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Atrium > Mathematics, Physics & Philosophy: String structures on M vs. Spin structures on LM

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  1.  
    • 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 BString\mathbf{B}String is defined as the homotopy pullback

    BString → * ↓ ↓ BSpin →12p 1 B 3U(1) \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,−][S^1,-] to the above diagram we get the homotopy commutative diagram (does internal hom preserve homotopy limits?)

    [S 1,BString] → * ↓ ↓ [S 1,BSpin] →[S 1,12p 1] [S 1,B 3U(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)] }

    Now we consider the fiber integration/transgression morphis exp2πi∫ S 1:[S 1,B 3U(1)]→B 2U(1)exp 2\pi i\int_{S^1}:[S^1,\mathbf{B}^3U(1)]\to \mathbf{B}^2U(1):

    [S 1,BString] → * ↓ ↓ [S 1,BSpin] →[S 1,12p 1] [S 1,B 3U(1)] ↘ exp2π∫ S 1 mmB 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

    [S 1,BString] → * ↓ ↓ [S 1,BSpin] →exp2πi∫ S 1[S 1,12p 1] B 2U(1) \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,−][S^1,-] commutes with B\mathbf{B} we can rewrite as

    [S 1,BString] → * ↓ ↓ B[S 1,Spin] →exp2πi∫ S 1[S 1,12p 1] B 2U(1) \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

    ℒBString → * ↓ ↓ BℒSpin →exp2πi∫ S 1[S 1,12p 1] B 2U(1) \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 ℒSpin\mathcal{L}Spin. By the universal property of the homotopy pullback, the above homotopy commutative diagram therefore factors as

    ℒBString ↘ BℒSpin˜ → * ↓ ↓ BℒSpin →exp2πi∫ S 1[S 1,12p 1] B 2U(1) \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 ℒSpin˜\widetilde{\mathcal{L}Spin} is the canonical U(1)U(1)-central extension of the loop group ℒSpin\mathcal{L}Spin. We are done: the morphism ℒBString→BℒSpin˜\mathcal{L}\mathbf{B}String\to \mathbf{B}\widetilde{\mathcal{L}Spin}, factoring the natural projection ℒBString→ℒBSpin≅BℒSpin˜\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 MM is a lift of the Spin structure M→BSpinM\to \mathbf{B}Spin to M→BStringM\to\mathbf{B}String. Applying the internal hom [S 1,−][S^1,-] to this lift we obtain a lift of the morphism ℒM→ℒBSpin≅BℒSpin\mathcal{L}M\to \mathcal{L}\mathbf{B}Spin\cong \mathbf{B}\mathcal{L}Spin to a morphism ℒM→ℒBString\mathcal{L}M\to \mathcal{L}\mathbf{B}String. Since the projection ℒBString→BℒSpin˜\mathcal{L}\mathbf{B}String\to \mathbf{B}\widetilde{\mathcal{L}Spin} factors through BℒSpin˜\mathbf{B}\widetilde{\mathcal{L}Spin} we get a lift of the natural morphism ℒM→BℒSpin\mathcal{L}M\to \mathbf{B}\mathcal{L}Spin to a morphism ℒM→BℒSpin˜\mathcal{L}M\to \mathbf{B}\widetilde{\mathcal{L}Spin}. But this is precisely the definition of a Spin structure on ℒM\mathcal{L}M.

  2. 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(ℒX;ℤ)→H n(X;ℤ)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 H^ n+1(ℒX;ℤ)→H^ n(X;ℤ)\hat{H}^{n+1}(\mathcal{L}X;\mathbb{Z})\to \hat{H}^{n}(X;\mathbb{Z}), it is only the latter to be the π 0\pi_0 of a morphism of smooth stacks [S 1,B nU(1) conn]→B n−1U(1) conn[S^1,\mathbf{B}^n U(1)_{conn}]\to \mathbf{B}^{n-1} U(1)_{conn}.

    This is manifest in the n=1n=1 case where the evident holonomy map [S 1,BU(1) conn]→U(1)[S^1,\mathbf{B} U(1)_{conn}]\to U(1) which maps a U(1)U(1)-connection on a manifold XX to its holonomy as a function ℒX→U(1)\mathcal{L}X\to U(1) has no analogue without the conn{}_conn subscript. What really happens at a integral cohomology level is the following: one chooses an arbitary conenction on a U(1)U(1)-bundle representing a class in H 2(X,ℤ)=π 0(Maps(X,BU(1))H^2(X,\mathbb{Z})=\pi_0(Maps(X,BU(1)) and uses the holonomy of this connection to define a map ℒX→U(1)\mathcal{L}X\to U(1). Since the space of U(1)U(1)-connections on a fixed bundle is contractible, the homotopy type of the map ℒX→U(1)\mathcal{L}X\to U(1) is well defined and so one has a well defined element in the set π 0(Maps(ℒX,U(1))=π 0(Maps(ℒX,K(ℤ,1))=H 1(ℒX,ℤ)\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 π 0(Maps(X,BU(1))→π 0(Maps(ℒX,U(1))\pi_0(Maps(X,BU(1))\to \pi_0(Maps(\mathcal{L}X,U(1)) is not induced by a morphism of stacks ℒBU(1)→U(1)\mathcal{L}\mathbf{B}U(1)\to U(1).

