Virasoro Representations and Modular Invariance in Rational CFTs

Hello! Welcome back to our study of computational methods in 2D Conformal Field Theory.

In our last lesson, we mastered Wick's theorem, a powerful tool for calculating correlation functions in free theories. We saw how to systematically compute n-point functions by summing over all possible pairings of fields. This gave us insight into the "dynamic" observables of a CFT.

Today, we shift our focus back to the structure of the theory's spectrum. We previously examined the full partition function on a torus and its modular invariance. Now, we will dissect this partition function into its fundamental building blocks: Virasoro characters. These objects encode the precise structure of each representation of the conformal algebra. We will then see how the deep principle of modular invariance, when applied to these characters, places powerful constraints on the allowed operator content of a consistent theory, especially for the well-behaved class of Rational Conformal Field Theories (RCFTs).

Your learning objective is to: Define characters of Virasoro representations and use modular invariance to constrain the operator spectrum of a rational CFT.

1. Virasoro Characters: Counting States

In group theory, a character is a trace of a group element in a given representation. In CFT, the concept is analogous. A Virasoro character is a generating function that "counts" the number of states at each energy level within a single representation of the Virasoro algebra.

Definition:
For a representation $R$ of the Virasoro algebra, its character is defined as the trace over all states $∣ ψ ⟩$ in that representation:

χ_{R} (τ) = Tr_{R} (q^{L_{0} - c /24}) = ∣ ψ ⟩ \in R \sum ⟨ ψ ∣ q^{L_{0} - c /24} ∣ ψ ⟩

where $q = e^{2 π i τ}$ and $τ$ is a complex variable we will soon identify with the modulus of a torus. The shift by $c /24$ is the familiar Casimir energy contribution when mapping the theory from the plane to the cylinder.

If the representation $R$ is a highest-weight representation built on a primary state $∣ h ⟩$ with conformal dimension $h$ , the character can be written more explicitly:

χ_{h} (τ) = q^{h - c /24} k = 0 \sum \infty N_{k} q^{k}

Here, $N_{k}$ is the number of linearly independent descendant states at level $k$ (where level is the eigenvalue of $L_{0} - h$ ).

Week 7 1 Modular Invariance

The following resource provides a clear introduction to the concept of a Virasoro character as a state-counting function.

Please read the first two paragraphs of Section 1, 'Modular Invariance'. Focus on the definition of the character in equation (1) and its product form in equation (2), which counts the partitions of integers corresponding to descendant states.

Characters of Verma and Degenerate Modules

For a generic Verma module $V_{h}$ , there are no null states. The number of states at level $k$ , $N_{k}$ , is simply the number of ways to partition the integer $k$ , denoted $p (k)$ . This leads to a beautiful product formula for the character involving the Dedekind eta function, $η (τ)$ :

χ_{V_{h}} (τ) = \frac{q ^{h - c /24}}{\prod _{n = 1}^{\infty} ( 1 - q ^{n} )} = \frac{q ^{h}}{η ( τ )}

where $η (τ) = q^{1/24} \prod_{n = 1}^{\infty} (1 - q^{n})$ .

A Rational CFT is characterized by having a finite number of primary fields. The representations associated with these fields are typically degenerate, meaning they contain null states. To find the character of a degenerate representation, we use the principle of inclusion-exclusion. We start with the character of the Verma module and subtract the character of the submodule generated by the null vector.

For a representation with a single null state at level $N$ , which itself is a primary of a submodule with dimension $h + N$ , the character is:

χ_{h}^{deg} (τ) = χ_{V_{h}} (τ) - χ_{V_{h + N}} (τ) = \frac{q ^{h} - q ^{h + N}}{η ( τ )}

This subtraction correctly removes the null state and all of its descendants.

2. The Torus Partition Function and Modular Invariance

As we've seen before, a CFT can be placed on a torus. A torus is topologically defined by identifying the opposite sides of a parallelogram in the complex plane, specified by two periods, which we can normalize to 1 and $τ$ , where $τ$ is a complex number with $Im (τ) > 0$ called the modular parameter or modulus.

