The free loop space LM and transgression

Intuition Beginner

Pick a shape — a sphere, a doughnut, any smooth surface in space. Now think about every possible loop you could draw on it: every closed string that starts somewhere, wanders around, and returns to its start. The collection of all those loops is itself a kind of shape. It is enormous, because each point of it is a whole loop, but it bends and connects like any other space. This collection is called the free loop space.

Loops can be nudged. Slide a loop a little, and you get a nearby loop. So the loop space has its own notion of "nearby," its own directions to move in, its own geometry. A direction at one loop is a recipe that says how to push each point of that loop.

Visual Beginner

Think of one fat dot in the loop space. That single dot is not a point in the original shape — it stands for one entire loop drawn on the shape. Move the dot a little, and the whole loop wiggles. The whole picture of loops, all sitting together, forms the big new space.

There is also a turning move. Rotate where each loop "starts" around the circle, and you get the same loop relabelled. This spinning is a symmetry that lives on the loop space and acts on every loop at once.

Worked example Beginner

Take the flat plane as the base shape, and take one specific loop: the unit circle, traced once. A direction to move this loop is an arrow attached to each point of the circle. Suppose at every point the arrow pushes straight outward, away from the center. Following that direction, the circle grows: it becomes a bigger circle.

Now suppose instead the arrows all point the same way, say to the right. Following that direction, the whole circle slides to the right without changing size. Both are single directions in the loop space, and each turns one loop into a nearby loop.

What this tells us: a "point" of the loop space is a loop, and a "direction" there is a field of little arrows along that loop telling each part of it where to go.

Check your understanding Beginner

Formal definition Intermediate+

Let $M$ be a smooth finite-dimensional manifold. The free loop space is the set of smooth maps from the circle into $M$,

$$LM=C^{\infty }(S^1,M),$$

with no basepoint condition imposed. It carries the structure of a Fréchet manifold modelled on Fréchet spaces of smooth sections ^{[Brylinski Ch. 3]}. A chart around a loop $\gamma \in LM$ is built from the pullback bundle ${\gamma }^{\ast }TM\to S^1$: an exponential map sends a section $X\in \Gamma (S^1,{\gamma }^{\ast }TM)$ to the loop $\theta \mapsto {\exp }_{\gamma (\theta )}(X(\theta ))$, a diffeomorphism from a neighbourhood of the zero section onto a neighbourhood of $\gamma$. The model space $\Gamma (S^1,{\gamma }^{\ast }TM)$ is a Fréchet space under the seminorms of uniform convergence of all derivatives.

The tangent space at $\gamma$ is exactly this section space,

$$T_{\gamma }LM=\Gamma (S^1,{\gamma }^{\ast }TM),$$

a vector field along the loop. The circle $S^1$ acts smoothly on $LM$ by loop rotation, $(s\cdot \gamma )(\theta )=\gamma (\theta +s)$. The evaluation map

$$\mathrm{ev}:S^1\times LM\to M,\mathrm{ev}(\theta ,\gamma )=\gamma (\theta ),$$

is smooth; it is the universal device for comparing geometry on $M$ with geometry on $LM$.

A non-example: the based loop space $\Omega M=\{\gamma :\gamma (0)=m_0\}$ is the fibre of the evaluation $LM\to M$, $\gamma \mapsto \gamma (0)$, and carries no loop-rotation action because rotation moves the basepoint.

Key theorem with proof Intermediate+

Definition (transgression). Fibre integration over the circle factor of $S^1\times LM$ defines, for each $k$, a linear map

$$\tau ={\int }_{S^1}{\mathrm{ev}}^{\ast }:{\Omega }^{k+1}(M)\to {\Omega }^k(LM),\tau (\alpha )={\int }_{S^1}{\mathrm{ev}}^{\ast }\alpha ,$$

where ${\int }_{S^1}$ integrates out the $S^1$-direction of the pulled-back form, leaving a form on $LM$ of degree one lower ^{[Brylinski Ch. 3]}.

