# nLab Sullivan model of free loop space

Contents

### Context

#### Rational homotopy theory

and

rational homotopy theory

# Contents

## Idea

In rational homotopy theory, given a rational topological space modeled by a Sullivan model dg-algebra, there is an explicit description of the Sullivan model of its free loop space.

This is a special class of cases of the general notion of Sullivan models of mapping spaces.

## Construction

###### Proposition

Let $(\wedge^\bullet V, d_X)$ be a semifree dg-algebra being a minimal Sullivan model of a rational simply connected space $X$. Then a Sullivan model for the free loop space $\mathcal{L} X$ is given by

$(\wedge^\bullet( V \oplus s V ), d_{\mathcal{L}X}) \,,$

where

• $s V$ is the graded vector space obtained from $V$ by shifting degrees down by one: $deg(s v) = deg(v)-1$;

• $d_{\mathcal{L}X}$ is defined on elements $v$ of $V$ by

$d_{\mathcal{L}X} v \coloneqq d v$

and on elements $s v$ of $s V$ by

$d_{\mathcal{L}X} s v \coloneqq - s ( d v ) \,,$

where on the right $s \colon V \to s V$ is extended as a graded derivation $s \colon \wedge^2 V \to \wedge^\bullet (V \oplus s V)$.

This is due to (Vigué-Sullivan 76). Review includes (Felix-Halperin-Thomas 00, p. 206, Hess 06, example 2.5, Félix-Oprea-Tanre 08, theorem 5.11).

###### Remark

The formula in prop. is the same as that for the Weil algebra of the L-infinity algebra of wich $(\wedge^\bullet V,d_X)$ is the Chevalley-Eilenberg algebra, except that here $s$ shifts down whereas for the Weil algebra it shifts up.

## Properties

### Homotopy quotient by $S^1$ and cyclic homology

###### Proposition

Given a Sullivan model $(\wedge^\bullet (V \oplus s V), d_{\mathcal{L}X})$ for a free loop space as in prop. , then a Sullivan model for the cyclic loop space, i.e. for the homotopy quotient $\mathcal{L} X // S^1$ with respect to the canonical circle group action that rotates loops (i.e. for the Borel construction $\mathcal{L}X \times_{S^1} E S^1$) is given by

$(\wedge^\bullet( V\oplus s V \oplus \langle \omega_2\rangle ), d_{\mathcal{L}X/S^1})$

where

• $\omega_2$ is in degree 2;

• $d_{\mathcal{L}X/S^1}$ is defined on generators $w \in V\oplus s V$ by

$d_{\mathcal{L}X/S^1} w \;\coloneqq\; d_{\mathcal{L}X} w + \omega_2 \wedge s w \,.$

Moreover, the canonical sequence of morphisms of dg-algebras

$(\wedge \omega_2, d = 0) \longrightarrow (\wedge^\bullet( V\oplus s V \oplus \langle \omega_2\rangle ), d_{\mathcal{L}X/S^1}) \longrightarrow (\wedge^\bullet( V\oplus s V ), d_{\mathcal{L}X})$

is a model for the rationalization of the homotopy fiber sequence

$\mathcal{L}X \longrightarrow \mathcal{L}X / / S^1 \longrightarrow B S^1$

which exhibits the infinity-action (by the discussion there) of $S^1$ on $\mathcal{L}X$.

This is due to (Vigué-Burghelea 85, theorem A).

### Relation to Hochschild homology and cyclic homology

Let $X$ be a simply connected topological space.

The ordinary cohomology $H^\bullet$ of its free loop space is the Hochschild homology $HH_\bullet$ of its singular chains $C^\bullet(X)$:

$H^\bullet(\mathcal{L}X) \simeq HH_\bullet( C^\bullet(X) ) \,.$

Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space $\mathcal{L}X/^h S^1$ is the cyclic homology $HC_\bullet$ of the singular chains:

$H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( C^\bullet(X) )$

(Loday 11)

If the coefficients are rational, and $X$ is of finite type then by prop. and prop. , and the general statements at rational homotopy theory, the cochain cohomology of the above minimal Sullivan models for $\mathcal{L}X$ and $\mathcal{l}X/S^1$ compute the rational Hochschild homology and cyclic homology of (the cochains on) $X$, respectively.

In the special case that the topological space $X$ carries the structure of a smooth manifold, then the singular cochains on $X$ are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that

$H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.$
$H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( \Omega^\bullet(X) ) \,.$

This is known as Jones' theorem (Jones 87)

An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.

## Examples

### The 4-sphere and twisted de Rham cohomology

###### Example

Let $X = S^4$ be the 4-sphere. The corresponding rational n-sphere has minimal Sullivan model

$(\wedge^\bullet \langle g_4, g_7 \rangle, d)$

with

$d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4 \,.$

Hence prop. gives for the rationalization of $\mathcal{L}S^4$ the model

$( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} )$

with

\begin{aligned} d_{\mathcal{L}S^4} h_3 & = 0 \\ d_{\mathcal{L}S^4} \omega_4 & = 0 \\ d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 \\ \end{aligned}

and prop. gives for the rationalization of $\mathcal{L}S^4 / / S^1$ the model

$( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 / / S^1} )$

with

\begin{aligned} d_{\mathcal{L}S^4 / / S^1} h_3 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_2 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_4 & = h_3 \wedge \omega_2 \\ d_{\mathcal{L}S^4 / / S^1} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4 / / S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6 \end{aligned} \,.
###### Proposition

Let $\hat \mathfrak{g} \to \mathfrak{g}$ be a central Lie algebra extension by $\mathbb{R}$ of a finite dimensional Lie algebra $\mathfrak{g}$, and let $\mathfrak{g} \longrightarrow b \mathbb{R}$ be the corresponding L-∞ 2-cocycle with coefficients in the line Lie 2-algebra $b \mathbb{R}$, hence (FSS 13, prop. 3.5) so that there is a homotopy fiber sequence of L-∞ algebras

