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 null submodules; quotienting the maximal proper submodule gives the irreducible representation used in irreducible characters and conformal families. This page constructs the first two levels explicitly, distinguishes null from singular vectors, and states the Kac determinant and unitarity classification with their hypotheses. The full proofs of those two general theorems are outside its scope.

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 nonzero 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 parameters $c,h$ may be complex in the algebraic construction. The ordered PBW monomials form a basis of the abstract Verma module $V_{c,h}$ :

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 $p(N)$ . Relations in a physical quotient do not invalidate this PBW independence in the abstract module.

Normalize the symmetric bilinear contravariant form $B$ by

B(|h;c\rangle,|h;c\rangle)=1, \qquad B(Xv,w)=B(v,\omega(X)w),

where the linear anti-involution obeys $\omega(L_n)=L_{-n}$ , fixes the central generator, and reverses products: $\omega(XY)=\omega(Y)\omega(X)$ . Its radical consists of vectors orthogonal to the whole module. The bra-ket matrix elements below denote this algebraic form unless a Hermitian interpretation is specified.

For real $c,h$ , the Gram entries in this PBW basis are real. Conjugating the coefficients of the first argument instead gives a Hermitian form with $L_n^\dagger=L_{-n}$ and $\langle h;c|h;c\rangle=1$ . This form can be indefinite or positive semidefinite; positivity is an additional requirement, not a property of every Verma module. Contravariance extends the radial-inner-product argument algebraically; the singular-vector orthogonality argument is given in Di Francesco, Mathieu, and Sénéchal 1997, §7.1.3, pp. 204–205, Eqs. (7.17)–(7.18).

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. These low-level forms and determinant agree with Di Francesco, Mathieu, and Sénéchal 1997, §7.2.1, pp. 205–207, Eqs. (7.19), (7.23)–(7.26); the source orders the two level-two basis vectors oppositely, which exchanges the diagonal entries but leaves the determinant unchanged.

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 $L_2$ , hence by every positive mode. The denominator requires $h\ne-1/2$ ; the degeneracy polynomial is 9 at $h=-1/2$ for any $c$ , so no solution is lost. The vector is nonzero in the abstract PBW module. It is a singular vector: a nonzero positive-level descendant that is itself highest weight. It generates a proper invariant submodule.

The submodule is null by contravariance. At the singular vector’s level, move the negative modes in any basis vector to positive modes acting on $|\chi_2\rangle$ , giving zero. Distinct levels are orthogonal because $L_0$ is self-adjoint for the contravariant form and their eigenvalues differ. Finally, contravariance gives $B(u,L_{-n}\chi_2)=B(L_nu,\chi_2)=0$ , so every descendant is also in the radical. This proves orthogonality; it does not say every descendant is highest weight. The same argument appears in Di Francesco, Mathieu, and Sénéchal 1997, §7.1.3, pp. 204–205, Eqs. (7.17)–(7.18).

Three related notions should be separated. A norm in the table means the Hermitian form at real $c,h$ ; radical and singular-vector conditions also make sense algebraically.

Object	Condition	Consequence before quotienting
Zero-norm state	$\langle\psi\lvert\psi\rangle=0$	Need not be orthogonal to all states in an indefinite form
Null state	Orthogonal to the whole module	Lies in the radical of the contravariant form
Singular vector	Nonzero positive-level descendant annihilated by every $L_{n>0}$	Generates a highest-weight submodule contained in the radical

For a positive-semidefinite Hermitian form, Cauchy–Schwarz makes zero norm equivalent to radical membership. Singularity remains a stronger condition. For example, take $c=1,h=0$ and let $|0\rangle$ denote the abstract highest vector before quotienting. The Gram form is positive semidefinite, as follows level by level by continuity from the positive forms at $c>1,h>0$ described below. Set

|\chi\rangle=L_{-1}|0\rangle, \qquad |d\rangle=L_{-2}|\chi\rangle.

Here $|\chi\rangle$ is singular and $|d\rangle$ is null, but

L_1|d\rangle =3L_{-1}|\chi\rangle =3L_{-1}^{2}|0\rangle\ne0.

