Toroidal Partition Functions and Modular Transformations

1. Putting a CFT on a Torus

A torus can be constructed by taking a parallelogram in the complex plane and identifying opposite edges. Without loss of generality, we can scale and rotate this parallelogram so that one side corresponds to the interval $[0,1]$ on the real axis, and the other side is represented by a complex number $\tau$ in the upper half-plane, $\text{Im}(\tau )>0$.

This means any point $z$ in the plane is identified with $z+n+m\tau$ for all integers $n,m$. The complex number $\tau$ is called the modular parameter, and it uniquely specifies the shape (the complex structure) of the torus.

So, how do we define a QFT on this geometry? There are two equivalent pictures: the path integral and the Hamiltonian formalism. The latter provides a more direct link to the operator spectrum we've been studying.

To understand the Hamiltonian picture, we can view the torus as a cylinder of circumference $L=1$ and length $\text{Im}(\tau )$, where the two ends are glued together after a spatial twist of $\text{Re}(\tau )$. The propagation along the cylinder's length corresponds to Euclidean time evolution. The partition function is then a trace over the theory's Hilbert space, weighted by the appropriate evolution operator.

As you've seen, the partition function on a torus with modular parameter $\tau$ is given by a trace over the Hilbert space $\mathcal{H}$ of the theory:

where $q=e^{2\pi i\tau }$ and $\bar{q}=e^{-2\pi i\bar{\tau }}$.

Let's break this down:

• Trace ${\text{Tr}}_{\mathcal{H}}$: Summing over all states in the theory.
• $L_0,{\bar{L}}_0$: These operators measure the conformal dimensions (or energies) of the states.
• $c/24,\bar{c}/24$: This is the Casimir energy correction, arising from the conformal transformation from the plane (where the vacuum has zero energy) to the cylinder.
• $q,\bar{q}$: These complex variables encode the geometry of the torus. Since $\text{Im}(\tau )>0$, we have $|q|<1$, ensuring the sum over states converges.

The partition function is a cornerstone of CFT. It packages the entire spectrum of the theory—all primary operators and their descendants—into a single function of the torus's geometry.

2. Modular Invariance: The Defining Constraint

The crucial insight is that different values of $\tau$ can represent the exact same torus. The lattice of points $\mathbb{Z}+\tau \mathbb{Z}$ can be generated by different pairs of basis vectors. For example, the basis $(1,\tau )$ generates the same lattice as $(1,\tau +1)$. Similarly, the basis $(\tau ,-1)$ also generates the same lattice.

Since the partition function is a physical property of the theory on the torus, it must be independent of our choice of basis. This means $Z$ must be invariant under these reparameterizations of $\tau$. These transformations form the modular group, which is isomorphic to $PSL(2,\mathbb{Z})$. This group is generated by two fundamental transformations.

The requirement of modular invariance is that the partition function remains unchanged under the action of the generators of the modular group:

1. T-transformation: $\tau \to \tau +1$

   $$Z(\tau +1,\bar{\tau }+1)=Z(\tau ,\bar{\tau })$$

   This corresponds to a "Dehn twist": cutting the torus along one cycle, twisting one end by $2\pi$, and re-gluing.

2. S-transformation: $\tau \to -1/\tau$

   $$Z(-1/\tau ,-1/\bar{\tau })=Z(\tau ,\bar{\tau })$$

   This transformation is more dramatic; it exchanges the two fundamental cycles of the torus. It essentially swaps the roles of "space" and "Euclidean time".

This principle is extraordinarily powerful. It asserts that a consistent CFT must make sense on any torus and that the physics doesn't depend on how we choose to draw its fundamental cycles.