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

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

  3. 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 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string} is defined as the homotopy pullback

    𝔰𝔱𝔯𝔦𝔫𝔤 → * ↓ ↓ 𝔰𝔬 →μ 3 R[2] \array{ \mathfrak{string} &\to& *\\ \downarrow && \downarrow \\ \mathfrak{so} &\stackrel{\mu_3}{\to}& \mathbf{R}[2] }

    where R[2]\mathbf{R}[2] is the chain complex consisting of ℝ\mathbb{R} in degree 2, and μ 3\mu_3 is the canonical 3-cocycle on the Lie algebra 𝔰𝔬\mathfrak{so}, seen as an L ∞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 1S^1 with values in the given (higher) Lie algebras gives a fibration diagram

    Ω •(S 1,𝔰𝔱𝔯𝔦𝔫𝔤) → * ↓ ↓ Ω •(S 1,𝔰𝔬) →Ω •(S 1,μ 3) Ω •(S 1,ℝ)[2] \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})[2] }

    where the evident isomorphism Ω •(S 1,ℝ[2])=Ω •(S 1,ℝ)[2]\Omega^\bullet(S^1,\mathbb{R}[2])= \Omega^\bullet(S^1,\mathbb{R})[2] has been used. It should be remarked that since μ 3\mu_3 is a nonlinear L ∞L_\infty-morphism, the L ∞L_\infty-morphism Ω •(S 1,μ 3)\Omega^\bullet(S^1,\mu_3) is not just “act with μ 3\mu_3 on the 𝔰𝔬\mathfrak{so}-part of an element in Ω •(S 1,𝔰𝔬)\Omega^\bullet(S^1,\mathfrak{so}). I will write the explict expresion of the L intyL_\inty-morphism Ω •(S 1,μ 3)\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 1S^1 gives a morphism of chian complexes (and so a morphism of abelian L ∞L_\infty-algebras)

    ∫ S 1:Ω •(S 1,ℝ)[2]→ℝ[1] \int_{S^1}:\Omega^\bullet(S^1,\mathbb{R})[2]\to \mathbb{R}[1]

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

    Ω •(S 1,𝔰𝔱𝔯𝔦𝔫𝔤) → * ↓ ↓ Ω •(S 1,𝔰𝔬) →Ω •(S 1,μ 3) Ω •(S 1,ℝ)[2] ↘ ∫ S 1 mmR[1] \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})[2]\\ & & & \searrow^{\int_{S^1}}\\ & & & \phantom{mm} \mathbf{R}[1] }

    to get the homotopy commutative diagram

    Ω •(S 1,𝔰𝔱𝔯𝔦𝔫𝔤) → * ↓ ↓ Ω •(S 1,𝔰𝔬) →∫ S 1Ω •(S 1,μ 3) R[1] \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}[1] }

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

    Ω •(S 1,𝔰𝔱𝔯𝔦𝔫𝔤) ↘ Ω •(S 1,𝔰𝔬)˜ → * ↓ ↓ Ω •(S 1,𝔰𝔬) →∫ S 1Ω •(S 1,μ 3 R[1] \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}[1] }

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

    In a follow up post this morphism and the L ∞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

    Ω •(S 1,𝔰𝔱𝔯𝔦𝔫𝔤)→Ω •(S 1,𝔰𝔬)˜ \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

    𝔰𝔬^→Ω •(S 1,𝔰𝔬)˜. \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,B 3U(1)]→B 2U(1)[S^1,\mathbf{B}^3U(1)] \to \mathbf{B}^2U(1), we should have a morphism [S 1,B 2(BU(1) conn)]→B 2U(1)[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 c:BG→B 3U(1)\mathbf{c}:\mathbf{B}G\to \mathbf{B}^3U(1) actually lifts to a morphism Bwzw^:BG→B 2(BU(1) conn)\mathbf{B}\hat{\mathbf{wzw}}:\mathbf{B}G\to \mathbf{B}^2(\mathbf{B}U(1)_{conn}).

    But via the equivalence (Ω 2→B 2U(1) conn)≃B(BU(1) conn)(\Omega^2 \to \mathbf{B}^2U(1)_{conn})\simeq \mathbf{B}(\mathbf{B}U(1)_{conn}), the stack B 2(BU(1) conn)\mathbf{B}^2(\mathbf{B}U(1)_{conn}) should be equivalently be presented by B(Ω 2→B 2U(1) conn)\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

    Bwzw^:BG→B(Ω 2→B 2U(1) conn) \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 GG a compact simply connected Lie group the Wess-Zumino-Witten gerbe wzw:BG→B 3U(1)\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 Bwzw^\mathbf{B}\widehat{\mathbf{wzw}}?)

