Minimal Models, Fusion, and the Ising Operator Algebra

Virasoro minimal models have a finite set of irreducible chiral representations at central charges $c=1-6(p'-p)^2/(pp')$, with coprime integers $2\le p<p'$. Their null-vector quotients lead to finite fusion rules. When the null submodules decouple in correlation functions, the corresponding BPZ equations constrain the operator-product channels. A full local theory also needs a consistent pairing of its two chiral sectors and crossing-consistent OPE coefficients.

We state the minimal-model spectrum and fusion rule, then derive the operator algebra of the diagonal Ising CFT. The general construction is outside this lesson; we use Lesson 27’s Kac labels and irreducible quotients and Lesson 26’s null-decoupling conditions. The Ising theory has three bulk primary families,

$$[\mathbf 1], \qquad [\sigma], \qquad [\varepsilon],$$

which obey the finite fusion rules

$$[\sigma]\times[\sigma]=[\mathbf 1]+[\varepsilon], \qquad [\sigma]\times[\varepsilon]=[\sigma], \qquad [\varepsilon]\times[\varepsilon]=[\mathbf 1].$$

These formulas should be read as statements about whole conformal families, not only about the primary fields at the top. For example, $[\mathbf 1]$ contains $\mathbf 1$, $T$, $\bar T$, $T\bar T$, and their descendants; $[\varepsilon]$ contains $\varepsilon$ and its derivatives. Fusion specifies which conformal families can appear in an OPE.

From Kac labels to a finite spectrum

The Kac formula gives a chiral conformal weight $h_{r,s}$. For the diagonal minimal models used here, the antiholomorphic weight is the same, $\bar h=h$, so the full scaling dimension is

$$\Delta=h+\bar h=2h.$$

Thus the Ising spin field has $h_\sigma=\bar h_\sigma=1/16$ and full dimension $\Delta_\sigma=1/8$. The energy field has $h_\varepsilon=\bar h_\varepsilon=1/2$ and full dimension $\Delta_\varepsilon=1$.

The minimal-model central charges and Kac weights are

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

and

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

Here $p$ and $p'$ are coprime integers greater than one. We take $p<p'$ for definiteness. The minimal-model spectrum is obtained by restricting the Kac labels to the finite rectangle

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

and then imposing the identification

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

There is no fixed point of this reflection: one would need both $p$ and $p'$ even, contradicting coprimality. The number of distinct chiral irreducible sectors, and of bulk primary families in the diagonal model, is therefore

$$\boxed{ N_{p,p'}={1\over2}(p-1)(p'-1). }$$

These are irreducible representations obtained after quotienting the appropriate null submodules. Their finite set closes under the minimal-model fusion rules; this does not prohibit other CFTs at the same central charge from using different representation spaces. The construction and label identification are developed in Di Francesco et al. 1997, §7.3.3, pp. 216–218, Eqs. (7.65–7.73).

The nontrivial unitary minimal series, with $0<c<1$, is the special case

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

with

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

The unitary classification is developed in Di Francesco et al. 1997, §7.2.3, pp. 210–211, with the positive-level coset construction in Di Francesco et al. 1997, §18.3, pp. 807–810. The first few models are

First nontrivial unitary minimal models and their central charges

p	Central charge c	Diagonal model
3	1/2	Ising
4	7/10	Tricritical Ising
5	4/5	Diagonal M(5,6); the critical three-state Potts CFT instead uses a non-diagonal D-series invariant

The endpoint $p=2$, $p'=3$ gives the trivial $c=0$ theory. Nonunitary minimal models also occur, but our worked example lies in the nontrivial unitary series. The Ising and tricritical identifications appear in Di Francesco et al. 1997, §§7.4.2–7.4.3, pp. 221–222; the different Potts pairing is explained in Di Francesco et al. 1997, §§10.7.2–10.7.3, pp. 365–368, Eq. (10.150).

