Degenerate Fields, Kac Labels, and Constraints

The previous page showed the first miracle of two-dimensional conformal field theory: a null descendant is not merely a state-counting accident. Once inserted into a correlator, the null relation becomes a differential equation. For a level-two degenerate field this is the BPZ equation.

This page turns that example into a classification tool. Degenerate Virasoro representations occur at special conformal weights. Those special weights are indexed by two positive integers, the Kac labels $(r,s)$. The labels are not decoration: they guarantee a singular vector at level $rs$, determine the corresponding BPZ differential equation, and restrict the local channels in an operator product expansion. At special rational central charges, coincident Kac weights can give the same module additional singular vectors, sometimes at a lower level.

The thread is:

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

The end result is the beginning of rational conformal field theory. A generic irrational CFT has infinitely many primaries or even a continuous spectrum. A minimal model has finitely many Virasoro primary representations because null submodules are quotiented out and the remaining representations close under fusion and modular consistency.

Modes acting on local fields

Highest-weight and Virasoro-mode normalization

We work in one holomorphic sector. 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.

Let $O(0)$ be a local holomorphic field. The stress-tensor OPE can be read as a generating function for 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 negative modes create descendants:

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

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 nontrivial descendant of the identity is

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

This is why the stress tensor belongs to the identity module. It is also why $T$ is not an ordinary primary: the $T(z)T(w)$ OPE contains the 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. For an ordinary primary field,

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

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

The Laurent coefficients in $T(z)O(0)$ are Virasoro descendants. For a primary, the positive modes vanish, while $L_{-1}$ gives a derivative and lower modes generate higher descendants.

Degenerate modules

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

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

At level $N=n_1+\cdots+n_k$, the number of basis states before imposing relations is the partition number $p(N)$. For example,

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

A module is degenerate if some descendant is also highest-weight. That is, 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 called a null state or singular vector. It generates its own Verma submodule inside $\mathcal V_{c,h}$. Passing to the irreducible representation means quotienting by that submodule.

The practical consequence is huge. If a primary field $\psi$ has a decoupled null vector at level $N$, then correlators containing $\psi$ obey an $N$th-order differential equation. The level-two case from the previous page gives a second-order BPZ equation. A $(1,3)$ or $(3,1)$ degenerate field gives a third-order equation. In general, a primary with Kac labels $(r,s)$ has a singular vector at level

$$N=rs.$$

For generic $c$, this is the first singular vector in the Verma module. At rational minimal-model values, the same irreducible module can carry a second singular vector, and coincidences among Kac weights can make the phrase “first null level” label-dependent.

Positivity and Gram matrices

Radial quantization turns local operators into states. The norm of a smeared state must be nonnegative in a unitary CFT:

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

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. In a unitary CFT with a unique $SL(2)$-invariant vacuum, the weight-zero state is the vacuum and this is translation invariance; in nonunitary, logarithmic, or noncompact theories, weight-zero sectors can require additional care.

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 level-two null state appears exactly when this determinant vanishes. For $h\ne0$, this gives

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

with null vector

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

At level two, the descendant space is two-dimensional before quotienting. The Gram determinant vanishes precisely when a linear combination of $L_{-2}|h\rangle$ and $L_{-1}^2|h\rangle$ becomes null.

The level-two calculation is the first visible piece of a much larger theorem. Positivity places inequalities on every Gram matrix. Zeros of those determinants mark reducible modules. For nontrivial unitary positive-energy Virasoro representations with $0<c<1$, enforcing all levels at once restricts both the central charge and the allowed highest weights to the discrete unitary minimal series.

The Kac determinant and Kac labels

The determinant of the level-$N$ Gram matrix factorizes beautifully. Up to a nonzero normalization constant,

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

Here $p(k)$ is the number of integer partitions of $k$, with $p(0)=1$. The zeros are the Kac weights $h_{r,s}(c)$. If

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

then the Verma module has a singular vector at level

$$rs.$$

To write these weights compactly, introduce

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

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 are exactly

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

The first few labels are therefore:

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

The Kac labels $(r,s)$ form a lattice of degenerate highest-weight representations. The label guarantees a singular vector at level $rs$; at generic central charge this is the first one.

The formula may look mysterious the first time one meets it. It is best understood as a map from representation theory to algebraic geometry: the space of possible highest weights $(c,h)$ contains hypersurfaces on which the Gram matrix becomes singular. The Kac labels name those hypersurfaces.

