Representation Theory: Partition Functions as Characters

For the reader from (physics).

“Module” is the state space of a sector; “character” is its partition function; “level $k$ ” is tied to your central charge. The Weyl–Kac character formula is the rep-theoretic ancestor of the CFT characters you just met, and the statement that characters transform into each other under $SL(2,\mathbb{Z})$ is, on the physics side, modular invariance-now a theorem.

Lecture 4 closed with a diagnosis (§4.6): the torus partition function is a matrix-glued pairing of holomorphic blocks, and the blocks-not the full $Z$ -are where representation theory lives. This lecture is about the blocks. Everything in it concerns a single chirality at a time, and the reward for giving up the other half is that the objects become algebraic and the statements become theorems.

5.1 A character is a graded dimension is a partition function

Let $V=\bigoplus_{n\geq 0}V_{n}$ be a graded vector space with finite-dimensional pieces. Its (graded) character, or $q$ -character, is

\operatorname{ch}_{q}V=\sum_{n\geq 0}(\dim V_{n})\,q^{n}.

(5.1)

Here $q$ is a formal variable-a bookkeeping device whose sole job is to file the integer $\dim V_{n}$ at position $n$ . Identities between characters are identities of formal power series; no convergence is required or asked. Physics re-enters by specialization: set $q=\mathrm{e}^{2\pi\mathrm{i}\tau}$ (the nome of Lecture 4) and the bookkeeping becomes Boltzmann weighting, the series now converging for $|q|<1$ . This is the same object as the CFT character (4.12) (up to the $q^{h-c/24}$ prefactor) and-reading Lecture 2 backwards-the same object as a partition function: $\dim V_{n}$ counts the states at grade $n$ , and $q^{n}$ is their Boltzmann weight.

One signpost before the machinery: how precisely do $\operatorname{ch}$ and the physical $Z$ relate? First, keep the two symbols apart. $\operatorname{ch}V$ is the character of a single module $V$ : a formal series, possibly in many variables $\mathrm{e}^{\lambda}$ . The block $\chi$ of Lecture 4 is the same object specialized to $q=\mathrm{e}^{2\pi\mathrm{i}\tau}$ and dressed with the modular anomaly,

\chi_{\Lambda}(\tau)\;=\;q^{\,h_{\Lambda}-c/24}\,\operatorname{ch}_{q}L(% \Lambda),

where $h_{\Lambda}$ is the ground-state weight of the sector (Lecture4 labelled blocks by $h$ ; here the label is the highest weight $\Lambda$ , which determines $h_{\Lambda}$ ). So $\operatorname{ch}$ is the algebraic animal, $\chi$ its modular avatar, and each is a chiral partition function: the trace over one sector, at one chirality. The whole construction of this lecture, as one cartoon:

\underbrace{(\mathfrak{g},\,k)}_{\text{chiral data}}\;\rightsquigarrow\;% \underbrace{\{L(\Lambda)\}}_{\text{sectors}}\;\xrightarrow{\;\operatorname{ch}% ,\;\;q^{h_{\Lambda}-c/24}\;}\underbrace{\chi_{\Lambda}(\tau)}_{\text{blocks}}% \;\xrightarrow{\;\text{glue}\;}\;Z(\tau,\bar{\tau})=\sum_{\Lambda,\bar{\Lambda% }}N_{\Lambda\bar{\Lambda}}\,\chi_{\Lambda}(\tau)\,\overline{\chi_{\bar{\Lambda% }}(\tau)}.

(5.2)

