May 17, 2005

Nonabelian Weak Deligne Hypercohomology

Posted by Urs Schreiber

What I described last time is really best thought of in the context of what I propose to call nonabelian weak Deligne hypercohomolgy.

Unless I am hallucinating the following is the correct formalism to generalize the well-known Deligne hypercohomology formulation of strict abelian $p$-gerbes to weak and nonabelian $p$-gerbes.

Consider two $p$-algebroids represented by dg-algebras $\left({d}^{A},{\bigwedge }^{•}{A}^{*}\right)$ and $\left({d}^{B},{\bigwedge }^{•}{B}^{*}\right)$ and let $I$ be some countable set.

Recall that the Čech complex

(1)$C=C\left(I\right)={\oplus }_{n=1}^{\infty }{C}_{n}\left(I\right)$

is the free abelian group generated by all tuples of elements of I,

(2)${C}_{n}=〈\left\{\left({i}_{1}{i}_{2}\cdots {i}_{n}\right)\mid {i}_{1},{i}_{2},\dots ,{i}_{n}\in I\right\}〉\phantom{\rule{thinmathspace}{0ex}},$

equipped with the boundary operator

(3)$\begin{array}{ccc}\delta :{C}_{n}& \to & {C}_{n-1}\\ \left({i}_{1}\dots {i}_{n}\right)& ↦& \sum _{m=1}^{n}\left(-1{\right)}^{\left(m+1\right)}\left({i}_{i}\dots {\stackrel{̂}{i}}_{m}\dots {i}_{n}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

Let

(4)${T}_{n}=\left\{f:\stackrel{•}{\bigwedge }{B}^{*}\to \stackrel{\left(•-n\right)}{\bigwedge }{A}^{*}\right\}$

be the space of maps from ${\bigwedge }^{•}{B}^{*}$ to ${\bigwedge }^{•}{A}^{*}$ of degree $-n$.

Let

(5)$\Omega =\left\{{\omega }_{n}:{C}_{n}\to {T}_{n}\right\}$

be the space of linear maps from the Čech complex to dg-algebra maps.

On $\Omega$ the Čech boundary operator acts as

(6)$\left(\stackrel{˜}{\delta }\omega \right)\left({i}_{1}\dots {i}_{n}\right)=\omega \left(\delta \left({i}_{1}\dots {i}_{n}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

Also the operator $Q$ which I introduced last time acts as

(7)$\left(Q\omega \right)\left({i}_{1}\dots {i}_{n}\right)={d}^{B}\circ \omega \left({i}_{1}\dots {i}_{n}\right)+\left(-1{\right)}^{n}\omega \left({i}_{1}\dots {i}_{n}\right)\circ {d}^{A}\phantom{\rule{thinmathspace}{0ex}}.$

Both these operators make $\Omega$ into a complex. On the resulting double complex we have the total differential

(8)$D=\stackrel{˜}{\delta }+\left(-1{\right)}^{n}Q\phantom{\rule{thinmathspace}{0ex}}.$

I believe that the infinitesimal cocycle laws and gauge transformations of a nonabelian weak $p$-bundle with $p$-connection (or equivalently a nonabelian weak $p-1$-gerbe) are specified by the cohomology class of a cocycle of $D$, i.e. that a $D$-closed element

(9)$\omega \in \Omega$

with

(10)$D\omega =0$

specifies the infinitesimal cocycle relations of such a $p$-bundle/$\left(p-1\right)$-gerbe and that the shift

(11)$\omega \to \omega +D\lambda$

specifies an infinitesimal gauge transformation.

As a first sanity check, note that if ${d}^{B}=0$, i.e. when the target $p$-algebroid is strict and abelian, the above $D$ reduces to the usual abelian Deligne coboundary operator (up to signs that I mixed up, possibly). Also, the content of the last entry can be seen to be a special case of this.

A more detailed discussion follows.

(A little more details are given in section 3.3 and section 3.5.4 here.)

Posted at May 17, 2005 1:46 PM UTC

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