Null States and BPZ Differential Equations

A Virasoro descendant can itself be a highest-weight vector. When a CFT sets its null submodule to zero, the stress-tensor Ward identity converts that relation into a differential equation for correlation functions. We derive the level-two Belavin–Polyakov–Zamolodchikov (BPZ) equation and use its local exponents to identify possible holomorphic OPE channels.

We use Lesson 25’s highest-weight states and descendant basis, together with Lesson 21’s plane-vacuum Ward identities.

The Poincaré–Birkhoff–Witt monomials are linearly independent in the Verma module for every $(c,h)$. What changes at special values is that this module becomes reducible: a nonzero positive-level descendant can itself be a highest-weight state. Such a state is called a singular vector; the submodule it generates is null with respect to the standard contravariant form and is quotiented out in the irreducible module. These vectors become zero in the quotient. In a CFT realization that imposes the corresponding null-field relation, this gives a differential equation for correlators.

The simplest nontrivial example occurs at level two. A relation of the form

$$\left(L_{-2}-\kappa L_{-1}^{2}\right)|\psi\rangle=0$$

holds when the corresponding singular vector is set to zero. In that realization, an insertion of $L_{-2}\psi$ equals $\kappa\partial_z^2\psi$. Computing the same insertion with the stress-tensor Ward identity gives the BPZ equation.

Verma modules and null descendants

We first work in the abstract holomorphic Verma module $V(c,h)$; the algebraic parameters $c,h$ may be complex. The Virasoro algebra is

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

A primary state $|h\rangle$ obeys

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

A CFT primary field of holomorphic weight $h$ obeys

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

Many CFT books write the holomorphic conformal weight as $\Delta$. On this page $h$ denotes the holomorphic weight, while $\Delta=h+\bar h$ is reserved for the full scaling dimension when needed. The antiholomorphic sector has the same construction with barred modes.

Given a highest-weight state $|h\rangle$, the Virasoro lowering operators $L_{-n}$ with $n>0$ generate descendants:

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

The level of this descendant is

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

and the $L_0$ eigenvalue is $h+N$. For example, level one has one state,

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

while level two has two natural basis states,

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

A Verma module is the vector space spanned by all such descendants before imposing any additional null relations. A nonzero positive-level descendant $|\chi\rangle$ is a singular vector if it obeys

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

Normalize the symmetric contravariant bilinear form $B$ by

$$B(|h\rangle,|h\rangle)=1, \qquad B(L_nu,v)=B(u,L_{-n}v).$$

Its radical $\mathcal N$ consists of vectors orthogonal to every vector in $V(c,h)$. To pair $|\chi\rangle$ with a descendant at the same level, move the negative modes of that descendant across $B$: the resulting positive modes annihilate $|\chi\rangle$. Different levels are orthogonal because $L_0$ is self-adjoint with respect to $B$. Contravariance then shows that every descendant of $|\chi\rangle$ also lies in $\mathcal N$.

For a Verma module, $\mathcal N$ is the maximal proper submodule, and $V(c,h)/\mathcal N$ is its irreducible highest-weight quotient. Indeed, a proper submodule cannot produce the highest-weight vector under positive modes; otherwise it would contain the entire Verma module. The same pairing argument therefore puts every proper submodule inside $\mathcal N$. Quotienting by the submodule generated by just one singular vector imposes that relation but need not remove all of $\mathcal N$; several singular generators may be needed. See Di Francesco et al. 1997, §7.1.3, pp. 204–205 and the minimal-model example in Di Francesco et al. 1997, §8.1.1, pp. 240–241.

For real $c,h$, the same Gram matrix defines a Hermitian radial form when complex conjugation is included, with $L_n^\dagger=L_{-n}$. Positivity is an additional assumption of a unitary theory. In an indefinite Hermitian form, zero norm alone need not mean orthogonality to every state. A singular vector lies in the radical, but its descendants need not themselves be singular.

