03.10.04 · modern-geometry / conformal-field-theory

Minimal Models, the Kac Formula, and Null Vectors

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Anchor (Master): Belavin-Polyakov-Zamolodchikov 1984; Friedan-Qiu-Shenker 1984; Kac 1979; Feigin-Fuchs 1982; Itzykson-Drouffe Vol. 2 Ch. 9-10

Intuition Beginner

A drum head can only vibrate in certain shapes. Pluck it and you do not get every conceivable wiggle: you get a discrete list of allowed modes, each with its own pitch. The shape of the drum decides the list. Conformal field theories in two dimensions behave the same way. The infinite symmetry of the theory acts like the rim of the drum, and for special theories it forces the list of allowed "field shapes" to be short and finite.

These special theories are the minimal models. Each one has only a handful of basic fields, called primaries, and each field carries a number called its scaling weight that says how strongly it responds to a change of scale. The remarkable fact is that you can write down the complete list of allowed weights from a single small formula with two whole-number labels.

The reason the list closes off is a kind of resonance cancellation. Inside the symmetry there are combinations of operations that should produce a new field but instead produce nothing at all. Such a combination is a null vector: it looks like a state but has zero size. When a would-be field is secretly null, it drops out of the theory, and the surviving fields form a tidy finite table.

Visual Beginner

Picture a rectangular grid of dots. The columns are counted by one whole number and the rows by another. Each dot stands for one candidate field, and its position tells you the field's scaling weight through the Kac formula. For a minimal model the grid is finite: it has a fixed width and a fixed height set by two integers that define the model. A reflection of the grid maps each dot to a partner dot carrying the same weight, so the genuinely different fields are only half of the cells. That short, reflection-folded list is the entire field content of the theory.

Worked example Beginner

Take the simplest interesting case, the one tied to the magnet-at-its-critical-point model called the Ising model. It is built from the two defining integers $3$ and $4$ . The scaling-weight formula uses these to produce a small set of allowed values.

The three physically distinct fields turn out to have weights $0$ , $1/16$ , and $1/2$ . The weight $0$ field is the identity, the do-nothing field that is always present. The weight $1/2$ field is the energy density, the thing that measures how much local order costs. The weight $1/16$ field is the spin itself, the up-or-down arrow whose correlations decay slowly at the critical point.

From these three numbers you can read off how each field's correlations fall with distance. A field of weight $h$ has its two-point correlation fall like distance to the power $- 4 h$ in the combined holomorphic-and-antiholomorphic sense. So the spin correlation falls like distance to the power $- 1/4$ , matching the long-known critical exponent of the two-dimensional Ising magnet.

What this tells us: a single pair of integers, fed through one formula, reproduces the exact critical exponents of a real statistical system. The finiteness of the field list is not a convenience. It is the reason these exponents are exactly computable rather than merely estimated.

Check your understanding Beginner

Formal definition Intermediate+

Work with a single chiral copy of the Virasoro algebra, with generators $L_{n}$ for $n \in Z$ and central element $c$ acting as a scalar, satisfying $[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{c}{12} m (m^{2} - 1) δ_{m + n, 0} .$ A highest-weight vector $∣ h ⟩$ obeys $L_{0} ∣ h ⟩ = h ∣ h ⟩$ and $L_{n} ∣ h ⟩ = 0$ for all $n > 0$ . The Verma module $V (c, h)$ is the free module generated from $∣ h ⟩$ by the lowering operators, with basis the descendants $L_{- n_{1}} L_{- n_{2}} \dots L_{- n_{k}} ∣ h ⟩, n_{1} \geq n_{2} \geq \dots \geq n_{k} \geq 1,$ graded by level $N = \sum_{i} n_{i}$ , the $L_{0}$ -eigenvalue offset $h + N$ . The number of basis states at level $N$ is the partition number $p (N)$ .

The contravariant (Shapovalov) form is the unique bilinear form $⟨ \cdot, \cdot ⟩$ with $⟨ h ∣ h ⟩ = 1$ for which $L_{n}$ is adjoint to $L_{- n}$ . A null (or singular) vector is a non-zero descendant $∣ χ ⟩$ at some level $N > 0$ that is itself highest-weight: $L_{n} ∣ χ ⟩ = 0$ for all $n > 0$ . A singular vector has zero norm and is orthogonal to the entire module, so it generates a proper submodule. The irreducible highest-weight module $M (c, h)$ is the quotient of $V (c, h)$ by its maximal proper submodule, the sum of all submodules generated by singular vectors.

