Virasoro Generators, Descendants, and Central Charge

Radial quantization turns stress-tensor modes into more than notation. In two dimensions, the holomorphic stress tensor $T(z)$ generates an infinite-dimensional algebra of local conformal transformations. A primary field is the starting point of a representation of this algebra, and its descendants are obtained by acting with the negative modes $L_{-1},L_{-2},\ldots$.

There is one genuinely quantum ingredient: the stress tensor has a singular OPE with itself that contains a c-number fourth-order pole. The coefficient is the central charge $c$. It measures the short-distance strength of stress-tensor fluctuations, controls the central extension of the Virasoro algebra, and later becomes the coefficient of the Schwarzian derivative in the transformation law of $T$.

Required background. Lesson 22 supplies radial quantization, the state–operator map, and the stress-tensor mode convention used below.

Stress-tensor modes as local generators

Virasoro modes and central-charge normalization

We work in one holomorphic sector. The antiholomorphic sector has generators $\bar L_n$ and central charge $\bar c$. We normalize modes by

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

A primary field $\mathcal O$ of weights $(h,\bar h)$ obeys

$$T(z)\mathcal O(w,\bar w) \sim {h\mathcal O(w,\bar w)\over (z-w)^2} +{\partial_w\mathcal O(w,\bar w)\over z-w},$$

and similarly for $\bar T(\bar z)$. The central charge is normalized by

$$T(z)T(w) \sim {c/2\over (z-w)^4} +{2T(w)\over (z-w)^2} +{\partial T(w)\over z-w}.$$

The definition

$$L_n={1\over 2\pi i}\oint dz\,z^{n+1}T(z)$$

says that $L_n$ is a contour integral of the conserved holomorphic current associated with the vector field

$$v_n(z)=z^{n+1}\partial_z.$$

The three modes $L_{-1},L_0,L_1$ correspond to translations, dilatations/rotations, and special conformal transformations on the plane. The modes with all other $n$ generate local conformal transformations that are not globally well-defined on the Riemann sphere but are perfectly meaningful as contour operations around operator insertions.

A Virasoro mode is the residue of $z^{n+1}T(z)$ around an insertion. In radial quantization, circular contours are equal-time slices, so $L_n$ acts as an operator on the state created at the origin.

To see how the modes act on a local field at the origin, insert the mode expansion into a product with $\mathcal O(0)$:

$$T(z)\mathcal O(0) = \sum_{n\in\mathbb Z} z^{-n-2}(L_n\mathcal O)(0).$$

For a primary field, the OPE is

$$T(z)\mathcal O(0) \sim {h\mathcal O(0)\over z^2} +{\partial\mathcal O(0)\over z} +\text{regular terms}.$$

Comparing coefficients gives

$$\boxed{ L_0\mathcal O(0)=h\mathcal O(0), \qquad L_{-1}\mathcal O(0)=\partial\mathcal O(0), \qquad L_n\mathcal O(0)=0\quad(n>0). }$$

The modes with $n>0$ annihilate a primary insertion at the origin. This is why primary states are often called highest-weight states: in radial quantization, the positive modes lower the $L_0$ eigenvalue, and a primary is a state that cannot be lowered further inside its conformal family.

The negative modes generate descendants. The first one is special:

$$L_{-1}\mathcal O=\partial\mathcal O.$$

But the next one is not simply a second derivative. The field

$$(L_{-2}\mathcal O)(0) ={1\over 2\pi i}\oint_0 dz\,{1\over z}T(z)\mathcal O(0)$$

is an independent stress-tensor descendant. It is related to the finite part of the composite operator $T\mathcal O$, whereas $L_{-1}^2\mathcal O=\partial^2\mathcal O$ is a translation descendant. This distinction becomes crucial in minimal models, where different descendants can become linearly dependent through null-state relations.

Primary states and descendant towers

Radial quantization converts a local operator into a state:

$$|\mathcal O\rangle=\mathcal O(0)|0\rangle.$$

A holomorphic primary of weight $h$ gives a state satisfying

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

The descendants are

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

The level is

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

The commutator with $L_0$ is

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

so a level-$N$ descendant has holomorphic weight $h+N$:

$$L_0\left(L_{-n_1}\cdots L_{-n_k}|\mathcal O\rangle\right) =(h+N)L_{-n_1}\cdots L_{-n_k}|\mathcal O\rangle.$$

A primary state $|h\rangle$ generates a tower of descendants. Level $N$ consists of all products of negative modes whose indices sum to $N$. Generically these states form a Verma module; in special theories some combinations are null.

