Partition Function of a Free Boson on a Torus

Hello! Welcome to the next lesson in our exploration of computational methods in 2D CFT.

In our last lesson, we introduced a fundamental object: the partition function on a torus, $Z (τ, τ ˉ)$ . We established that this function, which encodes the entire spectrum of a CFT, must be invariant under the modular group $P S L (2, Z)$ generated by the T-transformation ( $τ \to τ + 1$ ) and the S-transformation ( $τ \to - 1/ τ$ ). This principle of modular invariance provides one of the most powerful constraints on any consistent 2D CFT.

Today, we will move from principle to practice. Our goal is to perform our first full partition function calculation for the simplest non-trivial CFT: the free, non-compact boson. We will compute its partition function on the torus and then explicitly verify that the result satisfies the demands of modular invariance. This exercise will solidify the abstract concepts from our previous lesson into a concrete computational skill.

Your learning objective is to compute the partition function for a free boson on a torus and verify its modular invariance.

1. Anatomy of the Free Boson Partition Function

Let's begin with the general formula for the partition function on a torus:

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

For a single free boson, the central charge is $c = c ˉ = 1$ . The Hilbert space $H$ consists of states built by acting with creation operators $α_{- n}$ ( $n > 0$ ) on momentum eigenstates $∣ p ⟩$ . The trace is a sum over all of these states.

It is convenient to separate the trace into two independent parts:

The Oscillator Contribution: The trace over the Fock space of states created by the harmonic oscillator modes $α_{- n}$ and $α ˉ_{- n}$ .
The Zero-Mode Contribution: The trace (in this case, an integral) over the continuous center-of-mass momentum modes, $p$ .

We will calculate each contribution in turn.

2. The Oscillator Contribution and the Dedekind $η$ -function

Let's first compute the trace over the oscillator states. The part of $L_{0}$ that counts oscillator energy is $L_{0}^{osc} = \sum_{n = 1}^{\infty} α_{- n} α_{n}$ . A state like $(α_{- n})^{k} ∣0 ⟩$ has energy $k n$ .

The full oscillator Fock space is a tensor product of the spaces for each mode $n = 1, 2, \dots$ . Therefore, the trace over the full space is the product of traces over each mode. For a single mode $n$ , the contribution to the partition function is a geometric series:

Tr_{n} (q^{n a^{†} a}) = k = 0 \sum \infty (q^{n})^{k} = \frac{1}{1 - q ^{n}}

The full contribution from all left-moving oscillator modes is the product over all $n$ :

Tr_{osc} (q^{L_{0}^{osc}}) = n = 1 \prod \infty \frac{1}{1 - q ^{n}}

Now, let's combine this with the Casimir energy term $q^{- c /24}$ .

6. String Interactions

The following reading from David Tong's string theory notes performs exactly this calculation. It shows how the trace over oscillators leads to a famous object in number theory: the Dedekind eta function.

Please read the part of Section 6.4.2 titled 'The One-Loop Partition Function', starting from the paragraph 'Let’s compute this for the free string.' and ending just before the derivation of the zero-mode contribution (before the sentence 'We’re still left with the contribution from the zero mode p...'). Focus on how the trace over oscillator states for a single scalar field results in the product formula.

As you've read, the contribution to the partition function from the oscillator modes of a single scalar field (including the Casimir energy) is given by:

Tr_{osc} (q^{L_{0} - c /24}) = q^{- 1/24} n = 1 \prod \infty \frac{1}{1 - q ^{n}}

This combination defines the inverse of the Dedekind eta function, $η (τ)$ :

η (τ) = q^{1/24} n = 1 \prod \infty (1 - q^{n}) where q = e^{2 π i τ}

The total contribution from both left-moving ( $q$ ) and right-moving ( $q ˉ$ ) oscillators is therefore $∣ η (τ) ∣^{- 2}$ .

3. The Zero-Mode Contribution

Next, we must account for the center-of-mass momentum. In a non-compact theory, the momentum $p$ is a continuous variable. The zero-mode part of the Virasoro generators for a free boson is given by $L_{0}^{zero} = L ˉ_{0}^{zero} = \frac{α ^{'}}{4} p^{2}$ . For consistency with most CFT literature, we will work in units where the string scale is set to $α^{'} = 2$ . This gives $L_{0}^{zero} = p^{2} /2$ .

