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• CommentRowNumber1.
• CommentAuthordomenico_fiorenza
• CommentTimeJun 4th 2013
• (edited Jun 4th 2013)

Here’s a sketchy derivation of the fact that a String structure on a manifold induces a Spin structure on its loop space. As to be expected from the supposed naturality of the construction, everything happens at the level of stacks (or at least, it should, modulo a few details I’ve not checked).

To begin with, the (higher) stack $\mathbf{B}String$ is defined as the homotopy pullback

$\array{ \mathbf{B}String &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathbf{B}^3U(1) }$

Applying the internal hom $[S^1,-]$ to the above diagram we get the homotopy commutative diagram (does internal hom preserve homotopy limits?)

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}Spin] &\stackrel{[S^1,\frac{1}{2}p_1]}{\to}& [S^1,\mathbf{B}^3U(1)] }$

Now we consider the fiber integration/transgression morphis $exp 2\pi i\int_{S^1}:[S^1,\mathbf{B}^3U(1)]\to \mathbf{B}^2U(1)$:

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}Spin] &\stackrel{[S^1,\frac{1}{2}p_1]}{\to}& [S^1,\mathbf{B}^3U(1)]\\ & & & \searrow^{exp 2\pi\int_{S^1}}\\ & & & \phantom{mm} \mathbf{B}^2U(1) }$

to get the homotopy commutative diagram

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}Spin] &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$

which, assuming $[S^1,-]$ commutes with $\mathbf{B}$ we can rewrite as

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}[S^1,Spin] &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$

i.e., as

$\array{ \mathcal{L}\mathbf{B}String &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}\mathcal{L}Spin &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$

and the bottom horizontal arrow is the canonical 2-cocycle on the loop group $\mathcal{L}Spin$. By the universal property of the homotopy pullback, the above homotopy commutative diagram therefore factors as

$\array{ \mathcal{L}\mathbf{B}String\\ & \searrow\\ && \mathbf{B}\widetilde{\mathcal{L}Spin} &\to& *\\ && \downarrow && \downarrow \\ && \mathbf{B}\mathcal{L}Spin &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$

where $\widetilde{\mathcal{L}Spin}$ is the canonical $U(1)$-central extension of the loop group $\mathcal{L}Spin$. We are done: the morphism $\mathcal{L}\mathbf{B}String\to \mathbf{B}\widetilde{\mathcal{L}Spin}$, factoring the natural projection $\mathcal{L}\mathbf{B}String\to \mathcal{L}\mathbf{B}Spin\cong \mathbf{B}\widetilde{\mathcal{L}Spin}$ is the universal morphism inducing the transgression from String structureson a Spin manifold to Spin structures on its loop space.

Namely, by definition, a String structur on $M$ is a lift of the Spin structure $M\to \mathbf{B}Spin$ to $M\to\mathbf{B}String$. Applying the internal hom $[S^1,-]$ to this lift we obtain a lift of the morphism $\mathcal{L}M\to \mathcal{L}\mathbf{B}Spin\cong \mathbf{B}\mathcal{L}Spin$ to a morphism $\mathcal{L}M\to \mathcal{L}\mathbf{B}String$. Since the projection $\mathcal{L}\mathbf{B}String\to \mathbf{B}\widetilde{\mathcal{L}Spin}$ factors through $\mathbf{B}\widetilde{\mathcal{L}Spin}$ we get a lift of the natural morphism $\mathcal{L}M\to \mathbf{B}\mathcal{L}Spin$ to a morphism $\mathcal{L}M\to \mathbf{B}\widetilde{\mathcal{L}Spin}$. But this is precisely the definition of a Spin structure on $\mathcal{L}M$.

1. From a behind the scenes email exchange with Urs I see I’ve been forgetting the crucial role of connections in the above. Namely, while it is true that the fiber integration/transgression morphism in integral cohomology $H^{n+1}(\mathcal{L}X;\mathbb{Z})\to H^{n}(X;\mathbb{Z})$ refines to a fiber integration/transgression morphism in ordinary differential cohomology $\hat{H}^{n+1}(\mathcal{L}X;\mathbb{Z})\to \hat{H}^{n}(X;\mathbb{Z})$, it is only the latter to be the $\pi_0$ of a morphism of smooth stacks $[S^1,\mathbf{B}^n U(1)_{conn}]\to \mathbf{B}^{n-1} U(1)_{conn}$.

