8 Sugawara construction

8.1 The basic bosonic Sugawara construction

Throughout this section, let \(\mathbb {K}\) be a field of characteristic zero.

If a vector space \(V\) has a representation of the Heisenberg algebra on a vector space \(V\), where the central element \(K\) (see Definition 29), acts as \(\mathrm{id}_V\), then the basis elements \((J_k)_{k \in \mathbb {Z}}\) (see Definition 29) are linear operators \(\mathsf{J}_k \colon V \to V\) satisfying the commutation relations

\begin{align*} \mathrm{\textrm{(HeiComm)}} \qquad [\mathsf{J}_k, \mathsf{J}_l] \; = \; \mathsf{J}_k \circ \mathsf{J}_l - \mathsf{J}_l \circ \mathsf{J}_k \; = \; k \, \delta _{k+l,0} \; \mathrm{id}_V . \end{align*}

Below we will assume such operators being fixed, and satisfying furthermore the local truncation condition on \(V\): for any fixed \(v \in V\) we have \(\mathsf{J}_k \, v = 0\) for \(k \gg 0\), i.e.,

\begin{align*} \mathrm{\textrm{(HeiTrunc)}} \qquad \forall v \in V , \; \exists N, \; \forall k \ge N, \quad \mathsf{J}_k \, v = 0 . \end{align*}

Definition 37 Normal ordering

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For \(k,l \in \mathbb {Z}\), we denote the normal ordered product of the operators \(\mathsf{J}_k\) and \(\mathsf{J}_l\) by

\begin{align*} {\mathbb {:} \mathsf{J}_k \, \mathsf{J}_l \mathbb {:}} \; := \; \begin{cases} \mathsf{J}_k \circ \mathsf{J}_l & \text{ if } k \le l \\ \mathsf{J}_l \circ \mathsf{J}_k & \text{ if } k \, {\gt} \, l . \end{cases}\end{align*}

Lemma 38 Alternative normal ordering

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm). Then for any \(k,l \in \mathbb {Z}\) we have

\begin{align*} {\mathbb {:} \mathsf{J}_k \, \mathsf{J}_l \mathbb {:}} \; = \; \begin{cases} \mathsf{J}_k \circ \mathsf{J}_l & \text{ if } k {\lt} 0 \\ \mathsf{J}_l \circ \mathsf{J}_k & \text{ if } k \ge 0 . \end{cases}\end{align*}

Proof ▶

Straightforward using the commutation relations (HeiComm).

Lemma 39 Local truncation for normal ordered products

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc). Then for any \(v \in V\) there exists an \(N\) such that whenever \(\max \{ k,l\} \ge N\) we have \({\mathbb {:} \mathsf{J}_{k} \, \mathsf{J}_{l} \mathbb {:}} \, v = 0\).

Proof ▶

Fixing \(v \in V\), the local truncation condition (HeiTrunc) gives the existence of an \(N\) such that \(\mathsf{J}_{k} \, v = 0\) for \(k \ge N\). It is then clear by inspection of Definition 37 that \({\mathbb {:} \mathsf{J}_{k} \, \mathsf{J}_{l} \mathbb {:}} \, v = 0\) when \(\max \{ k,l\} \ge N\).

Lemma 40 Local finite support for homogeneous normal ordered products

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc). Then for any \(n \in \mathbb {Z}\) and any \(v \in V\), there are only finitely many \(k \in \mathbb {Z}\) such that \({\mathbb {:} \mathsf{J}_{n-k} \, \mathsf{J}_k \mathbb {:}} \, v \ne 0\).

Proof ▶

Straightforward from Lemma 39.

Definition 41 Sugawara operators

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc). Then for any \(n \in \mathbb {Z}\), a linear operator

\begin{align*} \mathsf{L}_n \colon V \to V \end{align*}

can be defined by the formula

\begin{align*} \mathsf{L}_n \, v := \frac{1}{2} \sum _{k \in \mathbb {Z}} {\mathbb {:} \mathsf{J}_{n-k} \, \mathsf{J}_k \mathbb {:}} \, v \qquad \text{ for } v \in V \end{align*}

(the sum only has finitely many terms by Lemma 40).

We call the operators \((\mathsf{L}_n)_{n \in \mathbb {Z}}\) the Sugawara operators.