The full partition function of the theory on this torus is the trace over the entire Hilbert space $H$ :

Z (τ, τ ˉ) = Tr_{H} (q^{L_{0} - c /24} q ˉ^{L ˉ_{0} - c ˉ /24})

The Hilbert space of a CFT can be decomposed into a direct sum of irreducible representations of the left and right Virasoro algebras:

H = i, j ⨁ N_{ij} H_{i} \otimes H ˉ_{j}

where $N_{ij}$ are non-negative integers specifying how many times the representation pair $(H_{i}, H ˉ_{j})$ appears in the spectrum. The partition function can thus be expressed as a bilinear combination of characters:

Z (τ, τ ˉ) = i, j \sum N_{ij} χ_{i} (τ) χ ˉ_{j} (τ ˉ)

The fundamental principle of modular invariance states that the partition function must be independent of the choice of basis vectors for the torus lattice, as long as they describe the same torus. All such equivalent choices of $τ$ are related by the action of the modular group, $S L (2, Z)$ :

τ \mapsto τ^{'} = \frac{a τ + b}{c τ + d} where (a c b d) \in S L (2, Z)

This means $Z (τ, τ ˉ) = Z (τ^{'}, τ ˉ^{'})$ .

The modular group is generated by two transformations:

T-transformation: $τ \mapsto τ + 1$
S-transformation: $τ \mapsto - 1/ τ$

Fundamental Domain of the Modular Group in the Upper Half-Plane — A depiction of the upper half-plane tessellated by the action of the modular group $SL(2,\mathbb{Z})$. All points in the upper half-plane can be mapped into the shaded fundamental domain by a unique modular transformation. Requiring the partition function to be invariant under these transformations provides powerful constraints.

3. Constraints from Modular Invariance

The requirement that $Z (τ, τ ˉ)$ be invariant under $T$ and $S$ transformations imposes strong constraints on the allowed spectrum ${N_{ij}}$ and the conformal dimensions ${h_{i}, h ˉ_{j}}$ of a rational CFT.

Constraint from T-invariance

Under $T : τ \to τ + 1$ , we have $q \to e^{2 π i (τ + 1)} = q e^{2 π i}$ . A character transforms with a simple phase:

χ_{h} (τ + 1) = Tr ((q e^{2 π i})^{L_{0} - c /24}) = e^{2 π i (h - c /24)} χ_{h} (τ)

For the full partition function $Z (τ, τ ˉ)$ to be invariant under $T$ , each term in the sum $\sum N_{ij} χ_{i} (τ) χ ˉ_{j} (τ ˉ)$ must be individually invariant. This requires the total phase to be 1:

e^{2 π i (h_{i} - c /24)} e^{- 2 π i (h ˉ_{j} - c ˉ /24)} = 1 ⟹ (h_{i} - h ˉ_{j}) - \frac{c - c ˉ}{24} \in Z

For a unitary theory, we have $c = c ˉ$ . The spectrum must also contain the identity operator with $(h, h ˉ) = (0, 0)$ . Applying the condition to this operator is trivial, but for any other operator $(h_{i}, h ˉ_{j})$ that couples to it (or exists in the theory), we must have:

h_{i} - h ˉ_{j} \in Z

This means the conformal spin of any operator in the spectrum must be an integer.

Week 7 1 Modular Invariance

The following resource provides an excellent and concise derivation of this constraint on conformal spin.

Please read Section 2.1, 'Mutual locality'. The key argument is in the three paragraphs following equation (22). Pay close attention to how invariance of the character under τ → τ+1 leads to the condition in equation (23).

Constraint from S-invariance

The $S$ transformation, $τ \to - 1/ τ$ , is far more powerful. It mixes different characters. Under an S-transformation, a character of one representation transforms into a linear combination of all characters in the theory:

χ_{i} (- 1/ τ) = j \sum S_{ij} χ_{j} (τ)

The matrix $S_{ij}$ is the modular S-matrix, a unitary and symmetric matrix ( $S = S^{T}, S^{†} S = I$ ) whose entries are determined by the CFT data (the central charge and conformal dimensions).

The invariance of the partition function $Z (- 1/ τ, - 1/ τ ˉ) = Z (τ, τ ˉ)$ now becomes:

i, j \sum N_{ij} χ_{i} (- 1/ τ) χ ˉ_{j} (- 1/ τ ˉ) = k, l \sum (i, j \sum N_{ij} S_{ik} S_{j l}^{*}) χ_{k} (τ) χ ˉ_{l} (τ ˉ) = k, l \sum N_{k l} χ_{k} (τ) χ ˉ_{l} (τ ˉ)

(Note: for a real S-matrix, $S_{j l}^{*} = S_{j l}$ ). This implies a powerful algebraic constraint on the matrix $N$ of multiplicities:

N_{k l} = i, j \sum N_{ij} S_{ik} S_{j l} or in matrix form N = S N S

This equation is often called the modular bootstrap equation.