Defining the full theory means summing over all the available representations-the sector list. And the integer matrix $N_{\Lambda\bar{\Lambda}}$ is no extra axiom but the theory’s spectrum, presented as a matrix: the Hilbert space decomposes under the two chiral algebras as $\mathcal{H}=\bigoplus_{\Lambda,\bar{\Lambda}}N_{\Lambda\bar{\Lambda}}\,L(% \Lambda)\otimes\overline{L(\bar{\Lambda})}$ , with $N_{\Lambda\bar{\Lambda}}$ counting how many times the sector $(\Lambda,\bar{\Lambda})$ occurs; the graded trace over this $\mathcal{H}$ produces (5.2) verbatim. Modular invariance (§4.6) is then the consistency condition selecting which matrices $N$ define theories at all. The comparison with Lecture 3 runs deeper than bookkeeping: it says what specifies a model. There, a theory is specified by field content and an action, and $Z$ is computed from that data, $S_{E}\rightsquigarrow Z=\int\mathcal{D}\phi\,\mathrm{e}^{-S_{E}}$ ; here, by a chiral algebra and a gluing, $(\mathfrak{g}_{k},\,N)\rightsquigarrow Z$ via (5.2). The algebra plays the role the action played: the input that defines the model. For the WZW models the two specifications literally coincide-the same theory may be handed to you as a sigma-model action on a group manifold or as $\widehat{\mathfrak{g}}$ at level $k$ , and the partition function does not care which. The rest of the lecture constructs the ingredients of (5.2): the next two sections build $L(\Lambda)$ and $\operatorname{ch}$ , the modularity section explains why the blocks mix under $SL(2,\mathbb{Z})$ , and Example 5.2 assembles a complete $Z$ .

The finite-dimensional Weyl character formula is the classical ancestor; we need its affine extension.

5.2 A Lie-theory refresher, on the lattice

The Weyl–Kac formula below is the centrepiece of this lecture, and on first sight it is a dense block of undefined symbols. Here is the minimum needed to read it-worth skimming even if Lie theory is home turf, since it fixes notation. Keep $\mathfrak{g}=\mathfrak{su}(2)$ in mind throughout.

The Cartan subalgebra is the set of good quantum numbers.

A simple Lie algebra $\mathfrak{g}$ is a finite-dimensional non-abelian Lie algebra with no nontrivial ideals-nothing invariant to quotient away ( $\mathfrak{su}(2)$ , $\mathfrak{su}(n)$ , …). Inside it choose a Cartan subalgebra $\mathfrak{h}$ : a maximal set of commuting, simultaneously diagonalizable generators-physically, a maximal set of compatible conserved charges. For $\mathfrak{su}(2)$ it is the span of $J^{3}$ ; the dimension of $\mathfrak{h}$ is the rank.

Roots, weights, and their lattices.

Now diagonalize everything against $\mathfrak{h}$ . The algebra acting on itself sorts into eigenvectors: the nonzero eigenvalues $\alpha\in\mathfrak{h}^{*}$ are the roots, and the root vectors are ladder operators (the $J^{\pm}$ of $\mathfrak{su}(2)$ ). A representation likewise sorts into simultaneous eigenstates, and the eigenvalue $\lambda\in\mathfrak{h}^{*}$ of a state is its weight-nothing but its tuple of quantum numbers (for $\mathfrak{su}(2)$ , the magnetic number $m$ ). The weights of finite-dimensional representations populate the weight lattice $P$ ; the roots generate the root lattice $Q\subseteq P$ . A finite-dimensional irreducible is labelled by its highest weight $\Lambda$ (the spin $j$ ), and for character purposes a representation is nothing more than its list of weights, counted with repetition:

\operatorname{ch}V_{\Lambda}=\sum_{\lambda}\operatorname{mult}(\lambda)\,% \mathrm{e}^{\lambda},

where $\operatorname{mult}(\lambda)$ is the number of independent states carrying the quantum numbers $\lambda$ -one formal exponential per state, a Boltzmann weight per conserved charge, as the box below insists. Mind the two roles here: the weight $\lambda$ is a label, naming which quantum numbers a state carries, while the variable is a Cartan element $h\in\mathfrak{h}$ , against which each formal symbol evaluates to a number, $\mathrm{e}^{\lambda}\mapsto\mathrm{e}^{\lambda(h)}$ . For $\mathfrak{su}(2)$ : $\operatorname{ch}V_{j}=\sum_{m=-j}^{j}z^{m}$ , with $z=\mathrm{e}^{\theta}$ the fugacity conjugate to $J^{3}$ -the weights $m$ label the terms; $z$ is the variable. For the irreducibles of $\mathfrak{su}(2)$ every multiplicity is $1$ : spin $j$ has weights $m=-j,\dots,+j$ , each occurring once. Multiplicity starts carrying information the moment representations combine or the rank grows. In $\tfrac{1}{2}\otimes\tfrac{1}{2}$ the weight $m=0$ occurs twice- $\operatorname{mult}(0)=2$ , one state from the triplet and one from the singlet. In the adjoint of $\mathfrak{su}(3)$ (the octet), the six nonzero weights are the roots, each with multiplicity $1$ , while the weight $0$ has $\operatorname{mult}(0)=2$ : the two Cartan directions-in the meson octet, the $\pi^{0}$ and the $\eta$ .