The level-one warm-up

At level one the only descendant is $L_{-1}|h\rangle$. It is primary if it is killed by all $L_n$ with $n>0$. It is enough to check $L_1$, since higher positive modes are even easier:

$$L_1L_{-1}|h\rangle=[L_1,L_{-1}]|h\rangle=2L_0|h\rangle=2h|h\rangle.$$

Therefore

$$L_{-1}|h\rangle\quad\text{is null only if}\quad h=0.$$

Thus $L_{-1}|h\rangle$ is a nonzero singular vector in the abstract Verma module precisely at $h=0$. In the identity representation its image is set to zero:

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

In local language, this is just

$$\partial \mathbf 1=0.$$

The identity module does not begin with a genuine level-one state. Its first descendant not forced to vanish by global invariance is

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

which corresponds to the stress tensor $T(0)|0\rangle$. This is why the stress tensor lives in the identity module but is not itself the identity.

The identity relation and stress-tensor descendant are described in Di Francesco et al. 1997, §7.3.1, p. 214. At level two, requiring a descendant to be highest weight gives a relation between $h$ and the central charge $c$.

The level-two null vector

In the abstract Verma module, seek a nonzero level-two singular vector. A vector proportional only to $L_{-1}^2|h\rangle$ cannot be singular: the two commutators below would require both $2h+1=0$ and $h=0$. We may therefore normalize the coefficient of $L_{-2}$ to one:

$$|\chi\rangle=(L_{-2}+aL_{-1}^{2})|h\rangle.$$

We want $|\chi\rangle$ to be primary:

$$L_n|\chi\rangle=0\qquad(n>0).$$

It is enough to impose $L_1|\chi\rangle=0$ and $L_2|\chi\rangle=0$. The higher conditions follow from commutators, because for example $L_3$ is proportional to $[L_1,L_2]$.

First compute the $L_1$ condition. Since $L_1|h\rangle=0$,

$$L_1L_{-2}|h\rangle=[L_1,L_{-2}]|h\rangle=3L_{-1}|h\rangle.$$

Also

$$\begin{aligned} L_1L_{-1}^{2}|h\rangle &=[L_1,L_{-1}^{2}]|h\rangle \\ &=\left(2L_0L_{-1}+2L_{-1}L_0\right)|h\rangle \\ &=2(2h+1)L_{-1}|h\rangle. \end{aligned}$$

Therefore

$$L_1|\chi\rangle=\bigl[3+2a(2h+1)\bigr]L_{-1}|h\rangle.$$

For $h=-1/2$, this coefficient is $3$ independently of $a$, so no vector of this form is singular. For $h\ne-1/2$, the condition $L_1|\chi\rangle=0$ gives

$$\boxed{ a=-{3\over 2(2h+1)}. }$$

Now impose the $L_2$ condition. The Virasoro algebra gives

$$L_2L_{-2}|h\rangle=[L_2,L_{-2}]|h\rangle =\left(4L_0+{c\over2}\right)|h\rangle =\left(4h+{c\over2}\right)|h\rangle,$$

and

$$L_2L_{-1}^{2}|h\rangle=6h|h\rangle.$$

Thus

$$L_2|\chi\rangle=\left(4h+{c\over2}+6ah\right)|h\rangle.$$

Substituting the value of $a$ gives the level-two null-vector condition

$$4h+{c\over2}-{9h\over 2h+1}=0.$$

Equivalently,

$$\boxed{ 16h^2+(2c-10)h+c=0. }$$

When this quadratic relation holds, the null state is

$$\boxed{ |\chi\rangle=\left(L_{-2}-{3\over 2(2h+1)}L_{-1}^{2}\right)|h\rangle. }$$

This agrees with Di Francesco et al. 1997, §7.3.1, pp. 211–212, Eqs. (7.42)–(7.44). The two roots are

$$h={5-c\pm\sqrt{(c-1)(c-25)}\over16}.$$

For the Ising value $c=1/2$, these are

$$h={1\over16}, \qquad h={1\over2}.$$

These are the holomorphic weights of the spin field $\sigma$ and the energy field $\varepsilon$ in the diagonal Ising CFT. Their antiholomorphic weights are the same, so the corresponding full scaling dimensions are $1/8$ and $1$; see Di Francesco et al. 1997, §7.4.2, p. 221, Eqs. (7.83)–(7.84).