Following ^{[jimmyqin Virasoro representations, degenerate states and minimal models]}, a representation is called degenerate when $V (c, h)$ contains a singular vector. The conformal weights at which this happens are the Kac weights. Parametrise the central charge by two coprime positive integers $p, p^{'}$ with $p, p^{'} \geq 2$ , $c = 1 - \frac{6 ( p - p ^{'} ) ^{2}}{p p ^{'}},$ and define, for integers $1 \leq r$ , $1 \leq s$ , $h_{r, s} (c) = \frac{( r p ^{'} - s p ) ^{2} - ( p - p ^{'} ) ^{2}}{4 p p ^{'}} .$ A minimal model $M (p, p^{'})$ is the conformal field theory whose chiral primary content is exactly the finite set of irreducible modules $M (c, h_{r, s})$ with $1 \leq r \leq p - 1$ and $1 \leq s \leq p^{'} - 1$ , subject to the identification $(r, s) \sim (p - r, p^{'} - s)$ , which leaves $\frac{1}{2} (p - 1) (p^{'} - 1)$ distinct primaries. This finite list is the Kac table. The unitary discrete series is the subfamily $p^{'} = p + 1$ , that is $p = m$ , $p^{'} = m + 1$ with $m \geq 3$ integer, giving $c = 1 - \frac{6}{m ( m + 1 )}, h_{r, s} = \frac{( ( m + 1 ) r - m s ) ^{2} - 1}{4 m ( m + 1 )} .$

Counterexamples to common slips

A generic Verma module $V (c, h)$ with $c$ and $h$ off the Kac locus has no singular vectors and is already irreducible; degeneracy is the exceptional, measure-zero situation, not the default.
The Kac formula labels candidate degenerate weights, but only the rectangular $(p, p^{'})$ window with the reflection identification gives a consistent closed model. Taking all $(r, s)$ with $r, s \geq 1$ overcounts: most lie outside the table or are reflection-duplicates.
Unitarity is strictly stronger than degeneracy. Every minimal model is degenerate, but only the $p^{'} = p + 1$ subfamily has a positive-definite contravariant form on each irreducible module. The three-state Potts point sits inside a non-adjacent $(p, p^{'})$ but is still unitary because it equals a $c = 4/5$ member of the discrete series; non-unitary minimal models such as the Lee-Yang $M (2, 5)$ have a primary of negative weight.

Key theorem with proof Intermediate+

Theorem (Kac determinant formula). Let $V (c, h)$ be the Virasoro Verma module and let $M_{N} (c, h)$ be the Gram matrix of the contravariant form restricted to level $N$ , in the partition basis. Then up to a positive basis-dependent constant $K_{N}$ , $det M_{N} (c, h) = K_{N} r, s \geq 1 r s \leq N \prod (h - h_{r, s} (c))^{p (N - r s)},$ where $h_{r, s} (c)$ is the Kac weight and $p (\cdot)$ is the partition function. In particular the form degenerates at level $N$ precisely when $h = h_{r, s} (c)$ for some $r, s$ with $r s \leq N$ , and the order of vanishing is $p (N - r s)$ .

Proof. Fix $c$ and treat $h$ as a formal variable. Each entry of $M_{N} (c, h)$ is a polynomial in $h$ obtained by commuting raising past lowering operators using the Virasoro relations, so $det M_{N}$ is a polynomial in $h$ . Bound its degree by tracking the leading power: the diagonal entry $⟨ h ∣ L_{n_{1}} \dots L_{n_{k}} L_{- n_{k}} \dots L_{- n_{1}} ∣ h ⟩$ has top term $(2 h)^{k} \prod_{i} n_{i}$ up to lower order, and the highest total degree over the symmetric group of pairings is $\sum_{N^{'} \leq N} p (N^{'})$ , which equals $\sum_{r s \leq N} p (N - r s)$ by a partition bookkeeping identity. This matches the total degree of the product on the right, so it suffices to show every factor $(h - h_{r, s})$ divides $det M_{N}$ with at least the claimed multiplicity, and the leading coefficients agree.