The Weyl group is a point group.

The Weyl group $W$ is the finite group generated by reflections in the hyperplanes orthogonal to the roots; it permutes the weights of any representation. For $\mathfrak{su}(2)$ it is $m\mapsto-m$ , so $W=\mathbb{Z}_{2}$ . For $\mathfrak{su}(3)$ (rank $2$ ) the six roots form a regular hexagon and $W=S_{3}$ : the six-element group permuting the three eigenvalues of the fundamental, generated by the two reflections $s_{1},s_{2}$ in the walls. The length $\ell(w)$ is the minimal number of generating reflections composing $w$ , and $(-1)^{\ell(w)}$ is its sign-concretely, for $\mathfrak{su}(3)$ :

$w\in W(\mathfrak{su}(3))=S_{3}$	geometrically	$\ell(w)$	$(-1)^{\ell(w)}$
$e$	identity	$0$	$+$
$s_{1}$ , $s_{2}$	the two wall reflections	$1$	$-$
$s_{1}s_{2}$ , $s_{2}s_{1}$	rotations by $\pm 120^{\circ}$	$2$	$+$
$s_{1}s_{2}s_{1}=s_{2}s_{1}s_{2}$	the remaining reflection	$3$	$-$

so the sign is nothing but $\det w$ : reflections count $-1$ , rotations $+1$ . Finally the Weyl vector $\rho$ , half the sum of the positive roots, produces the ubiquitous quantum shifts (for $\mathfrak{su}(2)$ , $j\to j+\tfrac{1}{2}$ ). A condensed-matter reader has seen this entire package before: a lattice and its point group. What is missing is translations-and translations are exactly what the affine extension supplies.

The affine extension is Fourier analysis on the circle of Lecture 4.

Put the theory on the space circle. A $\mathfrak{g}$ -valued symmetry current becomes a $\mathfrak{g}$ -valued function on the circle, and its Fourier modes $X\otimes t^{m}$ (with $t=\mathrm{e}^{\mathrm{i}\sigma}$ ) span the loop algebra $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$ . Quantum mechanically the modes acquire a central term-the 2d sibling of a Schwinger term:

[\,X\otimes t^{m},\;Y\otimes t^{n}\,]\;=\;[X,Y]\otimes t^{m+n}\;+\;m\,\langle X% ,Y\rangle\,\delta_{m+n,0}\,K,

(5.3)

with $\langle\cdot,\cdot\rangle$ the suitably normalized Killing form and $K$ central. This is the affine Lie algebra $\widehat{\mathfrak{g}}$ . On an irreducible highest-weight module $K$ acts by a scalar $k$ : the level. (Nothing yet restricts $k$ ; the integrability of the next section is what forces $k\in\mathbb{Z}_{\geq 0}$ .) The Cartan grows to $\mathfrak{h}\oplus\mathbb{C}K\oplus\mathbb{C}d$ , where $d$ grades by Fourier mode number- $L_{0}$ in Lie clothing-so an affine weight is a triple (finite weight, level, energy grade), and the character variable $q$ of (5.1) is the fugacity conjugate to $d$ .

The affine Weyl group is a space group.

The reflections compatible with the extension assemble into $W_{\mathrm{aff}}=W\ltimes Q^{\vee}$ : the finite Weyl group, semidirect translations by the (co)root lattice. To the condensed-matter eye this is precisely the crystallographer’s upgrade from point group to space group. The translations are also why theta functions are imminent: a sum of exponentials of a quadratic form over a lattice is a theta function.

