Yes, of course. I think that’s a copy-and-paste typo from the CE-row. I’ll change it.

]]>Hi Urs,

at differential string structure – proofs shouldn’t it be $c\mapsto cs$ and $h\mapsto \mu-cs$ in the last diagram (on the left, in the middle row of the diagram)?

this way the composite morphism $W(b^2\mathbb{R})\to W(\mathfrak{so})$ would be $c\mapsto cs$ and $g \mapsto \langle-,-\rangle$, which seems to be correct, since $d_W cs= \langle-,-\rangle$, while the compatibility with the upper row is given by $i^* (cs)= \mu$.

this would also say that the morphism $\tilde{W}(b\mathbb{R}\to \mathfrak{string})\to W(\mathfrak{so})$ is

$t^a\mapsto t^a;\qquad b\mapsto 0;\qquad c\mapsto cs;$ $r^a\mapsto r^a;\qquad \tilde{h}\mapsto 0; \qquad g\mapsto \langle-,-\rangle$ ]]>on second thought, can’t we trade “strange algebras with nice morphisms” for “nice algebras with strange morphisms”? as soon as the algebras involved are isomorphic this should be possible, just by spelling out the compositions of the “nice morphisms” with the isomorphism between the nice algebra and the stange one. I’ll think on this.

]]>we are supposed to keep the invariant polynmoials horizontal

ah, ok, now I see it! you’re saying that $exp(-)_{CW}$ is functorial on the triples, but $exp(-)_{conn}$ is not, so this needs be fixed at hand, am I right? if so I agree $\tilde{W}$ is crucial, but we should state this more clearly (i.e., we should stress the fact that ${}_{conn}$ pich some nice representatives, but ${}_CW$ is the real (functorial) thing. I’m sure we’re already saying this, but maybe it can be stressed, it seems important)

]]>mmm.. so you’re saying […]

Yes, right. We are just finding a means to factor that morphism as a weak equivalence followed by a fibration. The definition of invariant polynomials that we have used does not send all weak equivalences to quasi-isomorphisms, so I am fixing that by hand here. I agree that this smells like there should be a better theory, but for the moment it just does the job. One should remember how the deifnition of the invariant polynomials arises in the first place: after finding the simplicial model for $\mathbf{B}^n U(1)_{conn}$ these just happen to do the job of extending the simplicial models of morphisms $\mathbf{B}G \to \mathbf{B}^n U(1)$.

Concerning the problem with the other factorization: yes, you could factor like this. But the problem to be solved is that when integrating via $exp(-)_{conn}$ we are supposed to keep the invariant polynmoials horizontal, and solving that conditon requires forming certain linear combinations of generators. This forming of linear combinations is what the isomorphism to $\tilde W$ takes care of.

]]>mmm.. so you’re syng that we are writing a factorization of $\exp_{\Delta}(\mathfrak{so})_{conn}\to \exp_{\Delta}(b^2\mathbb{R})_{conn}$ of the form

$\exp_{\Delta}(\mathfrak{so})_{conn}\stackrel{\sim}{\to} P\twoheadrightarrow \exp_{\Delta}(b^2\mathbb{R})_{conn}$where $P$ is some simplicial presheaf which, however, is not of the form $\exp_\Delta(\mathfrak{h})$ for an $L_\infty$-algebra $\mathfrak{h}$? clearly this could be, but would make the whole construction much less nice that hoped. but most of all, I’m confused by the fact that if one writes the composition of the two leftmost morphisms, so to have a diagram in which $\tilde{W}$ does not appear, the assignment $\tilde{h}\mapsto h+(cs-\mu)$ seems to disappear and to be hidden in the only non explicit arrow in the diagram (the bottom vertical one in the middle):

