Degenerate Fields, Kac Labels, and Constraints

A Virasoro Verma module becomes reducible at special highest weights. The Kac determinant identifies these weights, and the Kac labels $(r,s)$ locate singular descendants. When a CFT sets the corresponding null submodule to zero, its correlation functions obey differential constraints. This connects representation theory to the possible operator-product channels.

We compute the level-two Gram matrix, state the general determinant and unitary-series results with their hypotheses, and use degenerate fields to constrain fusion. The full Kac-determinant and unitarity proofs are outside this lesson. We use Lesson 25’s highest-weight states and descendant basis and Lesson 26’s null quotients and BPZ equation.

The thread is:

$$\text{Virasoro module}\quad\longrightarrow\quad \text{Gram matrix}\quad\longrightarrow\quad \text{Kac determinant}\quad\longrightarrow\quad \text{degenerate fields}\quad\longrightarrow\quad \text{fusion constraints}.$$

At the minimal-model values of the central charge, additional null relations allow a finite set of irreducible Virasoro representations to close under fusion. A full local CFT also requires a consistent pairing of the holomorphic and antiholomorphic sectors.

Modes acting on local fields

We work in one holomorphic sector, suppressing antiholomorphic coordinates when considering full local fields. 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 $|h\rangle$ satisfies

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

The holomorphic stress tensor is expanded as

$$T(z)=\sum_{n\in\mathbb Z}{L_n\over z^{n+2}}, \qquad L_n={1\over2\pi i}\oint_0 dz\,z^{n+1}T(z).$$

We write $h$ for a holomorphic conformal weight. The full scaling dimension is $\Delta=h+\bar h$ when the antiholomorphic sector is included. Contours are counterclockwise and enclose no other insertion; products are radially ordered.

For a local field $O$ at the origin, the Laurent expansion of the stress-tensor OPE defines the local Virasoro descendants:

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

Equivalently,

$$(L_nO)(0)={1\over2\pi i}\oint_0 dz\,z^{n+1}T(z)O(0).$$

If $O$ is primary of weight $h$, then

$$T(z)O(0)\sim {hO(0)\over z^2}+{\partial O(0)\over z},$$

so

$$L_0O=hO, \qquad L_{-1}O=\partial O, \qquad L_nO=0\quad(n>0).$$

The singular terms determine the modes $n\ge-1$. The regular terms give $L_{-2}O,L_{-3}O,\ldots$; iterating the local mode construction also generates composite descendants such as

$$L_{-2}O, \quad L_{-1}^2O, \quad L_{-3}O, \quad L_{-2}L_{-1}O, \quad\ldots$$

The identity field is special. Since it is invariant under the global conformal group,

$$L_{-1}\mathbf 1=L_0\mathbf 1=L_1\mathbf 1=0.$$

The first descendant not forced to vanish by global invariance is

$$L_{-2}\mathbf 1=T.$$

Thus the stress tensor belongs to the identity module, although its state need not be nonzero in every physical quotient. For $c\ne0$, it is not an ordinary primary because its self-OPE contains a fourth-order central pole:

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

Under an infinitesimal conformal transformation generated by a holomorphic vector field $\epsilon(z)$,

$$\boxed{ \delta_\epsilon T =\epsilon\partial T+2(\partial\epsilon)T+{c\over12}\partial^3\epsilon. }$$

The last term is the local form of the Virasoro central extension; this obstruction vanishes at $c=0$. For an ordinary primary field,

$$\delta_\epsilon O=\epsilon\partial O+h(\partial\epsilon)O,$$

with no $\partial^3\epsilon$ term.

Degenerate modules

A Verma module $\mathcal V_{c,h}$ is generated by all products of lowering operators acting on $|h\rangle$:

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

The Poincaré–Birkhoff–Witt basis uses ordered products with $n_1\ge\cdots\ge n_k$. At level $N=n_1+\cdots+n_k$, its dimension is the partition number $p(N)$ for every $(c,h)$, before any null quotient. For example,

$$\begin{array}{c|c} N & \text{basis states} \\ \hline 0 & |h\rangle \\ 1 & L_{-1}|h\rangle \\ 2 & L_{-2}|h\rangle,\ L_{-1}^2|h\rangle \\ 3 & L_{-3}|h\rangle,\ L_{-2}L_{-1}|h\rangle,\ L_{-1}^3|h\rangle. \end{array}$$

The Verma module is degenerate if a descendant is itself highest weight. Thus there is a nonzero state $|\chi\rangle$ at positive level $N$ such that

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

Such a state is a singular vector. For arbitrary complex $c,h$, 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 the entire Verma module. Moving lowering modes across $B$ shows that a positive-level singular vector is orthogonal to every state: the resulting positive modes annihilate it, and different levels are orthogonal. Contravariance puts all of its descendants in $\mathcal N$ as well. Such radical vectors are called null; a null descendant need not itself be singular.