3. Decomposing the Partition Function into Characters

To understand the consequences of modular invariance, we first decompose the partition function. The Hilbert space $\mathcal{H}$ is built from the irreducible representations of the Virasoro algebra. Recall that these are the Verma modules (or their quotients by null states), each built on a primary state.
The full Hilbert space is a direct sum of these representations, pairing left-movers and right-movers:

Here, ${\mathcal{V}}_{h_i}$ is the representation associated with the primary of weight $h_i$, and $M_{ij}$ is a matrix of non-negative integers indicating how many times the left-moving representation $i$ is paired with the right-moving representation $j$. The matrix $M_{ij}$ defines the spectrum of local operators in the theory.

The trace in the partition function can thus be decomposed. We define the Virasoro character of a representation ${\mathcal{V}}_h$ as the trace over just that representation:

where $\text{dim}({\mathcal{V}}_h^{(N)})$ is the number of states at level $N$. The character is a generating function for the states in a single conformal family.

The full partition function is then a sum over these characters, weighted by the spectrum matrix $M_{ij}$:

4. Consequences of Modular Invariance

The characters ${\chi }_i(\tau )$ are not themselves modular invariant. They transform in a well-defined way. The invariance of the full partition function $Z$ then constrains the allowed spectrum $M_{ij}$.

A. T-invariance ($\tau \to \tau +1$)

Under a T-transformation, $q=e^{2\pi i\tau }\to e^{2\pi i(\tau +1)}=e^{2\pi i}q$. The character transforms with a simple phase:

The full partition function then transforms as:

For $Z$ to be invariant, the phase factor must be 1 for every non-zero term in the sum. Assuming $c=\bar{c}$ (for a non-chiral theory), this requires:

This means that all local operators in the theory must have integer spin $s=h-\bar{h}$. This is a remarkably strong conclusion derived purely from geometric consistency.

B. S-invariance ($\tau \to -1/\tau$)

Under S-transformation, the characters mix in a more complicated way:

where $S_{ik}$ is a unitary matrix known as the modular S-matrix. For the partition function to be invariant, the spectrum matrix $M$ must commute with the S-matrix transformation: $M=SM{S}^{†}$.

This condition implies that if a particular representation ${\mathcal{V}}_i$ is in the spectrum, its modular S-transform must also be constructible from the representations in the spectrum. This often forces the inclusion of multiple conformal families.

A classic example is a theory of a free fermion ($c=1/2$). On a torus, a fermion field can have periodic (P) or anti-periodic (A) boundary conditions along each of the two cycles. This leads to four "spin structures": (A,A), (A,P), (P,A), and (P,P). Each corresponds to a different sector of the theory with a different partition function. It turns out that none of these are modular invariant on their own. The modular S-transformation mixes them. To construct a modular invariant partition function, one must sum the contributions from different sectors in a specific way. This is a general feature: modular invariance dictates the complete, consistent set of operators and their pairings.

Conclusion

In this lesson, we have taken a significant step by placing our CFT on a torus. This has allowed us to define the partition function, a single object that contains the theory's full operator spectrum.

Key Takeaways:

• The partition function $Z(\tau ,\bar{\tau })=\text{Tr}(q^{L_0-c/24}{\bar{q}}^{{\bar{L}}_0-\bar{c}/24})$ encodes the spectrum of a 2D CFT as a function of the torus's shape, defined by the modular parameter $\tau$.
• Physics must be independent of how we parameterize the torus. This leads to the constraint of modular invariance: $Z(\tau )$ must be invariant under the transformations $\tau \to \tau +1$ (T) and $\tau \to -1/\tau$ (S).
• The partition function is a sum of products of Virasoro characters, $Z=\sum M_{ij}{\chi }_i{\bar{\chi }}_j$, which are traces over individual conformal families.
• Modular invariance imposes powerful constraints on the operator spectrum $M_{ij}$. T-invariance forces local operators to have integer spin, while S-invariance dictates which conformal families must coexist in a consistent theory.

In our next lesson, we will make this concrete by performing our first major partition function calculation: we will compute the partition function for a free boson on a torus and explicitly verify its modular invariance. This will showcase all the principles we have discussed today in a practical setting.