Two-Dimensional CFT: the Partition Function on the Torus

Put a 2d CFT on a spatial circle of circumference $2\pi$. Its states live in a Hilbert space $\mathscr{H}$ acted on by two copies of the Virasoro algebra-generators $L_n$ for the left-movers, ${\overline{L}}_n$ for the right-and the zero modes do most of this chapter’s work. The one statement to anchor everything, especially if the lingo is new, is this: the Hamiltonian of the theory on the circle is

$$H=L_0+{\overline{L}}_0-\frac{c}{12},P=L_0-{\overline{L}}_0,$$

with $P$ the momentum, generating translation around the circle. $L_0$ and ${\overline{L}}_0$ are nothing exotic: they are the left- and right-moving halves of the energy. And the shift $-c/12$ is the Casimir energy of Lecture 3, one $-c/24$ for each half. Evolving in Euclidean time by $\beta$ and tracing makes a torus: one cycle is the space circle, the other the time circle. Allowing the time evolution to also translate along the space circle by an angle $\theta$ (a “twist”) combines the two real parameters into one complex modular parameter $\tau ={\tau }_1+\mathrm{i}{\tau }_2\in ℍ$, with $\beta =2\pi {\tau }_2$ and $\theta =2\pi {\tau }_1$, and we set $q={\mathrm{e}}^{2\pi \mathrm{i}\tau }$. The literature calls this $q$ the nome of the torus-an old word from elliptic-function theory, worth recognizing but nothing mysterious: it is the natural expansion variable for anything living on the torus, and, physically, a Boltzmann weight once more. Indeed $|q|={\mathrm{e}}^{-\beta }$, so $q^{L_0}$ suppresses a state by ${\mathrm{e}}^{-\beta }$ and rotates its phase by $\theta$ per unit of $L_0$. Mind that it is not the thermal $q={\mathrm{e}}^{-\beta \mathrm{\hslash }\omega }$ of L2, which weighted the quanta of one chosen oscillator; the two coincide only once the tower is graded by $L_0$ in integer steps. The torus is the quotient $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$.

Why two copies of the Virasoro algebra, and why the constant talk of holomorphy? In two Euclidean dimensions, conformal geometry and complex analysis are the same subject: pack the two worldsheet coordinates into one complex $z$, and a map preserves angles precisely where it is holomorphic (or antiholomorphic)-the Cauchy–Riemann equations are the conformality condition. The 2d conformal “group” is therefore infinite-dimensional: every holomorphic $z\mapsto z+ϵ(z)$ is an infinitesimal symmetry, one generator ${\mathrm{\ell }}_n=-z^{n+1}{\partial }_z$ per Fourier mode of $ϵ$. Quantum mechanically their algebra acquires a central term, measured by the same $c$ that set the Casimir energy a paragraph ago:

$$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1){\delta }_{m+n,0},$$

(4.1)

one copy for the holomorphic side and an independent copy ${\overline{L}}_n$ for the antiholomorphic. Everything in the theory accordingly splits into left-movers (functions of $z$) and right-movers (functions of $\overline{z}$), coupled only through the spectrum-and the split survives the trace: a quantity built from left-movers alone depends on the modulus only through $q={\mathrm{e}}^{2\pi \mathrm{i}\tau }$, never $\overline{q}$, so chiral objects will be holomorphic functions of $\tau$. That innocuous remark is the engine of §4.6.

The partition function is the graded trace

$$\boxed{Z(\tau ,\overline{\tau })={\mathrm{Tr}}_{\mathscr{H}}{q}^{L_0-\frac{c}{24}}{\overline{q}}^{{\overline{L}}_0-\frac{c}{24}}.}$$

(4.2)

Compare (1)-and note this is an identity, not an analogy: unpacking $q={\mathrm{e}}^{2\pi \mathrm{i}\tau }$,

$$q^{L_0-\frac{c}{24}}{\overline{q}}^{{\overline{L}}_0-\frac{c}{24}}={\mathrm{e}}^{-\beta H}{\mathrm{e}}^{\mathrm{i}\theta P},$$

so (4.2) is the thermal trace of Lecture 2, refined by a conjugate potential for the momentum. Inverse temperature has been complexified into $\tau$.