For the zeros, suppose $h = h_{r, s} (c)$ with $r s \leq N$ . The defining property of the Kac weight is that $V (c, h_{r, s})$ contains a singular vector $∣ χ_{r, s} ⟩$ at level $r s$ . Existence at level $r s$ is the Feigin-Fuchs construction: write $h_{r, s}$ using the substitution $c = 1 - 6 (b - b^{- 1})^{2}$ and $h = \frac{1}{4} (b - b^{- 1})^{2} + \frac{1}{4} α^{2}$ with $α = α_{r, s} = - \frac{1}{2} ((r - 1) b - (s - 1) b^{- 1})$ shifted to the screening value; the free-field (vertex-operator) realisation produces an explicit screened operator annihilated by all $L_{n > 0}$ at exactly level $r s$ . This singular vector is orthogonal to all of $V$ , so it lies in the kernel of $M_{r s}$ , forcing $det M_{r s} (c, h_{r, s}) = 0$ . Its descendants populate the kernel at higher levels: at level $N$ the submodule generated by $∣ χ_{r, s} ⟩$ contributes $p (N - r s)$ independent null directions, since the singular vector behaves as a new highest-weight vector with its own Verma tower. Each null direction lowers the rank of $M_{N}$ by one, so $(h - h_{r, s})$ divides $det M_{N}$ to order at least $p (N - r s)$ . Summing the required multiplicities over all $(r, s)$ with $r s \leq N$ saturates the degree bound, so equality holds and the leading coefficient is the positive constant $K_{N}$ . $■$

Bridge. This determinant is the foundational reason the minimal-model field list is finite: a primary survives only where its Verma module is not killed by an over-abundance of null directions, and the rectangular Kac table is exactly the locus where the singular vectors close up consistently. The construction builds toward the conformal bootstrap, because each singular vector becomes a differential operator annihilating correlators; putting these together, the level-two null vector of $h_{1, 2}$ and $h_{2, 1}$ yields a second-order differential equation whose solutions are the conformal blocks. The unitarity sign analysis below generalises the determinant from "where does it vanish" to "where is it positive", and this is exactly the question Friedan-Qiu-Shenker answered. The same null-vector decoupling appears again in the Coulomb-gas screening construction of the next unit, where the singular vectors are realised concretely by contour integrals of screening charges, and the determinant's zero structure is dual to the fusion rules that the bootstrap then enforces.

Exercises Intermediate+

Exercise (hard, symbolic). For a degenerate primary with a level-two singular vector $\lvert\chi\rangle=\bigl(L_{-2}+\eta L_{-1}^2\bigr)\lvert h\rangle$, impose $L_1\lvert\chi\rangle=0$ and $L_2\lvert\chi\rangle=0$ to solve for $\eta$ and the constraint on $h$. Give $\eta$ in terms of $h$.

Hint

Use $[L_{1}, L_{- 2}] = 3 L_{- 1}$ , $[L_{1}, L_{- 1}] = 2 L_{0}$ , $[L_{2}, L_{- 2}] = 4 L_{0} + \frac{c}{2}$ , $[L_{2}, L_{- 1}] = 3 L_{1}$ .

Answer

$η = - \frac{3}{2 ( 2 h + 1 )}$ . From $L_{1} ∣ χ ⟩ = (3 + 2 η (2 h + 1)) L_{- 1} ∣ h ⟩ = 0$ one gets $η = - 3/ [2 (2 h + 1)]$ . Substituting into $L_{2} ∣ χ ⟩ = (4 h + \frac{c}{2} + 6 η h) ∣ h ⟩ = 0$ yields the consistency relation $2 h (2 h + 1) \cdot \frac{c}{2} + \dots$ , equivalently $16 h^{2} - (10 - 2 c) h + c = 0$ , whose roots are $h_{1, 2}$ and $h_{2, 1}$ . Feedback if correct: this is the BPZ level-two condition. Feedback if wrong: recompute $L_{1}$ acting on $L_{- 1}^{2}$ carefully.

Advanced results Master

The unitary discrete series is singled out by demanding that the contravariant form be positive-definite on every irreducible module in the table. The complete answer is the Friedan-Qiu-Shenker classification.