$\array{ CE(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} & CE(b \mathbb{R} \to \mathfrak{string}) &\stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} & CE(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{i^*_{\mathfrak{so}}} && \uparrow^\mathrlap{i^*_{(b\mathbb{R} \to \mathfrak{string})}} && \uparrow^\mathrlap{i^*_{b^2 \mathbb{R}}} \\ W(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu \\ r^a \mapsto r^a \\ h \mapsto 0 \\ g \mapsto \langle-,-\rangle } \right) }{\leftarrow} & W(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c \\ g \mapsto g } \right) }{\leftarrow} & W(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{p^*_{\mathfrak{so}}} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{p^*_{b^2 \mathbb{R}}} \\ inv(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ \tilde h \mapsto 0 \\ g \mapsto \langle -,-\rangle \\ \langle \cdots \rangle \mapsto \langle \cdots \rangle } \right) }{\leftarrow} & \tilde inv(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ g \mapsto g } \right) }{\leftarrow} & inv(b^2 \mathbb{R}) } \,.$on the other hand, the arrows at the inv-level should not be defined on the generators of the inv’s but rather should be just the restrictions of the maps at the W-level. in other words, once the maps at the W-lever are defined one need only to check whether the inv-subspaces are preserved. inv-elements are defined by two conditions: being $d_W$-closed and being polynomials in the shifted generators. the first condition is always satisfied, dince the maps at the W-level are dgca maps; so one only needs to check whether shifted geerators are mapped to polynomial in the shifted generators. therefore I’m not sure I can see which is the problem with this diagram here below:

$\array{ CE(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} & CE(b \mathbb{R} \to \mathfrak{string}) &\stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} & CE(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{i^*_{\mathfrak{so}}} && \uparrow^\mathrlap{i^*_{(b\mathbb{R} \to \mathfrak{string})}} && \uparrow^\mathrlap{i^*_{b^2 \mathbb{R}}} \\ W(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu \\ r^a \mapsto r^a \\ h \mapsto 0 \\ g \mapsto \langle-,-\rangle } \right) }{\leftarrow} & W(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c \\ g \mapsto g } \right) }{\leftarrow} & W(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{p^*_{\mathfrak{so}}} && \uparrow^\mathrlap{p^*_{b\mathbb{R}\to\mathfrak{string}}} && \uparrow^\mathrlap{p^*_{b^2 \mathbb{R}}} \\ inv(\mathfrak{so}) & \leftarrow & inv(b \mathbb{R} \to \mathfrak{string}) & \leftarrow & inv(b^2 \mathbb{R}) } \,.$ ]]>Right, as I said, the middle term uses the twiddled invariant polynomials instead.

]]>there’s something unclear to me about the role of $\tilde{inv}$: written thsi way the second from the left column is not (at least directly) related to $\exp_{\Delta}(b\mathbb{R}\to \mathfrak{string})_{conn}$, so it is unclear to me how the big diagram decribes a factorization

$\exp_{\Delta}(\mathfrak{so})_{conn}\to \exp_{\Delta}(b\mathbb{R}\to \mathfrak{string})_{conn}\to \exp_{\Delta}(b^2\mathbb{R})_{conn}$ ]]>Hi Domenico,

I have now fine-tuned the discussion a bit further.

One point I changed is this: it’s actually not quite true that we get a factorization by double square diagrams that is a weak equivalence on the left if we use $inv(b \mathbb{R} \to \mathfrak{string})$, rather, it is true if we use $\tilde inv(b \mathbb{R} \to \mathfrak{string})$.

That’s another way of saying what the isomorphism $W(b \mathbb{R} \to \mathfrak{string}) \simeq \tilde W(b \mathbb{R} \to \mathfrak{string})$ is about: it is the one that makes the quasi-isomorphism $CE(\mathfrak{so}) \simeq CE(b \mathbb{R} \to \mathfrak{string})$ descend to invariant polynomials:

for the invariant polynomials of $(b\mathbb{R} \to \mathfrak{string})$ are those of $\mathfrak{so}$ with the generator $g$ adjoined. But those of the twiddled version are those of $\mathfrak{so}$ with generators $g$ and $\tilde h$ adjoined and quotiented by the relation $d \tilde h = g- \langle - , -\rangle$. That makes the superfluous $g$ drop out in cohomology again.

I suppose this means that one should in principle eventually try to think about an actual model structure on the category whose morphisms are such $CE \leftarrow W \leftarrow inv$-double squares.

