## December 27, 2004

### Re: Categorified Gauge Theory

#### Posted by Urs Schreiber

Over on sci.physics.research Thomas Larsson is trying to understand how drastic the restriction of vanishing fake curvature in a 2-bundle/nonabelian gerbe really is. Does it imply that the nonabelian 2-bundle/gerbe can be ‘reduced’ in some sense to an abelian 2-bundle/gerbe?

To recall some notions, we need the following:

A crossed module $\left(G,H,t,\alpha \right)$, where $G$ and $H$ are Lie groups and $t:H\to G$ is a homomorphisms and $\alpha :G\to \mathrm{Aut}\left(H\right)$ an action of $G$ on $H$, together with its differential version $\left(𝔤,𝔥,\mathrm{dt},d\alpha \right)$. Given a good cover $\left\{{U}_{i}{\right\}}_{i\in I}$ of the base space $B$ there are $𝔤$-valued 1-forms ${A}_{i}$ and $𝔥$-valued 2-forms ${B}_{i}$ on each ${U}_{i}$ which induce a local connection 1-form

(1)${𝒜}_{\left(A,B\right)}={\oint }_{A}\left(B\right)$

on any path space ${𝒫}_{s}^{t}\left({U}_{i}\right)$. Here ${\oint }_{A}\left({\omega }_{1},\dots ,{\omega }_{n}\right)$ denotes the differential form on path space obtained by pulling back the target space forms ${\omega }_{j}$ to a given path and integrating them over a parameter $n$-simplex.

Given a bigon in ${U}_{i}$, i.e. a thin homotopy equivalence class of a smooth parametrized surface with two corners, there is a curve in path space mapping to that bigon and the surface holonomy of that bigon can be defined to be the ordinary holonomy of $𝒜$ along that curve. This notion of surface holonomy can be shown to compute 2-group holonomy and induce on the ${A}_{i}$ and ${B}_{i}$ the transformation laws of a nonabelian gerbe with connection and curving – but only if if the ‘fake curvature’ vanishes: ${F}_{A}+\mathrm{dt}\left(B\right)=0$. (This is for strict 2-groups and gets modified for coherent ones.)

Does this imply that we can compute the nonabelian surface holonomy of closed surfaces by integrating an abelian 3-form over a 3-volume?

In order to answer apply the nonabelian Stokes theorem on path space. The curvature of $𝒜$ can be shown to be

(2)${ℱ}_{𝒜}=d𝒜+𝒜\wedge 𝒜=-{\oint }_{A}\left({d}_{A}B\right)-{\oint }_{A}\left(d\alpha \left({T}_{a}\right)\left(B\right),{\left({F}_{A}+\mathrm{dt}\left(B\right)\right)}^{a}\right)$

i.e. for vanishing fake curvature

(3)${ℱ}_{𝒜}=-{\oint }_{A}\left({d}_{A}B\right)\phantom{\rule{thinmathspace}{0ex}}.$

This takes values in the abelian subalgebra $\mathrm{ker}\left(\mathrm{dt}\right)\subset 𝔥$.

If the 2-bundle/nonabelian gerbe induced an ordinary bundle on path space this would imply that the structure group of this bundle could be reduced to an abelian one. But this is not the case. Maybe a similar reduction is still possible, but I do not see how it would work. To see the subtleties, we can derive the nonabelian volume integral that computes the nonabelian surface holonomy at the boundary of its integration domain:

For starters, restrict attention to the case that the surface in question is the boundary $\partial V$ of a 3-dimensional submanifold $V$ in a single ${U}_{i}$. The 3-fold $V$ comes from a surface $\Sigma$ in path space and the path space holonomy over the boundary of that surface is by the nonabelian Stokes theorem given by the integral

(4)${\int }_{\Sigma }{T}_{𝒜}\left(ℱ\right)$

where ${T}_{𝒜}$ denotes the parallel transport of $ℱ$ along a curve of a foliation of $\Sigma$ (all in path space itself).

There is an implicit integral over the paths that are points in this integral. If you write that out the whole thing becomes roughly the integral of an abelian 3-form ${H}_{i}={d}_{{A}_{i}}{B}_{i}$ over $V$ but ${H}_{i}$ here at every point is parallel transported with ${A}_{i}$ to the $\sigma$-origin and with $𝒜$ to a $\tau$-origin, where $𝒜$ itself involves lots of $\sigma$-integrals.

In general, this does not seem to have any simple expression in terms of an ordinary integral. Of course, when the adjoint action of $𝔥$ on itself is trivial the whole thing simplifies a lot. And this action indeed is trivial for the crossed modules that I know of.

All that would remain in that case is the parallel transport with respect to ${A}_{i}$. Only if we also assume that the action of $G$ on $\mathrm{ker}\left(\mathrm{dt}\right)\subset 𝔥$ is trivial does the whole integral reduce to the ordinary

(5)${\int }_{V}{H}_{i}\phantom{\rule{thinmathspace}{0ex}}.$

Note that this assumed that we can work in a single patch in the first place. In the general case where surfaces in different patches have to be glued together using the 2-bundle/gerbe cocylce transition laws, things get more involved.

