The Virasoro Algebra and Central Charge

Hello! In our last lesson, we discovered that the classical symmetry algebra of a 2D conformal theory is infinite-dimensional, described by two copies of the Witt algebra. We saw its generators $L_{n}$ satisfying the commutation relation $[L_{n}, L_{m}] = (n - m) L_{n + m}$ .

Today, we will investigate what happens to this beautiful structure when we move from a classical to a quantum theory. As you know from your study of quantum field theory, classical symmetries can acquire subtle but profound modifications upon quantization, a phenomenon known as a quantum anomaly. This is precisely what happens here.

Our goal is to construct the Virasoro algebra as the central extension of the 2D conformal algebra and interpret the central charge. We will see how the quantum nature of the theory introduces a new term into the algebra, and we will explore the deep physical meaning of the constant, $c$ , that governs this term.

The Stress-Energy Tensor and its Quantum OPE

In a 2D CFT, the generators of conformal transformations are encoded in the stress-energy tensor. Due to holomorphic factorization, we can focus on the holomorphic component, $T (z)$ , and its anti-holomorphic counterpart, $T ˉ (z ˉ)$ . The generators of the Witt algebra, which we previously wrote as differential operators, are realized in the quantum theory as the Laurent expansion modes of the operator $T (z)$ :

T (z) = n \in Z \sum L_{n} z^{- n - 2}

Here, the $L_{n}$ are now operators acting on the Hilbert space of the theory. The commutation relations of these operators are determined by the Operator Product Expansion (OPE) of $T (z)$ with itself.

In any quantum field theory, products of local operators at the same point are singular. The OPE is the tool we use to handle this by systematically characterizing the singularities as two operators approach each other. The operator $T (z)$ has a scaling dimension of 2. Dimensional analysis then suggests that the most singular term in its OPE with itself, $T (z) T (w)$ , will scale as $(z - w)^{- 4}$ .

4. Introducing Conformal Field Theory

The following section from David Tong's notes, '4.4 The Central Charge', provides a clear introduction to the OPE of the stress-energy tensor and the definition of the central charge. It also includes a concrete calculation for the free boson, which you've already seen is a fundamental CFT.

Please read section 4.4, 'The Central Charge' (pages 82-84). Focus on the general form of the T(z)T(w) OPE, the definition of the central charge 'c', and the argument that 'c' measures the number of degrees of freedom. Note how the presence of the (z-w)⁻⁴ term implies that T(z) is not a primary operator.

As you've just read, the OPE of the stress-energy tensor with itself in a general 2D CFT takes the form:

T (z) T (w) = \frac{c /2}{( z - w ) ^{4}} + \frac{2 T ( w )}{( z - w ) ^{2}} + \frac{\partial _{w} T ( w )}{z - w} + \dots

The terms with $(z - w)^{- 2}$ and $(z - w)^{- 1}$ are expected for an operator of dimension 2. The crucial new feature is the most singular term, $\frac{c /2}{( z - w ) ^{4}}$ . This term is the quantum anomaly. The constant $c$ , known as the central charge, is a defining characteristic of a CFT.

The Virasoro Algebra as a Central Extension

This new term in the OPE has a direct consequence for the algebra of the generators $L_{n}$ . The classical Witt algebra gets modified by an additional term proportional to $c$ . This new algebra is called the Virasoro algebra, and mathematically it is a central extension of the Witt algebra.

Let's briefly formalize what a central extension is, as this is a concept you will encounter frequently in advanced physics.

An algebra for string theory -- The Virasoro algebra.

This video by mathematician Michael Penn provides a rigorous definition of a central extension of a Lie algebra. His perspective will resonate with your preference for mathematical structure.

Please watch the first 6 minutes and 36 seconds. Focus on the definition of a central extension using a short exact sequence of Lie algebras: 0 → A → G' → G → 0. In our case, G is the Witt algebra, A is a one-dimensional abelian algebra (C), and G' will be the Virasoro algebra.

The term is "central" because the new piece, which will be proportional to the identity operator multiplied by $c$ , commutes with all elements of the algebra. The uniqueness of this extension for the Witt algebra (up to the value of $c$ ) is why it is so fundamental.

Derivation of the Virasoro Algebra

Now for the central calculation of this lesson. We will derive the commutation relations of the $L_{n}$ operators starting from the $T (z) T (w)$ OPE. The key is a contour integral identity that relates the commutator to the OPE:

[L_{m}, L_{n}] = \oint_{∣ w ∣ = R} \frac{d w}{2 π i} w^{n + 1} \oint_{∣ z - w ∣ = r} \frac{d z}{2 π i} z^{m + 1} T (z) T (w)

This expression, which involves a contour integral for $z$ around $w$ , picks out exactly the singular terms in the OPE.

Introduction to conformal field theory, Lecture 9

The following video segment from Tobias Osborne's lecture series carries out this calculation in full detail. It's a classic and essential computation in CFT.

Please watch from 49:35 to 1:21:40. The first part (until 1:03:22) re-derives the T-T OPE for the free boson, which serves as a concrete example. The core derivation of the Virasoro algebra from the general OPE starts at 1:09:11. Follow the contour integral calculation carefully to see how each term of the OPE contributes to the final commutation relation.