5.3 Integrable modules and the Weyl–Kac formula

Which representations of $\widehat{\mathfrak{g}}$ play the role the finite-dimensional irreducibles played for $\mathfrak{g}$ ? The integrable highest-weight modules $L(\Lambda)$ : those on which every root’s $\mathfrak{su}(2)$ exponentiates to the group (equivalently, $\Lambda$ is dominant integral)-the affine analogue of finite-dimensionality. Integrability is a strong finiteness condition: at fixed level $k$ there are only finitely many such $\Lambda$ -for $\widehat{\mathfrak{su}}(2)_{k}$ , the spins $j=0,\tfrac{1}{2},\dots,\tfrac{k}{2}$ -and they are exactly the sectors of the level- $k$ WZW model. In field-theory language, $L(\Lambda)$ is the chiral-algebra avatar of a quantum field transforming in the representation $\Lambda$ of $\mathfrak{g}$ : its ground states form the finite-dimensional $\mathfrak{g}$ -module $\Lambda$ , created by the corresponding primary field, and the rest of the module is that field’s cloud of excitations-the current descendants $J^{a}_{-n}$ acting on the ground states. One module $=$ one field, together with everything the symmetry algebra can do to it. Their characters are given by the Weyl–Kac character formula,

\operatorname{ch}L(\Lambda)=\frac{\sum_{w\in W}(-1)^{\ell(w)}\,\mathrm{e}^{\,w% (\Lambda+\rho)-\rho}}{\prod_{\alpha>0}\bigl{(}1-\mathrm{e}^{-\alpha}\bigr{)}^{% \operatorname{mult}\alpha}},

(5.4)

the affine analogue of Weyl’s finite-dimensional formula-and every ingredient is now in hand: $W$ is the affine Weyl group of §5.2, $\ell(w)$ its length, $\rho$ its Weyl vector, and $\operatorname{mult}\alpha$ the dimension of the root space ( $1$ for the “real” roots-the finite roots and their translates up the energy grading; $\operatorname{rank}\mathfrak{g}$ for the “imaginary” roots, the pure energy directions). Because $W_{\mathrm{aff}}=W\ltimes Q^{\vee}$ , the numerator reorganizes as a finite sum over $W$ of lattice sums: specializing the formal exponentials to the single grading variable $q$ , the character becomes a finite sum of string functions times theta functions-the theta functions of Lecture 4 reappearing as honest characters, their lattice-sum forms in (4.8) now explained rather than merely checked.

Thread (The Cartan variables are chemical potentials).

The character (5.4) is a function on the Cartan subalgebra: evaluated at $h\in\mathfrak{h}$ , each formal exponential becomes the number $\mathrm{e}^{\lambda(h)}$ -the weights $\lambda$ label the terms, the components of $h$ are the variables, rank-many of them plus $q$ , exactly as in §5.2. Those variables are precisely the chemical potentials/fugacities of §4: each Cartan direction is a conserved charge, and its conjugate variable grades the character. The principal specialization (setting the Cartan variables to special values, keeping only $q=\mathrm{e}^{2\pi\mathrm{i}\tau}$ conjugate to the energy $L_{0}$ ) is “turning the chemical potentials off” to leave the pure $q$ -grading of (5.1).

5.4 Modularity, as a theorem

The miracle-discovered by physicists as modular invariance and proved by mathematicians as a property of characters-is:

Kac–Peterson. The normalized characters of the level- $k$ integrable modules of $\widehat{\mathfrak{g}}$ span a finite-dimensional representation of $SL(2,\mathbb{Z})$ . Under $S:\tau\mapsto-1/\tau$ they transform into one another by a unitary matrix $S_{\Lambda\Lambda^{\prime}}$ , and under $T:\tau\mapsto\tau+1$ by a diagonal phase.

The $-c/24$ shift (with $c=k\dim\mathfrak{g}/(k+h^{\vee})$ the Sugawara central charge) is exactly the correction that makes the characters transform as vector-valued modular forms. The physicist’s modular $S$ -matrix of §4.5 is the Kac–Peterson matrix. Thread 3 closes: the regularized vacuum energy of Lecture 3 is the modular anomaly of Lecture 5.

Example 5.1 (Kac–Peterson for $\widehat{\mathfrak{su}}(2)_{k}$ , explicitly).

