7 Sugawara construction

7.1 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 30 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 31 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 32 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 30 that \({\mathbb {:} \mathsf{J}_{k} \, \mathsf{J}_{l} \mathbb {:}} \, v = 0\) when \(\max \{ k,l\} \ge N\).

Lemma 33 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 32.

Definition 34 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 33).

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

Lemma 35 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 33, 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 36 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 35 and the commutator formula \([A,BC] = B[A,C] + [A,B]C\).

Lemma 37 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 35, 31, and 36, the commutation relations (HeiComm), and the commutator formula \([A,BC] = B[A,C] + [A,B]C\) again.

Lemma 38 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 39 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 37 and 38, among other observations.

Theorem 40 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 39 and a comparison with the Lie brackets in the basis of Definition 25.