Kac labels and the Coulomb-gas parametrization

It is useful to package the special weights by two integers. For $c\le1$, choose the real branches

$$\alpha_\pm =\sqrt{{1-c\over24}}\pm\sqrt{{25-c\over24}}, \qquad \alpha_+\alpha_-=-1.$$

For other values of $c$, the same formulas are understood by analytic continuation with a consistent branch choice.

The Kac determinant organizes the degenerate weights as follows; its general proof is outside this lesson. The parametrization is Di Francesco et al. 1997, §7.2.1, pp. 207–208, Eqs. (7.28)–(7.30):

$$\boxed{ h_{r,s} ={c-1\over24}+{1\over4}\left(r\alpha_+ + s\alpha_-\right)^2, \qquad r,s\in\mathbb Z_{>0}. }$$

The two level-two solutions are

$$h_{1,2}, \qquad h_{2,1}.$$

The subscripts on $\psi_{1,2}$ are Kac labels $(1,2)$.

For $c=1/2$ one finds

$$\alpha_+={2\over\sqrt3}, \qquad \alpha_-=-{\sqrt3\over2},$$

and hence

$$h_{1,1}=0, \qquad h_{1,2}={1\over16}, \qquad h_{2,1}={1\over2}.$$

Here we need only the two level-two weights $h_{1,2}$ and $h_{2,1}$. Degeneracy of the Verma module does not by itself select a physical spectrum or its OPE coefficients.

Null-field decoupling

Use the normalized plane identity vacuum, with separated primary insertions and no additional operator or boundary at infinity. Correlators are radially ordered. All small contours are counterclockwise; on the chosen compatible branch domain, the weighted stress-tensor correlator is single-valued along the deformations and has no other singularities. These are the plane-vacuum conditions of Lesson 21.

Let $\psi$ be a primary whose CFT representation imposes the level-two null relation

$$\left(L_{-2}-\kappa L_{-1}^{2}\right)|\psi\rangle=0, \qquad \kappa={3\over2(2h_\psi+1)}.$$

In local operator language this becomes

$$\left(L_{-2}\psi\right)(z)-\kappa\partial_z^2\psi(z)=0$$

inside separated correlators in this quotient realization. This null-vector decoupling is the extra input beyond the Virasoro commutators; we do not assume that every realization of the same formal $(c,h)$ module has already taken this quotient:

$$\left\langle \left[\bigl(L_{-2}\psi\bigr)(z)-\kappa\partial_z^2\psi(z)\right] \prod_{i=1}^N\phi_i(z_i) \right\rangle=0.$$

The power of this identity is that the two terms can be evaluated in very different ways. The $L_{-1}^2$ term is just a second derivative with respect to the insertion point $z$. The $L_{-2}$ term is computed by inserting the stress tensor and using the Ward identity.

Here $L_{-2}$ acting on a field at $z$ is the local descendant mode, centered at $z$. It is not the origin-mode commutator $[L_{-2},\psi(z)]$. Its contour representation is

$$\bigl(L_{-2}\psi\bigr)(z) ={1\over2\pi i}\oint_z {dw\over w-z}\,T(w)\psi(z).$$

Therefore, for

$$G(z;z_i)=\left\langle\psi(z)\prod_{i=1}^N\phi_i(z_i)\right\rangle,$$

we have

$$\left\langle\bigl(L_{-2}\psi\bigr)(z)\prod_i\phi_i(z_i)\right\rangle ={1\over2\pi i}\oint_z {dw\over w-z}\, \left\langle T(w)\psi(z)\prod_i\phi_i(z_i)\right\rangle.$$