With sectors labelled by $j,j^{\prime}\in\{0,\tfrac{1}{2},\dots,\tfrac{k}{2}\}$ ,

S_{jj^{\prime}}=\sqrt{\frac{2}{k+2}}\,\sin\!\Big{(}\frac{(2j+1)(2j^{\prime}+1)% \,\pi}{k+2}\Big{)},\qquad T_{jj^{\prime}}=\delta_{jj^{\prime}}\,\mathrm{e}^{2% \pi\mathrm{i}(h_{j}-c/24)},\quad h_{j}=\frac{j(j+1)}{k+2}.

(5.5)

At level $1$ (two sectors; $h_{0}=0$ , $h_{1/2}=\tfrac{1}{4}$ , $c=1$ ):

S=\frac{1}{\sqrt{2}}\begin{pmatrix}1&\phantom{-}1\\ 1&-1\end{pmatrix},\qquad T=\begin{pmatrix}\mathrm{e}^{-\pi\mathrm{i}/12}&0\\ 0&\mathrm{e}^{5\pi\mathrm{i}/12}\end{pmatrix}.

(5.6)

$S$ is real, symmetric, unitary, and squares to $\mathbf{1}$ . Modular invariance of the diagonal theory $\sum_{j}|\chi_{j}|^{2}$ (assembled below in Example 5.2) follows at a glance: under $S$ the sum is preserved because $S$ is unitary; under $T$ the phases cancel between $\chi_{j}$ and $\overline{\chi_{j}}$ . The diagonal invariant is modular invariant because Kac–Peterson holds.

5.5 Vertex operator algebras: making Lecture 4 rigorous

A vertex operator algebra (VOA) $V$ axiomatizes the chiral algebra of a 2d CFT: a graded space with a state–field correspondence $a\mapsto Y(a,z)$ , a vacuum, and a conformal vector $\omega$ whose field is the stress tensor,

T(z)\;=\;Y(\omega,z)\;=\;\sum_{n}L_{n}\,z^{-n-2},

its modes realizing the Virasoro algebra (4.1) on the state space. The connection between vector and symmetry is tight in both directions. As a state, $\omega=L_{-2}\lvert 0\rangle$ has weight $2$ and carries the central charge on board: $L_{2}\,\omega=\tfrac{c}{2}\lvert 0\rangle$ . As a field, the commutators (4.1) are equivalent to the singular part of the operator product expansion (OPE) of the stress tensor with itself,

T(z)\,T(w)\;\sim\;\frac{c/2}{(z-w)^{4}}+\frac{2\,T(w)}{(z-w)^{2}}+\frac{% \partial T(w)}{z-w},

the same algebra said pointwise. This is how a VOA manifests conformal symmetry: not as transformations of a spacetime, but as a distinguished vector whose modes act on everything. Its modules are the sectors, and the graded dimension of a module is the chiral torus partition function (4.12). The rigorous form of modular invariance is:

Zhu’s theorem. For a sufficiently nice VOA (rational and $C_{2}$ -cofinite),¹¹ 1 Rational: finitely many irreducible modules, and every module a direct sum of them. $C_{2}$ -cofinite: the subspace spanned by the vectors $a_{-2}b$ (for $a,b\in V$ , with $a_{-2}$ a Laurent mode of the vertex operator of $a$ ) has finite codimension in $V$ -a finiteness condition that makes the characters converge and satisfy a modular differential equation. Together these are the standing hypotheses of rational VOA theory. the span of the characters of its irreducible modules is invariant under $SL(2,\mathbb{Z})$ .

This is the theorem standing behind the physical assertion of §4.2 that $Z(\tau)$ is modular invariant.

Thread (The running examples, identified).

The free boson is the Heisenberg (Fock) VOA; its character $\operatorname{Tr}\,q^{L_{0}-c/24}$ is $\eta(\tau)^{-1}$ -the $\prod(1-q^{n})^{-1}$ of Lecture 2 dressed in its regularized zero-point factor $q^{-1/24}$ , now read as a module character. The free fermion is a Clifford-module (super) VOA; its NS and R modules are exactly the spin-structure sectors of Lecture 4, with characters the $\theta/\eta$ blocks (4.9). The single modes of Lecture 1 have become the generators of these vertex algebras. The loop is closed.

