The geometric central extension of the loop group LG

Intuition Beginner

A loop in a group $G$ is a way of assigning a group element to every point of a circle, smoothly. Two such loops can be multiplied point by point, so the loops themselves form a group, written $LG$. This is the loop group, and it is enormous: it has infinitely many independent directions of motion.

Here is the surprise. When you try to keep careful track of how loops combine, the bookkeeping does not quite close. Combining two spinning loops leaves behind a phase, a circular dial reading, that the naive product forgets.

To make the algebra honest, you add that dial as a genuine new coordinate. The enlarged group is the central extension. Its extra circle commutes with every loop, so it never changes which loop you see; it only remembers the phase.

Visual Beginner

Picture the loop group as a vast surface of all possible loops. Above each loop, attach a small circle of phases, like a dial sitting over every point. As you slide one loop past another and multiply, the dials turn by an amount set by how the loops wind around each other.

The whole picture is a circle bundle over the loop group. Projecting down forgets the dial and returns the plain loop. The remarkable fact is that the rule for turning the dials is fixed up to one whole number: the level. You may turn the dial once, twice, three times, but never a fractional amount. The phase is quantized.

Worked example Beginner

Take $G$ to be the rotation group of a sphere in three dimensions and look at loops inside it. A loop is a circle's worth of rotations, one rotation for each angle on $S^1$.

Combine two such loops by rotating at each angle separately. The plain product is again a loop. But the geometric data of how the first loop drags the second around contributes a winding count, and that count feeds the dial.

The number that controls how fast the dial turns is the level. At level one the dial makes one full turn for the basic winding; at level two it turns twice as fast. There is no level one-half. What this tells us: the allowed phase rules are labelled by a single integer, and that integer is the only choice you get to make.

Check your understanding Beginner

Formal definition Intermediate+

Let $G$ be a compact, simple, simply-connected Lie group with Lie algebra $\mathfrak{g}$, and fix the basic invariant inner product $⟨\cdot ,\cdot ⟩$ on $\mathfrak{g}$ normalised so that the longest coroot has length squared $2$. The loop group is

$$LG=C^{\infty }(S^1,G),$$

a group under pointwise multiplication. It is an infinite-dimensional Fréchet Lie group whose Lie algebra is the loop algebra $L\mathfrak{g}=C^{\infty }(S^1,\mathfrak{g})$ with pointwise bracket ^{[Pressley-Segal §4]}.

A central extension of $LG$ by the circle $U(1)$ is a short exact sequence of Lie groups

$$1\to U(1)\to \widetilde{LG}\to LG\to 1$$

in which the image of $U(1)$ lies in the centre of $\widetilde{LG}$. The extension is smooth when $\widetilde{LG}\to LG$ is a principal $U(1)$-bundle and multiplication is smooth. The associated complex line bundle $\mathcal{L}\to LG$ carries a multiplicative structure: a coherent family of isomorphisms ${\mathcal{L}}_{\gamma }\otimes {\mathcal{L}}_{{\gamma }^{\prime }}\cong {\mathcal{L}}_{\gamma {\gamma }^{\prime }}$ realising the group law on total spaces.

The generator of $H^3(G,\mathbb{Z})\cong \mathbb{Z}$ is represented by the level-one WZW gerbe of 03.06.09, whose Dixmier-Douady class is the basic generator. Multiplying by an integer $k$ gives the level-$k$ class. Transgression 03.11.04 sends a class in $H^3(G,\mathbb{Z})$ to a class in $H^2(LG,\mathbb{Z})$, the isomorphism class of a line bundle on $LG$; the level-$k$ gerbe transgresses to the line bundle of the level-$k$ central extension ${\widetilde{LG}}_k$.

Key theorem with proof Intermediate+

Theorem (Lie-algebra shadow of the geometric extension). The smooth central extension ${\widetilde{LG}}_k$ has Lie algebra ${\widetilde{L\mathfrak{g}}}_k=L\mathfrak{g}\oplus \mathbb{R}c$, with $c$ central and bracket

$$[(\xi ,ac),(\eta ,bc)]=([\xi ,\eta {]}_{L\mathfrak{g}},\omega (\xi ,\eta )c),\omega (\xi ,\eta )=\frac{k}{2\pi }{\int }_{S^1}⟨\xi (\theta ),{\eta }^{\prime }(\theta )⟩d\theta .$$

This $\omega$ is the level-$k$ Pressley-Segal / Kac-Moody affine cocycle, and it is the curvature, integrated against the loop, of the transgressed line bundle.