This is manifest in the $n=1$ case where the evident holonomy map $[S^1,\mathbf{B} U(1)_{conn}]\to U(1)$ which maps a $U(1)$-connection on a manifold $X$ to its holonomy as a function $\mathcal{L}X\to U(1)$ has no analogue without the ${}_conn$ subscript. What really happens at a integral cohomology level is the following: one chooses an arbitary conenction on a $U(1)$-bundle representing a class in $H^2(X,\mathbb{Z})=\pi_0(Maps(X,BU(1))$ and uses the holonomy of this connection to define a map $\mathcal{L}X\to U(1)$. Since the space of $U(1)$-connections on a fixed bundle is contractible, the homotopy type of the map $\mathcal{L}X\to U(1)$ is well defined and so one has a well defined element in the set $\pi_0(Maps(\mathcal{L}X,U(1))=\pi_0(Maps(\mathcal{L}X,K(\mathbb{Z},1))=H^1(\mathcal{L}X,\mathbb{Z})$. But the map $\pi_0(Maps(X,BU(1))\to \pi_0(Maps(\mathcal{L}X,U(1))$ is not induced by a morphism of stacks $\mathcal{L}\mathbf{B}U(1)\to U(1)$.

So I will now revise the above argument dropping a few ${}_conn$ here and there.

Concerning $[X,-]$ preserving homotopy limits, Urs confirmed me this is correct.

2. So here’s what should happen when one makes connection come into play: having connections means we should have some background picture involving Lie algebras, and since we have higher connections here, also Lie algebras will be higher Lie algebras. So what follows is (or should be) a higher Lie algebras version of the above post.

To begin with, the Lie 2-algebra $\mathfrak{string}$ is defined as the homotopy pullback

$\array{ \mathfrak{string} &\to& *\\ \downarrow && \downarrow \\ \mathfrak{so} &\stackrel{\mu_3}{\to}& \mathbf{R} }$

where $\mathbf{R}$ is the chain complex consisting of $\mathbb{R}$ in degree 2, and $\mu_3$ is the canonical 3-cocycle on the Lie algebra $\mathfrak{so}$, seen as an $L_\infty$-morphism.

I still have to prove this in general, but at least in this particular case it is true that taking differential forms on $S^1$ with values in the given (higher) Lie algebras gives a fibration diagram

$\array{ \Omega^\bullet(S^1,\mathfrak{string}) &\to& *\\ \downarrow && \downarrow \\ \Omega^\bullet(S^1,\mathfrak{so}) &\stackrel{\Omega^\bullet(S^1,\mu_3)}{\to}& \Omega^\bullet(S^1,\mathbb{R}) }$

where the evident isomorphism $\Omega^\bullet(S^1,\mathbb{R})= \Omega^\bullet(S^1,\mathbb{R})$ has been used. It should be remarked that since $\mu_3$ is a nonlinear $L_\infty$-morphism, the $L_\infty$-morphism $\Omega^\bullet(S^1,\mu_3)$ is not just “act with $\mu_3$ on the $\mathfrak{so}$-part of an element in $\Omega^\bullet(S^1,\mathfrak{so})$. I will write the explict expresion of the $L_\inty$-morphism $\Omega^\bullet(S^1,\mu_3)$ later today in some follow up post to this one (in any case it is not hard to write out).

Now we observe that integration of differential forms on $S^1$ gives a morphism of chian complexes (and so a morphism of abelian $L_\infty$-algebras)

$\int_{S^1}:\Omega^\bullet(S^1,\mathbb{R})\to \mathbb{R}$

(this is essentially Stokes theorem for $S^1$). So we have

$\array{ \Omega^\bullet(S^1,\mathfrak{string}) &\to& *\\ \downarrow && \downarrow \\ \Omega^\bullet(S^1,\mathfrak{so}) &\stackrel{\Omega^\bullet(S^1,\mu_3)}{\to}& \Omega^\bullet(S^1,\mathbb{R})\\ & & & \searrow^{\int_{S^1}}\\ & & & \phantom{mm} \mathbf{R} }$