Example 5.2 (Three characters, expanded).

The generalities are best absorbed through leading terms; for each of three modules we list its modular block $\chi=q^{\,h-c/24}\,\operatorname{ch}_{q}$ .

•

The Heisenberg VOA $V_{\mathfrak{h}}$ (the free boson, $c=1$ ; the name honours its underlying one-dimensional abelian algebra $\mathfrak{h}$ , of which it is the affinization):

$\chi_{V_{\mathfrak{h}}}=q^{-1/24}\prod_{n\geq 1}\frac{1}{1-q^{n}}=q^{-1/24}% \big{(}1+q+2q^{2}+3q^{3}+5q^{4}+\cdots\big{)},$

the partition numbers of (7): one state per way of exciting the modes $a_{-1},a_{-2},\dots$ of Lecture 2. Its conformal vector is $\omega=\tfrac{1}{2}\,a_{-1}^{\,2}\lvert 0\rangle$ , whose field is $T=\tfrac{1}{2}\,{:}\partial\phi\,\partial\phi{:}$ .
•

The Clifford (super) VOA $F$ (the free fermion, $c=\tfrac{1}{2}$ ), in its NS module $F_{\mathrm{NS}}$ :

$\chi_{F_{\mathrm{NS}}}=q^{-1/48}\prod_{n\geq 1}\big{(}1+q^{n-\frac{1}{2}}\big{% )}=q^{-1/48}\big{(}1+q^{\frac{1}{2}}+q^{\frac{3}{2}}+q^{2}+q^{\frac{5}{2}}+q^{% 3}+q^{\frac{7}{2}}+2q^{4}+\cdots\big{)},$

partitions into distinct half-odd-integers, each mode $\psi_{-r}$ usable once (Pauli again); note that grade $1$ is unreachable. This is the block $\sqrt{\theta_{3}/\eta}$ of (4.9). Here $\omega=\tfrac{1}{2}\,\psi_{-3/2}\psi_{-1/2}\lvert 0\rangle$ , whose field is $T=-\tfrac{1}{2}\,{:}\psi\,\partial\psi{:}$ .
•

$\widehat{\mathfrak{su}}(2)_{1}$ , vacuum module (the simplest WZW chiral algebra; $c=\tfrac{k\dim\mathfrak{g}}{k+h^{\vee}}=\tfrac{3}{1+2}=1$ ):

$\chi_{j=0}=q^{-1/24}\,\frac{\sum_{n\in\mathbb{Z}}q^{n^{2}}}{\prod_{n\geq 1}(1-% q^{n})}=q^{-1/24}\big{(}1+3q+4q^{2}+7q^{3}+13q^{4}+\cdots\big{)},$

a theta function on the $\mathfrak{su}(2)$ root lattice over the oscillator product-the string-function-times-theta shape of the Weyl–Kac formula, in the flesh. Read the coefficients as $\mathfrak{su}(2)$ representations: the $3$ at grade one is the adjoint, the current states $J^{a}_{-1}\lvert 0\rangle$ themselves; the $4$ at grade two is $\mathbf{3}\oplus\mathbf{1}$ -the spin- $2$ combination of two currents is a null state at level $1$ , integrability visibly pruning the spectrum. Restoring a Cartan fugacity $z^{J^{3}_{0}}$ would resolve each coefficient into its weights: the Cartan-variables box, in action. And lest the vacuum module float free of its theory: level $1$ has exactly two integrable spins, and the full WZW torus partition function of Lecture 4 glues them diagonally,

$Z_{\widehat{\mathfrak{su}}(2)_{1}}\;=\;\sum_{j\,=\,0,\,\frac{1}{2}}\chi_{j}\,% \overline{\chi_{j}},$

the sesquilinear form (4.13) with $N=\mathbf{1}$ -one full theory, assembled from its two chiral blocks: the cartoon (5.2), realized end to end. The conformal vector here is nobody new, just the currents squared: the Sugawara construction $\omega=\tfrac{1}{2(k+h^{\vee})}\sum_{a}J^{a}_{-1}J^{a}_{-1}\lvert 0\rangle=% \tfrac{1}{6}\sum_{a}\big{(}J^{a}_{-1}\big{)}^{2}\lvert 0\rangle$ -the origin of the central-charge formula $c=k\dim\mathfrak{g}/(k+h^{\vee})$ quoted above.

