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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 27th 2012
   • (edited Jan 27th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am wondering about a way to understand _[[generalized complex geometry]]_ as a special case of twisted nonabelian cohomology along the lines indicated at [[twisted differential c-structures]]. $\,$ Let $X$ be a smooth manifold of [[dimension]] $d$. Consider the [[smooth infinity-groupoid|smooth moduli stacks]] $\mathbf{B} \; (O(d)\times O(d))$ and $\mathbf{B} O(d,d)$. Let the twisting characteristic map be the evident $$ \mathbf{c} \; : \; \mathbf{B} \; (O(d) \times O(d)) \to \mathbf{B} O(d,d) \,. $$ Notice that the [[homotopy fiber]] of this is the coset manifold $O(d,d)/(O(d) \times O(d))$; we have a [[fiber sequence]] $$ O(d,d)/(O(d) \times O(d)) \to \mathbf{B}\, (O(d) \times O(d)) \to \mathbf{B}\, O(d,d) $$ in [[Smooth∞Grpd]]. We also have an essentially canonical map of stacks $$ T X \otimes T^*X \; : \; X \to \mathbf{B} O(d,d) \,. $$ With this, the groupoid of $T X \otimes T^*X$-twisted $\mathbf{c}$-structures is the homotopy pullback in $$ \array{ \mathbf{c}Struc_{T X \otimes T* X}(X) &\to& * \\ \downarrow &\swArrow_\simeq^e& \downarrow^{\mathrlap{T X \otimes T* X}} \\ \mathbf{H}(X, \mathbf{B} \, (O(d) \times O(d))) & \stackrel{\mathbf{H}(X, \mathbf{c})}{\to} & \mathbf{H}(X, \mathbf{B} O(d,d)) } \,. $$ Objects of this groupoid are pairs consisting of an $O(d) \times O(d)$-principal bundle (or equivalently its associated [[Courant algebroid]]) and a "generalized vielbein" $e$ (as for instance on [page 6 here](https://xxx.lanl.gov/pdf/0807.4527v1.pdf#page=6)). Morphisms are $O(d)\times O(d)$-gauge transformations, acting on both in the evident way. Alternatively, forming the internal such pullback yields the moduli stack of generalized complex geometry structures (metric and flat $B$-field) on $X$. $\,$ It seems. $\,$ In particular, in the case that we restrict to a subset $U \subset X$ for which $T U$ is trivial, we have be the fiber sequence above an equivalence $$ \begin{aligned} \mathbf{c}Struc_{T X \otimes T* X}( U ) & \simeq \mathbf{H}(U , O(d,d)/(O(d) \times O(d))) \\ & \simeq C^\infty( U, \; O(d,d)/( O(d) \times O(d) ) ) \end{aligned} \,. $$ This is all for flat $B$-field. For general $B$-fields there is an evident generalization of the above. But I won't talk about that for the moment.

   I am wondering about a way to understand generalized complex geometry as a special case of twisted nonabelian cohomology along the lines indicated at twisted differential c-structures.

   $\,$

   Let $X$ be a smooth manifold of dimension $d$.

   Consider the smooth moduli stacks $\mathbf{B} \; (O(d)\times O(d))$ and $\mathbf{B} O(d,d)$. Let the twisting characteristic map be the evident

   $$\mathbf{c} \; : \; \mathbf{B} \; (O(d) \times O(d)) \to \mathbf{B} O(d,d) \,.$$

   Notice that the homotopy fiber of this is the coset manifold $O(d,d)/(O(d) \times O(d))$; we have a fiber sequence

   $$O(d,d)/(O(d) \times O(d)) \to \mathbf{B}\, (O(d) \times O(d)) \to \mathbf{B}\, O(d,d)$$

   in Smooth∞Grpd.

   We also have an essentially canonical map of stacks

   $$T X \otimes T^*X \; : \; X \to \mathbf{B} O(d,d) \,.$$

   With this, the groupoid of $T X \otimes T^*X$-twisted $\mathbf{c}$-structures is the homotopy pullback in

   $$\array{ \mathbf{c}Struc_{T X \otimes T* X}(X) &\to& * \\ \downarrow &\swArrow_\simeq^e& \downarrow^{\mathrlap{T X \otimes T* X}} \\ \mathbf{H}(X, \mathbf{B} \, (O(d) \times O(d))) & \stackrel{\mathbf{H}(X, \mathbf{c})}{\to} & \mathbf{H}(X, \mathbf{B} O(d,d)) } \,.$$

   Objects of this groupoid are pairs consisting of an $O(d) \times O(d)$-principal bundle (or equivalently its associated Courant algebroid) and a “generalized vielbein” $e$ (as for instance on page 6 here). Morphisms are $O(d)\times O(d)$-gauge transformations, acting on both in the evident way.

   Alternatively, forming the internal such pullback yields the moduli stack of generalized complex geometry structures (metric and flat $B$-field) on $X$.