to get the homotopy commutative diagram

$\array{ \Omega^\bullet(S^1,\mathfrak{string}) &\to& *\\ \downarrow && \downarrow \\ \Omega^\bullet(S^1,\mathfrak{so}) &\stackrel{\int_{S^1}\Omega^\bullet(S^1,\mu_3)}{\to}& \mathbf{R} }$

where the bottom horizontal arrow extends to the dgla of all differential forms $\Omega^\bullet(S^1,\mathfrak{g})$ the canonical 2-cocycle on the loop Lie algebra $\Omega^0(S^1,\mathfrak{g})$. By the universal property of the homotopy pullback, the above homotopy commutative diagram therefore factors as

$\array{ \Omega^\bullet(S^1,\mathfrak{string})\\ & \searrow\\ && \widetilde{\Omega^\bullet(S^1,\mathfrak{so})} &\to& *\\ && \downarrow && \downarrow \\ && \Omega^\bullet(S^1,\mathfrak{so}) &\stackrel{\int_{S^1}\Omega^\bullet(S^1,\mu_3}{\to}& \mathbf{R} }$

where $\widetilde{\Omega^\bullet(S^1,\mathfrak{so})}$ is a canonical central extension of the dgla $\Omega^\bullet(S^1,\mathfrak{so}$, whose degree zero part is the affine lie algebra $\hat{\mathfrak{so}}$. We are done: we have obtained by abstract nonsense a canonical $L_\infty$-morphism $\Omega^\bullet(S^1,\mathfrak{string})\to \widetilde{\Omega^\bullet(S^1,\mathfrak{so})}$, factoring the natural projection $\Omega^\bullet(S^1,\mathfrak{string})\to \Omega^\bullet(S^1,\mathfrak{so})$.

In a follow up post this morphism and the $L_\infty$-algebras involved will be spelled out in detail.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 6th 2013
• (edited Jun 6th 2013)

Hi Domenico,

thanks for further pushing this. We talked about it in Bayrischzell and I promised to come back to it, but of course I got a bit distracted.

So one thing one could further explore here is the kernel of the map

$\Omega^\bullet(S^1 , \mathfrak{string}) \to \widetilde{\Omega^\bullet(S^1, \mathfrak{so})}$

or maybe rather the homotopy pullback of that along the inclusion of the affine Lie algebra

$\widehat \mathfrak{so} \to \widetilde{\Omega^\bullet(S^1, \mathfrak{so})} \,.$

That (pullback of the) kernel is a measure for how much the transgression loses information…

• CommentRowNumber5.
• CommentAuthordomenico_fiorenza
• CommentTimeJun 10th 2013
• (edited Jun 10th 2013)

maybe I’ve solved:

while we do not have a morphism $[S^1,\mathbf{B}^3U(1)] \to \mathbf{B}^2U(1)$, we should have a morphism $[S^1,\mathbf{B}^2(\mathbf{B}U(1)_{conn})] \to \mathbf{B}^2U(1)$. So what we would need to complete the above argument is that the characteristic morphism $\mathbf{c}:\mathbf{B}G\to \mathbf{B}^3U(1)$ actually lifts to a morphism $\mathbf{B}\hat{\mathbf{wzw}}:\mathbf{B}G\to \mathbf{B}^2(\mathbf{B}U(1)_{conn})$.

But via the equivalence $(\Omega^2 \to \mathbf{B}^2U(1)_{conn})\simeq \mathbf{B}(\mathbf{B}U(1)_{conn})$, the stack $\mathbf{B}^2(\mathbf{B}U(1)_{conn})$ should be equivalently be presented by $\mathbf{B}(\Omega^2 \to \mathbf{B}^2U(1)_{conn})$, and (if I’m not confused here) it seems to me that the datum of a morphism

$\mathbf{B}\hat{\mathbf{wzw}}:\mathbf{B}G\to \mathbf{B}(\Omega^2 \to \mathbf{B}^2U(1)_{conn})$

is precisely what Konrad Waldorf calls a multiplicative bundle gerbe with connection. So we do know that for $G$ a compact simply connected Lie group the Wess-Zumino-Witten gerbe $\mathbf{wzw}:\mathbf{B}G\to \mathbf{B}^3U(1)$ is actually a multiplicative bundle gerbe with connection, and indeed this property plays a fundamental role in Konrad’s construction.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 10th 2013

Hi,

yes, exactly, precisely by that construction of multiplicative bundle gerbes with connection equipped with that “relaxed” version of morphism, namely the one which does not need to respect the 2-form connection. (Maybe that morphism is better denoted $\mathbf{B}\widehat{\mathbf{wzw}}$?)