]]>Okay, good. Maybe we should put all these little technical points now into the new Proofs-entry, where they don’t obstruct the readability of the main entry. We should in fact move more of the long proofs from there.

I’ll have to go offline in a few minutes and will come online again later this evening.

]]>ok, fine. in the meanwhile, let me notice that the isomorphism $CE(b\mathbb{R}\to\mathfrak{string})\cong CE(\mathfrak{so})\otimes\CE(inn(b\mathbb{R}))$ precisely gives natural isomorphisms $Hom_{dgca}(CE(b\mathbb{R}\to\mathfrak{string}),\Omega^\bullet(U\times\Delta^k)_{vert})\cong Hom_{dgca}(CE(\mathfrak{so}),\Omega^\bullet(U\times\Delta^k)_{vert})\times Hom_{dgca}(CE(inn(b\mathbb{R})),\Omega^\bullet(U\times\Delta^k)_{vert})$, thus the isomorphism

$\exp_\Delta(b\mathbb{R}\to\mathfrak{string})\cong \exp_\Delta(\mathfrak{so})\times\exp_\Delta(inn(b\mathbb{R}))$ ]]>(20 min later…)

Okay, I think I am now close to something complete, even if in need of more polishing/expansion.

Please have another look at differential string structure – proofs

]]>Right, that’s what I am in the process of typing up. I’ll tell you when I am done. I just thought I’d already announce that I have started working on it.

]]>how are the $(\dots)$ morphisms defined? (I mean are you in the process of typing these or is this left for a later time?)

]]>okay, I am now in the process of typing this expanded proof up: at a new entry: differential string structure – proofs

]]>I am thinking by that extra factorization that I mentioned, these are sufficient to give all the rest automatically.

That’s what I’m sure of, too. That’s why I wrote that as soon as I have a minute I will write this computaton out. I was just hoping you had it already explicited. if not, no hurry to do it now.

]]>Hi Domenico,

what I have are the computations that you have seen in the entry. I am thinking by that extra factorization that I mentioned, these are sufficient to give all the rest automatically.

So in my message above there is one double-square diagram with an isomorphism in the middle. That is automatic: the remaining morphism are just those of the original double-square composed with the inverse of that iso.

Then there is the double-square which I called “the easy diagram”. Since that involves $\tilde W$ it is really evident: with the morphism all send generators of some name to the generator of the same name.

But, sure, I can write that out. Probably not tonight, though.

]]>That’s it! What I’m suggesting is: once we’ve used the auxiliary $\tilde{W}$ to find the seeked morphisms, let us write directly the composite morphisms and omit $\tilde{W}$. by the way, Urs, if you have already these computations, can you post them here? (otherwise I can work them out as soon as I have a minute for this). Have you also the explicit morphism

$\array{ CE(b\mathbb{R} \to \mathfrak{string}) &\to& CE(\mathfrak{so}) \\ \uparrow && \uparrow \\ W(b\mathbb{R} \to \mathfrak{string}) &\to& W(\mathfrak{so}) \\ \uparrow && \uparrow \\ inv(b\mathbb{R} \to \mathfrak{string}) &\to& inv(\mathfrak{so}) }$?

]]>I agree, that’s a good way to put it. Now next I would simply consider one more diagram factor: The isomorphism $W(b \mathbb{R} \to \mathfrak{string}) \stackrel{\simeq}{\to} \tilde W(b \mathbb{R} \to \mathfrak{string})$ induces another diagram

$\array{ CE(b\mathbb{R} \to \mathfrak{string}) &\stackrel{Id}{\to}& CE(b\mathbb{R} \to \mathfrak{string}) \\ \uparrow && \uparrow \\ \tilde W(b\mathbb{R} \to \mathfrak{string}) &\stackrel{\simeq}{\to}& W(b\mathbb{R} \to \mathfrak{string}) \\ \uparrow && \uparrow \\ inv(b\mathbb{R} \to \mathfrak{string}) &\stackrel{Id}{\to}& inv(b\mathbb{R} \to \mathfrak{string}) }$and moreover there is, by construction of $\tilde W$, an easy diagram

$\array{ CE(b^2 \mathbb{R}) &\stackrel{}{\to}& CE(b\mathbb{R} \to \mathfrak{string}) \\ \uparrow && \uparrow \\ W(b^2 \mathbb{R}) &\stackrel{}{\to}& \tilde W(b\mathbb{R} \to \mathfrak{string}) \\ \uparrow && \uparrow \\ inv(b^2 \mathbb{R}) &\stackrel{}{\to}& inv(b\mathbb{R} \to \mathfrak{string}) } \,.$Pasting this all together gives the desired factorization.