The irreducible highest-weight representation is $\mathcal V_{c,h}/\mathcal N$, where $\mathcal N$ is the maximal proper submodule. Quotienting by just one singular vector’s submodule need not remove the whole radical. These distinctions and the need for intersecting singular submodules are developed in Di Francesco et al. 1997, §7.1.3, pp. 204–205 and Di Francesco et al. 1997, §8.1.1, pp. 240–241.

When a CFT imposes the null-field relation, the stress-tensor Ward identity turns it into a differential constraint of order at most $N$. The usual $N$th-order BPZ equation requires a nonzero coefficient of the highest derivative. The level-two example in Lesson 26 has this property; generically the $(1,3)$ and $(3,1)$ fields give third-order equations. More generally, the Kac weight $h_{r,s}(c)$ guarantees a singular vector in the Verma module at level

$$N=rs.$$

For generic $c$, this is the first singular-vector level. At minimal-model values, the same Verma module can have another singular vector, possibly at a lower level for the chosen label. Their images vanish in the irreducible quotient. Further null relations can reduce the independent solutions of an individual BPZ equation. The general labels and determinant are stated below; the differential constraints are discussed in Di Francesco et al. 1997, §§7.3.1–7.3.2, pp. 211–215.

Positivity and Gram matrices

Now take real $c,h$. With complex conjugation included, the same matrices define a Hermitian radial form with $L_n^\dagger=L_{-n}$ and $\langle h|h\rangle=1$. Positivity is an additional unitary assumption; an indefinite Hermitian form can have zero-norm vectors outside its radical.

For example, let the smeared operator below create a well-defined state from the unit-normalized plane vacuum, with smooth test function supported strictly inside the unit radial circle. The dagger means the radial adjoint, including reflection to the outside of the circle. Unitarity requires

$$\left\langle \left(\int d^2z\, f(z,\bar z)O(z,\bar z)\right)^\dagger \left(\int d^2w\, f(w,\bar w)O(w,\bar w)\right) \right\rangle\ge0.$$

For a highest-weight module, this positivity is tested level by level by the Gram matrix of descendant inner products.

At level one,

$$\langle h|L_1L_{-1}|h\rangle=2h.$$

Thus unitarity requires

$$h\ge0.$$

If $h=0$, then $L_{-1}|h\rangle$ is null and its image vanishes in the unitary quotient. This identifies the holomorphic vacuum representation, not necessarily the full CFT vacuum. For example, the antiholomorphic stress state has $(h,\bar h)=(0,2)$ when $\bar c\ne0$. A full primary with $h=\bar h=0$ is the identity state only under the assumption of a unique globally conformally invariant vacuum. The physical interpretation of the radial form is developed in Lesson 25.

At level two, use the basis

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

The Gram matrix is

$$M_2= \begin{pmatrix} \langle h|L_2L_{-2}|h\rangle & \langle h|L_2L_{-1}^2|h\rangle \\ \langle h|L_1^2L_{-2}|h\rangle & \langle h|L_1^2L_{-1}^2|h\rangle \end{pmatrix} = \begin{pmatrix} 4h+{c\over2} & 6h \\ 6h & 4h(2h+1) \end{pmatrix}.$$

Hence

$$\det M_2 =2h\left[16h^2+(2c-10)h+c\right].$$

A zero determinant means that the level-two form has a nonzero kernel. It need not signal a new singular vector at that level. For $h\ne0$, the kernel condition becomes

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

and the corresponding vector is singular:

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

The quadratic cannot vanish at $h=-1/2$: its value there is $9$, so the denominator is harmless on this locus. At $h=0$ and $c\ne0$, however, $M_2=\operatorname{diag}(c/2,0)$ and its kernel is the descendant $L_{-1}^2|0\rangle$ of the level-one singular vector. It is not a new singular vector, since in the abstract Verma module

$$L_1L_{-1}^2|0\rangle=2L_{-1}|0\rangle\ne0.$$

At $c=h=0$, the quadratic locus also passes through this point and the displayed level-two singular combination exists. These low-level forms agree with Di Francesco et al. 1997, §7.2.1, pp. 206–207, Eqs. (7.23)–(7.26); the source reverses our two basis vectors.