Deform the contour away from $z$. Regularity of the plane vacuum at infinity gives $\langle T(w)\psi(z)\prod_i\phi_i(z_i)\rangle=O(w^{-4})$; the kernel $(w-z)^{-1}$ adds another inverse power. The contour at infinity vanishes, so the contour around $z$ equals minus the sum of the other counterclockwise contours. The OPE

$$T(w)\phi_i(z_i)\sim {h_i\phi_i(z_i)\over(w-z_i)^2}+{\partial_{z_i}\phi_i(z_i)\over w-z_i}$$

gives the counterclockwise residue

$$\mathop{\rm Res}_{w=z_i} {\langle T(w)\psi(z)\prod_j\phi_j(z_j)\rangle\over w-z} =-{h_i\over(z-z_i)^2}G -{1\over z-z_i}\partial_{z_i}G.$$

The double pole differentiates $(w-z)^{-1}$, producing its minus sign. Taking minus the sum of these residues gives

$$\boxed{ \left\langle\bigl(L_{-2}\psi\bigr)(z)\prod_i\phi_i(z_i)\right\rangle = \sum_{i=1}^N \left( {h_i\over(z-z_i)^2}+{1\over z-z_i}\partial_{z_i} \right)G(z;z_i). }$$

Combining this with null-vector decoupling gives the BPZ equation

$$\boxed{ \left[ \kappa\partial_z^2 - \sum_{i=1}^N \left( {h_i\over(z-z_i)^2}+{1\over z-z_i}\partial_{z_i} \right) \right] G(z;z_i)=0, \qquad \kappa={3\over2(2h_\psi+1)}. }$$

This is Di Francesco et al. 1997, §7.3.1, p. 212, Eqs. (7.45)–(7.47), with the overall equation multiplied by $-1$. Global conformal invariance supplies first-order Ward identities; imposing the null-field relation supplies the additional second-order equation.

Four-point functions and the cross ratio

For the four-point reduction, let the brackets denote a chiral conformal block on a fixed branch. A full nonchiral correlator pairs holomorphic and antiholomorphic blocks; its insertion at infinity would also include the factor $\bar R^{2\bar h_3}$. We analyze one holomorphic block, as in Di Francesco et al. 1997, §8.3, p. 247. Move three primary insertions to $0$, $1$, and $\infty$ and define

$$G(z)=\lim_{R\to\infty}R^{2h_3} \left\langle\psi(z)\phi_1(0)\phi_2(1)\phi_3(R)\right\rangle.$$

The remaining coordinate $z$ is the cross ratio. The BPZ equation becomes a Fuchsian ordinary differential equation whose possible singular points are

$$z=0, \qquad z=1, \qquad z=\infty.$$

These singularities are regular when present; exceptional parameters can make a point removable. The operator at infinity is defined by the displayed normalized limit after deriving the finite-insertion Ward equation. This is the setting in which the equation reduces to hypergeometric form.

For the unprefactored chiral block above, let $\partial_0$ and $\partial_1$ mean derivatives with respect to those insertion points before setting them to $0$ and $1$. After taking the normalized $R\to\infty$ limit, the global Ward identities give

$$\partial_0G=(z-1){dG\over dz}+A G, \qquad \partial_1G=-z{dG\over dz}-A G,$$

where

$$A=h_\psi+h_1+h_2-h_3.$$

Indeed, translation and dilation give

$$\partial_0G+\partial_1G+G'=0, \qquad \partial_1G+zG'+AG=0.$$

The term $-h_3$ in $A$ comes from $R\partial_R$ acting on $R^{-2h_3}$ together with the insertion’s $+h_3$ Ward term. Substituting the two eliminated derivatives into the BPZ equation gives

$$\boxed{ \left[ \kappa {d^2\over dz^2} +{2z-1\over z(z-1)}{d\over dz} -{h_1\over z^2} -{h_2\over(z-1)^2} +{h_\psi+h_1+h_2-h_3\over z(z-1)} \right]G(z)=0. }$$

This $G$ is the function called $H$ in Di Francesco et al. 1997, §8.3.3, pp. 252–254, Eqs. (8.62)–(8.71), with $h_0=h_\psi$ and $1/t=\kappa$. Changing the prefactor of a reduced block changes the derivative and potential terms.