Proof. Differentiating the principal $U(1)$-bundle ${\widetilde{LG}}_k\to LG$ at the identity produces a one-dimensional central extension of $L\mathfrak{g}$, so the bracket has the displayed shape for some alternating bilinear $\omega :L\mathfrak{g}\times L\mathfrak{g}\to \mathbb{R}$. By 03.11.01 the bracket satisfies the Jacobi identity exactly when $\omega$ is a 2-cocycle. We verify $\omega$ is alternating: integration by parts on the circle gives ${\int }_{S^1}⟨\xi ,{\eta }^{\prime }⟩d\theta =-{\int }_{S^1}⟨{\xi }^{\prime },\eta ⟩d\theta$, so $\omega (\xi ,\eta )=-\omega (\eta ,\xi )$. The cocycle condition

$$\omega ([\xi ,\eta ],\zeta )+\omega ([\eta ,\zeta ],\xi )+\omega ([\zeta ,\xi ],\eta )=0$$

reduces, after integrating by parts and using $\mathrm{ad}$-invariance $⟨[\xi ,\eta ],\zeta ⟩=⟨\xi ,[\eta ,\zeta ]⟩$, to a cyclic sum of $⟨\xi ,[\eta ,\zeta {]}^{\prime }⟩$-type terms that cancel in pairs. The level appears because the transgression of the level-$k$ gerbe scales the curvature two-form linearly in $k$, and the integral pairing with the loop returns the prefactor $k/2\pi$ ^{[Brylinski Ch. 7]}. The factor $1/2\pi$ normalises the period of the dial so that the central $U(1)$ closes up after one turn.

Bridge. This builds toward the positive-energy representation theory of ${\widetilde{LG}}_k$ and appears again in 03.10.03, where the same level-$k$ line bundle is the geometric carrier of the WZW action. The foundational reason the level is an integer is that $\omega$ is the de Rham shadow of an integral cohomology class on $G$: the central insight is that the algebraic 2-cocycle of 03.11.01 is exactly the curvature of the transgressed bundle of 03.11.04, so the group-level quantization $k\in \mathbb{Z}$ generalises the bare alternating form into honest geometry. Putting these together, the bridge is that integrality on $G$ forces integrality of the level on $LG$.

Exercises Intermediate+

Advanced results Master

Positive-energy representations. The smooth central extension ${\widetilde{LG}}_k$ is the object whose unitary representations of positive energy realise the level-$k$ integrable highest-weight representations of the affine Kac-Moody algebra $\hat{\mathfrak{g}}$ 03.11.02. Positive energy means the rotation group $\mathrm{Rot}(S^1)$ acting on loops is implemented by an operator with spectrum bounded below; such representations exist only for the centrally extended group, never for $LG$ itself, because the rotation action is anomalous at the bare level. The level $k$ must be a nonnegative integer for a nonzero positive-energy representation to exist, and the finite list of highest weights at fixed level is the fusion data of the corresponding rational conformal field theory.

Pressley-Segal classification. For $G$ compact, simple, and simply-connected, the smooth central extensions of $LG$ by $U(1)$ are classified up to isomorphism by the level $k\in \mathbb{Z}\cong H^3(G,\mathbb{Z})$. The basic extension is the generator $k=1$; every other arises by raising the basic line bundle to the tensor power $k$. Simple-connectedness of $G$ removes ${\pi }_1(G)$-torsion contributions and makes the transgression map an isomorphism onto the relevant cohomology, so the integer is a complete invariant ^{[Pressley-Segal §4]}.

Synthesis. The level is the single integer organising the whole structure, and the central insight is that it lives simultaneously in three guises: as the generator count of $H^3(G,\mathbb{Z})$ on the group 03.06.09, as the curvature normalisation of the transgressed line bundle on the loop space 03.11.04, and as the Lie-algebra cocycle scale of 03.11.01. This is exactly the statement that the geometric extension is the integral lift of the algebraic one; the foundational reason positive-energy representations require the extension is that the bare rotation action is anomalous, and the central circle absorbs that anomaly. Putting these together, the Pressley-Segal classification generalises the cohomological count $H^2(\mathfrak{g};K)$ of the finite-dimensional theory to the loop setting, and the bridge is that integrality of the source class on $G$ is dual to quantization of the level on $LG$.

Full proof set Master

Proposition (uniqueness of the level-one cocycle up to scale). For $G$ compact, simple, and simply-connected, the space of continuous Lie-algebra 2-cocycles on $L\mathfrak{g}$ modulo coboundaries is one-dimensional, spanned by ${\omega }_1(\xi ,\eta )=\frac{1}{2\pi }{\int }_{S^1}⟨\xi ,{\eta }^{\prime }⟩d\theta$.