At level $0$ there is the primary itself. At level $1$ there is only

$$L_{-1}|h\rangle.$$

At level $2$ there are two natural states,

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

At level $3$ there are three,

$$L_{-3}|h\rangle, \qquad L_{-2}L_{-1}|h\rangle, \qquad L_{-1}^3|h\rangle.$$

In a generic highest-weight representation, the number of descendants at level $N$ is the number $p(N)$ of integer partitions of $N$. This combinatorics is one of the reasons the Virasoro algebra is powerful: it turns a continuum field theory problem into a highly constrained representation-theory problem.

Why the stress tensor is special

For an ordinary primary field of weight $h$, the stress-tensor OPE has only a second- and first-order pole. Since $T$ has weight $2$, one might guess

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

This is the classical transformation law of a quadratic differential. It says that under $z\mapsto z+\epsilon(z)$,

$$\delta_\epsilon T =\epsilon\partial T+2(\partial\epsilon)T.$$

Quantum mechanically this is incomplete. The product $T(z)T(w)$ has a c-number singularity:

$$\boxed{ T(z)T(w) \sim {c/2\over (z-w)^4} +{2T(w)\over (z-w)^2} +{\partial T(w)\over z-w}. }$$

The first term is not another local field. It is proportional to the identity operator. Its coefficient $c$ is the central charge.

The $T(z)T(w)$ OPE contains the primary-like terms required by the weight of $T$, plus a fourth-order identity pole. With the convention shown here, the two-point function is $\langle T(z)T(w)\rangle=c/[2(z-w)^4]$.

The vacuum one-point function on the plane is usually set to zero:

$$\langle T(z)\rangle=0.$$

Then the $T T$ OPE immediately gives

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

Thus $c$ is the normalization of the stress-tensor two-point function after the stress tensor itself has already been normalized by the Ward identity. It is not removed by rescaling $T$, because rescaling $T$ would also rescale the generator of conformal transformations and spoil the standard transformation law of every operator.

A more invariant way to say the same thing is this: the stress tensor is not a primary field when $c\ne0$. It is quasiprimary, because the global modes $L_{-1},L_0,L_1$ still act on it as expected, but under general local conformal transformations it acquires an anomalous c-number term. The next page derives that term as a Schwarzian derivative.

The central term from contour algebra

The Virasoro commutator follows directly from the $T T$ OPE. Start with

$$[L_m,L_n] = {1\over(2\pi i)^2}\oint_0 dw\,w^{n+1} \oint_w dz\,z^{m+1}T(z)T(w),$$

where the inner contour around $w$ computes the singular part of the OPE. Insert

$$T(z)T(w) \sim {c/2\over (z-w)^4} +{2T(w)\over (z-w)^2} +{\partial T(w)\over z-w}.$$

The double-pole term gives

$${1\over2\pi i}\oint_w dz\,{z^{m+1}2T(w)\over (z-w)^2} =2(m+1)w^mT(w).$$

The simple-pole term gives

$${1\over2\pi i}\oint_w dz\,{z^{m+1}\partial T(w)\over z-w} =w^{m+1}\partial T(w).$$

Multiplying by $w^{n+1}$ and integrating around the origin,

$${1\over2\pi i}\oint dw\, \left[2(m+1)w^{m+n+1}T(w)+w^{m+n+2}\partial T(w)\right].$$

The second term is integrated by parts:

$${1\over2\pi i}\oint dw\,w^{m+n+2}\partial T(w) =-(m+n+2){1\over2\pi i}\oint dw\,w^{m+n+1}T(w).$$

Thus the noncentral contribution is

$$\bigl(2m+2-m-n-2\bigr)L_{m+n} =(m-n)L_{m+n}.$$

For the fourth-order pole,

$${1\over2\pi i}\oint_w dz\,{z^{m+1}\over (z-w)^4} ={1\over 3!}\partial_w^3 w^{m+1} ={m(m^2-1)\over6}w^{m-2}.$$