3. (right: I’m editing above accordingly)

• CommentRowNumber8.
• CommentAuthordomenico_fiorenza
• CommentTimeJun 10th 2013
• (edited Jun 11th 2013)

another comment (which is relevant to the above discussion but I’ll only be able to expand and make this connection explicit tomorrow as I have a free minute): given a morphism

$\mathbf{B}G\to \mathbf{B}^2U(1)$

we can form two distinct natural homotopy pullbacks: one is

$\array{ \mathbf{B}\tilde{G} &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}G &\to& \mathbf{B}^2U(1) }$

and the other is

$\array{ \mathbf{B}H &\to& \mathbf{B}^2\mathbb{R}\\ \downarrow && \downarrow \\ \mathbf{B}G &\to& \mathbf{B}^2U(1) }$

The two are part of a larger diagram:

$\array{ \mathbf{B}\mathbb{R}&\to& \mathbf{B}U(1)&\to&\mathbf{B}\tilde{G} &\to& *\\ \downarrow &&\downarrow&&\downarrow &&\downarrow\\ *&\to&\mathbf{B}^2\mathbb{Z}&\to&\mathbf{B}H &\to& \mathbf{B}^2\mathbb{R}\\ &&\downarrow &&\downarrow && \downarrow \\ && *&\to&\mathbf{B}G &\to& \mathbf{B}^2U(1) }$

Under topological realization $\mathbf{B}\tilde{G}$ and $\mathbf{B}H$ become equivalent, however they are not equivalent as smooth stacks. However, the fiber sequence

$\array{ \mathbf{B}\tilde{G} &\to& *\\ \downarrow &&\downarrow\\ \mathbf{B}H &\to& \mathbf{B}^2\mathbb{R} }$

shows that, if $X$ is a smooth manifold, then there are no obstructions to lifting a morphism $X\to \mathbf{B}H$ to a morphism $X\to \mathbf{B}\tilde{G}$. Namely, the obstruction to the lift is a class in $H^2(X,\underline{\mathbb{R}})=0$ since $\underline{\mathbb{R}}$ is a fine sheaf. Moreover, the equivalence classes of lifts are classified by $H^1(X,\underline{\mathbb{R}})$ which is again zero by the same reason, and so the lift is unique up to equivalence. In other words, if $X$ is a manifold we have a natural isomorphism of sets o equivalence classes of bundles over $X$

$H^1(X,\underline{\tilde{G}})\to H^1(X,\underline{H})$

This can equivalently be seen from the long exact sequence of homotopy groups associated with the fibration sequence

$\array{ \mathbf{H}(X,\mathbf{B}\tilde{G}) &\to& *\\ \downarrow &&\downarrow\\ \mathbf{H}(X,\mathbf{B}H) &\to& \mathbf{H}(X,\mathbf{B}^2\mathbb{R}) }$

The connection to $Spin$ and $String$ (which I’ll expand tomorrow) is that making multiplicative bundle gerbes come into play, the natural central extension of the loop group $\mathcal{L}Spin$ one gets has the form

$\array{ \mathbf{B}\mathbb{Z} &\to& \widetilde{\mathcal{L}Spin}_{\mathbb{Z}}\\ \downarrow && \downarrow \\ * &\to& \mathcal{L}Spin }$

rather than

$\array{ U(1) &\to& \widetilde{\mathcal{L}Spin}\\ \downarrow && \downarrow \\ * &\to& \mathcal{L}Spin }$

However, by the above argument, for $X$ a smooth manifold one has a natural isomorphism

$H^1(X,\underline{\widetilde{\mathcal{L}Spin}})\to H^1(X,\underline{\widetilde{\mathcal{L}Spin}}_{\mathbb{Z}})$
• CommentRowNumber9.
• CommentAuthordomenico_fiorenza
• CommentTimeJun 11th 2013
• (edited Jun 11th 2013)

so, as promised, here is the relation of the above post to Spin and String structures.