Theorem (Friedan-Qiu-Shenker, unitarity in $c < 1$ ). A Virasoro highest-weight module with $0 \leq c < 1$ is unitary if and only if $c = 1 - \frac{6}{m ( m + 1 )}, m = 3, 4, 5, \dots, and h = h_{r, s} = \frac{( ( m + 1 ) r - m s ) ^{2} - 1}{4 m ( m + 1 )}$ for integers $1 \leq r \leq m - 1$ , $1 \leq s \leq m$ . For $c \geq 1$ unitarity holds for all $h \geq 0$ ; for $c < 0$ no minimal model is unitary.

The necessity half is a determinant argument: outside the discrete series, the Kac determinant changes sign as $h$ crosses Kac weights at low level, so some level carries a negative-norm state. The sufficiency half is the harder direction, established by the coset (GKO) construction realising each discrete-series model as $s u (2)_{k} \times s u (2)_{1} / s u (2)_{k + 1}$ with $m = k + 2$ , whose manifest positivity is inherited from unitary affine modules.

The decoupling of singular vectors turns into computational power. A primary $ϕ_{r, s}$ with a level- $r s$ null vector obeys a linear differential equation of order $r s$ for its correlators. The level-two case generates the BPZ equation: inserting $ϕ_{2, 1}$ (or $ϕ_{1, 2}$ ) into an $n$ -point function and demanding the null combination annihilate it gives a hypergeometric ordinary differential equation whose solution space is two-dimensional, the conformal blocks. Fusion of degenerate fields is correspondingly truncated: the operator product of $ϕ_{1, 2}$ with $ϕ_{r, s}$ contains only $ϕ_{r, s - 1}$ and $ϕ_{r, s + 1}$ , and analogously for $ϕ_{2, 1}$ , so repeated fusion never leaves the finite Kac table. This closure is what makes a minimal model a genuinely finite, solvable conformal field theory.

The smallest members organise the universality classes of two-dimensional critical statistical mechanics. The $m = 3$ model, $c = 1/2$ , is the critical Ising model with primaries ${0, 1/16, 1/2}$ realised by free Majorana fermions. The $m = 4$ model, $c = 7/10$ , is the tricritical Ising model, with the extra relevant fields encoding the vacancy field of the dilute Ising magnet. The three-state Potts critical point sits at $c = 4/5$ , the $m = 5$ member, as a $Z_{3}$ -symmetric extension whose chiral algebra is enlarged beyond the bare Virasoro algebra; its order parameter has weight $h = 1/15$ . The $m = 2$ "model" is empty ( $c = 0$ , only the identity), consistent with there being no nontrivial unitary CFT below Ising.

Synthesis. Putting these together, the determinant, the unitarity classification, and the differential equations are three faces of one structure: the singular vectors. The central insight is that a zero of the Kac determinant is simultaneously a place where a state decouples, where positivity can fail, and where a correlator must satisfy a differential equation, so the foundational reason the discrete series exists is that the same null vectors that prune the field list also enforce the positivity that unitarity demands. This is exactly why $c = 1 - 6/ [m (m + 1)]$ is both the unitarity boundary and the rational locus: the boundary generalises the single-field decoupling of the vacuum's $L_{- 1} ∣ 0 ⟩$ into a lattice of decouplings filling a finite table. The fusion truncation is dual to the differential-equation order, and this duality builds toward the bootstrap closure of the next unit and appears again in the rational-CFT and modular-tensor-category framework, where the same finite table becomes the object set of a modular fusion category.

Full proof set Master

Proposition 1 (the vacuum singular vector). In any conformal field theory the $SL (2, C)$ -invariant vacuum $∣ 0 ⟩$ has $h = 0 = h_{1, 1}$ and satisfies $L_{- 1} ∣ 0 ⟩ = 0$ .

Proof. The vacuum is invariant under the global conformal group generated by $L_{- 1}, L_{0}, L_{1}$ , hence $L_{- 1} ∣ 0 ⟩ = 0$ . This vector sits at level one with $h = 0$ , and the level-one Gram entry computed above is $2 h$ , which vanishes at $h = 0$ , confirming $L_{- 1} ∣ 0 ⟩$ is a singular vector. In the Kac parametrisation $h_{1, 1} = ((p^{'} - p)^{2} - (p - p^{'})^{2}) / (4 p p^{'}) = 0$ for every $(p, p^{'})$ , so the identity always heads the table. $■$

Proposition 2 (level-two degenerate weights). A level-two singular vector $(L_{- 2} + η L_{- 1}^{2}) ∣ h ⟩$ exists exactly when $h$ is a root of $16 h^{2} - (10 - 2 c) h + c = 0$ , and these roots are $h_{1, 2}$ and $h_{2, 1}$ .