The trace over these zero-modes becomes an integral over all possible momenta:

Z_{zero} = \int \frac{d p}{2 π} ⟨ p ∣ q^{L_{0}^{zero}} q ˉ^{L ˉ_{0}^{zero}} ∣ p ⟩ = \int \frac{d p}{2 π} e^{2 π i τ (p^{2} /2)} e^{- 2 π i τ ˉ (p^{2} /2)}

The exponent simplifies nicely:

2 π i (τ - τ ˉ) \frac{p ^{2}}{2} = 2 π i (2 i Im τ) \frac{p ^{2}}{2} = - 2 π (Im τ) p^{2}

This leaves us with a standard Gaussian integral:

Z_{zero} = \int \frac{d p}{2 π} e^{- 2 π (Im τ) p^{2}} = \frac{1}{2 π} \frac{π}{2 π ( Im τ )} = \frac{1}{4 π Im τ}

Ignoring the constant numerical factor (which can be absorbed into the path integral measure), the crucial dependence on the modular parameter is:

Z_{zero} (τ) \propto \frac{1}{Im τ}

You can find this result as well in the resource from David Tong, section 6.4.2.

4. Assembling the Partition Function and Verifying Modular Invariance

Combining the oscillator and zero-mode contributions, the full partition function for a single, non-compact free boson is:

Z_{boson} (τ, τ ˉ) = \frac{1}{Im τ} \frac{1}{∣ η ( τ ) ∣ ^{2}}

This is a celebrated result. Now, we must subject it to the ultimate test: is it modular invariant? To check this, we need the transformation properties of the eta function, which were established by mathematicians in the 19th century, long before CFT was conceived.

6. String Interactions

The modular properties of the Dedekind eta function are a classic result. The following reading states them clearly.

Please read the short subsection titled 'Modular Invariance' within Section 6.4.2. Focus on the two identities given for \eta( au+1) and \eta(-1/ au).

We have the transformation rules:

T-transformation: $η (τ + 1) = e^{iπ /12} η (τ)$
S-transformation: $η (- 1/ τ) = - i τ η (τ)$

Let's test the invariance of our partition function $Z_{boson}$ .

A. Invariance under T: $τ \to τ + 1$

The imaginary part is unchanged: $Im (τ + 1) = Im τ$ .
The squared modulus of the eta function transforms as: $∣ η (τ + 1) ∣^{2} = ∣ e^{iπ /12} η (τ) ∣^{2} = ∣ e^{iπ /12} ∣^{2} ∣ η (τ) ∣^{2} = 1 \cdot ∣ η (τ) ∣^{2} = ∣ η (τ) ∣^{2}$

Since both the numerator and the denominator are invariant, $Z_{boson} (τ + 1) = Z_{boson} (τ)$ . The partition function is T-invariant.

B. Invariance under S: $τ \to - 1/ τ$

This is more involved. Let's transform each piece separately.

The imaginary part transforms as: $Im (- 1/ τ) = Im (- \frac{τ ˉ}{τ τ ˉ}) = Im (- \frac{Re τ - i Im τ}{∣ τ ∣ ^{2}}) = \frac{Im τ}{∣ τ ∣ ^{2}}$ Therefore, the zero-mode factor transforms as: $\frac{1}{Im ( - 1/ τ )} = \frac{1}{Im τ /∣ τ ∣ ^{2}} = \frac{∣ τ ∣}{Im τ}$
The eta function's squared modulus transforms as: $∣ η (- 1/ τ) ∣^{2} = ∣ - i τ η (τ) ∣^{2} = ∣ - i τ ∣^{2} ∣ η (τ) ∣^{2}$ The modulus of $- i τ$ is $∣ - i τ ∣ = ∣ τ ∣$ . So, $∣ - i τ ∣^{2} = ∣ τ ∣$ . $∣ η (- 1/ τ) ∣^{2} = ∣ τ ∣∣ η (τ) ∣^{2}$
Now, we assemble the transformed partition function: $Z_{boson} (- 1/ τ) = \frac{1}{Im ( - 1/ τ )} \frac{1}{∣ η ( - 1/ τ ) ∣ ^{2}} = (\frac{∣ τ ∣}{Im τ}) (\frac{1}{∣ τ ∣∣ η ( τ ) ∣ ^{2}}) = \frac{1}{Im τ} \frac{1}{∣ η ( τ ) ∣ ^{2}}$