Fusion rules of minimal models

The OPE of two primary families has the form

$$[\phi_i]\times[\phi_j] = \sum_k N_{ij}^{\phantom{ij}k}[\phi_k],$$

where the fusion coefficients $N_{ij}^{\phantom{ij}k}$ are nonnegative integers. The chiral fusion rule for irreducible Virasoro minimal-model representations can be written directly in terms of Kac labels. In the diagonal A-series, the same labels pair left and right sectors to give the bulk primary families used below:

$$\boxed{ [\phi_{r_1,s_1}]\times[\phi_{r_2,s_2}] = \sum_{\substack{r=|r_1-r_2|+1\\ \mathrm{step}\ 2}}^{r_{\max}} \sum_{\substack{s=|s_1-s_2|+1\\ \mathrm{step}\ 2}}^{s_{\max}} [\phi_{r,s}], }$$

with

$$r_{\max}=\min(r_1+r_2-1,2p-r_1-r_2-1),$$

and

$$s_{\max}=\min(s_1+s_2-1,2p'-s_1-s_2-1).$$

The phrase “step 2” means that $r$ and $s$ run in jumps of two. This parity condition resembles adding $SU(2)$ spins, but the second upper bound is essential. The full rule is given in Di Francesco et al. 1997, §7.3.3, p. 217, Eqs. (7.70–7.71). Its finite-algebra derivation uses the null relations to impose polynomial identities on the elementary fusion generators; for models containing both elementary degenerate fields, see Di Francesco et al. 1997, §8.4.3, pp. 259–264, Eq. (8.131). We use the stated rule rather than reproduce that construction.

For fusion with $(1,3)$, all three channels survive when $3\le s\le p'-3$ and $1\le r\le p-1$. In that interior range, the rule becomes

$$(1,3)\times(r,s) \sim (r,s-2)\oplus(r,s)\oplus(r,s+2).$$

At a boundary, apply the exact lower and upper endpoints first. Removing only labels outside the Kac window is insufficient: the upper bound can also exclude an in-window label. For example, in Ising $(1,3)\times(1,3)$ retains $s=1$ but removes both $s=3$ and $s=5$, as calculated below. Equivalent surviving labels are then identified by

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

The fusion rule is stronger than dimensional analysis. Dimensional analysis only says what powers of $|z|$ could occur once a field is present. Fusion says which fields are present at all.

The Ising minimal model

The Ising CFT is

$$\mathcal M(3,4), \qquad c=1-{6(4-3)^2\over 3\cdot4}={1\over2}.$$

The allowed Kac-table window is

$$1\le r\le2, \qquad 1\le s\le3.$$

Before the reflection identification, the six entries are

$$\begin{array}{c|ccc} & s=1 & s=2 & s=3 \\ \hline r=1 & 0 & {1\over16} & {1\over2} \\ r=2 & {1\over2} & {1\over16} & 0. \end{array}$$

The reflection

$$(r,s)\sim(3-r,4-s)$$

pairs the entries into three equivalence classes:

$$\begin{aligned} \mathbf 1&\leftrightarrow (1,1)\sim(2,3), &&h=0,\\ \sigma&\leftrightarrow (1,2)\sim(2,2), &&h={1\over16},\\ \varepsilon&\leftrightarrow (1,3)\sim(2,1), &&h={1\over2}. \end{aligned}$$

The lattice order spin has $\sigma$ as its leading continuum scaling field, with a nonuniversal conversion factor. It is odd under the global $\mathbb Z_2$ spin-flip symmetry. The even field $\varepsilon$ is the thermal scaling operator; its signed relation to a lattice bond is fixed by the course’s thermal convention below.

The full diagonal scaling dimensions are

$$\Delta_{\mathbf 1}=0, \qquad \Delta_\sigma={1\over8}, \qquad \Delta_\varepsilon=1.$$

These are the numbers that control decay of two-point functions on the plane:

$$\langle \sigma(z,\bar z)\sigma(0)\rangle ={1\over |z|^{1/4}}, \qquad \langle \varepsilon(z,\bar z)\varepsilon(0)\rangle ={1\over |z|^2},$$

in the standard normalization.

