\cdots\mathcal O_N\rangle.$$

Define the current-inserted correlator $G_J^\mu(q;p_1,\ldots,p_N)$ with Fourier phase $e^{-iq\cdot x-i\sum_k p_k\cdot x_k}$. Fourier transformation then gives

$$\boxed{ q_\mu G_J^\mu(q;p_1,\ldots,p_N) = -\sum_{k=1}^N e_k\,G(p_1,\ldots,p_k+q,\ldots,p_N). }$$

Reversing the Fourier phase or changing the generator convention reverses the displayed overall sign, but not the momentum-shift structure. For the amputated proper vertex of a field with charge $e$, the corresponding Ward–Takahashi identity is

$$q_\mu\Gamma^\mu(p+q,p) =e\left[S^{-1}(p+q)-S^{-1}(p)\right].$$

The Fourier transform of current conservation turns the divergence of a current insertion into a sum over contact terms. In momentum space, contact terms appear as charge actions on the external insertions.

After continuation back to Lorentzian signature, integrate the local Ward identity over space. The left-hand side becomes a commutator with the charge,

$$Q=\int d^{D-1}x\,J^0(t,\mathbf x),$$

so formally

$$\langle0|[Q,\mathcal O_1\cdots\mathcal O_N]|0\rangle = \sum_k\langle0|\mathcal O_1\cdots[Q,\mathcal O_k]\cdots\mathcal O_N|0\rangle.$$

If the symmetry is unbroken, then

$$Q|0\rangle=0,$$

and the Ward identity is a selection rule. If the vacuum is not invariant, the same formula cannot be interpreted by simply moving $Q$ through the correlator and killing the vacuum. In infinite volume the global charge may itself fail to exist as a normalizable operator, so the precise argument uses a regulated charge $Q_R$ or, equivalently, the local current Ward identity.

For a Lorentz-invariant theory with a spontaneously broken continuous global symmetry, the current has a soft massless pole. With the one-particle normalization absorbed into $F$,

$$\langle0|J_\mu(0)|\pi(q)\rangle=iFq_\mu,$$

and current exchange contributes

$$\langle J_\mu(q)\mathcal O(-q)\rangle \sim {Fq_\mu\over q^2+i\epsilon}\langle \pi(q)|\mathcal O(0)|0\rangle.$$

Multiplication by $q^\mu$ cancels the pole and leaves a finite soft contribution. This is the Ward-identity core of Goldstone’s theorem.

For an unbroken symmetry, $Q|0\rangle=0$ and the integrated Ward identity gives selection rules. For a broken continuous global symmetry, regulated charges have soft spectral weight and the current correlator contains a Goldstone pole.

This comparison is useful for conformal field theory. The global conformal generators annihilate the CFT vacuum, so their Ward identities are honest constraints on correlators. Negative Virasoro modes acting on a primary do not break the vacuum symmetry; they generate local descendant states inside a representation. Broken internal charges are different: the current has physical soft spectral weight. Similar-looking commutators therefore have different physical meanings depending on the state of the vacuum.

The hypotheses matter. The symmetry must be genuine and global rather than gauged, the infinite-volume limit must be taken, and the current must be anomaly-free. Moreover, under the standard locality and spectral assumptions a continuous internal symmetry cannot spontaneously break in $1+1$ dimensions (Coleman’s theorem). On this two-dimensional CFT page, the Goldstone discussion is therefore a comparison with higher-dimensional relativistic QFT, not a claim that the CFT vacuum breaks such a symmetry.

Summary

The global conformal generators $L_{-1}$, $L_0$, and $L_1$ form an $sl_2$ subalgebra with no central extension. Their action on a primary field is

$$[L_n,\phi(z)]=\left(z^{n+1}\partial_z+(n+1)h z^n\right)\phi(z), \qquad n=-1,0,1.$$

Vacuum invariance turns this local transformation rule into global Ward identities for correlation functions. These identities fix the coordinate dependence of two- and three-point functions and are the first layer of conformal kinematics.

Radial quantization repackages the same information in representation language. A primary state obeys $L_n|h\rangle=0$ for $n>0$, while negative modes generate descendants. The vacuum module is special: global descendants vanish, and the first descendant not kinematically excluded is $L_{-2}|0\rangle=T(0)|0\rangle$, whose norm is $c/2$.

Finally, the same Ward-identity logic appears for ordinary conserved currents. If a well-defined charge annihilates the vacuum, integrated Ward identities become selection rules. In a phase with spontaneous breaking, the local current Ward identity instead requires soft massless spectral weight. This physical Goldstone mechanism is distinct from the representation-theoretic null relations studied next.

Common pitfalls

The most common sign trap is the relation between vector fields and mode labels. With the convention $L_n={1\over2\pi i}\oint z^{n+1}T$, the action on primaries is $[L_n,\phi]=(z^{n+1}\partial+(n+1)hz^n)\phi$, and the algebra is $[L_n,L_m]=(n-m)L_{n+m}+\cdots$. Some texts absorb a minus sign into the vector field basis.

A second pitfall is to confuse global and local conformal transformations. The modes $L_{-1},L_0,L_1$ are globally defined on the sphere and have no Schwarzian anomaly. Generic $L_n$ are local conformal generators in a coordinate patch; they are essential in radial quantization, but they are not all globally nonsingular transformations of the sphere.