Multiplying by $c/2$ and integrating over $w$ gives

$${c\over 12}m(m^2-1)\delta_{m+n,0}.$$

Therefore

$$\boxed{ [L_m,L_n] =(m-n)L_{m+n} +{c\over12}m(m^2-1)\delta_{m+n,0}. }$$

This is the Virasoro algebra. The central term vanishes for $m=-1,0,1$, so the global conformal subalgebra remains

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

The infinite-dimensional extension is quantum mechanically projective: the symmetry generators close up to a central c-number.

Current algebra detour: Schwinger terms

Central terms can look suspicious the first time one meets them. They are not arbitrary decorations. They are the local remnants of short-distance singularities.

A useful analogy comes from two-dimensional current algebra. Let $J_0$ and $J_1$ be the charge and spatial-current components of a conserved current,

$$\partial_0J_0+\partial_1J_1=0.$$

A naive equal-time calculation may suggest that current commutators vanish. In a quantum field theory this is too quick: products of currents at the same point are singular, and a two-dimensional chiral current algebra with nonzero level contains a c-number derivative of a delta function,

$$[J_0(x),J_1(y)] \sim i k\,\partial_x\delta(x-y),$$

where the sign and factor of $2\pi$ depend on the current and Fourier-transform conventions. This is a Schwinger term. It is the equal-time version of the central term in the chiral current algebra.

For a nontrivial chiral current algebra, the short-distance level becomes a derivative-of-delta Schwinger term at equal time. Unitarity makes the level nonnegative. The Virasoro central term is the analogous c-number term for the stress tensor.

The positivity reason is simple in spirit. For a Fourier mode of the charge density,

$$J_0(q)=\int dx\,e^{-iqx}J_0(x),$$

spectral decomposition gives

$$\sum_n |\langle0|J_0(q)|n\rangle|^2\ge0.$$

In a chiral current algebra, conservation relates the matrix elements of $J_0$ and $J_1$, while the same short-distance coefficient appears in the current two-point function, its OPE, and the equal-time contact term. With the conventional normalization,

$$J(z)J(w)\sim {k\over (z-w)^2}.$$

The stress tensor has the analogous singularity one derivative higher:

$$T(z)T(w)\sim {c/2\over (z-w)^4}+\cdots.$$

For unitary theories, this interpretation makes the sign of $c$ transparent. The stress-tensor state

$$|T\rangle=T(0)|0\rangle$$

is the same as $L_{-2}|0\rangle$ up to the conventional state-operator map. Its norm is

$$\langle T|T\rangle = \langle0|L_2L_{-2}|0\rangle ={c\over2}.$$

Unitarity therefore implies

$$c\ge0.$$

More generally,

$$\langle0|L_nL_{-n}|0\rangle ={c\over12}n(n^2-1), \qquad n\ge2.$$

So the central charge is not a conventionless aesthetic flourish. It is a measurable coefficient in the stress-tensor two-point function and a positivity-controlled central extension of the local conformal algebra.

Free-field checks

The normalization above gives familiar values:

$$\text{real free boson:}\quad c=1, \qquad \text{Majorana fermion:}\quad c={1\over2}, \qquad \text{Dirac fermion:}\quad c=1.$$

The Ising CFT contains a Majorana fermion $\psi(z)$ with

$$\psi(z)\psi(w)\sim {1\over z-w}.$$

With the holomorphic stress tensor normalized as

$$T(z)=-{1\over2}:\psi\partial\psi:(z),$$

Wick contraction gives

$$T(z)T(w) \sim {1/4\over (z-w)^4} +{2T(w)\over (z-w)^2} +{\partial T(w)\over z-w}.$$

Comparing with

$$T(z)T(w) \sim {c/2\over (z-w)^4}+\cdots$$

gives

$$c={1\over2}.$$

This is one of the cleanest ways to see that the Ising model at criticality is not merely “some” conformal field theory. Its stress-tensor normalization is that of one chiral Majorana fermion in each sector, equivalently half the value of a chiral complex fermion.

