Nonabelian Weak Deligne Hypercohomology

Posted by Urs Schreiber

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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.

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

Recall that the Čech complex

(1)

C = C(I) = \oplus_{n=1}^\infty C_n(I)

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

(2)

C_n = \left\langle \left\{ (i_1 i_2 \cdots i_n) | i_1,i_2,\dots,i_n \in I \right\} \right\rangle \,,

equipped with the boundary operator

(3)

\array{ \delta : C_n &\to& C_{n-1} \\ (i_1 \dots i_n) &\mapsto& \sum_{m=1}^n (-1)^{(m+1)} (i_i \dots \hat i_m \dots i_n) } \,.

Let

(4)

T_n = \left\{ f: \bigwedge^\bullet B^* \to \bigwedge^{(\bullet-n)} A^* \right\}

be the space of maps from $\bigwedge^\bullet B^*$ to $\bigwedge^\bullet A^*$ 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)

(\tilde \delta \omega)(i_1\dots i_n) = \omega(\delta(i_1\dots i_n)) \,.

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

(7)

(Q\omega)(i_1\dots i_n) = \mathbf{d}^B \circ \omega(i_1\dots i_n) + (-1)^n \omega(i_1\dots i_n) \circ \mathbf{d}^A \,.

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

(8)

D = \tilde \delta + (-1)^n Q \,.

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/ $(p-1)$ -gerbe and that the shift

(11)

\omega \to \omega + D\lambda

specifies an infinitesimal gauge transformation.

As a first sanity check, note that if $\mathbf{d}^B = 0$ , 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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May 17, 2005