Rational central charge and the minimal-model window

The Kac formula simplifies when $\alpha_+^2$ is rational. Write

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

where $p$ and $p'$ are coprime integers greater than one. 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 coprime integers $p,p'>1$, these formulas describe the Kac table of a Virasoro minimal model once null-submodule quotients, fusion closure, and a consistent left–right pairing are imposed. 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 unitary minimal series is the special sequence

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

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 nontrivial primary weights are

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

corresponding to the identity, spin, and energy fields.

For $c<1$, unitarity is extremely restrictive: the allowed central charges form the discrete sequence $c_p=1-6/[p(p+1)]$, accumulating at $c=1$. The first point, $p=3$, is the Ising model.

This discreteness is one of the deepest consequences of the Virasoro algebra. In higher-dimensional CFT, conformal symmetry constrains correlators strongly but usually does not quantize the operator spectrum. In two dimensions, the infinite-dimensional chiral algebra can do exactly that.

Degenerate fields as fusion constraints

The same Kac labels also constrain OPEs. It is helpful to temporarily 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 $\phi_\alpha$:

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

Because $\psi_{1,2}$ satisfies a second-order BPZ equation, the local OPE near $z=0$ has only two independent local exponents. Those two exponents correspond to intermediate weights

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

Thus, schematically,

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

Similarly,

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

For a general degenerate field $\psi_{r,s}$, repeated application gives the schematic fusion rule

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

This formula should be read as a representation-theoretic selection rule. A particular CFT may omit some fields or have vanishing OPE coefficients because of additional symmetries, locality constraints, or the finite Kac-table identifications of a minimal model.

A level-two degenerate insertion has only two local OPE channels. In the $\alpha$ parametrization, $\psi_{1,2}$ shifts a generic primary by $\pm\alpha_-$, while $\psi_{2,1}$ shifts it by $\pm\alpha_+$.

When both fields are degenerate, the shifts close on a discrete set of Kac labels. This is the algebraic origin of the finite fusion rules of the minimal models.

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

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

The three primary fields of the Ising model are therefore identified as

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

The level-two degeneracy of $\sigma$ implies that

$$\sigma\times\sigma$$

has only two local channels. The exponent calculation on the previous page identified these channels as

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

Thus

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

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 Ising fusion rules. The point is not that we guessed them from lattice spins. The Virasoro representation theory forces the same algebra.

Summary

A local field $O$ is primary if the stress-tensor OPE has only the standard second- and first-order poles. The negative Virasoro modes generate descendants; special descendants can become singular vectors. The Kac weight $h_{r,s}(c)$ guarantees such a vector at level $rs$, while rational central charges may produce additional singular vectors in the same 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 field does two things at once: it creates a null relation in the Hilbert space, and it restricts the OPE channels in position space. For the level-two fields,

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

At the minimal-model rational central charges, after quotienting null submodules and imposing fusion closure and a consistent left–right spectrum, these constraints close on finite Kac tables. The next page uses this structure to build the minimal models and the Ising operator algebra systematically.

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 $(p,p')$.

A null state is stronger than a state of zero expectation value. In a unitary theory it has zero norm and is orthogonal to all physical states. In the Verma module the singular vector is a nonzero algebraic state; it becomes zero only after one passes to the irreducible quotient by the submodule it generates.

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

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

Substitute the OPE:

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

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

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 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

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

Equivalently, since the quadratic has two roots and both $h_{1,2}$ and $h_{2,1}$ are level-two degenerate weights, these are precisely the two roots.

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

Use the schematic fusion 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 and further reading

V. G. Kac, “Contravariant form for infinite-dimensional Lie algebras and superalgebras,” is the original representation-theoretic source of the determinant formula.

A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nuclear Physics B 241 (1984), introduced the conformal-bootstrap use of degenerate fields and BPZ equations.

D. Friedan, Z. Qiu, and S. Shenker, “Conformal invariance, unitarity, and critical exponents in two dimensions,” Physical Review Letters 52 (1984), established the unitary minimal series.

P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Chapters 6–8, gives a detailed account of Kac determinants, degenerate representations, fusion rules, and minimal models.

P. Ginsparg, “Applied Conformal Field Theory,” Les Houches lectures, is a compact and readable guide to the same circle of ideas.

Further reading

This lesson follows the manuscript’s route from the level-two Gram matrix to Kac labels and fusion shifts. For maintained reference accounts, see Highest-weight modules, null states, and the Kac determinant and Descendant Gram matrices.