Proof. Apply $L_{1}$ : using $[L_{1}, L_{- 2}] = 3 L_{- 1}$ and $[L_{1}, L_{- 1}] = 2 L_{0}$ , one finds $L_{1} (L_{- 2} + η L_{- 1}^{2}) ∣ h ⟩ = (3 + 2 η (2 h + 1)) L_{- 1} ∣ h ⟩$ , so highest-weight requires $η = - 3/ [2 (2 h + 1)]$ . Apply $L_{2}$ : using $[L_{2}, L_{- 2}] = 4 L_{0} + \frac{c}{2}$ and $[L_{2}, L_{- 1}] = 3 L_{1}$ acting on the highest-weight vector gives $L_{2} (L_{- 2} + η L_{- 1}^{2}) ∣ h ⟩ = (4 h + \frac{c}{2} + 6 η h) ∣ h ⟩$ . Substituting $η$ and clearing denominators yields $16 h^{2} - (10 - 2 c) h + c = 0$ . Solving with $c = 1 - 6 (p - p^{'})^{2} / (p p^{'})$ and matching to the Kac formula identifies the two roots as $h_{1, 2}$ and $h_{2, 1}$ , the two weights whose Kac labels have $r s = 2$ . $■$

Proposition 3 (fusion truncation for $ϕ_{1, 2}$ ). The decoupling of the level-two null vector forces the operator product expansion $ϕ_{1, 2} \times ϕ_{r, s}$ to contain only the primaries $ϕ_{r, s - 1}$ and $ϕ_{r, s + 1}$ .

Proof. The null vector of $ϕ_{1, 2}$ gives a second-order differential equation for any correlator containing $ϕ_{1, 2}$ . Insert $ϕ_{1, 2} (z) ϕ_{r, s} (w)$ into a correlator with a third primary $ϕ_{r^{'}, s^{'}}$ at infinity. The indicial equation of the second-order ordinary differential equation at the merging point $z \to w$ has exactly two exponents, fixing the two allowed fusion channels by their leading powers. Computing the exponents from the difference of conformal weights $h_{r^{'}, s^{'}} - h_{1, 2} - h_{r, s}$ shows the admissible $ϕ_{r^{'}, s^{'}}$ are precisely those with $r^{'} = r$ and $s^{'} = s \pm 1$ . Any other channel would require a third independent solution of a second-order equation, which cannot occur. Hence the fusion stays inside the Kac table and the model closes on finitely many primaries. $■$

Connections Master

The Virasoro algebra and its central extension that anchor every definition here are built in 03.11.03 and 03.10.02; this unit specialises their generic representation theory to the degenerate locus, so the highest-weight machinery of 03.10.02 is the direct upstream dependency and the present Kac table is its sharpest consequence.
The Verma-module and singular-vector language is the Virasoro instance of the general highest-weight theory of 07.06.06; the contravariant form here is the Virasoro analogue of the Shapovalov form, and the Kac determinant is the Virasoro counterpart of the Shapovalov determinant for finite-dimensional and Kac-Moody Lie algebras.
The identification of the $m = 3$ model with the critical Ising universality class connects to the free-fermion solution in 08.14.02 and to the conformal-criticality framework of 08.06.02; the exact exponents $β = 1/8$ and $η = 1/4$ that statistical mechanics extracts are the scaling weights $h \in {1/2, 1/16}$ computed here, so the two routes meet on the same numbers.
The null-vector decoupling that yields the BPZ differential equation is the foundation of the Coulomb-gas and bootstrap construction in the co-produced unit 03.10.05; there the singular vectors become explicit screened vertex operators and the conformal blocks become Dotsenko-Fateev contour integrals, closing the solvability programme this unit opens.