Theorem (transgression is a chain map of degree $-1$). For every $\alpha \in {\Omega }^{k+1}(M)$,

$$d_{LM}\tau (\alpha )=\tau (d_M\alpha ).$$

Hence $\tau$ descends to a homomorphism $H^{k+1}(M)\to H^k(LM)$, and a closed form transgresses to a closed form.

Proof. Write $\omega ={\mathrm{ev}}^{\ast }\alpha \in {\Omega }^{k+1}(S^1\times LM)$. Decompose any form on the product by its number of $d\theta$-legs: $\omega =d\theta \wedge {\omega }_{(1)}+{\omega }_{(0)}$, where ${\omega }_{(1)}$ and ${\omega }_{(0)}$ contain no $d\theta$. Fibre integration retains only the leg along $S^1$:

$${\int }_{S^1}\omega ={\int }_{S^1}d\theta {\omega }_{(1)}=:\tau (\alpha ),$$

a form on $LM$ of degree $k$. The total differential splits as $d=d\theta \wedge {\partial }_{\theta }+d_{LM}$. Apply it to $\omega$ and project onto terms with exactly one $d\theta$-leg whose remaining legs lie along $LM$. The $d_{LM}$ acting on ${\omega }_{(0)}$ contributes no $d\theta$-leg and drops out of the integral; the term $d\theta \wedge {\partial }_{\theta }{\omega }_{(0)}$ integrates to zero because ${\partial }_{\theta }{\omega }_{(0)}$ is an exact $\theta$-derivative integrated over the closed circle. What survives is

$${\int }_{S^1}(d\omega )=-d_{LM}{\int }_{S^1}\omega +{\int }_{S^1}d\theta \wedge {\partial }_{\theta }(\cdots ),$$

and the boundaryless circle kills every total $\theta$-derivative by the fundamental theorem of calculus on $S^1$. Collecting signs from commuting $d_{LM}$ past the single $d\theta$, one obtains ${\int }_{S^1}{d}_M({\mathrm{ev}}^{\ast }\alpha )=d_{LM}{\int }_{S^1}{\mathrm{ev}}^{\ast }\alpha$. Since ${\mathrm{ev}}^{\ast }$ commutes with $d$, the left side is $\tau (d_M\alpha )$ and the right side is $d_{LM}\tau (\alpha )$. $\square$

Bridge. This fibre-integration identity builds toward 03.11.05 (the loop group extension), where transgression of an invariant form on a Lie group produces the cocycle of the affine central extension, and it appears again in 03.07.16 (the $B$-field as a gerbe connection), where the same circle integration carries a gerbe on $M$ to a line bundle on $LM$. The foundational reason a degree-$(k+1)$ form on $M$ yields a degree-$k$ form on $LM$ is exactly the loss of one degree to the integrated circle direction; this is dual to the way a path integral trades a bulk field for a boundary phase. Putting these together, the central insight is that transgression is the geometric shadow of the $S^1$-action on $LM$, and the bridge is the evaluation map $\mathrm{ev}$ that couples the two spaces.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has neither a Fréchet-manifold layer nor the smooth structure on $C^{\infty }(S^1,M)$ via charts from $\Gamma (S^1,{\gamma }^{\ast }TM)$.

A formalization would build the model Fréchet spaces of smooth sections, the exponential charts around a loop, fibre integration ${\int }_{S^1}{\mathrm{ev}}^{\ast }$, the chain-map property $d\tau =\tau d$, and the induced map on de Rham cohomology. Closedness of the transgressed 2-form would then be a corollary of the chain-map lemma applied to a closed 3-form.

Advanced results Master

Fix a closed 3-form $H\in {\Omega }^3(M)$ with integral periods. Its transgression ${\omega }_H=\tau (H)\in {\Omega }^2(LM)$ is a closed 2-form on the loop space, with pointwise formula

$$({\omega }_H{)}_{\gamma }(X,Y)={\int }_{S^1}{H}_{\gamma (\theta )}(\dot{\gamma }(\theta ),X(\theta ),Y(\theta ))d\theta ,X,Y\in \Gamma ({\gamma }^{\ast }TM).$$

The single $d\theta$-leg of ${\mathrm{ev}}^{\ast }H$ feeds the loop velocity $\dot{\gamma }$ into the first slot of $H$, and fibre integration over the circle assembles the result.