   $\,$

   It seems.

   $\,$

   In particular, in the case that we restrict to a subset $U \subset X$ for which $T U$ is trivial, we have be the fiber sequence above an equivalence

   $$\begin{aligned} \mathbf{c}Struc_{T X \otimes T* X}( U ) & \simeq \mathbf{H}(U , O(d,d)/(O(d) \times O(d))) \\ & \simeq C^\infty( U, \; O(d,d)/( O(d) \times O(d) ) ) \end{aligned} \,.$$

   This is all for flat $B$-field. For general $B$-fields there is an evident generalization of the above. But I won’t talk about that for the moment.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 27th 2012
   • (edited Jan 27th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor completeness one should first look at the analous statement for ordinary Riemannian geometry. I have added this now to _[[vielbein]]_: a discussion of Riemannian structures as in terms of $\mathbf{c}$-twisted nonabelian cohomology, for $\mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n)$. Of course here you may feel that this is a bit of overkill. But I think it is good to see the pattern start here. By just following our nose from here we easily find [[generalized complex geometry]], then [[exceptional generalized geometry]] and (in principle) so on.

   For completeness one should first look at the analous statement for ordinary Riemannian geometry.

   I have added this now to vielbein: a discussion of Riemannian structures as in terms of $\mathbf{c}$-twisted nonabelian cohomology, for $\mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n)$.

   Of course here you may feel that this is a bit of overkill. But I think it is good to see the pattern start here. By just following our nose from here we easily find generalized complex geometry, then exceptional generalized geometry and (in principle) so on.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 28th 2012
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   Author: David_Corfield
   Format: MarkdownItexWith talk of a 'pattern' and 'following our nose', is there some preferred restricted class of $c$ for which this construction is typically of most interest, or is it rather wide open to all characteristic classes?

   With talk of a ’pattern’ and ’following our nose’, is there some preferred restricted class of $c$ for which this construction is typically of most interest, or is it rather wide open to all characteristic classes?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> is there some preferred restricted class of $\mathbf{c}$ for which this construction is typically of most interest, or is it rather wide open to all characteristic classes? First I should say, by the way, that this $\mathbf{c}$-twisted cohomology is just ordinary cohomology with coefficients in $\mathbf{c}$ in the slice $\infty$-topos over the codomain of $\mathbf{c}$. Therefore the question is essentially more generally: "is there a preferred coefficient object for cohomology". And phrased this way I think the answer is "no, all of them are interesting for something". Because, after all, cohomology is given by hom-spaces and what would it mean to have hom-objects with "preferred" codomains? On the other hand, in every single application there are of course coefficient objects of more and of less interest. Just recently in our 5-brane work we had occasion to single out those $\mathbf{c}$ with the property that under geometric realization they become identities. Here now, in the application to "$G$-structures", it happens to be those $\mathbf{c}$ that come from group inclusions which are of interest. So eventually I think an interesting question would be a sort of converse: which properties of $\mathbf{c}$ make it the twist that controls which kind of application?

   is there some preferred restricted class of $\mathbf{c}$ for which this construction is typically of most interest, or is it rather wide open to all characteristic classes?

   First I should say, by the way, that this $\mathbf{c}$-twisted cohomology is just ordinary cohomology with coefficients in $\mathbf{c}$ in the slice $\infty$-topos over the codomain of $\mathbf{c}$.

   Therefore the question is essentially more generally: “is there a preferred coefficient object for cohomology”. And phrased this way I think the answer is “no, all of them are interesting for something”. Because, after all, cohomology is given by hom-spaces and what would it mean to have hom-objects with “preferred” codomains?

   On the other hand, in every single application there are of course coefficient objects of more and of less interest. Just recently in our 5-brane work we had occasion to single out those $\mathbf{c}$ with the property that under geometric realization they become identities.

   Here now, in the application to “$G$-structures”, it happens to be those $\mathbf{c}$ that come from group inclusions which are of interest.

   So eventually I think an interesting question would be a sort of converse: which properties of $\mathbf{c}$ make it the twist that controls which kind of application?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 31st 2012
   • (edited Jan 31st 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have typed up some rudiments of the above story in * [section 4.4.1](http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=322) _Reduction of structure groups in terms of twisted cohomology_; * [section 4.4.2](http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=323) _Orthogonal/Riemannian structures in terms of twisted cohomology_; * [section 4.4.3](http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=325) _Type II generalized geometry in terms of twisted nonabelian cohomology_ . More to be done, but it's a start.

   I have typed up some rudiments of the above story in

   • section 4.4.1 Reduction of structure groups in terms of twisted cohomology;

   • section 4.4.2 Orthogonal/Riemannian structures in terms of twisted cohomology;

   • section 4.4.3 Type II generalized geometry in terms of twisted nonabelian cohomology .

   More to be done, but it’s a start.