After performing the contour integrals and applying the residue theorem, we arrive at the celebrated Virasoro algebra:

[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{c}{12} m (m^{2} - 1) δ_{m + n, 0}

The first term is the Witt algebra we already know. The second term is the central extension. A similar, commuting Virasoro algebra exists for the anti-holomorphic generators $L ˉ_{n}$ , with its own central charge $c ˉ$ .

The factor of $m (m^{2} - 1)$ might seem complex, but it is the unique form (up to an overall constant) that is consistent with the Jacobi identity for the Lie algebra. The factor of $1/12$ is a convention, but it simplifies many other formulas in CFT.

Test your understanding!

The global conformal algebra on the plane is generated by $L_{- 1}, L_{0}, L_{1}$ . Show that this subalgebra is unaffected by the central extension, i.e., the commutator of any two of these generators does not have a central term. What Lie algebra do they form?

Show answer

We need to check the central term $\frac{c}{12} m (m^{2} - 1) δ_{m + n, 0}$ for pairs from ${L_{- 1}, L_{0}, L_{1}}$ .
For $[L_{1}, L_{- 1}]$ , we have $m = 1, n = - 1$ . The delta function $δ_{1 - 1, 0}$ is non-zero. The term is $\frac{c}{12} (1) (1^{2} - 1) = 0$ .
For $[L_{1}, L_{0}]$ or $[L_{- 1}, L_{0}]$ , $m + n \neq = 0$ , so the central term is zero.
Since the central term vanishes for all commutators within this set, the subalgebra is not centrally extended. It remains the Witt subalgebra for $n, m \in {- 1, 0, 1}$ .
The commutators are:
$[L_{1}, L_{0}] = (1 - 0) L_{1} = L_{1}$
$[L_{0}, L_{- 1}] = (0 - (- 1)) L_{- 1} = L_{- 1}$
$[L_{1}, L_{- 1}] = (1 - (- 1)) L_{0} = 2 L_{0}$
This is the algebra of $sl (2, C)$ , the Lie algebra of the global conformal group, as expected.

The Physical Interpretation of the Central Charge

The central charge $c$ is arguably the most important number that characterizes a CFT. It has several deep physical interpretations.

A Measure of Degrees of Freedom: As mentioned in the reading from Tong, $c$ effectively counts the number of massless degrees of freedom in the theory. A free real boson has $c = 1$ . A free complex boson has $c = 2$ . A free Majorana fermion has $c = 1/2$ . A theory composed of multiple non-interacting sectors has a total central charge that is the sum of the individual charges.
The Conformal Anomaly: The central charge quantifies the failure of the stress-energy tensor to transform like a primary operator. Under a finite conformal transformation $z \to w (z)$ , $T (z)$ transforms anomalously:
$T_{n e w} (w) = (\frac{\partial z}{\partial w})^{2} T_{o l d} (z) + \frac{c}{12} {z; w}$
The anomalous term is proportional to $c$ and a function called the Schwarzian derivative, ${z; w}$ . This anomaly is also responsible for the non-vanishing trace of the stress-energy tensor on a curved background, $⟨ T_{μ}^{μ} ⟩ \propto c R$ , where $R$ is the Ricci scalar.
Casimir Energy: When a CFT is defined on a spatial circle of circumference $L$ (a cylinder in spacetime), the vacuum state has a non-zero energy due to quantum fluctuations, known as the Casimir energy. This energy is universal and directly proportional to the central charge:
$E_{C a s imi r} = - \frac{π c}{6 L}$
This provides a direct physical, and in principle measurable, consequence of $c$ .
Unitarity Constraint: In any physically sensible (unitary) quantum field theory, the norm of any state in the Hilbert space must be non-negative. This powerful constraint has a simple but profound consequence for the central charge.

4. Introducing Conformal Field Theory

The following brief excerpt from David Tong's notes provides a simple proof that unitarity requires the central charge to be positive.

Please read the two short bullet points in section 4.5.4, 'Consequences of Unitarity' (page 100). Focus on the proof for c > 0, which involves calculating the norm of the state L₋ₙ|0⟩.

The proof you just read shows that for any non-trivial unitary CFT, we must have $c > 0$ . This is a fundamental constraint that carves out the space of allowed physical theories.

Conclusion

Today, we've taken a major step from the classical description of 2D conformal symmetry to the quantum one, culminating in the Virasoro algebra.

Key Takeaways:

The quantum generators of conformal symmetry, $L_{n}$ , are the Laurent modes of the stress-energy tensor $T (z)$ .
Upon quantization, the classical Witt algebra receives a quantum correction, becoming the Virasoro algebra: $[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{c}{12} m (m^{2} - 1) δ_{m + n, 0}$
This is a central extension of the Witt algebra, characterized by the central charge $c$ . The mathematical origin of this term is in the OPE of the stress tensor with itself.
The central charge $c$ is a fundamental quantity with deep physical meaning: it counts the degrees of freedom, quantifies the conformal anomaly, and determines the universal Casimir energy. Unitarity requires $c > 0$ .

We now have the fundamental symmetry algebra that governs all 2D CFTs. The next logical step is to study its representations. Just as the representations of the Poincaré group classify particles in QFT, the representations of the Virasoro algebra will allow us to classify the states and operators in a CFT. In our next lesson, we will begin this program by constructing Verma modules, the highest-weight representations of the Virasoro algebra.

Recommended resources

The Stress-Energy Tensor and its Quantum OPE

The Virasoro Algebra as a Central Extension

Derivation of the Virasoro Algebra

The Physical Interpretation of the Central Charge

Conclusion