Away from the singular points, a second-order equation has a two-dimensional local solution space. Near a regular singular point, repeated or resonant exponents can require logarithmic Frobenius solutions. The equation alone does not fix the physical correlator: the spectrum, other null relations, OPE coefficients, antiholomorphic pairing, single-valuedness and crossing impose further conditions.

Near $z=0$, let

$$G(z)\sim z^p.$$

The leading terms give the indicial equation

$$\kappa p(p-1)+p-h_1=0.$$

In OPE language the exponent is

$$p=h_{\text{int}}-h_\psi-h_1,$$

where $h_{\text{int}}$ is the weight of the intermediate primary appearing in the OPE $\psi\times\phi_1$.

For the Ising spin field, $c=1/2$ and

$$h_\psi=h_1={1\over16}, \qquad \kappa={4\over3}.$$

The indicial equation becomes

$${4\over3}p(p-1)+p-{1\over16}=0,$$

with the following possible exponents and intermediate weights:

Holomorphic channels allowed by the Ising spin-field BPZ equation, with hψ = h₁ = 1/16.

OPE exponent p	Intermediate weight hint	Ising family of that weight
−1/8	0	Identity 1
3/8	1/2	Energy ε

The indicial equation supplies necessary weights; it does not prove that both OPE coefficients are nonzero. The independent Ising operator algebra does contain both families, giving

$$\sigma\times\sigma=\mathbf 1+\varepsilon.$$

The nonzero Ising channels are stated in Di Francesco et al. 1997, §7.4.2, p. 221, Eq. (7.85).

More generally, the level-two $(1,2)$ constraint permits the nominal neighboring labels $(r,s-1)$ and $(r,s+1)$ in its fusion with a Kac primary labelled $(r,s)$. This is a selection rule. A particular theory must still supply the allowed fields and nonzero OPE coefficients; field identifications, other null relations and Kac-table boundaries can remove nominal channels. The distinction is explicit in Di Francesco et al. 1997, §7.3.1, pp. 213–214, Eqs. (7.50)–(7.54).

Setting the singular vector to zero gives an algebraic relation in the quotient state space. The BPZ equation is the same relation written in position space. The allowed OPE channels are encoded in the local exponents of that differential equation.

Null-state quotients are not gauge fixing

The null-state quotient is already an intrinsic statement about a stand-alone CFT representation. Generic Virasoro descendants are genuine states and operators; one does not remove them merely because Virasoro generators implement local conformal transformations. The irreducible quotient removes the maximal proper submodule $\mathcal N$, which can contain several singular submodules.

When a CFT is coupled to two-dimensional gravity, or used as worldsheet matter in string theory, diffeomorphism and Weyl gauge fixing introduces constraints and a ghost sector. In a consistent BRST quantization, physical states are defined by a cohomology problem involving the combined matter-plus-ghost system. BRST-exact states are gauge redundancies, but that construction is not identical to quotienting a matter Verma module by a BPZ null submodule.

Summary

A Virasoro Verma module is freely generated by the lowering modes $L_{-n}$. At special values of $(c,h)$ it becomes reducible because a nonzero descendant is itself highest weight. Its singular vectors generate null submodules. The irreducible highest-weight representation is obtained by quotienting by the full maximal proper submodule.

At level one, $L_{-1}|h\rangle$ is singular precisely when $h=0$; in the identity module its vanishing reflects $\partial\mathbf 1=0$. At level two, the null vector is

$$\left(L_{-2}-{3\over2(2h+1)}L_{-1}^{2}\right)|h\rangle,$$

provided

$$16h^2+(2c-10)h+c=0.$$

The two solutions are the Kac weights $h_{1,2}$ and $h_{2,1}$. For a primary $\psi$ whose CFT realization imposes that null relation, the separated plane-vacuum Ward identity gives the BPZ equation

$$\left[ {3\over2(2h_\psi+1)}\partial_z^2 - \sum_i \left( {h_i\over(z-z_i)^2}+{1\over z-z_i}\partial_{z_i} \right) \right] \left\langle\psi(z)\prod_i\phi_i(z_i)\right\rangle=0.$$

For four insertions, the local exponents restrict possible OPE weights. They leave the physical spectrum and nonzero OPE coefficients to be determined.