Positivity must hold at every level, not just the first two. For nontrivial unitary positive-energy Virasoro representations with $0<c<1$, the all-level condition restricts the central charge and highest weights to the discrete series below.

The Kac determinant and Kac labels

In the ordered PBW basis and with the normalization of $B$ above, the Kac determinant formula states that the level-$N$ determinant factorizes as follows. The omitted nonzero constant depends on $N$ and the basis normalization, not on $c,h$:

$$\boxed{ \det M_N(c,h) \propto \prod_{\substack{r,s\ge1\\rs\le N}} \left(h-h_{r,s}(c)\right)^{p(N-rs)}. }$$

Here $p(k)$ is the number of integer partitions of $k$, with $p(0)=1$. The powers account for the descendants of singular vectors; at coincident weights they do not count independent primitive singular generators. The general determinant and degenerate-weight theorem are stated in Di Francesco et al. 1997, §7.2.1, pp. 207–209, Eqs. (7.28)–(7.36). In particular, if

$$h=h_{r,s}(c),$$

then the Verma module has a singular vector at level

$$rs.$$

To write these weights compactly, choose the following real square-root branch for $c\le1$. For other complex values the formulas require a consistent analytic continuation; branch changes can exchange the labels:

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

Equivalently,

$$c=1-6(\alpha_++\alpha_-)^2.$$

Define

$$h(\alpha)={c-1\over24}+{\alpha^2\over4}.$$

Then the Kac weights are

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

The two level-two roots, counted with multiplicity, are exactly

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

For generic $c$, the first few labels give the following singular-vector levels. Here “identity module” refers to the irreducible holomorphic $h=0$ representation, and the BPZ descriptions assume the corresponding null decoupling and a nonzero highest-derivative coefficient:

$$\begin{array}{c|c|c} \text{label} & \text{singular-vector level} & \text{interpretation} \\ \hline (1,1) & 1 & \text{identity module} \\ (1,2),(2,1) & 2 & \text{level-two BPZ fields} \\ (1,3),(3,1) & 3 & \text{third-order BPZ fields} \\ (2,2) & 4 & \text{level-four degenerate field}. \end{array}$$

Each equation $h=h_{r,s}(c)$ describes a reducibility curve in the $(c,h)$ parameter space. Where curves meet, the same highest weight can have several Kac labels and several null constraints.

Rational parameters and the minimal-model window

For the minimal-model series, choose a rational value of $\alpha_+^2$ by setting

$$\alpha_+=\sqrt{p'\over p}, \qquad \alpha_-=-\sqrt{p\over p'},$$

where $2\le p<p'$ are coprime integers. It is the rationality of $\alpha_+^2$, not arbitrary rationality of $c$, that is used here. Then

$$\boxed{ c_{p,p'}=1-{6(p'-p)^2\over pp'} }$$

and

$$\boxed{ h_{r,s}^{(p,p')} ={ (p'r-ps)^2-(p'-p)^2 \over 4pp'}. }$$

For these coprime integers, the formulas describe a Virasoro minimal-model Kac table after taking the null quotients. The usual finite window is

$$1\le r\le p-1, \qquad 1\le s\le p'-1,$$

with the identification

$$(r,s)\sim(p-r,p'-s).$$

The minimal-model construction also requires fusion closure and a consistent left–right pairing. The window and reflection are given in Di Francesco et al. 1997, §7.3.3, pp. 216–218, Eqs. (7.65)–(7.73); that chapter’s integer labels $(p,p')$ are reversed relative to ours.

For real $c,h$ and $0<c<1$, the irreducible positive-energy highest-weight representation is unitary precisely for the following central charges and weights. In the finite window above, set

$$p'=p+1, \qquad p=3,4,5,\ldots.$$

It gives

$$\boxed{ c_p=1-{6\over p(p+1)} }$$

and

$$\boxed{ h_{r,s}^{(p)} ={\bigl((p+1)r-ps\bigr)^2-1\over4p(p+1)}. }$$

The first member is the Ising CFT:

$$p=3, \qquad c={1\over2}.$$

Its three holomorphic primary weights are

$$0, \qquad {1\over16}, \qquad {1\over2},$$

corresponding to the identity, spin, and energy fields.

