Minimal Models, Fusion, and the Ising Operator Algebra

The previous page introduced the Kac labels $(r,s)$ and explained why degenerate Virasoro representations are so restrictive. A degenerate primary carries null vectors; null vectors become BPZ differential equations; BPZ equations restrict the possible OPE channels. When the central charge is rational in the right way, these restrictions close on a finite set of representations. This is the beginning of the minimal models.

This page turns that representation-theoretic mechanism into a concrete operator algebra. The main example is the two-dimensional Ising CFT. Its three primary families,

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

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 is the bookkeeping system for which conformal families are allowed to appear in an OPE.

From Kac labels to a finite spectrum

Chiral and full dimensions

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 Kac weights at rational central charge 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

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

The number of distinct primary fields is therefore

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

The Virasoro minimal model $\mathcal M(p,p')$ keeps a finite Kac-table window and identifies reflected labels. The reflection halves the number of primary fields.

This finite table is not an arbitrary truncation. Outside the table, null-vector constraints and crossing consistency make the corresponding representations redundant or inconsistent with the minimal operator algebra. The finite spectrum is the two-dimensional analog of a small closed multiplication table: repeatedly take OPEs, and you never generate a new primary outside the table.

The unitary minimal series is the special case

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

with

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

The first few are

$$\begin{array}{c|c|c} p & c_p & \text{model} \\ \hline 3 & 1/2 & \text{Ising} \\ 4 & 7/10 & \text{tricritical Ising} \\ 5 & 4/5 & \text{diagonal }\mathcal M(5,6)\text{; the Potts CFT uses a }D\text{-series invariant}. \end{array}$$

The nonunitary minimal models are also important, but the Ising course thread runs through the unitary series.

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 is the same selection rule familiar from adding $SU(2)$ spins: not every integer between the endpoints appears.

A particularly useful special case is fusion with the degenerate field $(1,3)$. Away from boundaries, the rule becomes

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

At the edges of the Kac table, channels that leave the allowed window are removed, and identified labels must be folded back by

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

Fusion with $(1,3)$ acts like a short nearest-neighbor rule in the $s$ direction of the Kac table. The finite window truncates channels at the boundary.

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 Ising model $\mathcal M(3,4)$ has three primary families after the Kac-table reflection: the identity $\mathbf 1$, the spin field $\sigma$, and the energy field $\varepsilon$.

In lattice language, $\sigma$ is the continuum limit of the Ising spin. It is odd under the global $\mathbb Z_2$ spin-flip symmetry. The field $\varepsilon$ is the continuum limit of the local energy-density deviation from criticality. It is even under $\mathbb Z_2$.

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 $s$ label has only the allowed value $s=2$ after boundary truncation. 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. The result is

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

The Ising fusion algebra closes on three primary families. The energy field is a simple current of order two, while the spin field obeys $[\sigma]^2=[\mathbf 1]+[\varepsilon]$.

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. Normalize the two-point functions as displayed above and choose the sign of $\varepsilon$ so that $C_{\sigma\sigma\varepsilon}>0$. In this standard convention, the Ising 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 OPE $\sigma\times\sigma$ has two channels. Multiplying by another spin field sends both fusion paths back into the spin family, so the fusion ring gives $[\sigma]^3=2[\sigma]$. This counts fusion paths; it is not a literal identity between coincident local fields.

The coefficient $1/2$ in the formulas above is OPE data in the stated normalization, not a fusion coefficient. Rescaling $\sigma$ or $\varepsilon$ changes numerical OPE coefficients, while the statement that the energy family appears—and that no other primary family appears—is invariant.

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 therefore independent of parenthesization. For the ordinary sphere four-point function, however, the vacuum boundary condition projects the total product onto $[\mathbf 1]$. The coefficient of $[\mathbf 1]$ in $[\sigma]^4$ is two, corresponding to the two allowed intermediate families $[\mathbf 1]$ and $[\varepsilon]$. The coefficient $2[\varepsilon]$ in the unprojected fusion product belongs to a different choice of total output; it is not two extra blocks in the same vacuum correlator.

The full crossing equation is stronger than this counting statement: it equates the actual conformal-block decompositions in different channels through the fusing matrix and constrains OPE coefficients in a fixed normalization. In the Ising model this crossing problem is simple enough to solve exactly, but the conceptual lesson is broader. A minimal model is not merely a list of dimensions; it is a finite, associative, crossing-consistent operator algebra.

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 fusion multiplicity two means there are two independent fusion paths to the spin family, and 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.

This distinction matters when comparing the exact Ising CFT to the more familiar Landau–Ginzburg description. A convenient schematic normalization 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 the critical point, $r$ is tuned to its critical value, and the leading infrared scaling field in the $\mathbb Z_2$-odd microscopic variable $\phi$ is the CFT spin field $\sigma$. 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 product $\phi^3$ with either $\sigma$ or a specific descendant. What fusion does establish is that no new primary family beyond $[\sigma]$ is required in the odd triple product.

Ignoring the identity perturbation, which only shifts the vacuum energy, the two nontrivial relevant primary deformations of the Ising fixed point are

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

The coefficient $t$ measures the temperature deviation from criticality, and $H$ is the magnetic field. Their relevance follows from

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

The Ising fixed point has two relevant primary deformations: the energy field $\varepsilon$, corresponding to temperature, and the spin field $\sigma$, corresponding to magnetic field. A Landau–Ginzburg scalar action represents the same universality class after tuning to criticality.

This is one of the cleanest examples of what “universality” means in CFT language. The microscopic field, the lattice spin, and the scalar Landau–Ginzburg variable are not literally the same object at all scales. At the fixed point, they project onto the same scaling field $\sigma$ plus less relevant corrections.

Summary

The Virasoro minimal models are rational CFTs whose primary fields form a finite Kac table. For coprime $p<p'$,

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

The finite spectrum is

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

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 controls OPE channels, four-point conformal-block decompositions, and the way the Landau–Ginzburg scalar description flows to the exact Ising CFT.

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 primary fields in $\mathcal M(3,4)$ 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

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

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

Assuming the quantum dimensions satisfy $d_i d_j=\sum_k N_{ij}^{\phantom{ij}k}d_k$ 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 and further reading

A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nuclear Physics B 241 (1984), is the original BPZ paper introducing the minimal-model conformal bootstrap.

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

P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Chapters 7–12, gives a systematic treatment of minimal models, fusion, Ising correlators, and modular invariance.

P. Ginsparg, “Applied Conformal Field Theory,” Les Houches lectures, gives a concise and readable overview of Kac tables, minimal models, and the Ising example.

A. M. Polyakov, Gauge Fields and Strings, especially the discussion of statistical systems, conformal field theory, and operator products, gives the broader field-theoretic setting for the course.

Further reading

This lesson follows Polyakov’s Kac-table-to-Ising operator-algebra sequence. For a maintained reference account, see Minimal models and fusion rules.