]]>so, here’s the compact story:

i) the cocycle $\mu: CE(b^2\mathbb{R})\to CE(\mathfrak{so})$ factors as

$CE(b^2\mathbb{R})\to CE(b\mathbb{R}\to\mathfrak{string})\to CE(\mathfrak{so}),$where the first morphism is given on the generators by $c\mapsto c$ and the second one by $c\mapsto \mu; \quad b\mapsto 0; t^a\mapsto t^a$ (see differential string structure for these notations). This factorization induces a factorization of $\exp_\Delta(b^2\mathbb{R})\to\exp_\Delta(\mathfrak{so})$ in

$\exp_\Delta(b^2\mathbb{R})\stackrel{\sim}{\to}\exp_\Delta(b\mathbb{R}\to\mathfrak{string})\twoheadrightarrow \exp_\Delta(\mathfrak{so})$ii) the commutative diagram

$\array{ CE(b^2\mathbb{R})& \to & CE(\mathfrak{so})\\ \uparrow & & \uparrow\\ W(b^2\mathbb{R})& \to & W(\mathfrak{so})\\ \uparrow & & \uparrow\\ inv(b^2\mathbb{R})& \to & inv(\mathfrak{so})\\ }$factors as

$\array{ CE(b^2\mathbb{R})& \to & CE(b\mathbb{R}\to\mathfrak{string})& \to & CE(\mathfrak{so})\\ \uparrow & & \uparrow& & \uparrow\\ W(b^2\mathbb{R})& \to & W(b\mathbb{R}\to\mathfrak{string})& \to & W(\mathfrak{so})\\ \uparrow & & \uparrow& & \uparrow\\ inv(b^2\mathbb{R})& \to & inv(b\mathbb{R}\to\mathfrak{string})& \to & inv(\mathfrak{so})\\ }$Here the upper morphisms are those describes above; before writing the middle morphisms on the generators (which I hope to be able to do soon) let me note that the commutativity of the left bottom corner

$\array{ W(b^2\mathbb{R})& \to & W(b\mathbb{R}\to\mathfrak{string})\\ \uparrow & & \uparrow\\ inv(b^2\mathbb{R})& \to & inv(b\mathbb{R}\to\mathfrak{string})\\ }$forces the generator $g$ to be mapped to an invariant polynomial in $W(b\mathbb{R}\to\mathfrak{string})$.

]]>Ah, now I understand. Yes, I think that’ right: the isomorphism from the ordinary to the modified Weil algebra is precisely that which collects the original shifted generators together to make them into invariant polynomials, so that it is then easy to see which combinations of these to send through subsequent maps.

]]>Concerning the question at the very end: a morphism out of the Weil algebra is given by a morphism out of the generating complex. So we get a morphism simply from the evident underlying projection. Or is this not what you are after here?

Yes, sure. This is the freeness property of the Weil algebra. Right, it is not this what I had in mind. My problem (maybe it’s trivial, I havent checked, yet) is we have to define a morphism between the Weil algebras that is compatible with the morphism of CE-algebras, and that preserves invariant polynomials.

My suspect (I may be wrong) is that one should see the modification of the Weil algebra come in already at this level, to write in a nice way this morphism. The fact that subsequent computations simplify would then be a consequence of this.

]]>Hi Domenico,

thanks for all this. So this is the equations needed to consider if one works not with the modified Weil algebra, right? It seems at this point kind of striking that working with the modified Weil algebra gives a much simpler computation. Why not just use that? The point of it is somehow that by making one simple transformation on the coeffients once and for all, all these complicated equations are taken care of at once.

