PSM and Algebroids, Part V

Posted by Urs Schreiber

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[Update: The following has become section 13 of hep-th/0509163.]

Last time I discussed how the functorial definition of a $p$ -bundle with $p$ -connection can locally be differentiated to yield morphisms between $p$ -algebroids. Now I think I have figured out the differential version of the transition law describing the transformation of these algebroid morphisms from one patch to the other. The result is a formalism that allows you to derive the infinitesimal cocylce relations of a nonabelian $p$ -gerbe with curving and connection, etc. using just a couple of elementary steps.

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As discussed here a $p$ -bundle with $p$ -connection is completely encoded in a $p$ -holonomy $p$ -functor from a Čech-extended $p$ -path $p$ -groupoid to the structure (gauge) $p$ -group(oid).

When differentiating this statement we obtain a morphism between $p$ -algebroids. These are conveniently handled in their dual incarnation as differential graded algebras (dg-algebras).

The source dg-algebra is essentially just that of the exterior bundle, namely

(1)

\left( \mathbf{d},\bigwedge^\bullet \Gamma\left(T^*U\right) \right) \,,

where $U\to M$ is a good cover of the base space. The target dg-algebra $(\mathbf{d}^\mathfrak{g},\bigwedge^\bullet V^*)$ comes from a complex

(2)

\mathfrak{g}^* \overset{\mathbf{d}^\mathfrak{g}}{\to} \mathfrak{h}^* \oplus \left( \bigwedge^2 \mathfrak{g}^* \right) \overset{\mathbf{d}^\mathfrak{g}}{\to} \cdots

where $\g^*$ and $\h^*$ are the spaces dual to the Lie algebras $\g$ , $\h$ , … that describe the target $p$ -group and where $\mathbf{d}^\mathfrak{g}$ is the dual to $D = \sum_i \hat l_i$ , where $\hat l_i$ are the coderivations that define the corresponding $L_{\infty}$ algebra.

Now, the original holonomy functor becomes a connection morphism between these two dg-algebras, i.e. a chain map between the corresponding complexes

(3)

\con : \bigwedge^\bullet V^* \to \bigwedge^\bullet \Gamma\left(T^* U\right) \,.

If we write

(4)

Q = (\mathbf{d},\mathbf{d}^\mathfrak{g})

for the operator that acts on

(5)

\bigwedge^\bullet V^* \oplus \bigwedge^\bullet \Gamma\left(T^* U\right)

as $\mathbf{d}$ or $\mathbf{d}^\mathfrak{g}$ , respectively, then the property of being a chain map is equivalent to being $Q$ -closed.

(6)

[Q,\mathrm{con}] = 0 \,.

An (infinitesimal) gauge transformation between two such connection morphisms is just a chain homotopy

(7)

\con \to \widetilde \con \,,

which means that the two connections differ by a $Q$ -exact term

(8)

\widetilde \mathrm{con} = \mathrm{con} + [Q,l] \,,

where

(9)

l : \bigwedge^\bullet V^* \to \bigwedge^{\bullet-1} \Gamma(T^* U) \,,

and the bracket denotes the graded commutator. Let me call this a 1-gauge transformation. This is the differential version of a natural transformation between two $p$ -holonomy $p$ -functors.

But there are higher-order gauge transformations, corresponding to higher order morphisms in the $p$ -groupoid. Given any two $(n-1)$ -gauge transformations $l_{n-1}$ and $\tilde l_{n-1}$ we can have an $n$ -gauge transformation going between them

(10)

l_{n-1} \overset{l_n}{\to} \tilde l_{n-1}

iff

(11)

\tilde l_{n-1} - l_n = [Q,l_n] \,,

where

(12)

l_n : \bigwedge^\bullet V^* \to \bigwedge^{\bullet-n} \Gamma(T^* U) \,,

In other words, gauge equivalence classes of $n$ -morphisms for these $p$ -algebroids represented as dg-algebras are nothing but $Q$ -cohomology classes at grade $n$ .

So now let a global connection be given by local connection morphisms $\mathrm{con}_i$ on every patch and let them (infinitesimally) be related by 1-gauge transformations

(13)

\mathrm{con}_i \overset{\mathfrak{g}_{ij}}{\to} \mathrm{con}_j \,.

Then the differential version of the big diagram in section 3.6.1 of these notes looks as follows:

This says that there is a 2-gauge transformation

(14)

\mathfrak{f}_{ijk} : \mathfrak{g}_{ij} \circ \mathfrak{g}_{jk} \to \mathfrak{g}_{ik}

implying that

(15)

\mathfrak{g}_{ij} + \mathfrak{g}_{jk} = \mathfrak{g}_{ik} + [Q,\mathfrak{f}_{ijk}] \,.

When one works out what this simple equation says in terms of components one ideed finds that it expresses the infinitesimal version of the familiar

(16)

t(f_{ijk})g_{ik} = g_{ij}g_{jk}

as well as the otherwise rather formidable cocycle relations for the ‘2-connection transition function’ in a 2-bundle (1-gerbe) with 2-connection. This is spelled out in section 3.4 of the notes that I mentioned above.

Posted at May 16, 2005 4:05 PM UTC

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