Deriving the Ising fusion rules

Use the minimal-model fusion formula. First,

$$\sigma\times\sigma=(1,2)\times(1,2).$$

For the $r$ label, only $r=1$ appears. For the $s$ label, the possible values are $s=1,3$. Hence

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

or

$$\boxed{[\sigma]\times[\sigma]=[\mathbf 1]+[\varepsilon].}$$

Next,

$$\sigma\times\varepsilon=(1,2)\times(1,3).$$

The $r$ label again remains $1$, while the exact endpoints leave only $s=2$. Thus

$$\boxed{[\sigma]\times[\varepsilon]=[\sigma].}$$

Finally,

$$\varepsilon\times\varepsilon=(1,3)\times(1,3).$$

Without the finite-table truncation, the $SU(2)$-type pattern would suggest $s=1,3,5$. The actual upper endpoint is

$$s_{\max}=\min(3+3-1,2\cdot4-3-3-1)=1,$$

so only $s=1$ survives. In particular, $s=3$ is removed even though it lies inside the Kac window. The result is

$$\boxed{[\varepsilon]\times[\varepsilon]=[\mathbf 1].}$$

This is the operator-algebra form of the Ising universality class. It knows, for instance, that two spin fields can produce an even field, but one spin field and one energy field must produce an odd field.

OPEs and conformal families

Fusion tells us which families appear. The OPE tells us how they appear at short distance. Keep the unit two-point functions above and the energy-field sign of Lesson 12: $\varepsilon=i\psi\bar\psi$, with positive coupling in $+\tau\int\varepsilon$ pointing toward the ordered phase. For the original order spin, Lesson 15’s signed thermal identification then gives $C_{\sigma\sigma\varepsilon}=-1/2$.

The four-spin function fixes $C_{\sigma\sigma\varepsilon}^2=1/4$. Di Francesco et al. choose $+1/2$ after allowing $\varepsilon\mapsto-\varepsilon$; maintaining the same thermal perturbation would also reverse its coupling. We retain the course’s energy sign. See Di Francesco et al. 1997, §12.3.3, pp. 450–451, Eqs. (12.63–12.64). The OPEs begin as

$$\boxed{ \sigma(z,\bar z)\sigma(0) = {1\over |z|^{1/4}} \left( \mathbf 1-{1\over2}|z|\,\varepsilon(0)+\cdots \right). }$$

The first term is the identity family. The second term is the energy family. The exponent $|z|$ inside the parenthesis is fixed by dimensions:

$$\Delta_\varepsilon-2\Delta_\sigma =1-{1\over4}={3\over4},$$

so the energy contribution is proportional to $|z|^{3/4}\varepsilon(0)$. Factoring out the identity singularity $|z|^{-1/4}$ leaves $|z|\varepsilon(0)$.

Similarly,

$$\boxed{ \varepsilon(z,\bar z)\varepsilon(0) ={1\over |z|^2}\left(\mathbf 1+\cdots\right), }$$

and

$$\boxed{ \varepsilon(z,\bar z)\sigma(0) =-{1\over2|z|}\left(\sigma(0)+\cdots\right). }$$

The ellipses include descendants of the displayed primaries. For example, the identity family in $\sigma\times\sigma$ includes the stress tensors $T$ and $\bar T$ at the next appropriate orders. The family notation is therefore more compact:

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

The coefficient $-1/2$ is OPE data in the stated normalization and energy convention, not a fusion coefficient. The nonnegative coefficient of $[\varepsilon]$ in the fusion rule counts a family; it does not assign a positive OPE coefficient. Rephasing $\varepsilon$ changes the signed OPE data while preserving its two-point function and the four-spin function.

Associativity and the spin four-point function

Fusion rules are not just pairwise selection rules. They must be associative. For four spin fields, one can first fuse the left pair or the middle pair:

$$([\sigma]\times[\sigma])\times([\sigma]\times[\sigma]) =([\mathbf 1]+[\varepsilon])\times([\mathbf 1]+[\varepsilon]).$$

Using the Ising fusion rules,

$$([\mathbf 1]+[\varepsilon])\times([\mathbf 1]+[\varepsilon]) =2[\mathbf 1]+2[\varepsilon].$$

On the other hand,

$$[\sigma]\times([\sigma]\times[\sigma])\times[\sigma] =[\sigma]\times([\mathbf 1]+[\varepsilon])\times[\sigma] =2[\sigma]\times[\sigma] =2[\mathbf 1]+2[\varepsilon].$$