The same torus can be described by different choices of which lattice vectors are the two cycles. The relabellings form the modular group $SL(2,\mathbb{Z})$, acting by

(4.3)

generated by

$$T:\tau \mapsto \tau +1,S:\tau \mapsto -\frac{1}{\tau }.$$

(4.4)

Since they describe the same torus, the partition function of a consistent theory must be invariant:

$$Z(\tau +1)=Z(\tau ),Z(-1/\tau )=Z(\tau ).$$

(4.5)

$T$-invariance is a statement about the spectrum modulo the $-c/24$ shift; $S$-invariance exchanges the space and time circles, relating high and low “temperature.” Modular invariance is the conformal descendant of the cyclicity of the trace (§1).

For a single non-compact free boson ($c=1$) the trace (4.2) factorizes into the zero-mode (momentum) integral and the oscillator trace (5). The result, quoted per unit target-space volume (the momentum integral also supplies an overall volume factor), is

$$Z_{\text{boson}}(\tau )=\frac{1}{\sqrt{\mathrm{Im}\tau }{|\eta (\tau )|}^2},\eta (\tau )=q^{1/24}\prod _{n\ge 1}\left(1-q^n\right).$$

(4.6)

There is the $-c/24$ thread again, as the $q^{1/24}$. The Dedekind eta is a modular form of weight $\frac{1}{2}$: $\eta (\tau +1)={\mathrm{e}}^{\mathrm{i}\pi /12}\eta (\tau )$ and $\eta (-1/\tau )=\sqrt{-\mathrm{i}\tau }\eta (\tau )$. The factor $1/\sqrt{\mathrm{Im}\tau }$ from the momentum integral is exactly what is needed to make (4.6) modular invariant. The oscillator product $\prod {(1-q^n)}^{-1}$ of Lecture 2 has become a modular object.

A free fermion on the torus must be assigned a boundary condition on each of the two cycles-periodic (Ramond, R) or antiperiodic (Neveu–Schwarz, NS)-by the rule of §3.2. That is $2\times 2=4$ choices: the four spin structures. We write $(a,b)$ with $a$ the space-cycle and $b$ the time-cycle condition; by §3.2, an R (periodic) time cycle means a ${(-1)}^F$ insertion in the trace. Each choice yields a trace that evaluates to a ratio of a Jacobi theta function and $\eta$:

$$\text{(NS, NS)}\sim \sqrt{\frac{{\theta }_3}{\eta }},\text{(NS, R)}\sim \sqrt{\frac{{\theta }_4}{\eta }},\text{(R, NS)}\sim \sqrt{\frac{{\theta }_2}{\eta }},\text{(R, R)}\sim \sqrt{\frac{{\theta }_1}{\eta }}\equiv 0,$$

(4.7)

the last vanishing because ${\theta }_1\equiv 0$ (a fermion zero mode; the same zero mode leaves the R ground states degenerate). Parallel to $\eta$ in (4.6), let us spell the Jacobi theta constants out as products,

$${\theta }_3=\prod _{n\ge 1}(1-q^n){\left(1+q^{n-\frac{1}{2}}\right)}^2,{\theta }_4=\prod _{n\ge 1}(1-q^n){\left(1-q^{n-\frac{1}{2}}\right)}^2,{\theta }_2=2q^{1/8}\prod _{n\ge 1}(1-q^n){\left(1+q^n\right)}^2$$

(4.8)

(equivalently the sums ${\sum }_{n\in \mathbb{Z}}{q}^{n^2/2}$, ${\sum }_{n\in \mathbb{Z}}{(-1)}^n{q}^{n^2/2}$ and ${\sum }_{n\in \mathbb{Z}}{q}^{{(n+1/2)}^2/2}$). The blocks are then the Lecture 2 products in thin disguise:

$$\sqrt{\frac{{\theta }_3}{\eta }}=q^{-\frac{1}{48}}\prod _{r=\frac{1}{2},\frac{3}{2},\mathrm{\dots }}\left(1+q^r\right),\sqrt{\frac{{\theta }_4}{\eta }}=q^{-\frac{1}{48}}\prod _{r=\frac{1}{2},\frac{3}{2},\mathrm{\dots }}\left(1-q^r\right),\sqrt{\frac{{\theta }_2}{\eta }}=\sqrt{2}{q}^{\frac{1}{24}}\prod _{n\ge 1}\left(1+q^n\right):$$