Lemma 42 Commutators of Sugawara operators as series

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc), and suppose that \(\mathsf{A} \colon V \to V\) is a linear operator. Then for any \(n \in \mathbb {Z}\), the action of the commutator \([\mathsf{L}_n, \mathsf{A}]\) on any \(v \in V\) is given by the series

\begin{align*} [\mathsf{L}_n, \mathsf{A}] \, v = \frac{1}{2} \sum _{k \in \mathbb {Z}} [{\mathbb {:} \mathsf{J}_{n-k} \, \mathsf{J}_k \mathbb {:}}, \mathsf{A}] \, v \end{align*}

where only finitely many of the terms are nonzero.

Proof ▶

Write

\begin{align*} [\mathsf{L}_n, A] \, v = \; & \mathsf{L}_n \, A \, v - A \, \mathsf{L}_n \, v \\ = \; & \frac{1}{2} \sum _{k \in \mathbb {Z}} {\mathbb {:} \mathsf{J}_{n-k} \, \mathsf{J}_k \mathbb {:}} \, \mathsf{A} \, v - \frac{1}{2} \mathsf{A} \sum _{k \in \mathbb {Z}} {\mathbb {:} \mathsf{J}_{n-k} \, \mathsf{J}_k \mathbb {:}} \, v . \end{align*}

By Lemma 40, only finitely many of the terms in both sums are nonzero and they may be rearranged to the asserted form of sum of commutators. The resulting sum only has finitely many nonzero terms and is therefore well-defined.

Lemma 43 Commutator of Sugawara operators with Heisenberg operators

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm) and the local truncation condition (HeiTrunc). Then for any \(n \in \mathbb {Z}\) and \(k \in \mathbb {Z}\), we have

\begin{align*} [\mathsf{L}_n, \mathsf{J}_k] \; = \; - k \, \mathsf{J}_{n+k} . \end{align*}

Proof ▶

Calculation, using Lemma 42 and the commutator formula \([A,BC] = B[A,C] + [A,B]C\).

Lemma 44 Commutator of Sugawara operators with normal ordered pairs

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\begin{align*} [\mathsf{L}_n, {\mathbb {:} \mathsf{J}_k \, \mathsf{J}_{m-k} \mathbb {:}}] = \; & -k \, {\mathbb {:} \mathsf{J}_{n+k} \, \mathsf{J}_{m-k} \mathbb {:}} - (m-k) \, {\mathbb {:} \mathsf{J}_{k} \, \mathsf{J}_{n+m-k} \mathbb {:}} \\ & \; + (n+k) \, \delta _{n+m,0} \Big( \mathbb {I}_{{-n \le k {\lt} 0}} - \mathbb {I}_{{0 \le k {\lt} - n}} \Big) \, \mathrm{id}_V . \end{align*}

where \(\mathbb {I}_{{\mathrm{condition}}}\) is defined as \(1\) if the condition is true and \(0\) otherwise.

Proof ▶

Calculation, using Lemmas 42, 38, and 43, the commutation relations (HeiComm), and the commutator formula \([A,BC] = B[A,C] + [A,B]C\) again.

Lemma 45 Auxiliary calculation

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For any \(n \in \mathbb {N}\), we have

\begin{align*} \sum _{l=0}^{n-1} (n-l) l = \frac{n^3 - n}{6} . \end{align*}

Proof ▶

Calculation (with induction).

Lemma 46 Virasoro commutation relations for Sugawara operators

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm) and the local truncation condition (HeiTrunc). Then for any \(n, m \in \mathbb {Z}\), we have

\begin{align*} [\mathsf{L}_n, \mathsf{L}_m] = \; & (n-m) \, \mathsf{L}_{n+m} + \delta _{n+m,0} \frac{n^3 - n}{12} \, \mathrm{id}_V . \end{align*}

Proof ▶

Calculation, using Lemmas 44 and 45, among other observations.

Theorem 47 Sugawara construction

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Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm) and the local truncation condition (HeiTrunc). Then there exists a representation of the Virasoro algebra \(\mathfrak {vir}\) with central charge \(c = 1\) on \(V\) (i.e., the central element \(C \in \mathfrak {vir}\) acts as \(c \, \mathrm{id}_V\) with \(c = 1\)) where the basis elements \(L_n\) of \(\mathfrak {vir}\) act by the Sugawara operators \((\mathsf{L}_n)_{n \in \mathbb {Z}}\).