Since the characteristic morphism $\mathbf{B}Spin\to \mathbf{B}^3U(1)$ factors as $\mathbf{B}Spin\to \mathbf{B}^2(\mathbf{B}U(1)_conn)\to\mathbf{B}^3U(1)$, we have a pasting of homotopy pullback diagrams

$\array{ \mathbf{B}String&\to& \mathbf{B}^2\Omega^1&\to&*\\ \downarrow && \downarrow && \downarrow\\ \mathbf{B}Spin&\to&\mathbf{B}^2(\mathbf{B}U(1)_conn)&\to&\mathbf{B}^3U(1) }$

Applying $[S^1,-]$ to the left homotopy pullpback, we get the homotopy pullback diagram

$\array{ [S^1,\mathbf{B}String]&\to& [S^1,\mathbf{B}^2\Omega^1]\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}Spin]&\to&[S^1,\mathbf{B}^2(\mathbf{B}U(1)_conn)] }$

We can paste on the right the homotopy commutative diagram

$\array{ [S^1,\mathbf{B}^2\Omega^1]&\to&\mathbf{B}^2\mathbb{R}\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}^2(\mathbf{B}U(1)_conn)]&\to&\mathbf{B}^2U(1) }$

to get the homotopy commutative diagram

$\array{ [S^1,\mathbf{B}String]&\to& [S^1,\mathbf{B}^2\Omega^1]&\to& \mathbf{B}^2\mathbb{R}\\ \downarrow && \downarrow &&\downarrow\\ [S^1,\mathbf{B}Spin]&\to&[S^1,\mathbf{B}^2(\mathbf{B}U(1)_conn)]&\to& \mathbf{B}^2U(1) }$

Therefore, if $\mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}}$ denotes the homotopy pullback

$\array{ \mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}}&\to & \mathbf{B}^2\mathbb{R}\\ \downarrow&&\downarrow\\ \mathbf{B}[S^1,Spin]&\to&\mathbf{B}^2U(1) }$

we get from the universal property of the homotopy pullback a canonical morphism

$[S^1,\mathbf{B}String]\to \mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}}$

If now $X$ is a smooth manifold, we have a natural morphism

$\mathbf{H}(X,\mathbf{B}String)\stakrel{[S^1,-]}{\to}\mathbf{H}([S^1,X],[S^1,\mathbf{B}String])\to\mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}})$

inducing a natural morphism

$\pi_0\mathbf{H}(X,\mathbf{B}String)\to \pi_0\mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}}).$

Also, by the defining homotopy commutative diagram of $\mathbf{B}\widetilde{\mathcal{L}Spin}$, i.e.,

we have a natural morphism

$\array{ \mathbf{B}\widetilde{\mathcal{L}Spin}}& \mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin})\to \mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}})$

inducing a natural morphism

$\pi_0\mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin})\to \pi_0\mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}})$

This way we get a span of morphisms of sets of equivalence classes

$\pi_0\mathbf{H}(X,\mathbf{B}String)\to \pi_0\mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}})\leftarrow \pi_0\mathbf{H}([S^1,X],\mathbf{B}\widetilde{\mathcal{L}Spin})$

and the question to be answered to conclude is “is the $\leftarrow$ in the above diagram an isomorphism?” (the answer would be yes if $[S^1,X]$ were an ordinary smooth manifold)

4. maybe the conclusion of the argument could be the following: for $X$ a finite dimensional smooth manifold, the smooth stack $[S^1,X]$ is represented by a Frechet manifold $\mathcal{L}X$, which happens to have a smooth partition of unit (see http://mathoverflow.net/questions/16104/which-frechet-manifolds-have-a-smooth-partition-of-unity). this in turn implies that the sheaf of smooth real valued functions on $\mathcal{L}X$ is acyclic and so it should follow that for $i=0,1$, one has $\pi_i\mathbf{H}([S^1,X],\mathbf{B}^2\mathbb{R})=H^{2-i}(\mathcal{L}X,\underline{\mathbb{R}})=0$.

done (?)

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 12th 2013

done (?)

I think so, yes.

5. ok, thanks. I’ll now spend some time behind the scenes to prepare a version of this which may be submitted to nPub