(4.9)

the fermionic partition function (6), mode by mode, with half-integer (NS) or integer (R) moding, a sign when ${(-1)}^F$ is inserted, a $\sqrt{2}$ for the R zero mode, and the sector ground-state energy out front. The leading powers in (4.9) are old friends: $q^{-1/48}$ for NS, $q^{+1/24}$ for R-the two regulator-box vacuum energies (3.8) and (3.7) of Lecture 3, in units of $2\pi /L$, now reading as $-\frac{c}{24}$ and $h-\frac{c}{24}$ for the $c=\frac{1}{2}$ Majorana fermion. The NS ground state is the true vacuum; the R ground state is not-it is a state of dimension $h=\frac{1}{16}$, created by the spin field of the Ising model, sitting at $\frac{1}{16}-\frac{1}{48}=+\frac{1}{24}$. The boundary conditions of Lecture 3 have become sector data, each sector with its own ground-state energy. Under the modular group, $T$ and $S$ permute the spin structures, hence permute ${\theta }_2,{\theta }_3,{\theta }_4$:

$$T:{\theta }_3\leftrightarrow {\theta }_4,{\theta }_2\to {\theta }_2;S:{\theta }_2\leftrightarrow {\theta }_4,{\theta }_3\to {\theta }_3.$$

(4.10)

A modular-invariant fermion partition function is therefore a specific sum over spin structures, $\frac{1}{2}\left(|{\theta }_3/\eta |+|{\theta }_4/\eta |+|{\theta }_2/\eta |\right)$ in schematic form-the content of the 2d Ising model at its critical point, the $c=\frac{1}{2}$ theory realized by a single Majorana fermion.

One more enlargement before the machinery of Lecture 5. Put the boson and the fermion together ($c=1+\frac{1}{2}=\frac{3}{2}$), and the combined theory carries more symmetry than two commuting Virasoros: alongside the stress tensor there is a fermionic current $G(z)$ of weight $\frac{3}{2}$-concretely $G\sim \psi \partial \varphi$-whose modes $G_r$ extend (4.1) to the $𝒩=1$ super-Virasoro algebra,

$$[L_m,G_r]=\left(\frac{m}{2}-r\right)G_{m+r},\{G_r,G_s\}=2L_{r+s}+\frac{c}{3}\left(r^2-\frac{1}{4}\right){\delta }_{r+s,0}.$$

(4.11)

This is superconformal symmetry: conformal transformations together with supersymmetries rotating the boson into the fermion, holomorphic like everything else in this chapter.¹¹ 1 Indeed the pattern of §4.1 persists one rung up. Adjoin to $z$ an anticommuting partner $\theta$ (with ${\theta }^2=0$), making the worldsheet a superspace with coordinates $(z|\theta )$; a function $f(z,\theta )=f_0(z)+\theta f_1(z)$ packs a boson and a fermion into one superfield. Superconformal transformations are exactly the superholomorphic maps of $(z|\theta )$-those preserving, up to scale, the odd derivative $D={\partial }_{\theta }+\theta {\partial }_z$, a square root of translations ($D^2={\partial }_z$) foreshadowing the square root of the Hamiltonian in Lecture 5. Mode-expanding one such map yields the $L_m$ and $G_r$ together: super-Virasoro is to superholomorphy what Virasoro is to holomorphy. We will not need superspace, but the hint is worth having. Being fermionic, $G$ inherits the spin-structure question just settled: its moding $r$ is half-integer in the NS sector and integer in the R sector-the same two choices, now inside the symmetry algebra itself. What the enlarged algebra buys (a square root of the Hamiltonian, and with it the sturdiest partition functions in these notes) is the closing business of Lecture 5.