Historical & philosophical context Master

The degenerate-representation structure was isolated by Belavin, Polyakov, and Zamolodchikov in 1984, who recognised that the null vectors of the Virasoro algebra convert the symmetry of a two-dimensional critical theory into differential equations for its correlators ^{[Belavin 1984]}. The determinant of the contravariant form had been computed earlier by Kac, announced in 1978 and proved with Feigin and Fuchs, as part of the representation theory of infinite-dimensional Lie algebras ^{[Kac 1979]}. Friedan, Qiu, and Shenker then identified the discrete unitary series $c = 1 - 6/ [m (m + 1)]$ later the same year, showing that the demand for a positive-definite inner product selects exactly the rational values realised by the Ising, tricritical-Ising, and three-state-Potts critical points ^{[Friedan 1984]}. Itzykson and Drouffe's second volume gathers this minimal-model machinery alongside the lattice and Monte Carlo methods of the surrounding chapters ^{[jimmyqin Virasoro representations and minimal models]}. The unitarity sufficiency was completed by Goddard, Kent, and Olive through the coset construction in 1986.

Bibliography Master

@article{BPZ1984,
  author  = {Belavin, A. A. and Polyakov, A. M. and Zamolodchikov, A. B.},
  title   = {Infinite conformal symmetry in two-dimensional quantum field theory},
  journal = {Nuclear Physics B},
  volume  = {241},
  pages   = {333--380},
  year    = {1984}
}

@article{FQS1984,
  author  = {Friedan, D. and Qiu, Z. and Shenker, S.},
  title   = {Conformal invariance, unitarity, and critical exponents in two dimensions},
  journal = {Physical Review Letters},
  volume  = {52},
  pages   = {1575--1578},
  year    = {1984}
}

@incollection{Kac1979,
  author    = {Kac, V. G.},
  title     = {Contravariant form for infinite-dimensional Lie algebras and superalgebras},
  booktitle = {Group Theoretical Methods in Physics},
  series    = {Lecture Notes in Physics},
  volume    = {94},
  pages     = {441--445},
  publisher = {Springer},
  year      = {1979}
}

@article{FeiginFuchs1982,
  author  = {Feigin, B. L. and Fuchs, D. B.},
  title   = {Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra},
  journal = {Functional Analysis and Its Applications},
  volume  = {16},
  pages   = {114--126},
  year    = {1982}
}

@article{GKO1986,
  author  = {Goddard, P. and Kent, A. and Olive, D.},
  title   = {Unitary representations of the Virasoro and super-Virasoro algebras},
  journal = {Communications in Mathematical Physics},
  volume  = {103},
  pages   = {105--119},
  year    = {1986}
}

@book{ItzyksonDrouffe1989,
  author    = {Itzykson, C. and Drouffe, J.-M.},
  title     = {Statistical Field Theory, Volume 2},
  publisher = {Cambridge University Press},
  year      = {1989}
}

@book{DiFrancesco1997,
  author    = {Di Francesco, P. and Mathieu, P. and S\'en\'echal, D.},
  title     = {Conformal Field Theory},
  publisher = {Springer},
  year      = {1997}
}

Prerequisites

03.10.02
03.11.03
07.06.06
08.06.02

Tier anchors

beginner: 3Blue1Brown / Strogatz visual symmetry-and-resonance pacing; a discrete vibrational-mode analogy
intermediate: Di Francesco-Mathieu-Sénéchal §7-8; Tong — String Theory §4-5; Ginsparg Les Houches §6-8
master: Belavin-Polyakov-Zamolodchikov 1984; Friedan-Qiu-Shenker 1984; Kac 1979; Feigin-Fuchs 1982; Itzykson-Drouffe Vol. 2 Ch. 9-10

References

raw/pdfs/string-theory.pdf · Conformal field theory: Virasoro representations, degenerate states and minimal models
images/fields-and-strings-2-contents-16-scaled__7d850192e2.jpg · Itzykson-Drouffe, Statistical Field Theory Vol. 2, Ch. 9-10 (conformal field theory) table of contents
Belavin, Polyakov, Zamolodchikov — Infinite conformal symmetry in two-dimensional quantum field theory · Nucl. Phys. B241 (1984) 333-380, §2-3 (degenerate representations, the Kac formula)
Friedan, Qiu, Shenker — Conformal invariance, unitarity, and critical exponents in two dimensions · Phys. Rev. Lett. 52 (1984) 1575-1578 (the unitary discrete series)
Kac — Contravariant form for infinite-dimensional Lie algebras and superalgebras · Lecture Notes in Physics 94 (1979) 441-445 (the determinant formula)

Estimated time

beginner: 20m
intermediate: 55m
master: 90m