Finally, descendants are not automatically independent. At special values of $h$ and $c$, linear combinations of descendants can be null. Those null states are not optional decoration; they are what make minimal models exactly solvable.

A final pitfall is to treat $Q|0\rangle\ne0$ literally in infinite volume. The global charge often has infrared-divergent norm in a broken phase; regulated charges and local Ward identities are the safe formulation of the Goldstone argument.

Exercises

Exercise 1: Global two-point function

Use the global Ward identities to derive the holomorphic two-point function of two primary fields.

Solution

Let

$$G(z_1,z_2)=\langle \phi_1(z_1)\phi_2(z_2)\rangle.$$

Translation invariance gives

$$(\partial_{z_1}+\partial_{z_2})G=0,$$

so $G=f(z_{12})$, where $z_{12}=z_1-z_2$. Scale covariance gives

$$(z_1\partial_{z_1}+z_2\partial_{z_2}+h_1+h_2)G=0.$$

Since $z_1\partial_{z_1}+z_2\partial_{z_2}=z_{12}\partial_{z_{12}}$, this implies

$$f(z_{12})=Cz_{12}^{-(h_1+h_2)}.$$

The special conformal identity gives

$$\left(z_1^2\partial_{z_1}+z_2^2\partial_{z_2}+2h_1z_1+2h_2z_2\right)G=0.$$

Using the form above, this equation reduces to

$$(h_1-h_2)(z_1-z_2)G=0.$$

Thus either $G=0$ or $h_1=h_2$. Therefore

$$\langle \phi_1(z_1)\phi_2(z_2)\rangle ={C_{12}\over z_{12}^{2h_1}}, \qquad C_{12}=0\ \text{unless}\ h_1=h_2.$$

Exercise 2: Stress-tensor state norm

Show that the norm of the stress-tensor state $L_{-2}|0\rangle$ is $c/2$.

Solution

Using radial quantization, $L_n^\dagger=L_{-n}$. Hence

$$\|L_{-2}|0\rangle\|^2 =\langle0|L_2L_{-2}|0\rangle.$$

Since $L_2|0\rangle=0$, we may replace $L_2L_{-2}$ by the commutator:

$$\langle0|L_2L_{-2}|0\rangle =\langle0|[L_2,L_{-2}]|0\rangle.$$

The Virasoro algebra gives

$$[L_2,L_{-2}]=4L_0+{c\over12}2(2^2-1)=4L_0+{c\over2}.$$

Because $L_0|0\rangle=0$,

$$\|L_{-2}|0\rangle\|^2={c\over2}.$$

This agrees with the normalization of $\langle T(z)T(w)\rangle$.

Exercise 3: Translation Ward identity

Let $G_N=\langle\phi_1(z_1)\cdots\phi_N(z_N)\rangle$. Derive the translation Ward identity directly from $L_{-1}|0\rangle=0$ and $[L_{-1},\phi_i]=\partial_i\phi_i$.

Solution

Since the vacuum is translation invariant,

$$\langle0|[L_{-1},\phi_1(z_1)\cdots\phi_N(z_N)]|0\rangle=0.$$

The commutator is a derivation:

$$[L_{-1},\phi_1\cdots\phi_N] = \sum_{i=1}^N \phi_1\cdots [L_{-1},\phi_i]\cdots\phi_N.$$

Using $[L_{-1},\phi_i]=\partial_{z_i}\phi_i$ gives

$$0= \sum_{i=1}^N \partial_{z_i}\langle\phi_1(z_1)\cdots\phi_N(z_N)\rangle.$$

Therefore

$$\boxed{\sum_i\partial_{z_i}G_N=0.}$$

This is the statement that a simultaneous translation of all insertion points does not change the correlator.

Exercise 4: Goldstone pole and finite divergence

In a spacetime dimension and setting where continuous symmetry breaking is allowed, assume a conserved current couples to a Goldstone boson as $\langle0|J_\mu(0)|\pi(q)\rangle=iFq_\mu$. Explain why a current correlator can have a finite divergence even though it contains a massless pole.

Solution

A Goldstone intermediate state contributes to a current correlator schematically as

$$\langle J_\mu(q)\mathcal O(-q)\rangle \sim {iFq_\mu\over q^2+i\epsilon}\langle\pi(q)|\mathcal O(0)|0\rangle.$$

Taking the divergence gives

$$q^\mu\langle J_\mu(q)\mathcal O(-q)\rangle \sim {iFq^2\over q^2+i\epsilon}\langle\pi(q)|\mathcal O(0)|0\rangle.$$

The factor $q^2$ from the divergence cancels the massless propagator, in the distributional sense appropriate to the Ward identity. Provided the remaining matrix element has a nonzero soft limit, the result approaches a finite nonzero contact term. Thus the current is conserved away from insertions while its correlator still records the broken symmetry through massless spectral weight.

References

• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996), Chapters 4–5.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer (1997), Chapters 5–6.
• P. Ginsparg, “Applied Conformal Field Theory,” in Fields, Strings and Critical Phenomena, Les Houches Session XLIX, Elsevier (1989), pp. 1–168.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers (1987), Chapter 9.
• S. Weinberg, The Quantum Theory of Fields, Volume II: Modern Applications, Cambridge University Press (1996), chapters on Ward identities and spontaneous symmetry breaking.