Common pitfalls

Null does not mean singular. A singular vector is nonzero and highest weight inside the abstract Verma module. Its whole descendant submodule lies in the radical, but those descendants need not be highest weight. In a physical quotient their images are zero. In an indefinite Hermitian form, a vector can have zero norm without belonging to the radical.

One relation need not give the irreducible module. Quotienting a chosen singular submodule implements that relation. Irreducibility requires quotienting the maximal proper submodule.

The operator $L_{-2}\psi$ is not generally equal to a second derivative. It becomes proportional to $\partial^2\psi$ only for a degenerate field whose state obeys a level-two null relation.

The Kac label $\psi_{1,2}$ is a pair of integers. It is not a fraction. The comma matters.

The BPZ equation for a reduced conformal block depends on the prefactor convention. The coordinate-invariant statement is the unreduced equation with derivatives with respect to all other insertion points.

Two local BPZ solutions mean at most two candidate holomorphic channels, not two automatically nonzero OPE coefficients. The spectrum, Kac-table identifications, and crossing-consistent OPE data decide which allowed channels are actually present.

Finally, BPZ null states should not be identified wholesale with gauge states. Matter Virasoro descendants are generally physical; worldsheet gauge reduction requires the separate BRST construction with ghosts.

Exercises

Exercise 1: The level-one singular vector

Work in the abstract Verma module with nonzero highest-weight vector $|h\rangle$. Show that the nonzero descendant $L_{-1}|h\rangle$ is singular precisely when $h=0$. Here “null primary” means this formal singular vector before taking the quotient.

Solution

A primary state satisfies $L_n|h\rangle=0$ for $n>0$ and $L_0|h\rangle=h|h\rangle$. At level one the only descendant is $L_{-1}|h\rangle$. It is primary if $L_nL_{-1}|h\rangle=0$ for all $n>0$.

The nontrivial condition is $n=1$:

$$L_1L_{-1}|h\rangle=[L_1,L_{-1}]|h\rangle=2L_0|h\rangle=2h|h\rangle.$$

Therefore $L_1L_{-1}|h\rangle=0$ only if $h=0$. For $n\ge2$,

$$[L_n,L_{-1}]=(n+1)L_{n-1},$$

and $L_{n-1}|h\rangle=0$ because $n-1>0$. Hence $h=0$ is the only condition.

Exercise 2: The level-two degeneracy condition

In the abstract Verma module, use the normalized ansatz $(L_{-2}+aL_{-1}^2)|h\rangle$ with $h\ne-1/2$ to derive the level-two null-vector condition. The main derivation has excluded a pure $L_{-1}^2$ singular vector and the exceptional value $h=-1/2$.

$$16h^2+(2c-10)h+c=0.$$

Solution

Start with

$$|\chi\rangle=(L_{-2}+aL_{-1}^2)|h\rangle.$$

The $L_1$ condition gives

$$L_1L_{-2}|h\rangle=3L_{-1}|h\rangle,$$

and

$$L_1L_{-1}^2|h\rangle=2(2h+1)L_{-1}|h\rangle.$$

Thus

$$3+2a(2h+1)=0,$$

so

$$a=-{3\over2(2h+1)}.$$

The $L_2$ condition gives

$$L_2L_{-2}|h\rangle=\left(4h+{c\over2}\right)|h\rangle,$$

and

$$L_2L_{-1}^2|h\rangle=6h|h\rangle.$$

Therefore

$$4h+{c\over2}+6ah=0.$$

Substituting $a$ gives

$$4h+{c\over2}-{9h\over2h+1}=0.$$

Multiplying by $2(2h+1)$ yields

$$16h^2+(2c-10)h+c=0.$$

Exercise 3: Null-field decoupling

Use separated primary insertions in the normalized plane identity vacuum, regular at infinity, and counterclockwise contours on a compatible branch domain with no additional singularities. Let $\psi$ be a primary whose CFT realization imposes