The abstract fusion product is independent of parenthesization. For a sphere four-point function with the vacuum at infinity, project the total product onto $[\mathbf 1]$. Its coefficient in $[\sigma]^4$ is two: the holomorphic conformal-block space has one block through $[\mathbf 1]$ and one through $[\varepsilon]$. The term $2[\varepsilon]$ in the unprojected product belongs to a different total output, not two extra blocks in this vacuum correlator. This fusion-path count is developed in Di Francesco et al. 1997, §10.8.2, pp. 376–377, Eqs. (10.179–10.184).

These are chiral blocks, not two independent full local correlators. A physical four-point function pairs holomorphic and antiholomorphic blocks into a single-valued function whose decompositions agree in different channels. This crossing condition constrains OPE data beyond the path count. The explicit Ising pairing is developed in Minimal models and fusion rules; its positive energy-field convention leaves the four-spin function and squared coefficient unchanged.

Triple spin products and the continuum equation

The fusion ring gives

$$[\sigma]^3=2[\sigma].$$

If we temporarily forget multiplicity and retain only which primary family can appear, a product of three nearby spin fields has support only in the spin family:

$$\operatorname{supp}(\sigma\times\sigma\times\sigma)=\{[\sigma]\}.$$

This is not a literal coincident-point identity $\sigma^3\propto\sigma$. The multiplicity two distinguishes the paths $(\sigma\sigma)_{\mathbf 1}\sigma$ and $(\sigma\sigma)_{\varepsilon}\sigma$, both ending in the spin family. A local composite of three coincident fields requires a point-splitting and subtraction prescription. Such a renormalized composite can mix with $\sigma$ and with its descendants.

For comparison, take a regularized Euclidean Landau–Ginzburg theory of one real scalar with local short-range interactions, a stable quartic coupling $u>0$, and $\mathbb Z_2$ symmetry at $H=0$. A schematic action is

$$S_{\rm LG}=\int d^2x\, \left[{1\over2}(\partial\phi)^2+{r\over2}\phi^2+{u\over4}\phi^4-H\phi\right].$$

Its classical equation of motion is

$$-\partial^2\phi+r\phi+u\phi^3-H=0.$$

At zero magnetic field, tune $r$ to its critical value. In the Ising scaling regime, the leading odd infrared component of $\phi$ is a nonuniversally normalized $\sigma$. The equation-of-motion operator is a particular renormalized, redundant combination of $\partial^2\phi$, $\phi$, and $\phi^3$; the fusion rule alone does not identify a bare $\phi^3$ with $\sigma$ or with a specific descendant. Inserting an equation of motion produces contact terms at other insertions, so its vanishing at separated points is not an unrestricted coincident-product identity. The effective scalar description and renormalized-product interpretation are discussed in Di Francesco et al. 1997, §7.4.7, pp. 231–235, Eqs. (7.115–7.119). This comparison does not prove the existence of the critical limit from the microscopic action.

Absorb the nonuniversal matching factors into continuum scaling couplings $t$ and $H$; these need not equal the parameters in the microscopic scalar action. Ignoring the identity perturbation, which only shifts the vacuum energy, the two nontrivial relevant primary deformations are

$$S_{\mathrm{Ising\ CFT}} +t\int d^2x\,\varepsilon(x) -H\int d^2x\,\sigma(x).$$

Here $t=\tau$ in the course’s thermal convention: at $H=0$, positive $t$ is ordered and negative $t$ is disordered. It is a signed thermal scaling field, not an identification with a positive multiple of $T-T_c$. The source $H$ couples to the order field and explicitly breaks $\mathbb Z_2$. Relevance follows from

$$\Delta_\varepsilon=1<2, \qquad \Delta_\sigma={1\over8}<2.$$

In two dimensions the coupling eigenvalue is $y=2-\Delta$, giving $y_t=1$ and $y_H=15/8$. The microscopic order spin and scalar variable have the same leading scaling operator $\sigma$, with different conversion factors and subleading corrections. This is the limited universality statement used here; the finite fusion algebra alone does not determine the RG trajectory.