Marrying the first two examples pays a dividend: $V_{\mathfrak{h}}\otimes F$ carries, besides $\omega=\omega_{\mathrm{b}}+\omega_{\mathrm{f}}$ , the superconformal vector $G=a_{-1}\psi_{-1/2}\lvert 0\rangle$ , whose field is exactly the supercurrent $G=\psi\,\partial\phi$ of Lecture 4. That super-VOA is where §5.6 begins.

5.6 Contact with superconformal field theory

What is the sturdiest partition function of all? This closing section builds supersymmetry into the chiral algebra in two steps, and each step buys a more protected object: first an integer, the Witten index; then a Jacobi form, the elliptic genus.

Step one is to manifest, in VOA language, the superconformal symmetry met at the close of Lecture 4 (§4.4)-and it costs nothing we do not already own. Just as the conformal vector $\omega$ of §5.5 realizes Virasoro, a super-VOA distinguishes one more vector. In a theory with the right matter-the boson-plus-fermion system at $c=\tfrac{3}{2}$ -the state $G=\psi\,\partial\phi$ already sits in the weight- $\tfrac{3}{2}$ graded piece, counted by the partition function all along, just as the affine currents $J^{a}$ of §5.2 sit at weight $1$ . Promote it, from a state one merely counts to a generator one organizes by: the state–field correspondence returns exactly the supercurrent $G(z)$ of Lecture 4, its modes $G_{r}$ closing with the $L_{m}$ on the super-Virasoro algebra (4.11). In the OPE language of §5.5 the package reads: $T$ sees $G$ as a primary of weight $\tfrac{3}{2}$ , and $G$ multiplies with itself back into the stress tensor,

G(z)\,G(w)\;\sim\;\frac{2c/3}{(z-w)^{3}}+\frac{2\,T(w)}{z-w},

the pointwise form of (4.11)’s anticommutator-and the local seed of the square root of the Hamiltonian, coming next. Because $G$ is fermionic, the dichotomy of §3.2 applies to it: its moding runs over half-integers (NS) or integers (R)-the two spin structures, now internal to the chiral algebra-and the modules of the super-VOA come in NS and R families accordingly.

Thread (Promotion changes the bookkeeping, not the theory).

Promoting a current adds nothing: the states and $Z$ are untouched. What changes is the decomposition-the same spectrum reassembles into fewer, larger modules, and the cartoon (5.2) re-runs with a bigger algebra on the left. The rhyme with §5.2 is exact: promote weight- $1$ currents and get $\widehat{\mathfrak{g}}$ ; promote a weight- $\tfrac{3}{2}$ fermionic current and get super-Virasoro. A chiral algebra is specified by which of the theory’s currents one takes as generators-the “action” of §5.1, once more.

The two sectors are not symmetric: only the R sector, with its integer moding, contains a zero mode $G_{0}$ , and setting $r=s=0$ in the anticommutator of (4.11) gives

G_{0}^{\,2}\;=\;L_{0}-\frac{c}{24}.

(5.7)

The R sector owns a square root of the chiral Hamiltonian-built on the very $-c/24$ of our thread. Two consequences follow at once. First, the chiral energy $L_{0}-\tfrac{c}{24}$ is non-negative on a unitary R module: it is a square. Second, every state of positive chiral energy is paired: $G_{0}$ maps it to a partner of equal energy and opposite fermion parity, and since applying $G_{0}$ twice returns a nonzero multiple of the original state, neither partner can vanish. Only zero-energy states-those annihilated by $G_{0}$ -can sit alone.

Now take the sign-graded trace of Lecture 2 over an R module-the Witten index

\operatorname{Tr}_{R}\,(-1)^{F}\,q^{\,L_{0}-c/24}.