The determinant-positivity argument gives the restriction; it is not proved by our level-two calculation. See Di Francesco et al. 1997, §7.2.3, pp. 210–211, Eq. (7.41). The positive-level $SU(2)$ coset construction realizes the allowed nontrivial series unitarily, establishing sufficiency; see Di Francesco et al. 1997, p. 798 and §18.3, pp. 807–810. Extending the formula to $p=2$ gives $c=h=0$, the trivial holomorphic representation. It is excluded by our $0<c<1$ scope.

The first three nontrivial charges are useful reference points:

First nontrivial unitary minimal-series central charges and examples of full local theories

p	Central charge c	Example at this charge
3	1/2	Ising, diagonal pairing
4	7/10	Tricritical Ising, diagonal pairing
5	4/5	Critical three-state Potts, non-diagonal pairing

The sequence increases toward $c=1$ as $p\to\infty$. A central charge alone does not determine the full local theory: the critical Potts spectrum at $c=4/5$ uses the non-diagonal $A_4,D_4$ modular invariant, unlike the diagonal minimal model at that charge. The Ising and tricritical identifications appear in Di Francesco et al. 1997, §§7.4.2–7.4.3, pp. 221–222, and the Potts distinction in Di Francesco et al. 1997, §§10.7.2–10.7.3, pp. 365–368, Eq. (10.150).

Degenerate fields as fusion constraints

Now suppose the CFT imposes the singular-vector relations of the inserted degenerate fields. Their BPZ equations constrain possible holomorphic OPE families. Label a generic primary by a momentum-like parameter $\alpha$:

$$h_\alpha={c-1\over24}+{\alpha^2\over4}.$$

The level-two degenerate fields have momenta

$$\alpha_{1,2}=\alpha_+ +2\alpha_-, \qquad \alpha_{2,1}=2\alpha_+ +\alpha_-.$$

Consider the OPE of $\psi_{1,2}$ with a generic primary $\phi_\alpha$:

$$\psi_{1,2}(z)\phi_\alpha(0).$$

The indicial equation of the second-order BPZ equation gives two possible exponents, counted with multiplicity. For generic nonresonant parameters they give two independent local solutions with different leading powers. At coincident or resonant roots, logarithmic solutions may occur, and $\alpha$ and $-\alpha$ already label the same weight. The possible intermediate weights are

$$h_{\alpha+\alpha_-}, \qquad h_{\alpha-\alpha_-}.$$

Thus the following schematic notation lists permitted families, not their coefficients or a guarantee that both occur:

$$\boxed{ \psi_{1,2}\times\phi_\alpha \sim \phi_{\alpha+\alpha_-}+\phi_{\alpha-\alpha_-}. }$$

Similarly,

$$\boxed{ \psi_{2,1}\times\phi_\alpha \sim \phi_{\alpha+\alpha_+}+\phi_{\alpha-\alpha_+}. }$$

For a general degenerate field $\psi_{r,s}$, the null-vector selection rule has the following form. Its higher-level derivation is outside this lesson; the degenerate-fusion constraints are developed in Di Francesco et al. 1997, §§7.3.1–7.3.2, pp. 213–215, Eqs. (7.54)–(7.61):

$$\boxed{ \psi_{r,s}\times\phi_\alpha \sim \sum_{i=0}^{r-1}\sum_{j=0}^{s-1} \phi_{\alpha+(r-1-2i)\alpha_+ +(s-1-2j)\alpha_-}. }$$

These are representation-theoretic selection rules. A particular CFT may omit a listed family or have a vanishing OPE coefficient. One must impose the other field’s null constraints as well, identify repeated weights, and respect the full theory’s spectrum and left–right pairing. The source explicitly allows zero coefficients and illustrates how an additional null condition removes a nominal channel on p. 215.

When both fields are degenerate, the candidate shifts can be expressed in Kac labels. At the minimal-model values, the further null constraints and finite-table identifications truncate them to the finite fusion rules.

Example: the Ising labels

For the Ising value $c=1/2$, the parameters may be chosen as

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

Then

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

In the diagonal Ising CFT, the three primary fields are identified as

$$\mathbf 1\leftrightarrow(1,1), \qquad \sigma\leftrightarrow(1,2), \qquad \varepsilon\leftrightarrow(2,1).$$

The decoupled level-two singular vector of $\sigma$ constrains the possible families in

$$\sigma\times\sigma$$

to at most two. The exponent calculation in Lesson 26 gives the candidate families

$$\mathbf 1 \qquad\text{and}\qquad \varepsilon.$$

Both occur in the Ising theory, whose fusion rule is

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

The energy field is also level-two degenerate, and its fusion with itself closes back to the identity:

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

Together with

$$\sigma\times\varepsilon=\sigma,$$

these are the actual Ising fusion rules, as given in Di Francesco et al. 1997, §7.4.2, p. 221, Eq. (7.85). They agree with the degenerate selection rules, while the model’s spectrum and nonzero couplings supply more information than any one indicial equation.