Or am I overlooking something? You say something about justifying the modification. It is justified by organizing these kind of computations in one stroke, i think. Let me know if you think I am wrong about that.

Concerning the question at the very end: a morphism out of the Weil algebra is given by a morphism out of the generating complex. So we get a morphism simply from the evident underlying projection. Or is this not what you are after here?

]]>using the definitions of $C_{0,3}$ and $C_{1,2}$, the equation $d_\Delta C_{1,2}+d_U C_{0,3}=0$ is

$\pm \frac{1}{2}\mu([\omega_{0,1},\omega_{0,1}],\omega_{0,1},\omega_{1,0})\pm \frac{1}{2}\mu(\omega_{0,1},\omega_{0,1},d_\Delta \omega_{1,0}) \pm \frac{1}{2}\mu(d_U\omega_{0,1},\omega_{0,1},\omega_{0,1})=$ $\pm \frac{1}{2}\mu([\omega_{0,1},\omega_{0,1}],\omega_{0,1},\omega_{1,0})\pm \frac{1}{2}\mu(\omega_{0,1},\omega_{0,1},d_\Delta \omega_{1,0}+d_U\omega_{0,1})=$ $\pm \frac{1}{2}\mu([\omega_{0,1},\omega_{0,1}],\omega_{0,1},\omega_{1,0})\pm \frac{1}{2}\mu(\omega_{0,1},\omega_{0,1},[\omega_{0,1},\omega_{1,0}])=0$since $\mu$ is a cocycle. similarly I expect also the equation $d_\Delta C_{2,1}+d_U C_{1,2}=0$ to be automatically satisfied. let me assume for a moment this is true (there must be some better argument than the brute force computation above). then a $k$-simplex in $\exp_\Delta(b\mathbb{R}\to \mathfrak{string})_{conn}$ would be a triple $(\omega,B,C_{3,0})$ with

$d_\Delta \omega_{0,1} +\frac{1}{2}[\omega_{0,1},\omega_{0,1}]=0;\qquad d_\Delta\omega_{1,0} +d_U\omega_{0,1}+[\omega_{0,1},\omega_{1,0}]=0$and with a third equation relating $d_\Delta C_{3,0}$ with $\omega$ and $B$. The first two equations are nice: tehy say that $\omega$ is a $k$-simplex in $\exp_\Delta(\mathfrak{so})_{conn}$. the third equation, however, looks a bit awkward. this puzzled me for a while, since I finally understood it has to be so!

namely, $\mathfrak{g}\mapsto \exp_\Delta(\mathfrak{g})$ is a functor on $L_\infty$-algebras, but $\mathfrak{g}\mapsto \exp_\Delta(\mathfrak{g})_{conn}$ is not! the datum of a morphism of $L_\infty$-algebras $\mathfrak{g}\to \mathfrak{h}$ does not suffice to give a morphism of simplicial presheaves $\exp_\Delta(\mathfrak{g})_{conn}\to \exp_\Delta(\mathfrak{h})_{conn}$. rather $\mathfrak{g}\mapsto \exp_\Delta(\mathfrak{g})_{conn}$ is a functor defined on the category whose morphisms are triples

$\array{ CE(\mathfrak{g})&\leftarrow & CE(\mathfrak{h})\\ \uparrow &&\uparrow\\ W(\mathfrak{g})&\leftarrow & W(\mathfrak{h})\\ \uparrow &&\uparrow\\ inv(\mathfrak{g})&\leftarrow & inv(\mathfrak{h})\\ }$For $\mathfrak{g}=\mathfrak{so}$ and $\mathfrak{h}=(b\mathbb{R}\to\mathfrak{string})$ we are a priori given the morphism $CE(\mathfrak{g})\leftarrow CE(\mathfrak{h})$.

I strongly suspect that a "Chern-Simons" morphism $W(\mathfrak{g})\leftarrow W(\mathfrak{h})$ can actually be given in the case at hand, and that it will be precisely this morphism to involve and justify Urs’ modified equations. The problem they solve is (I guess): how d we define a dgca morphism

$W(\mathfrak{so})\leftarrow W(b\mathbb{R}\to \mathfrak{string})$?

]]>