The last inequality follows from PBW independence. Thus $|d\rangle$ is not singular. Both states become zero in the physical irreducible quotient; the counterexample concerns their nonzero vectors in the abstract 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. Quotienting just one chosen singular submodule need not be sufficient. This is the irreducible construction; a physical realization that retains indecomposable structure requires additional data.

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

For nonzero complex $b$ , 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}.

The Kac determinant theorem states that, at level $N$ ,

\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 $A_N\ne0$ depends on the fixed ordered PBW basis and its normalization, not on $c,h$ . The exponent $p(N-rs)$ counts descendants of a singular vector arising at level $rs$ . 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. The general product, rather than a proof from the first two levels, is the sourced result in Di Francesco, Mathieu, and Sénéchal 1997, §7.2.1, pp. 207–208, Eqs. (7.28)–(7.35). Its parameter is $t=b^{-2}$ ; replacing $b$ by $b^{-1}$ exchanges $r$ and $s$ and leaves the full determinant unchanged.

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 $(r,s)\sim(p-r,p'-s)$ . The two generating singular vectors occur at levels $rs$ and $(p-r)(p'-s)$ ; their submodules can intersect. The unitary Virasoro minimal series is the special sequence $p'=p+1$ ; the other coprime pairs in this range give nonunitary minimal models. This submodule structure is described in Di Francesco, Mathieu, and Sénéchal 1997, §8.1.1, pp. 240–241, Eqs. (8.4)–(8.13). The source calls our smaller $p$ its $p'$ and our larger $p'$ its $p$ ; the label ranges and levels above use the present convention throughout.

Positivity and its limits

For a unitary irreducible representation, every quotient Gram matrix must be positive definite. Positivity of the normalized single-mode descendants requires $h\ge0$ and $c\ge0$ , as in Di Francesco, Mathieu, and Sénéchal 1997, §7.2.1, p. 205, Eq. (7.19). These conditions are necessary, not sufficient.

The classification of irreducible unitary highest-weight Virasoro representations is a sourced theorem, whose full proof is not given here. For real $c,h$ and $L_n^\dagger=L_{-n}$ , a positive-definite highest-weight representation exists precisely for $c\ge1,h\ge0$ , or for the discrete series

c=1-\frac{6}{m(m+1)}, \qquad h=h_{r,s}\ \text{with }p=m,\ p'=m+1, \qquad m=2,3,\ldots,

with the Kac label ranges and identification just stated. At $m=2$ , these give $c=h=0$ and the trivial one-dimensional irreducible representation. The theorem and its distinction from finite-level norm tests are discussed in Di Francesco, Mathieu, and Sénéchal 1997, §§7.2.2–7.2.3, pp. 209–211, Eq. (7.41). Sufficiency for the nontrivial discrete series follows from the unitary diagonal coset $(SU(2)_k\times SU(2)_1)/SU(2)_{k+1}$ at positive integer level $k$ , with $k+2=m\ge3$ : its branching characters reproduce all the minimal-series representations Di Francesco, Mathieu, and Sénéchal 1997, p. 798, §18.3, pp. 807–810, Eqs. (18.51)–(18.53), (18.78)–(18.80). This result concerns positive-energy representations, not the existence or classification of complete local CFTs with a consistent OPE and left/right pairing.

For the holomorphic identity/vacuum representation, $h=0$ and $L_{-1}|0\rangle$ is null in the vacuum Verma form; it is set to zero in the translation-invariant vacuum representation. A left weight $h=0$ alone does not identify a full left/right field with the vacuum. Treating the vacuum character as the unrestricted product $\prod_{n\ge1}(1-q^n)^{-1}$ overcounts states. After this level-one relation the remaining generating modes start at $L_{-2}$ , but further null relations can reduce them; in particular the $c=h=0$ irreducible quotient has no descendants.

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. The irreducible character requires the quotient by the maximal proper submodule, which may need more than one singular generator.

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

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