Summary

The Virasoro minimal models have finitely many irreducible chiral sectors. For coprime integers $2\le p<p'$,

$$c_{p,p'}=1-{6(p'-p)^2\over pp'}, \qquad h_{r,s}^{(p,p')} ={(p'r-ps)^2-(p'-p)^2\over4pp'}.$$

Their chiral labels obey

$$1\le r\le p-1, \qquad 1\le s\le p'-1, \qquad (r,s)\sim(p-r,p'-s),$$

so

$$N_{p,p'}={1\over2}(p-1)(p'-1).$$

For the Ising model $\mathcal M(3,4)$,

$$c={1\over2}, \qquad \mathbf 1:(h,\bar h)=(0,0), \qquad \sigma:(h,\bar h)=\left({1\over16},{1\over16}\right), \qquad \varepsilon:(h,\bar h)=\left({1\over2},{1\over2}\right).$$

The fusion algebra is

$$[\sigma]^2=[\mathbf 1]+[\varepsilon], \qquad [\sigma][\varepsilon]=[\sigma], \qquad [\varepsilon]^2=[\mathbf 1].$$

This finite algebra specifies OPE families and counts chiral fusion paths. A full local theory also requires the left–right pairing and crossing-consistent OPE coefficients. The Landau–Ginzburg comparison concerns renormalized infrared operators, rather than bare coincident products.

Common pitfalls

The Kac-table entries $h_{r,s}$ are chiral weights. In a diagonal model, the full scaling dimension is $\Delta=2h_{r,s}$.

Fusion rules do not determine numerical OPE coefficients by themselves. They determine which conformal families may appear. Coefficients require normalization choices and crossing equations.

The field $\sigma$ is not a free scalar field. It is a nontrivial primary of dimension $1/8$. The Landau–Ginzburg field $\phi$ has $\sigma$ as its leading odd infrared scaling field, but composites such as $\phi^3$ require renormalization and operator-mixing subtractions.

The identity family is not just the identity operator. It also contains descendants such as the stress tensor $T$ and composite descendants like $T\bar T$.

The Kac-label formula gives the chiral Virasoro fusion rules. The diagonal A-series pairs identical left and right representations; other modular invariants, such as the three-state Potts $D$-series invariant at $c=4/5$, reorganize the same chiral data into different full local CFTs.

The relation $[\sigma]^3=2[\sigma]$ is a statement in the fusion ring. It neither sets $\sigma^3=2\sigma$ as a local operator identity nor fixes a renormalized coincident triple product without a fusion channel and subtraction prescription.

Exercises

Exercise 1: Counting primaries in two unitary models

Compute the number of bulk primary families in the diagonal models $\mathcal M(3,4)$ and $\mathcal M(4,5)$.

Solution

The number of distinct primaries is

$$N_{p,p'}={1\over2}(p-1)(p'-1).$$

For $\mathcal M(3,4)$,

$$N_{3,4}={1\over2}(2)(3)=3.$$

For $\mathcal M(4,5)$,

$$N_{4,5}={1\over2}(3)(4)=6.$$

Thus the Ising model has three primaries, while the tricritical Ising model has six.

Exercise 2: Computing the Ising Kac weights

Use

$$h_{r,s}^{(p,p')} ={(p'r-ps)^2-(p'-p)^2\over4pp'}$$

to compute the three Ising weights $0$, $1/16$, and $1/2$.

Solution

For Ising, $p=3$ and $p'=4$, so

$$h_{r,s}={(4r-3s)^2-1\over48}.$$

For $(1,1)$,

$$h_{1,1}={(4-3)^2-1\over48}=0.$$

For $(1,2)$,

$$h_{1,2}={(4-6)^2-1\over48}={4-1\over48}={1\over16}.$$

For $(1,3)$,

$$h_{1,3}={(4-9)^2-1\over48}={25-1\over48}={1\over2}.$$

The remaining entries are identified by $(r,s)\sim(3-r,4-s)$ and repeat these weights.

Exercise 3: Deriving spin–spin fusion

Derive the Ising fusion rule

$$[\sigma]\times[\sigma]=[\mathbf 1]+[\varepsilon]$$

from the minimal-model fusion formula.

