Highest-Weight Modules, Null States, and the Kac Determinant

A Virasoro highest-weight representation begins with one state and all of its lowering-mode descendants. The contravariant, or Shapovalov, form detects when descendants become null; those vectors generate invariant submodules that must be quotiented before characters, OPE channels, or sewing sums are computed. This page constructs the first two levels explicitly and then states the Kac determinant with its parametrization and unitarity qualifications.

Required background. The Virasoro algebra and the stress tensor fix the mode and Hermiticity conventions. Descendant Gram matrices explain how radial inner products encode positivity and null states.

Helpful background. Characters and multiplet counting connect partitions of the level to graded traces.

Verma modules and the Shapovalov form

Fix a highest-weight state $|h;c\rangle$ satisfying

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

with central charge $c$. The Verma module $V_{c,h}$ is spanned by

$$L_{-\lambda}|h;c\rangle =L_{-\lambda_1}\cdots L_{-\lambda_k}|h;c\rangle, \qquad \lambda_1\ge\cdots\ge\lambda_k\ge1.$$

Its level is $|\lambda|=\sum_i\lambda_i$, so the number of unrestricted descendants at level $N$ is the partition number $p(N)$. In a unitary radial quantization, $L_n^\dagger=L_{-n}$ and $\langle h;c|h;c\rangle=1$. Algebraically, the same Gram matrices arise from the contravariant form defined by the anti-involution $L_n\mapsto L_{-n}$, even when that form is indefinite.

At level one there is one state and

$$G_1=\langle h|L_1L_{-1}|h\rangle=2h.$$

At level two, use the ordered basis

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

Repeated use of the Virasoro commutator gives

$$G_2= \begin{pmatrix} 4h+c/2 & 6h\\ 6h & 4h(2h+1) \end{pmatrix},$$

and therefore

$$\det G_2 =2h\left[16h^2+2(c-5)h+c\right].$$

This computation fixes all normalization factors: for example,

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

not $(2h)^2$, because the second commutator acts on a level-one state. Low-level forms and their relation to reducibility are developed in Di Francesco, Mathieu, and Sénéchal 1997, §§7.1–7.2.

Null vectors and irreducible quotients

When

$$16h^2+2(c-5)h+c=0,$$

the level-two vector

$$|\chi_{2}\rangle =\left( L_{-2}-\frac{3}{2(2h+1)}L_{-1}^2 \right)|h\rangle$$

is annihilated by $L_1$ and $L_2$ and is orthogonal to every state in the Verma module. It is a singular vector—another highest-weight vector inside $V_{c,h}$—and it generates a proper submodule.

Three related notions should be separated:

Object	Algebraic condition	What happens in the physical module
Zero-norm state	$\langle\psi\lvert\psi\rangle=0$	Need not be orthogonal to all states in an indefinite theory
Null state	Orthogonal to the whole module	Lies in the radical of the Shapovalov form
Singular vector	Annihilated by every $L_{n>0}$	Generates an invariant highest-weight submodule

In a positive-semidefinite highest-weight module, a zero-norm state is orthogonal to all states by Cauchy–Schwarz, so these distinctions collapse in the relevant subspace. They do not collapse in a general nonunitary module.

The irreducible highest-weight representation is the quotient

$$L_{c,h}=V_{c,h}/\mathcal N_{c,h},$$

where $\mathcal N_{c,h}$ is the maximal proper submodule. One must quotient the full submodule generated by each singular vector, not merely delete that vector at its first level. Nested and intersecting singular submodules are why a determinant zero alone does not immediately give the irreducible character.

For the Ising spin module, $c=1/2$ and $h=1/16$. Substitution gives

$$16h^2+2(c-5)h+c=0, \qquad |\chi_\sigma\rangle =\left(L_{-2}-\frac43L_{-1}^2\right)|\sigma\rangle.$$

Both $L_1|\chi_\sigma\rangle=0$ and $L_2|\chi_\sigma\rangle=0$ follow directly, providing a sensitive check on the $c/2$ entry of $G_2$ and the $4/3$ coefficient.