Summary

In the holomorphic sector, a field is primary if its stress-tensor OPE has only the standard second- and first-order singular terms. Negative Virasoro modes generate its descendants. The Kac weight $h_{r,s}(c)$ guarantees a singular vector at level $rs$ in the Verma module; at minimal-model values, further singular vectors can occur in that same Verma module.

The Kac determinant gives the global organizing formula:

$$\det M_N(c,h) \propto \prod_{rs\le N}\left(h-h_{r,s}(c)\right)^{p(N-rs)}.$$

With

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

the special weights are

$$h_{r,s}={c-1\over24}+{1\over4}(r\alpha_+ +s\alpha_-)^2.$$

A degenerate representation supplies a singular submodule. When the CFT quotients it out, the resulting null-field relation restricts possible OPE families. For the level-two fields, the selection rules are

$$\psi_{1,2}\times\phi_\alpha\sim\phi_{\alpha+\alpha_-}+\phi_{\alpha-\alpha_-}, \qquad \psi_{2,1}\times\phi_\alpha\sim\phi_{\alpha+\alpha_+}+\phi_{\alpha-\alpha_+}.$$

At the minimal-model values, the null quotients and fusion constraints lead to finite Kac tables. A full local theory also needs a consistent left–right spectrum and OPE coefficients satisfying crossing symmetry. Lesson 28 develops the minimal models and their fusion algebra.

Common pitfalls

The labels $(r,s)$ do not mean tensor indices, charges, or coordinate components. They label Virasoro degenerate representations and guarantee a singular vector at level $rs$. At special central charges, additional singular vectors may occur and the same weight can have more than one Kac label.

The formula $h_{r,s}=h_{p-r,p'-s}$ is not a universal identity for arbitrary $c$. It is a finite Kac-table identification in rational minimal models with specified coprime integers $(p,p')$.

Zero norm, radical-nullness and singularity are different notions. Positivity makes zero-norm vectors radical-null, but a null descendant need not be singular. Singular vectors are nonzero in the Verma module and vanish in its irreducible quotient by the maximal proper submodule.

The fusion formulas written in terms of $\alpha$ are selection rules. They do not by themselves determine the numerical OPE coefficients. Crossing symmetry and normalization conventions are still needed.

Exercises

Exercise 1: Reading primary data from the stress-tensor OPE

Use the stress-tensor OPE to show that a primary field $O$ obeys

$$L_{-1}O=\partial O, \qquad L_0O=hO, \qquad L_nO=0\quad(n>0).$$

Solution

For a primary field,

$$T(z)O(0)\sim {hO(0)\over z^2}+{\partial O(0)\over z}.$$

The mode action is

$$(L_nO)(0)={1\over2\pi i}\oint_0 dz\,z^{n+1}T(z)O(0).$$

For $n\ge-1$, regular OPE terms have zero residue, so substitute the singular terms:

$$(L_nO)(0) ={1\over2\pi i}\oint_0 dz\, \left(hO(0)z^{n-1}+\partial O(0)z^n\right).$$

The contour integral extracts the coefficient of $z^{-1}$. The first term contributes only for $n=0$, giving $hO$. The second contributes only for $n=-1$, giving $\partial O$. No pole occurs for $n>0$, so $L_nO=0$ for positive $n$.

Exercise 2: The level-two Gram determinant

Use the normalized contravariant form above, or the radial Hermitian form for real $c,h$ with $L_n^\dagger=L_{-n}$ and $\langle h|h\rangle=1$. Compute the level-two Gram matrix in the basis $L_{-2}|h\rangle$, $L_{-1}^2|h\rangle$ and show that

$$\det M_2=2h\left[16h^2+(2c-10)h+c\right].$$

Solution

Using the Virasoro algebra,

$$[L_2,L_{-2}]=4L_0+{c\over2},$$

so

$$\langle h|L_2L_{-2}|h\rangle=4h+{c\over2}.$$

Also

$$[L_2,L_{-1}]=3L_1,$$

so

$$L_2L_{-1}^2|h\rangle =3L_1L_{-1}|h\rangle=6h|h\rangle,$$

and hence the off-diagonal entries are $6h$. Finally,

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

so

$$\langle h|L_1^2L_{-1}^2|h\rangle =2(2h+1)\langle h|L_1L_{-1}|h\rangle =4h(2h+1).$$