This form is the source of Brylinski's symplectic structure on a space of knots. Let $M$ be an oriented Riemannian 3-manifold and $H={\mathrm{vol}}_M$ its volume form, which is closed. On the space $\mathcal{K}$ of unparametrised oriented embedded loops (knots), the reparametrisation directions $X=f\dot{\gamma }$ lie in the kernel of ${\omega }_H$, because $H(\dot{\gamma },\dot{\gamma },\cdot )=0$ by antisymmetry. Passing to the quotient by reparametrisation removes precisely this degeneracy, and ${\omega }_H$ descends to a closed, weakly non-degenerate 2-form on $\mathcal{K}$: the space of knots is symplectic 05.01.02, with symplectic form

$${\omega }_{\mathcal{K}}(X,Y)={\int }_{S^1}{\mathrm{vol}}_M(\dot{\gamma },X,Y)d\theta .$$

The flow of a natural Hamiltonian on $\mathcal{K}$ recovers the localised induction (binormal) evolution of vortex filaments, tying the construction to the Euler equations of an ideal fluid ^{[Brylinski Ch. 3]}.

The same machine, applied to $M=G$ a compact simple Lie group with $H$ the bi-invariant Cartan 3-form, transgresses to the 2-cocycle that defines the central extension of the loop group $LG$ 03.11.01, placing transgression at the root of the affine theory ^{[Pressley-Segal Ch. 3]}. The $S^1$-equivariant refinement — where one tracks the loop-rotation action and works in equivariant cohomology — sharpens transgression into a map of $S^1$-equivariant theories, the homotopy-theoretic shadow of which is the cyclic-homology picture of $LM$.

Synthesis. Transgression is the foundational reason that codimension-one geometric data on $M$ becomes geometric data of one degree lower on its loop space: this is exactly the statement that fibre integration ${\int }_{S^1}{\mathrm{ev}}^{\ast }$ is a chain map, and it generalises the elementary fact that integrating out one variable lowers form degree. The central insight is that a closed 3-form on $M$ becomes a closed 2-form on $LM$, and putting these together with weak non-degeneracy on the knot quotient $\mathcal{K}$ makes the space of unparametrised knots symplectic 05.01.02; this is dual to the line-bundle picture, where a gerbe on $M$ transgresses to a line bundle on $LM$ 03.07.16, the 2-form being the curvature of that bundle. The bridge from geometry to the affine Lie theory is the same circle integration: the Cartan 3-form on a Lie group transgresses to the cocycle of the loop-group central extension 03.11.01, so the symplectic, the line-bundle, and the central-extension stories are three readings of one transgression.

Full proof set Master

Proposition (kernel of the knot 2-form is the reparametrisation directions). Let $M^3$ be oriented with volume form $H={\mathrm{vol}}_M$, and let $\gamma$ be an immersed loop. A tangent field $X\in T_{\gamma }LM$ lies in the kernel of $({\omega }_H{)}_{\gamma }$ if and only if $X$ is everywhere tangent to the loop, $X(\theta )=f(\theta )\dot{\gamma }(\theta )$ for some smooth $f:S^1\to \mathbb{R}$.

Proof. Suppose $X=f\dot{\gamma }$. Then for every $Y$,

$$({\omega }_H{)}_{\gamma }(X,Y)={\int }_{S^1}f(\theta ){\mathrm{vol}}_M(\dot{\gamma },\dot{\gamma },Y)d\theta =0,$$

since ${\mathrm{vol}}_M$ is alternating and repeats $\dot{\gamma }$. Thus tangential fields lie in the kernel.