Cylinder preview

The central charge also appears when the plane is mapped to the cylinder. For

$$z=e^w, \qquad w=\tau+i\varphi,$$

the stress tensor does not transform as a purely classical quadratic differential. The quantum result is

$$T_{\rm cyl}(w)=z^2T_{\rm plane}(z)-{c\over24}.$$

The constant term is a Casimir-energy shift. For both holomorphic and antiholomorphic sectors, the cylinder Hamiltonian is

$$H_{\rm cyl}=L_0+\bar L_0-{c+\bar c\over24}.$$

The exponential map from the plane to the cylinder converts radial quantization into ordinary Euclidean time evolution. The central charge shifts the cylinder vacuum energy by $-(c+\bar c)/24$ on a unit-radius cylinder.

This formula will be derived from the Schwarzian derivative on the next page. For now, it is useful as a physical memory aid: $c$ controls both short-distance stress-tensor fluctuations on the plane and the finite-size vacuum energy on the cylinder.

Summary

The stress tensor is the generator of local conformal transformations. Its modes

$$L_n={1\over2\pi i}\oint dz\,z^{n+1}T(z)$$

act on primary fields and generate descendant towers. A primary state obeys

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

and descendants are obtained by applying $L_{-n}$ with $n>0$.

The stress tensor is special because its OPE with itself contains the identity pole

$$T(z)T(w) \sim {c/2\over (z-w)^4}+\cdots.$$

This pole is the central charge. It produces the Virasoro algebra

$$[L_m,L_n] =(m-n)L_{m+n} +{c\over12}m(m^2-1)\delta_{m+n,0}.$$

For unitary CFTs, $c$ is nonnegative because it is proportional to the norm of the stress-tensor state. For the critical Ising model, the Majorana fermion gives $c=1/2$.

Common pitfalls

Do not confuse $L_{-2}\mathcal O$ with $\partial^2\mathcal O$. The latter is $L_{-1}^2\mathcal O$; the former is a genuinely different stress-tensor descendant.

Do not treat $T$ as an ordinary primary field when $c\ne0$. It has primary-like terms in the OPE, but the fourth-order identity pole makes it anomalous under general local conformal maps.

Do not regard $c$ as removable by rescaling $T$. The normalization of $T$ is fixed by the Ward identity; once $T$ generates transformations correctly, $c$ is a physical coefficient of $\langle TT\rangle$.

Do not forget the antiholomorphic sector. A full local CFT has both $L_n$ and $\bar L_n$, and generally two central charges $c$ and $\bar c$. Parity-invariant unitary theories usually have $c=\bar c$.

Exercises

Exercise 1: Level of a Virasoro descendant

Let $|h\rangle$ be a primary state satisfying

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

Assuming

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

show that every level-$N$ descendant has $L_0$ eigenvalue $h+N$.

Solution

A general descendant has the form

$$|\chi\rangle=L_{-n_1}\cdots L_{-n_k}|h\rangle, \qquad n_i>0.$$

Using $[A,BC]=[A,B]C+B[A,C]$ repeatedly,

$$[L_0,L_{-n_1}\cdots L_{-n_k}] =(n_1+\cdots+n_k)L_{-n_1}\cdots L_{-n_k}.$$

Therefore

$$\begin{aligned} L_0|\chi\rangle &=L_{-n_1}\cdots L_{-n_k}L_0|h\rangle +[L_0,L_{-n_1}\cdots L_{-n_k}]|h\rangle \\ &=h|\chi\rangle+(n_1+\cdots+n_k)|\chi\rangle. \end{aligned}$$

Thus

$$L_0|\chi\rangle=(h+N)|\chi\rangle, \qquad N=n_1+\cdots+n_k.$$

Exercise 2: Virasoro algebra from the TT OPE

Use the OPE

$$T(z)T(w) \sim {c/2\over (z-w)^4} +{2T(w)\over (z-w)^2} +{\partial T(w)\over z-w}$$

and the mode definition

$$L_n={1\over2\pi i}\oint dz\,z^{n+1}T(z)$$

to derive

$$[L_m,L_n] =(m-n)L_{m+n} +{c\over12}m(m^2-1)\delta_{m+n,0}.$$

Solution

Compute the commutator by taking the $z$ contour around $w$:

$$[L_m,L_n] ={1\over(2\pi i)^2}\oint_0 dw\,w^{n+1} \oint_w dz\,z^{m+1}T(z)T(w).$$