The result is identical to our starting expression. $Z_{boson} (- 1/ τ) = Z_{boson} (τ)$ . The partition function is S-invariant.

We have successfully shown that the partition function of a free boson is fully modular invariant.

Test your understanding!

Imagine a hypothetical theory where the partition function was found to be just $Z_{h y p o} = 1/∣ η (τ) ∣^{2}$ , without the $1/ Im τ$ factor. Would this theory be consistent? Why or why not?

Show answer

No, this theory would not be consistent. While it would be T-invariant (since $∣ η (τ + 1) ∣^{2} = ∣ η (τ) ∣^{2}$ ), it would fail S-invariance. Under $τ \to - 1/ τ$ , the partition function would transform as $Z_{h y p o} (- 1/ τ) = 1/∣ η (- 1/ τ) ∣^{2} = 1/ (∣ τ ∣∣ η (τ) ∣^{2}) = (1/∣ τ ∣) Z_{h y p o} (τ)$ . Since this is not equal to $Z_{h y p o} (τ)$ , the theory violates modular invariance and is physically inconsistent as a 2D CFT. This highlights how both the zero-modes and oscillators are essential for consistency.

Alternative Derivation via Path Integrals

Given your background with path integrals, you might appreciate that the same result can be derived from a completely different starting point. Instead of the Hamiltonian operator formalism, one can compute the path integral $\int [d ϕ] e^{- S [ϕ]}$ for the boson field $ϕ$ on the torus. This involves evaluating the determinant of the Laplacian operator on the torus, which is an infinite product. This product is divergent and must be regularized, typically using zeta-function regularization.

Modular invariance and orbifolds

The following resource carries out this advanced calculation. It is not required reading, but given your mathematical interests, you may find it insightful to see how the same modular-invariant result emerges from the path integral formalism and zeta-function regularization.

Skim through Section 3.3.1, 'Detailed calculation in the path-integral formalism'. You don't need to follow every step of the calculation, but try to appreciate the overall strategy: the path integral is expressed as a determinant of the Laplacian, which is related to an infinite product over eigenvalues (17), this product is regularized using a zeta function G(s) (18), and a series of sophisticated mathematical manipulations (Poisson resummation, properties of the Gamma and Riemann zeta functions) ultimately yields the final result (42).

This alternative derivation underscores the robustness of the result. Whether we sum over states in the operator formalism or integrate over field configurations in the path integral formalism, the consistency of the theory forces the same modular-invariant structure.

Conclusion

In this lesson, we have successfully executed our first complete partition function calculation. By separating the contributions from the free boson's zero-modes and oscillator modes, we assembled the full partition function and rigorously verified its invariance under both T and S modular transformations.

Key Takeaways:

The partition function of a 2D CFT can be explicitly calculated by tracing over its Hilbert space.
For a free boson, the trace separates into an integral over continuous momenta (zero-modes) and a sum over discrete oscillator states.
The oscillator contribution is captured by the Dedekind eta function: $Z_{osc} = 1/∣ η (τ) ∣^{2}$ .
The zero-mode contribution provides the crucial $1/ Im τ$ factor.
The complete partition function $Z_{boson} = \frac{1}{Im τ ∣ η ( τ ) ∣ ^{2}}$ is fully modular invariant, with the transformations of the zero-mode and oscillator parts precisely canceling each other.

In our next lesson, we will apply these computational skills to another fundamental theory: the free fermion. We will compute correlation functions using a powerful tool called Wick's theorem, which is a cornerstone of perturbative quantum field theory and has a direct application in CFT.

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