## May 16, 2005

### PSM and Algebroids, Part V

#### Posted by urs

[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.

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)$\left(d,\stackrel{•}{\bigwedge }\Gamma \left({T}^{*}U\right)\right)\phantom{\rule{thinmathspace}{0ex}},$

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

(2)${𝔤}^{*}\stackrel{{d}^{𝔤}}{\to }{𝔥}^{*}\oplus \left(\stackrel{2}{\bigwedge }{𝔤}^{*}\right)\stackrel{{d}^{𝔤}}{\to }\cdots$

where ${g}^{*}$ and ${h}^{*}$ are the spaces dual to the Lie algebras $g$, $h$, … that describe the target $p$-group and where ${d}^{𝔤}$ is the dual to $D={\sum }_{i}{\stackrel{̂}{l}}_{i}$, where ${\stackrel{̂}{l}}_{i}$ 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)$con:\stackrel{•}{\bigwedge }{V}^{*}\to \stackrel{•}{\bigwedge }\Gamma \left({T}^{*}U\right)\phantom{\rule{thinmathspace}{0ex}}.$

If we write

(4)$Q=\left(d,{d}^{𝔤}\right)$

for the operator that acts on

(5)$\stackrel{•}{\bigwedge }{V}^{*}\oplus \stackrel{•}{\bigwedge }\Gamma \left({T}^{*}U\right)$

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

(6)$\left[Q,\mathrm{con}\right]=0\phantom{\rule{thinmathspace}{0ex}}.$

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

(7)$con\to \stackrel{˜}{con}\phantom{\rule{thinmathspace}{0ex}},$

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

(8)$\stackrel{˜}{\mathrm{con}}=\mathrm{con}+\left[Q,l\right]\phantom{\rule{thinmathspace}{0ex}},$

where

(9)$l:\stackrel{•}{\bigwedge }{V}^{*}\to \stackrel{•-1}{\bigwedge }\Gamma \left({T}^{*}U\right)\phantom{\rule{thinmathspace}{0ex}},$

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 $\left(n-1\right)$-gauge transformations ${l}_{n-1}$ and ${\stackrel{˜}{l}}_{n-1}$ we can have an $n$-gauge transformation going between them

(10)${l}_{n-1}\stackrel{{l}_{n}}{\to }{\stackrel{˜}{l}}_{n-1}$

iff

(11)${\stackrel{˜}{l}}_{n-1}-{l}_{n}=\left[Q,{l}_{n}\right]\phantom{\rule{thinmathspace}{0ex}},$

where

(12)${l}_{n}:\stackrel{•}{\bigwedge }{V}^{*}\to \stackrel{•-n}{\bigwedge }\Gamma \left({T}^{*}U\right)\phantom{\rule{thinmathspace}{0ex}},$

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)${\mathrm{con}}_{i}\stackrel{{𝔤}_{\mathrm{ij}}}{\to }{\mathrm{con}}_{j}\phantom{\rule{thinmathspace}{0ex}}.$

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)${𝔣}_{\mathrm{ijk}}:{𝔤}_{\mathrm{ij}}\circ {𝔤}_{\mathrm{jk}}\to {𝔤}_{\mathrm{ik}}$

implying that

(15)${𝔤}_{\mathrm{ij}}+{𝔤}_{\mathrm{jk}}={𝔤}_{\mathrm{ik}}+\left[Q,{𝔣}_{\mathrm{ijk}}\right]\phantom{\rule{thinmathspace}{0ex}}.$

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)$t\left({f}_{\mathrm{ijk}}\right){g}_{\mathrm{ik}}={g}_{\mathrm{ij}}{g}_{\mathrm{jk}}$

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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