The double pole gives

$${1\over2\pi i}\oint_w dz\,{2z^{m+1}T(w)\over(z-w)^2} =2(m+1)w^mT(w).$$

The simple pole gives

$${1\over2\pi i}\oint_w dz\,{z^{m+1}\partial T(w)\over z-w} =w^{m+1}\partial T(w).$$

Thus the noncentral part is

$${1\over2\pi i}\oint_0 dw\, \left[2(m+1)w^{m+n+1}T(w)+w^{m+n+2}\partial T(w)\right].$$

Integrating the second term by parts gives

$$-(m+n+2){1\over2\pi i}\oint_0dw\,w^{m+n+1}T(w).$$

Hence the coefficient is

$$2(m+1)-(m+n+2)=m-n,$$

so this part is $(m-n)L_{m+n}$.

For the central term,

$${1\over2\pi i}\oint_w dz\,{z^{m+1}\over(z-w)^4} ={1\over 3!}\partial_w^3w^{m+1} ={m(m^2-1)\over6}w^{m-2}.$$

Multiplying by $c/2$ and then by $w^{n+1}$, the $w$ integral is nonzero only when

$$m+n=0.$$

The coefficient is

$${c\over2}{m(m^2-1)\over6} ={c\over12}m(m^2-1).$$

Therefore

$$[L_m,L_n] =(m-n)L_{m+n} +{c\over12}m(m^2-1)\delta_{m+n,0}.$$

Exercise 3: Central charge of a free boson

Let a chiral free boson have the OPE

$$\varphi(z)\varphi(w)\sim -\log(z-w),$$

so that

$$\partial\varphi(z)\partial\varphi(w)\sim -{1\over (z-w)^2}.$$

With

$$T(z)=-{1\over2}:\partial\varphi\partial\varphi:(z),$$

show that the central charge is $c=1$.

Solution

The fourth-order pole in $T(z)T(w)$ comes from double contractions:

$$T(z)T(w) ={1\over4}:\partial\varphi\partial\varphi:(z):\partial\varphi\partial\varphi:(w).$$

There are two ways to contract the two $\partial\varphi$ fields at $z$ with the two $\partial\varphi$ fields at $w$. Each contraction contributes

$$\left(-{1\over(z-w)^2}\right)^2={1\over(z-w)^4}.$$

Therefore the identity singularity is

$$T(z)T(w)\sim {1\over4}\cdot 2\,{1\over(z-w)^4}+\cdots ={1/2\over(z-w)^4}+\cdots.$$

Comparing with

$$T(z)T(w)\sim {c/2\over(z-w)^4}+\cdots$$

gives

$$c=1.$$

Exercise 4: Norm of the stress-tensor state

Assume the vacuum is invariant under the global conformal generators,

$$L_n|0\rangle=0\quad(n\ge -1),$$

and that $L_n^\dagger=L_{-n}$. Use the Virasoro algebra to show

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

What does unitarity imply?

Solution

The norm is

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

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

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

The vacuum has $L_0|0\rangle=0$, so

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

A unitary Hilbert space has nonnegative norms. Therefore

$$c\ge0.$$

Further reading

• A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nuclear Physics B 241 (1984) 333–380, for Virasoro representation theory and the conformal bootstrap framework.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer, 1997), Chapters 5–7, for Virasoro modules, central charge, null states, and minimal models.
• P. Ginsparg, “Applied Conformal Field Theory,” in Fields, Strings and Critical Phenomena, Les Houches Session XLIX, edited by E. Brézin and J. Zinn-Justin (Elsevier, 1989), pp. 1–168, for stress-tensor modes, central charge, and cylinder quantization.
• J. Polchinski, String Theory, Volume 1 (Cambridge University Press, 1998), Chapter 2, for the Virasoro algebra and the cylinder interpretation of the central charge in worldsheet theory.
• A. M. Polyakov, Gauge Fields and Strings (Harwood Academic Publishers, 1987), Chapter 9, for the stress tensor and central charge in the random-surface and string-theory setting.