For the simplest case of a diagonal RCFT, the spectrum pairs each left-moving representation with its right-moving counterpart, so $N_{ij} = δ_{ij}$ . The modular bootstrap equation then simplifies to $I = S I S = S^{2}$ . Since we already know S is unitary, this condition is automatically satisfied. The problem then reduces to finding a set of characters whose S-matrix is symmetric ( $S = S^{T}$ ) and squares to the identity (or more generally, the charge conjugation matrix). This severely restricts which representations can exist in the theory.

Introduction to two-dimensional conformal field theory

Now let's see these concepts put together. The following text introduces the 'modular bootstrap' and formally states how the modular S-matrix relates characters and constrains the partition function.

Please read Section 4.3, 'Modular bootstrap', up to equation (4.16). Focus on Definition 4.10 (Torus partition function), Axiom 4.11 (Modular invariance), and how the character transformation under S (eq. 4.15) leads to a condition on the S-matrix for a diagonal CFT.

Test your understanding!

The Ising model at its critical point is described by a minimal model CFT with $c = 1/2$ . Its spectrum consists of three primary fields with dimensions $h \in {0, 1/16, 1/2}$ . This theory is known to have a diagonal modular invariant partition function:

Z (τ, τ ˉ) = ∣ χ_{0} (τ) ∣^{2} + ∣ χ_{1/16} (τ) ∣^{2} + ∣ χ_{1/2} (τ) ∣^{2}

A hypothetical theorist proposes a "truncated" Ising model containing only the identity ( $h = 0$ ) and the energy operator ( $h = 1/2$ ). Its proposed partition function is $Z^{'} (τ, τ ˉ) = ∣ χ_{0} (τ) ∣^{2} + ∣ χ_{1/2} (τ) ∣^{2}$ . Using the logic of modular invariance, explain why this theory is inconsistent.

Show answer

The modular S-matrix mixes all characters of the theory. The character $χ_{0}$ for the identity operator transforms under S as a linear combination of $χ_{0}$ , $χ_{1/16}$ , and $χ_{1/2}$ . Specifically, $χ_{0} (- 1/ τ) = S_{0, 0} χ_{0} (τ) + S_{0, 1/16} χ_{1/16} (τ) + S_{0, 1/2} χ_{1/2} (τ)$ .
If the theory only contained the fields with $h = 0$ and $h = 1/2$ , the partition function would be $Z^{'} = ∣ χ_{0} ∣^{2} + ∣ χ_{1/2} ∣^{2}$ . Upon an S-transformation, the term $∣ χ_{0} (- 1/ τ) ∣^{2}$ would generate a cross-term of the form $∣ S_{0, 1/16} χ_{1/16} (τ) ∣^{2}$ .
Since $S_{0, 1/16}$ is non-zero, this term must appear in the transformed partition function. However, the original partition function $Z^{'}$ does not contain any $χ_{1/16}$ term. For $Z^{'} (- 1/ τ, - 1/ τ ˉ)$ to equal $Z^{'} (τ, τ ˉ)$ , this term must be cancelled. But since all terms in the sum are positive ( $∣ χ ∣^{2}$ ), it cannot be cancelled. Therefore, the partition function $Z^{'}$ is not modular invariant. Modular invariance forces the inclusion of the $χ_{1/16}$ representation, demonstrating that the operator spectrum of an RCFT must close under the action of the modular group.

Conclusion

In this lesson, we have explored the deep connection between the operator spectrum of a 2D CFT and the geometry of the torus. By defining Virasoro characters as generating functions for states within a representation, we were able to express the torus partition function as a bilinear sum of these characters. The physical requirement that the theory be independent of our description of the torus—modular invariance—translates into a powerful set of algebraic constraints on this sum.

Key Takeaways:

Virasoro Characters $χ_{h} (τ) = Tr_{h} (q^{L_{0} - c /24})$ are generating functions that count the number of states at each level in a representation.
The torus partition function can be written as a sum of products of left- and right-moving characters, $Z = \sum N_{ij} χ_{i} χ ˉ_{j}$ .
Modular invariance requires the partition function to be invariant under the $S L (2, Z)$ transformations $τ \to τ + 1$ (T) and $τ \to - 1/ τ$ (S).
T-invariance constrains the conformal spins of all operators in the spectrum to be integers ( $h - h ˉ \in Z$ ).
S-invariance mixes characters via the modular S-matrix ( $χ_{i} (- 1/ τ) = \sum_{j} S_{ij} χ_{j} (τ)$ ) and leads to the bootstrap equation $N = S N S$ . For an RCFT, this severely restricts the possible operator content, forcing the spectrum to form a closed, finite set under modular transformations.

In our next module, we will step back from the specifics of 2D CFT to explore a more general and profound connection: the relationship between conformal field theories and the physics of critical phenomena and the renormalization group.

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