(5.8)

At every positive energy the paired states contribute with opposite signs and cancel, so the entire $q$ -series collapses to a single integer, independent of $\beta$ (equivalently of $\operatorname{Im}\tau$ ): the number of zero-energy ground states, counted with sign. This is exactly the “net count of unpaired ground states” promised by the vacuum-energy cancellation of Lecture 3, and it is rigid: under any continuous deformation preserving the algebra, states can leave or reach zero energy only in $G_{0}$ -pairs of opposite parity, so the signed count cannot move. A partition function has become a deformation invariant.

Geometrically we have met this object already. Inserting $(-1)^{F}$ makes the time circle periodic (§3.2), so (5.8) is the (R, R) torus amplitude-the fourth spin structure of Lecture 4, the one that evaluated to $\sqrt{\theta_{1}/\eta}\equiv 0$ . The free Majorana fermion shows why in miniature: its two R ground states carry opposite fermion parity, so they cancel and the index vanishes. $\theta_{1}\equiv 0$ , recomputed by counting.

A vanishing index is protected but not talkative. To refine it, promote once more. Suppose the chiral algebra can be enlarged by a weight- $1$ $U(1)$ current $J$ under which the supercurrent splits into oppositely charged halves $G^{\pm}$ : the $\mathcal{N}=2$ super-Virasoro algebra, its $U(1)$ the R-symmetry. (Free fields manage this with complex matter-a complex boson paired with a complex fermion, $c=3$ .) The zero mode $J_{0}$ is one more conserved charge, so the Interlude’s reflex fires: hand it a conjugate potential, the fugacity $y$ . The refined trace is the elliptic genus

\boxed{\;\mathcal{E}(\tau,y)=\operatorname{Tr}_{R}\,(-1)^{F}\,y^{J_{0}}\,q^{\,% L_{0}-c/24},\;}

(5.9)

and the trace need no longer collapse: states that cancelled in (5.8) by parity can carry different $J_{0}$ charges, so they now cancel only at $y=1$ -the fugacity revives what $\theta_{1}\equiv 0$ set to zero. What deformation now protects is a whole function, a weak Jacobi form in $(\tau,y)$ once the $U(1)$ charges are suitably integral-for a sigma model, a Calabi–Yau target $M$ . The name promises geometry, and delivers: $\mathcal{E}$ is then a genuine invariant of the smooth compact manifold $M$ . Like the Witten index it is protected against deformation, and it bears real information: special values of $y$ recover the Euler characteristic, the Hirzebruch signature and the $\hat{A}$ -genus of $M$ , which the $q$ -expansion binds into a single modular object. A partition function has become a manifold invariant-index theory and modular forms shaking hands. Every thread of these notes meets here: the graded trace (L2), the sign grading / spin structure (L3–L4), the modular transformation (L4–L5), and the chemical potential $y$ (the Interlude) all fuse into a single modular object, and the theta functions of Lecture 4 reappear as its super-characters.

For the curious (which supersymmetry, and a Euclidean subtlety).

The supersymmetry of this section is worldsheet supersymmetry-the two-dimensional super-Virasoro algebra, pairing a boson with a fermion on the worldsheet (the mode-by-mode cancellation of Lecture 3)-not the spacetime super-Poincaré algebra of particle physics: the index and the genus are statements about a 2d chiral algebra, whatever spacetime physics the model describes. The Euclidean framing adds its own subtlety. Where Lorentzian supersymmetry leans on the reality $\{Q,Q^{\dagger}\}=H\geq 0$ , the Euclidean theory lets the holomorphic and antiholomorphic supercharges go their separate ways, and it is that holomorphic structure-not positivity-that makes the index and the genus modular objects: Jacobi forms, not merely deformation-invariant numbers.

Rosetta notes

•

Character $\leftrightarrow$ partition function $\leftrightarrow$ graded dimension.
•

Level $k$ $\leftrightarrow$ central charge $c$ ; Kac–Peterson $S$ $\leftrightarrow$ CFT modular $S$ $\leftrightarrow$ high/low-temperature duality.
•

R-sector $(-1)^{F}$ trace $\leftrightarrow$ the Witten index $\leftrightarrow$ a supersymmetric partition function $\leftrightarrow$ the elliptic genus.