$$\left(L_{-2}-\kappa L_{-1}^2\right)|\psi\rangle=0.$$

Use the stress-tensor Ward identity to prove

$$\left[ \kappa\partial_z^2 - \sum_{i=1}^N \left({h_i\over(z-z_i)^2}+{1\over z-z_i}\partial_{z_i}\right) \right] \left\langle\psi(z)\prod_i\phi_i(z_i)\right\rangle=0.$$

Solution

Let

$$G(z;z_i)=\left\langle\psi(z)\prod_i\phi_i(z_i)\right\rangle.$$

The null relation gives

$$\left\langle(L_{-2}\psi)(z)\prod_i\phi_i(z_i)\right\rangle = \kappa\partial_z^2G.$$

Represent $L_{-2}$ by a contour integral around $z$:

$$\left\langle(L_{-2}\psi)(z)\prod_i\phi_i(z_i)\right\rangle ={1\over2\pi i}\oint_z{dw\over w-z} \left\langle T(w)\psi(z)\prod_i\phi_i(z_i)\right\rangle.$$

The contour at infinity vanishes under the stated hypotheses, so this contour equals minus the sum of counterclockwise contours around the other insertions. The OPE

$$T(w)\phi_i(z_i)\sim {h_i\phi_i(z_i)\over(w-z_i)^2}+{\partial_{z_i}\phi_i(z_i)\over w-z_i}$$

gives minus the counterclockwise residue

$${h_i\over(z-z_i)^2}G+{1\over z-z_i}\partial_{z_i}G.$$

Thus

$$\left\langle(L_{-2}\psi)(z)\prod_i\phi_i(z_i)\right\rangle =\sum_i\left({h_i\over(z-z_i)^2}+{1\over z-z_i}\partial_{z_i}\right)G.$$

Equating this expression with $\kappa\partial_z^2G$ proves the BPZ equation.

Exercise 4: Ising fusion exponents

For the Ising spin field, take $c=1/2$ and $h_\psi=h_1=1/16$. Use the BPZ equation near $z=0$ to find the two possible OPE exponents in $\psi(z)\phi_1(0)$.

Solution

For $h_\psi=1/16$,

$$\kappa={3\over2(2h_\psi+1)}={3\over2(9/8)}={4\over3}.$$

Near $z=0$, write $G(z)\sim z^p$. The leading terms in the BPZ equation give

$$\kappa p(p-1)+p-h_1=0.$$

Substituting $h_1=1/16$ and $\kappa=4/3$ gives

$${4\over3}p(p-1)+p-{1\over16}=0.$$

Multiplying by $48$,

$$64p^2-16p-3=0.$$

The roots are

$$p={3\over8}, \qquad p=-{1\over8}.$$

Since an OPE exponent has the form

$$p=h_{\text{int}}-h_\psi-h_1,$$

and $h_\psi+h_1=1/8$, these correspond to

$$h_{\text{int}}={1\over2}, \qquad h_{\text{int}}=0.$$

These are the two intermediate weights permitted by this BPZ equation. Independently, the Ising operator algebra has nonzero coefficients for both families, as stated in Di Francesco et al. 1997, §7.4.2, p. 221, Eq. (7.85):

$$\sigma\times\sigma=\mathbf 1+\varepsilon.$$

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI: 10.1007/978-1-4612-2256-9.

Further reading

• Belavin, A. A., A. M. Polyakov, and A. B. Zamolodchikov. “Infinite conformal symmetry in two-dimensional quantum field theory.” Nuclear Physics B 241 (1984), 333–380.
• Ginsparg, P. “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.