Conversely, suppose $({\omega }_H{)}_{\gamma }(X,Y)=0$ for all $Y$. Fix $\theta$ where $\dot{\gamma }(\theta )\ne 0$ (true everywhere for an immersion). The map $Y(\theta )\mapsto {\mathrm{vol}}_M(\dot{\gamma }(\theta ),X(\theta ),Y(\theta ))$ is the linear functional $Y\mapsto {\mathrm{vol}}_M(\dot{\gamma }\wedge X)(\cdot )$ contracted on its last slot. In the 3-dimensional fibre, ${\mathrm{vol}}_M(\dot{\gamma },X,\cdot )$ is the metric dual of the cross product $\dot{\gamma }\times X$. Vanishing against all $Y$ forces $\dot{\gamma }(\theta )\times X(\theta )=0$, hence $X(\theta )$ is parallel to $\dot{\gamma }(\theta )$. Smoothness of $X$ and nonvanishing of $\dot{\gamma }$ give a smooth $f$ with $X=f\dot{\gamma }$. Therefore the kernel is exactly the reparametrisation directions, and ${\omega }_H$ descends to a non-degenerate form on the quotient $\mathcal{K}$. $\square$

Proposition (rotation invariance of transgression). The transgressed form $\tau (\alpha )$ is invariant under the loop-rotation $S^1$-action on $LM$.

Proof. Rotation by $s$ on $LM$ is induced by the rotation $R_s(\theta )=\theta +s$ on the $S^1$-factor, under which $\mathrm{ev}\circ (R_s\times {\mathrm{rot}}_s)=\mathrm{ev}$. Since $R_s$ is an orientation-preserving diffeomorphism of $S^1$, fibre integration is invariant: ${\int }_{S^1}{R}_s^{\ast }={\int }_{S^1}$. Pulling back the defining integral by ${\mathrm{rot}}_s$ and changing the integration variable $\theta \mapsto \theta -s$ leaves $\tau (\alpha )$ unchanged. $\square$

Connections Master

• Loop group central extension 03.11.05. Transgression of the bi-invariant Cartan 3-form on a compact group $G$ produces the 2-cocycle defining the central extension $\widetilde{LG}\to LG$. The circle integration of this unit is the geometric origin of the affine cocycle, linking back to the algebraic central-extension machinery of 03.11.01.

• The $B$-field as a gerbe connection 03.07.16. A $U(1)$-gerbe-with-curving on $M$ transgresses to a $U(1)$-line bundle with connection on $LM$, whose first Chern class is the transgressed Dixmier-Douady class. The closed 2-form built here is the curvature of that transgressed bundle; the gerbe unit is the direct downstream consumer of this construction.

• Symplectic manifold 05.01.02. The transgression of a closed 3-form is a closed 2-form; on the reparametrisation quotient of loops it is weakly non-degenerate, making the space of knots a symplectic manifold. This unit supplies the infinite-dimensional symplectic geometry that 05.01.02 does not reach in the finite-dimensional setting.

• Central extension of a Lie algebra 03.11.01. The cocycle obtained by transgression realises, at the geometric level, the 2-cocycle that defines a central extension. The de Rham transgression here is the differential-geometric companion of the Lie-algebra cohomology there.

Historical & philosophical context Master

The systematic use of the free loop space as a smooth infinite-dimensional manifold, together with transgression as fibre integration over the circle, is due to Brylinski, who organised it as the geometry underlying gerbes, the Wess-Zumino term, and the affine loop-group extensions ^{[Brylinski Ch. 3]}. The space-of-knots symplectic structure, and its connection to the binormal evolution of vortex filaments in an ideal fluid, is one of his motivating examples and ties the abstract transgression to nineteenth-century vortex dynamics. The loop-group side — smooth structures on $LG$, the central extension, and the positive-energy representation theory selected by the rotation action — was developed in parallel by Pressley and Segal ^{[Pressley-Segal Ch. 3]}, whose treatment fixes the analytic conventions for the mapping-space topology used here.

Bibliography Master

• Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics 107, Birkhäuser, 1993. Ch. 3.
• Pressley, A. and Segal, G., Loop Groups, Oxford University Press, 1986. Ch. 3.
• Chen, K.-T., "Iterated path integrals", Bulletin of the American Mathematical Society 83 (1977), 831-879.
• Marsden, J. and Weinstein, A., "Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids", Physica D 7 (1983), 305-323.

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Infinite-dimensional Lie strand: the geometric source of loop-group extensions and gerbe transgression. Produced as the missing construction the gerbe units presuppose.