Proof. Expand loops in Fourier modes $\xi ={\sum }_n{\xi }_n{e}^{in\theta }$ with ${\xi }_n\in {\mathfrak{g}}_{\mathbb{C}}$. Simplicity of $\mathfrak{g}$ makes the invariant form $⟨\cdot ,\cdot ⟩$ unique up to scale, and a continuous invariant alternating bilinear form on $L\mathfrak{g}$ must be built from it. Invariance under the rotation $\theta \mapsto \theta +t$ forces the cocycle to pair mode $n$ with mode $-n$ only. The cocycle identity then constrains the $n$-dependence to be linear, giving $\omega ({\xi }_m^a,{\xi }_n^b)\propto m⟨\cdot ,\cdot ⟩{\delta }_{m+n,0}$, which is precisely ${\omega }_1$ after summation. Any coboundary $\lambda ([\xi ,\eta ])$ pairs mode $n$ with mode $-n$ with a constant in $n$, hence cannot reproduce the linear-in-$n$ part. Therefore the class of ${\omega }_1$ spans the quotient, and every cocycle is a real multiple of it. $\square$

Proposition (integrality of the level). The real multiple realised by a smooth central extension is an integer.

Proof. A smooth central extension is a principal $U(1)$-bundle whose Chern class lies in $H^2(LG,\mathbb{Z})$. The transgression map carries $H^3(G,\mathbb{Z})=\mathbb{Z}$ onto the line spanned by the integral lift of $[{\omega }_1]$, sending the generator to the level-one class. The curvature of an integral class integrates to $2\pi$ times an integer over any two-cycle; running this through the loop pairing returns the prefactor $k/2\pi$ with $k\in \mathbb{Z}$. Thus the scale of the cocycle is pinned to integer multiples of ${\omega }_1$. $\square$

Connections Master

• Transgression of gerbes to loop space 03.11.04 — the level-$k$ WZW gerbe on $G$ transgresses to the line bundle of ${\widetilde{LG}}_k$; the geometric extension is literally the total space of that transgressed bundle, and its curvature is the affine cocycle.

• Central extension of a Lie algebra 03.11.01 — the Pressley-Segal cocycle $\omega (\xi ,\eta )=\frac{k}{2\pi }{\int }_{S^1}⟨\xi ,{\eta }^{\prime }⟩d\theta$ is the Lie-algebra shadow of the group extension, fitting the abstract 2-cocycle template of that unit with the level as the scale.

• Dixmier-Douady class and $H^3(M,\mathbb{Z})$ 03.06.09 — the generator of $H^3(G,\mathbb{Z})\cong \mathbb{Z}$ is the level-one gerbe whose integer multiples set the level; integrality there forces quantization of the extension here.

• Infinite-dimensional Lie algebra representations 03.11.02 — the positive-energy representations of ${\widetilde{LG}}_k$ are exactly the integrable level-$k$ highest-weight modules of the affine Kac-Moody algebra, which exist only after the extension.

• WZW action 03.10.03 — the same level-$k$ line bundle is the geometric carrier of the Wess-Zumino-Witten action, where the integer level returns as the coefficient of the topological term.

Historical & philosophical context Master

The realisation that loop groups carry canonical central extensions, and that those extensions govern an entire representation theory, was crystallised by Pressley and Segal, who connected the analytic theory of positive-energy representations to the topology of $G$ ^{[Pressley-Segal §4]}. On the algebraic side the same extensions had appeared as Kac-Moody affine algebras, where the level is a representation-theoretic label. Brylinski's later geometric synthesis recast the construction through bundle gerbes and transgression, showing that the extension is not an algebraic accident but the loop-space shadow of a degree-three integral class on the group ^{[Brylinski Ch. 7]}.

The philosophical content is that an anomaly, the failure of a naive symmetry to act honestly, is the visible face of a hidden integer of topology. The loop group cannot act on a quantum theory without a phase ambiguity, and the only way to resolve the ambiguity consistently is to commit to one whole-number level. Quantization of the level is therefore not imposed by hand: it is the statement that a cohomology class on a compact group takes integer values ^{[Kac Ch. 7]}.

Bibliography Master

• Pressley, A. and Segal, G., Loop Groups, Oxford University Press, 1986. §4 (central extensions and their classification).
• Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser, 1993. Ch. 7 (transgression and the geometric central extension of $LG$).
• Kac, V. G., Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, 1990. Ch. 7 (affine algebras and the level).
• Segal, G., "Unitary representations of some infinite-dimensional groups", Communications in Mathematical Physics 80 (1981), 301-342.

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Infinite-dimensional Lie strand: the group-level lift of the algebraic central extension, supplying the geometry behind affine Kac-Moody representation theory and the WZW model.