Solution

In Ising, identify

$$\sigma=(1,2), \qquad \mathbf 1=(1,1), \qquad \varepsilon=(1,3).$$

Use the fusion formula with $(r_1,s_1)=(1,2)$ and $(r_2,s_2)=(1,2)$. The $r$ range starts at

$$|1-1|+1=1$$

and ends at

$$r_{\max}=\min(1+1-1,2p-1-1-1)=\min(1,3)=1.$$

Thus $r=1$. The $s$ range starts at

$$|2-2|+1=1$$

and ends at

$$s_{\max}=\min(2+2-1,2p'-2-2-1)=\min(3,3)=3.$$

With step two, $s=1,3$. Therefore

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

which is

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

Exercise 4: OPE powers in the two spin channels

For the scalar primaries of diagonal Ising, find the powers of $|z|$ multiplying the identity and energy fields in the OPE $\sigma(z,\bar z)\sigma(0)$.

Solution

The general OPE power for a field $O_k$ in $O_i(z)O_j(0)$ is

$$|z|^{\Delta_k-\Delta_i-\Delta_j}.$$

For the Ising spin field,

$$\Delta_\sigma={1\over8}.$$

For the identity channel, $\Delta_{\mathbf 1}=0$, so the power is

$$0-2\Delta_\sigma=-{1\over4}.$$

Thus the identity term behaves as $|z|^{-1/4}$. For the energy channel, $\Delta_\varepsilon=1$, so the power is

$$1-2\Delta_\sigma=1-{1\over4}={3\over4}.$$

Thus

$$\sigma(z,\bar z)\sigma(0) \sim |z|^{-1/4}\mathbf 1+C_{\sigma\sigma\varepsilon}|z|^{3/4}\varepsilon(0)+\cdots.$$

Equivalently, factoring out $|z|^{-1/4}$ gives a term proportional to $|z|\varepsilon(0)$.

Exercise 5: Triple-spin fusion support and multiplicity

Use the Ising fusion rules to show that the triple product of spin families contains only the spin family, and determine the fusion multiplicity.

Solution

Start from

$$[\sigma]^2=[\mathbf 1]+[\varepsilon].$$

Multiply by $[\sigma]$:

$$[\sigma]^3 =([\mathbf 1]+[\varepsilon])\times[\sigma] =[\mathbf 1]\times[\sigma]+[\varepsilon]\times[\sigma].$$

The identity acts trivially, and $[\varepsilon]\times[\sigma]=[\sigma]$, so

$$[\sigma]^3=[\sigma]+[\sigma].$$

Thus, in the fusion ring,

$$[\sigma]^3=2[\sigma].$$

If we ignore multiplicity and track only which primary families can appear, the triple spin product has support only in the spin family. No new primary family is generated.

Exercise 6: Quantum dimensions of the Ising fusion ring

Let the Ising fusion algebra be generated by $\sigma$ and $\varepsilon$ with

$$\sigma^2=\mathbf 1+\varepsilon, \qquad \varepsilon^2=\mathbf 1, \qquad \sigma\varepsilon=\sigma.$$

Use the positive quantum dimensions of unitary Ising. Assuming $d_i d_j=\sum_k N_{ij}^{\phantom{ij}k}d_k$ and $d_{\mathbf 1}=1$, find $d_\varepsilon$ and $d_\sigma$.

Solution

From

$$\varepsilon^2=\mathbf 1,$$

the dimension equation gives

$$d_\varepsilon^2=1.$$

Quantum dimensions are positive, so

$$d_\varepsilon=1.$$

From

$$\sigma^2=\mathbf 1+\varepsilon,$$

we get

$$d_\sigma^2=d_{\mathbf 1}+d_\varepsilon=1+1=2.$$

Thus

$$d_\sigma=\sqrt2.$$

The non-integer quantum dimension of $\sigma$ is another concise signature of the nontrivial Ising fusion category.

References

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

Further reading

Minimal models and fusion rules develops the explicit Ising conformal blocks and their full pairing. Highest-weight modules, null states, and the Kac determinant gives the representation-theory hypotheses behind the finite spectrum.