Thus

$$M_2= \begin{pmatrix} 4h+{c\over2} & 6h \\ 6h & 4h(2h+1) \end{pmatrix}.$$

Taking the determinant gives

$$\det M_2 =\left(4h+{c\over2}\right)4h(2h+1)-36h^2 =2h\left[16h^2+(2c-10)h+c\right].$$

Exercise 3: Matching the two level-two Kac weights

Show that the two roots, counted with multiplicity, of the level-two condition

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

are $h_{1,2}$ and $h_{2,1}$ in the parametrization

$$h_{r,s}={c-1\over24}+{1\over4}(r\alpha_+ +s\alpha_-)^2, \qquad c=1-6(\alpha_++\alpha_-)^2, \qquad \alpha_+\alpha_-=-1.$$

Solution

Let

$$h_{1,2}={c-1\over24}+{1\over4}(\alpha_+ +2\alpha_-)^2.$$

Using $c=1-6(\alpha_++\alpha_-)^2$ and $\alpha_+\alpha_-=-1$, expand $h_{1,2}$:

$$h_{1,2} =-{(\alpha_++\alpha_-)^2\over4} +{(\alpha_+ +2\alpha_-)^2\over4} ={2\alpha_+\alpha_-+3\alpha_-^2\over4} ={-2+3\alpha_-^2\over4}.$$

Substituting this expression, together with

$$c=1-6\left(\alpha_+^2+2\alpha_+\alpha_-+\alpha_-^2\right) =13-6\alpha_+^2-6\alpha_-^2,$$

and $\alpha_+=-1/\alpha_-$, into the quadratic gives zero. The same calculation with $\alpha_+$ and $\alpha_-$ exchanged proves the result for

$$h_{2,1}={c-1\over24}+{1\over4}(2\alpha_+ +\alpha_-)^2.$$

Moreover, $h_{1,2}+h_{2,1}=[-4+3(\alpha_+^2+\alpha_-^2)]/4=(5-c)/8$. This is the sum of the quadratic’s two roots by Vieta’s formula. Since $h_{1,2}$ is a root, the remaining root is $h_{2,1}$, including when they coincide.

Exercise 4: The first unitary minimal-series central charges

For the unitary minimal series $c_p=1-6/[p(p+1)]$, compute the central charges for $p=3,4,5$.

Solution

The formula is

$$c_p=1-{6\over p(p+1)}.$$

For $p=3$,

$$c_3=1-{6\over12}={1\over2}.$$

For $p=4$,

$$c_4=1-{6\over20}={7\over10}.$$

For $p=5$,

$$c_5=1-{6\over30}={4\over5}.$$

The first two are the Ising and tricritical-Ising central charges. The value $c=4/5$ is also the Virasoro central charge of the critical three-state Potts CFT, whose full local theory uses the non-diagonal $D$-series modular invariant rather than the diagonal $A$-series pairing.

Exercise 5: Fusing two level-two degenerate fields

Assuming the level-two null relation decouples, use the schematic selection rule

$$\psi_{1,2}\times\phi_\alpha \sim \phi_{\alpha+\alpha_-}+\phi_{\alpha-\alpha_-}$$

to determine the two possible momentum labels in $\psi_{1,2}\times\psi_{1,2}$.

Solution

The momentum label of $\psi_{1,2}$ is

$$\alpha_{1,2}=\alpha_+ +2\alpha_-.$$

Apply the fusion rule with $\alpha=\alpha_{1,2}$:

$$\alpha_{1,2}+\alpha_-= \alpha_+ +3\alpha_-,$$

and

$$\alpha_{1,2}-\alpha_-= \alpha_+ +\alpha_-.$$

These are the momenta corresponding to $(1,3)$ and $(1,1)$, respectively. Thus, at the level of Kac-label selection rules,

$$(1,2)\times(1,2)=(1,1)+(1,3),$$

before any finite-table identifications or truncations are imposed. In the Ising minimal model, $(1,3)$ is identified with the energy operator.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.

Further reading

Highest-weight modules, null states, and the Kac determinant develops the general Virasoro representation result. Descendant Gram matrices explains the broader method of extracting positivity constraints from conformal commutators.