Proof ▶

A direct consequence of the commutation relations in Lemma 46 and a comparison with the Lie brackets in the basis of Definition 25.

8.2 Charged Fock spaces (Heisenberg Verma modules)

Definition 48 Triangular decomposition of the Heisenberg algebra

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A triangular decomposition

\begin{align*} \mathfrak {hei}= \mathfrak {hei}^0 \oplus \mathfrak {hei}^+ \oplus \mathfrak {hei}^- \end{align*}

of \(\mathfrak {hei}\) is defined so that \(\mathfrak {hei}^0\) is spanned by \(J_0, K \in \mathfrak {hei}\), \(\mathfrak {hei}^+\) is spanned by \(J_k\) for \(k {\gt} 0\), and \(\mathfrak {hei}^-\) is spanned by \(J_k\) for \(k {\lt} 0\).

(Without further comment, for the Heisenberg algebra we always use this triangular decomposition.)

Definition 49 Charged Fock space (Heisenberg Verma module)

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Let \(\alpha \in \mathbb {K}\). The charged Fock space of charge \(\alpha \) is the Verma module \(\mathscr {V}^{\eta }\) associated to the linear functional \(\eta \colon \mathfrak {hei}^0 \to \mathbb {K}\) with

\begin{align*} \eta (J_0) = \alpha , \quad \eta (K) = 1 . \end{align*}

We denote the charged Fock space by \(\mathscr {F}^{\alpha }\). The highest weight vector \(\mathbb {v}^{\eta }\) is (also) called the vacuum vector of the charged Fock space, and we denote it by \(\mathbb {v}^{\alpha }\).

Lemma 50 Eventual commutation in the Heisenberg universal enveloping algebra

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Let \(a \in \mathscr {U}(\mathfrak {hei})\) be an arbitrary element of the universal enveloping algebra of the Heisenberg algebra \(\mathfrak {hei}\). Then there exists a \(k_0=k_0(a) \in \mathbb {Z}\) such that for all \(k \ge k_0\) we have

\begin{align*} J_k a = a J_k . \end{align*}

Proof ▶

Clear by the induction principle of the universal enveloping algebra and Lie brackets of the Heisenberg algebra basis elements.

Lemma 51 Local truncation condition for the charged Fock space

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Let \(\alpha \in \mathbb {K}\) and let \(v \in \mathscr {F}^{\alpha }\) be a vector in the charged Fock space with charge \(\alpha \). Then there exists a \(k_0 \in \mathbb {Z}\) such that for all \(k \ge k_0\) we have

\begin{align*} J_k \, v = 0 . \end{align*}

(In other words, the charged Fock space satisfies the local truncation condition (HeiTrunc) needed for the Sugawara construction.)

Proof ▶

Straightforward by Lemma 50 and the properties of the highest weight vector (cyclicity and annihilation by the upper part).

Definition 52 Virasoro action on the charged Fock space

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The charged Fock space of charge \(\alpha \) becomes a representation of the Virasoro algebra \(\mathfrak {vir}\) with central charge \(c = 1\) via the Sugawara construction (Theorem 47).

Lemma 53 The vacuum of the charged Fock space is a highest weight vector

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The vacuum vector \(\mathbb {v}^{\alpha }\) of the charged Fock space of charge \(\alpha \in \mathbb {K}\) satisfies

\begin{align*} L_0 \, \mathbb {v}^{\alpha } = \frac{\alpha ^2}{2} \, \mathbb {v}^{\alpha } \qquad \text{and} \qquad L_n \, \mathbb {v}^{\alpha } = 0 \text{ for all } n {\gt} 0 . \end{align*}

In particular (by the universal property of Verma modules) there exists a Virasoro-module map

\begin{align*} \mathscr {V}^{c=1,h=\alpha ^2/2} \to \mathscr {F}^{\alpha } \end{align*}

such that \(\mathbb {v}^{c=1,h=\alpha ^2/2} \mapsto \mathbb {v}^{\alpha }\).

Proof ▶

Calculation.