The Kac determinant

A convenient parametrization is

$$c=1-6\left(b-b^{-1}\right)^2,$$

with degenerate weights

$$h_{r,s}(c) =\frac14\left[ \left(\frac{r}{b}-sb\right)^2 -\left(\frac{1}{b}-b\right)^2 \right], \qquad r,s\in\mathbb Z_{>0}.$$

At level $N$, the Kac determinant has the form

$$\det G_N(c,h) =A_N \prod_{\substack{r,s\ge1\\rs\le N}} \left(h-h_{r,s}(c)\right)^{p(N-rs)},$$

where $A_N\neq0$ depends on the ordered descendant basis and normalization but not on $h$. The exponent $p(N-rs)$ counts descendants of the singular vector. At coincident Kac zeros, the embedding structure can be subtler than reading independent factors from this product; the determinant diagnoses reducibility, while submodule diagrams determine the quotient.

For coprime integers $2\le p<p'$, take $b^2=p/p'$. Then

$$c_{p,p'}=1-\frac{6(p'-p)^2}{pp'}, \qquad h_{r,s}=\frac{(p'r-ps)^2-(p'-p)^2}{4pp'}.$$

The minimal-model labels obey $1\le r\le p-1$, $1\le s\le p'-1$, with $(r,s)\sim(p-r,p'-s)$. A module at $h_{r,s}$ has singular vectors first appearing at levels $rs$ and $(p-r)(p'-s)$. The unitary Virasoro minimal series is the special sequence $p'=p+1$; generic coprime pairs define nonunitary minimal models. The determinant formula and null-state mechanism originate in the exact representation analysis of Belavin, Polyakov, and Zamolodchikov 1984, §§2–3, pp. 344–355.

Positivity and its limits

For a unitary representation, every quotient Gram matrix must be positive definite. The first levels already imply $h\ge0$ and $c\ge0$ for a nontrivial vacuum theory. These conditions are necessary, not sufficient. The full theorem classifies unitary highest-weight Virasoro representations: for $c\ge1$, unitarity is possible for $h\ge0$; for $0<c<1$, only the discrete series $c=1-6/[m(m+1)]$ with the allowed Kac weights survives. This statement assumes a highest-weight representation, positive-definite Hermitian form, $L_n^\dagger=L_{-n}$, and positive energy; it is not a classification of all two-dimensional QFTs.

At $h=0$, $L_{-1}|0\rangle$ is null because translation leaves the vacuum invariant. The vacuum Verma module must therefore be quotiented already at level one. Treating its character as the unrestricted product $\prod_{n\ge1}(1-q^n)^{-1}$ overcounts states; the first oscillator starts at $L_{-2}$ after the vacuum null relation.

Indefinite highest-weight forms, indecomposable extensions, and Jordan actions require additional representation data beyond this irreducible-quotient construction; those cases continue in Nonunitary, Logarithmic, and Noncompact Two-Dimensional CFT.

Exercises

Verify the Ising level-two null vector by acting with $L_1$ and $L_2$.

Solution

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

$$L_1|\chi\rangle =\left(3-2a(2h+1)\right)L_{-1}|h\rangle,$$

so $a=3/[2(2h+1)]$. Also

$$L_2|\chi\rangle =\left(4h+\frac c2-6ah\right)|h\rangle.$$

With $h=1/16$, $c=1/2$, one has $a=4/3$, and both coefficients vanish.

Common pitfalls

Deleting one null vector instead of its descendants. A singular vector generates a whole submodule. Characters and sewing sums require the irreducible quotient.

Inferring unitarity from a determinant zero. A zero diagnoses reducibility. Unitarity requires every non-null eigenvalue at every level to be positive after quotienting.

Changing the $b$ convention mid-calculation. The formulas above use $b^2=p/p'$ and $c=1-6(b-b^{-1})^2$. Replacing $b$ by $b^{-1}$ exchanges Kac labels; it is harmless only if done everywhere.

References

• Belavin, Alexander A., Alexander M. Polyakov, and Alexander B. Zamolodchikov. “Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory.” Nuclear Physics B 241, no. 2 (1984): 333–380. DOI.
• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.