  7. (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

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

    we can form two distinct natural homotopy pullbacks: one is

    BG˜ → * ↓ ↓ BG → B 2U(1) \array{ \mathbf{B}\tilde{G} &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}G &\to& \mathbf{B}^2U(1) }

    and the other is

    BH → B 2ℝ ↓ ↓ BG → B 2U(1) \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:

    Bℝ → BU(1) → BG˜ → * ↓ ↓ ↓ ↓ * → B 2ℤ → BH → B 2ℝ ↓ ↓ ↓ * → BG → B 2U(1) \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 BG˜\mathbf{B}\tilde{G} and BH\mathbf{B}H become equivalent, however they are not equivalent as smooth stacks. However, the fiber sequence

    BG˜ → * ↓ ↓ BH → B 2ℝ \array{ \mathbf{B}\tilde{G} &\to& *\\ \downarrow &&\downarrow\\ \mathbf{B}H &\to& \mathbf{B}^2\mathbb{R} }

    shows that, if XX is a smooth manifold, then there are no obstructions to lifting a morphism X→BHX\to \mathbf{B}H to a morphism X→BG˜X\to \mathbf{B}\tilde{G}. Namely, the obstruction to the lift is a class in H 2(X,ℝ̲)=0H^2(X,\underline{\mathbb{R}})=0 since ℝ̲\underline{\mathbb{R}} is a fine sheaf. Moreover, the equivalence classes of lifts are classified by H 1(X,ℝ̲)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 XX is a manifold we have a natural isomorphism of sets o equivalence classes of bundles over XX

    H 1(X,G˜̲)→H 1(X,H̲) 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

    H(X,BG˜) → * ↓ ↓ H(X,BH) → H(X,B 2ℝ) \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 SpinSpin and StringString (which I’ll expand tomorrow) is that making multiplicative bundle gerbes come into play, the natural central extension of the loop group ℒSpin\mathcal{L}Spin one gets has the form

    Bℤ → ℒSpin˜ ℤ ↓ ↓ * → ℒSpin \array{ \mathbf{B}\mathbb{Z} &\to& \widetilde{\mathcal{L}Spin}_{\mathbb{Z}}\\ \downarrow && \downarrow \\ * &\to& \mathcal{L}Spin }

    rather than

    U(1) → ℒSpin˜ ↓ ↓ * → ℒSpin \array{ U(1) &\to& \widetilde{\mathcal{L}Spin}\\ \downarrow && \downarrow \\ * &\to& \mathcal{L}Spin }

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

    H 1(X,ℒSpin˜̲)→H 1(X,ℒSpin˜̲ ℤ) 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 BSpin→B 3U(1)\mathbf{B}Spin\to \mathbf{B}^3U(1) factors as BSpin→B 2(BU(1) conn)→B 3U(1)\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

    BString → B 2Ω 1 → * ↓ ↓ ↓ BSpin → B 2(BU(1) conn) → B 3U(1) \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,−][S^1,-] to the left homotopy pullpback, we get the homotopy pullback diagram

    [S 1,BString] → [S 1,B 2Ω 1] ↓ ↓ [S 1,BSpin] → [S 1,B 2(BU(1) conn)] \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

    [S 1,B 2Ω 1] → B 2ℝ ↓ ↓ [S 1,B 2(BU(1) conn)] → B 2U(1) \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

    [S 1,BString] → [S 1,B 2Ω 1] → B 2ℝ ↓ ↓ ↓ [S 1,BSpin] → [S 1,B 2(BU(1) conn)] → B 2U(1) \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 BℒSpin˜ ℤ\mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}} denotes the homotopy pullback

    BℒSpin˜ ℤ → B 2ℝ ↓ ↓ B[S 1,Spin] → B 2U(1) \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,BString]→BℒSpin˜ ℤ [S^1,\mathbf{B}String]\to \mathbf{B}\widetilde{\mathcal{L}Spin}_{\mathbb{Z}}

    If now XX is a smooth manifold, we have a natural morphism

    H(X,BString)stakrel[S 1,−]→H([S 1,X],[S 1,BString])→H([S 1,X],BℒSpin˜ ℤ) \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

    π 0H(X,BString)→π 0H([S 1,X],BℒSpin˜ ℤ). \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 BℒSpin˜\mathbf{B}\widetilde{\mathcal{L}Spin}, i.e.,

    we have a natural morphism

    H([S 1,X],BℒSpin˜)→H([S 1,X],BℒSpin˜ ℤ) \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

    π 0H([S 1,X],BℒSpin˜)→π 0H([S 1,X],BℒSpin˜ ℤ) \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

    π 0H(X,BString)→π 0H([S 1,X],BℒSpin˜ ℤ)←π 0H([S 1,X],BℒSpin˜) \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][S^1,X] were an ordinary smooth manifold)

  10. maybe the conclusion of the argument could be the following: for XX a finite dimensional smooth manifold, the smooth stack [S 1,X][S^1,X] is represented by a Frechet manifold ℒX\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 ℒX\mathcal{L}X is acyclic and so it should follow that for i=0,1i=0,1, one has π iH([S 1,X],B 2ℝ)=H 2−i(ℒX,ℝ̲)=0\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.

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

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