The Hilbert space organizes into Virasoro representations $V_h$ labelled by conformal weight $h$. The trace over a single representation is its character,

$${\chi }_h(\tau )={\mathrm{Tr}}_{V_h}{q}^{L_0-c/24}=q^{h-c/24}\sum _{n\ge 0}(dim{V}_{h,n})q^n,$$

(4.12)

and the full partition function is a sesquilinear combination

$$Z(\tau ,\overline{\tau })=\sum _{h,\overline{h}}{N}_{h\overline{h}}{\chi }_h(\tau )\overline{{\chi }_{\overline{h}}(\tau )},N_{h\overline{h}}\in {\mathbb{Z}}_{\ge 0}.$$

(4.13)

In a rational theory there are finitely many ${\chi }_h$, and the modular $S$-transform acts on this finite vector by a matrix-the modular $S$-matrix $S_{h{h}^{\prime }}$. Modular invariance becomes a finite linear-algebra condition on the multiplicities $N_{h\overline{h}}$. Equation (4.12) is the doorway to Lecture 5: a character is manifestly a graded dimension.

Stare at the shape of (4.13). Every character ${\chi }_h(\tau )$ is holomorphic in $\tau$: built from one chirality alone, it sees $q$ and never $\overline{q}$, exactly as promised in §4.1. The partition function is therefore a finite sesquilinear form in holomorphic objects-chiral blocks and their conjugates, glued by a matrix of non-negative integers. Now recall the first structural fact of these notes (§4): independent systems multiply. If the left- and right-moving halves of the theory were independent systems, $Z$ would be a single product, $Z(\tau ,\overline{\tau })=f(\tau )\overline{f(\tau )}$. Equation (4.13) says that in general they are not: no such $f$ exists, and the matrix $N_{h\overline{h}}$ records exactly how the two chiral halves are correlated-which left-moving sectors occur, and paired with which right-moving ones. In information-theoretic terms $N$ is a joint distribution over sector labels, and the failure of $Z$ to factor is the statement that the chiral halves carry mutual information. The partition function has acquired internal structure: it is not one function but a pairing of two vectors of blocks.

Both running examples exhibit the failure. For the boson, (4.6) glues ${\eta }^{-1}$ to ${\overline{\eta }}^{-1}$ diagonally, and the zero-mode factor $1/\sqrt{\mathrm{Im}\tau }$-neither holomorphic nor antiholomorphic-ties the halves together. For the fermion, the invariant is a sum over spin structures, ${\sum }_i|{\theta }_i/\eta |$, correlating the left and right boundary conditions sector by sector. Modular invariance is the culprit: $S$ and $T$ shuffle the blocks among themselves, so (special theories aside) no single product $f\overline{f}$ is invariant-only matrix-glued combinations survive. The diagonal gluing $N_{h\overline{h}}={\delta }_{h\overline{h}}$ always works, because the $S$-matrix is unitary; but it is typically not alone, and distinct gluing matrices are inequivalent theories built from identical chiral ingredients. The chiral data does not determine the CFT: the pairing is extra physical information.

This is precisely why Lecture 5 studies the chiral half by itself. The blocks are characters of the chiral algebra-the affine Lie algebras and vertex operator algebras of the next lecture-and representation theory speaks about one chirality at a time. A full CFT is two chiral halves plus a modular-invariant gluing; the objects of Lecture 5 are the factors that (4.13) almost, but never quite, separates into. (For the reader keeping categorical score: the blocks span a finite-dimensional representation of $SL(2,\mathbb{Z})$, and a consistent $Z$ is an invariant vector in a product of two such-a first hint of the finale’s habit of valuing partition functions in vector spaces rather than numbers.)

• •

  $\tau \leftrightarrow$ complexified inverse temperature/geometry; $q^{L_0-c/24}\leftrightarrow {\mathrm{e}}^{-\beta H}$ with vacuum subtraction.

• •

  Spin structure $\leftrightarrow$ L3 boundary conditions $\leftrightarrow$ ${(-1)}^F$ insertions.

• •

  Modular $S$ $\leftrightarrow$ high/low-temperature duality $\leftrightarrow$ the Kac–Peterson transform (L5). The elliptic variable $y$ is the chemical potential of §4, now a Jacobi-form variable.

• •

  Chiral block/character $\leftrightarrow$ a module of the chiral algebra (L5); the gluing matrix $N_{h\overline{h}}$ $\leftrightarrow$ which left and right modules pair-the part of the theory the chiral halves alone cannot see.