$\hat \mathfrak{g} \longrightarrow \mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R}$

which is dually modeled by

$CE(\hat \mathfrak{g}) = ( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\; d_{\hat \mathfrak{g}} e = \omega_2) \,.$

For $X$ a space with Sullivan model $(A_X,d_X)$ write $\mathfrak{l}(X)$ for the corresponding L-∞ algebra, i.e. for the $L_\infty$-algebra whose Chevalley-Eilenberg algebra is $(A_X,d_X)$:

$CE(\mathfrak{l}X) = (A_X,d_X) \,.$

Then there is an isomorphism of hom-sets

$Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) ) \;\simeq\; Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) ) \,,$

with $\mathfrak{l}(S^4)$ from prop. and $\mathfrak{l}(\mathcal{L}S^4 //S^1)$ from prop. , where on the right we have homs in the slice over the line Lie 2-algebra, via prop. .

Moreover, this isomorphism takes

$\hat \mathfrak{g} \overset{(g_4, g_7)}{\longrightarrow} \mathfrak{l}(S^4)$

to

$\array{ \mathfrak{g} && \overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow} && \mathfrak{l}( \mathcal{L}X / S^1 ) \\ & {}_{\mathllap{\omega_2}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && b \mathbb{R} } \,,$

where

$\omega_4 = g_4 - h_3 \wedge e \;\,, \;\;\; h_7 = g_7 + \omega_6 \wedge e$

with $e$ being the central generator in $CE(\hat \mathfrak{g})$ from above, and where the equations take place in $\wedge^\bullet \hat \mathfrak{g}^\ast$ with the defining inclusion $\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast$ understood.

This is observed in (Fiorenza-Sati-Schreiber 16, FSS 16b), where it serves to formalize, on the level of rational homotopy theory, the double dimensional reduction of M-branes in M-theory to D-branes in type IIA string theory (for the case that $\mathfrak{g}$ is type IIA super Minkowski spacetime $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ and $\hat \mathfrak{g}$ is 11d super Minkowski spacetime $\mathbb{R}^{10,1\vert \mathbf{32}}$, and the cocycles are those of The brane bouquet).

###### Proof

By the fact that the underlying graded algebras are free, and since $e$ is a generator of odd degree, the given decomposition for $\omega_4$ and $h_7$ is unique.

Hence it is sufficient to observe that under this decomposition the defining equations

$d g_4 = 0 \,,\;\;\; d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4$

for the $\mathfrak{l}S^4$-valued cocycle on $\hat \mathfrak{g}$ turn into the equations for a $\mathfrak{l} ( \mathcal{L}S^4 / S^1 )$-valued cocycle on $\mathfrak{g}$. This is straightforward:

\begin{aligned} & d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \end{aligned}

as well as

\begin{aligned} & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e ) = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e) \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 \;\;\; and \;\;\; - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4 \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 + \omega_6 \wedge \omega_2 \;\;\; and \;\;\; d_\mathfrak{g} h_6 = h_3 \wedge \omega_4 \end{aligned}

### The 2-sphere

###### Example

Let $X = S^2$ be the 2-sphere. The corresponding rational n-sphere has minimal Sullivan model

$(\wedge^\bullet \langle g_3, g_2 \rangle, d)$

with

$d g_2 = 0\,,\;\;\;\; d g_3 = -\tfrac{1}{2} g_2 \wedge g_2 \,.$

Hence prop. gives for the rationalization of $\mathcal{L}S^2$ the model

$( \wedge^\bullet \langle \omega^A_2, \omega^B_2, h_1, h_3 \rangle , d_{\mathcal{L}S^2} )$

with

\begin{aligned} d_{\mathcal{L}S^2} h_1 & = 0 \\ d_{\mathcal{L}S^2} \omega^A_2 & = 0 \\ d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A \\ d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2 \end{aligned}

and prop. gives for the rationalization of $\mathcal{L}S^2 / / S^1$ the model

$( \wedge^\bullet \langle \omega^A_2, \omega^B_2, \omega^C_2 h_1, h_3 , d_{\mathcal{L}S^2 / / S^1} )$

with

\begin{aligned} d_{\mathcal{L}S^2} h_1 & = 0 \\ d_{\mathcal{L}S^2} \omega^A_2 & = \omega^C_2 \wedge h_1 \\ d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A \\ d_{\mathcal{L}S^2} \omega^C_2 & = 0 \\ d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2 + \omega^C_2 \wedge \omega^B_2 \end{aligned} \,.

## References

The original result is due to

Review is in

• Yves Félix, Steve Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

• Kathryn Hess, example 2.5 of Rational homotopy theory: a brief introduction (arXiv)

• Yves Félix, John Oprea, Daniel Tanre, Algebraic models in geometry, Oxford graduate texts in mathematics 17 (2008)

• A. Yu. Onishchenko and Th. Yu. Popelensky, Rational cohomology of the free loop space of a simply connected 4-manifold, J. Fixed Point Theory Appl. 12 (2012) 69–9 (DOI 10.1007/s11784-013-0100-0)

• Luc Menichi, Sullivan models and free loop space, A short introduction to Sullivan models, with the Sullivan model of a free loop space and the detailed proof of Vigué-Sullivan theorem on the Betti numbers of free loop space. Workshop on free loop space à Strasbourg, November 2008 (scanned notes pdf)

General background on Hochschild homology and cyclic homology is in

Last revised on March 4, 2019 at 10:05:21. See the history of this page for a list of all contributions to it.