So obviously vanishing fake curvature requires the 2-bundle to be ‘close’ to being abelian, in a sense. How close this really is is not clear to me yet. It seems that the non-abelianness of the 2-group connection becomes relevant mostly for ‘global’ problems, like surfaces that wrap cycles, where we cannot in principle work in a single patch ${U}_{i}$.

In this context one is reminded of the fact that the membranes attached to 5-branes described by this formalism are required to wrap nontrivial cycles, too.

Posted at December 27, 2004 4:39 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/487

### Re: Re: Categorified Gauge Theory

Since I am on vacation (fortunately not in Asia) and the meter is ticking, here is a fast comment. The algebra I wrote down on spr is the most general Lie algebra with an abelian ideal; an ideal means that [J,e]- e and [e,e] - e, and abelian that [e,e] = 0 (I cant find tilde on the Spanish keyboard). The algebra with an ideal is thus fixed by 3 data:

A Lie algebra g - that fixes f^ab_c

A g-module m - that fixes T^ua_v

An extension of g by m - that fixes k^ab_u.

Fix only up to a change of basis, of course. Since I dont really understand fancy, mathematical notation, I do thing in a fixed basis which I do understand.

Now, I think that I understand what you do, and I agree that it is correct and well-defined. What I question is that the label “non-abelian 2-form gauge theory”, when the 2-form connection does not connect anything (it´s not part of some covariant derivative, right?), and the 3-form curvature is not non-abelian.

### Re: Re: Categorified Gauge Theory

I am afraid that I am not sure what your discussion of ideals is supposed to imply. I guess that we indeed agree what it means for $H$ to take values in an abelian ideal of some algebra and for $g$ to act on this algebra and hence on that ideal by automorphisms.

Now to your other points: The 2-connection does ‘connect’! It can be regarded either as an ordinary connection on path space, locally, or as a 2-functor from bigons into a 2-group, just like an ordinary connection can be regarded as a functor from paths to a group.

The connection $𝒜$ on path space has an ordinary curvature 2-form

(1)${ℱ}_{𝒜}=-{\oint }_{A}\left(H\right)\phantom{\rule{thinmathspace}{0ex}},$

where $H={d}_{A}B$ is called the ‘curvature 3-form’ even though this is slight abuse of terminology.

I have tried to discuss the problem of determining if a nonabelian gerbe with vanishing $\mathrm{dt}\left(B\right)+{F}_{A}$ is ‘reducible’ to an abelian gerbe in the above post.

Currently all I can say is that

a) there is indeed non-vanishing reason to worry that such a nonabelian gerbe really is just an abelian gerbe

b) but that it is not obvious at all and that I don’t know yet, since

c) there are in fact arguments indicating that it is not the case, which I tried to sketch.

d) Finally, as noted by others (like Hendryk Pfeiffer) before, all this discussion is just a warm-up for the case where the structure 2-group is weak, which for several reasons is expected to be the case relevant for physics.

Tomorrow a big paper on this stuff will appear and after that I will have time and leisure to further investigate these questions.

Posted by: Urs Schreiber on December 29, 2004 11:01 AM | Permalink | PGP Sig | Reply to this

### Re: Re: Categorified Gauge Theory

I am afraid that I am not sure what your discussion of ideals is supposed to imply. I guess that we indeed agree what it means for H to take values in an abelian ideal of some algebra and for g to act on this algebra and hence on that ideal by automorphisms.

The purpose is to translate things into a notation which I feel comfortable with. It must be possible to express your formulas in a given basis, and I convinced myself that I wrote down the most general basis possible.

Now to your other points: The 2-connection does ‘connect’! It can be regarded either as an ordinary connection on path space, locally, or as a 2-functor from bigons into a 2-group, just like an ordinary connection can be regarded as a functor from paths to a group

I don’t understand this. The path space curvature is not abelian, but your 3-form curvature is, right? Isn’t this a discrepancy?

d) Finally, as noted by others (like Hendryk Pfeiffer) before, all this discussion is just a warm-up for the case where the structure 2-group is weak, which for several reasons is expected to be the case relevant for physics.

Here you need that some H^3(G,K) != 0, no? Even if some cases are not ruled out by various Whitehead theorems, like G a loop algebra, there is no guarantee that the H^3 will be nonzero.

I wonder if somebody has calculated these H^3s. Once I wrote down some H^2(G,K) for G a current algebra (it’s the same thing as finding abelian extensions), by restriction from Askar Dzhumadildaev’s classification for G an algebra of vector fields, but I don’t know about H^3. Pavel Zuzmanovich, who was Askar’s student and occasionally posts to spr, claims that the problem is difficult.

### Re: Re: Categorified Gauge Theory

The path space curvature is not abelian, but your 3-form curvature is, right? Isn’t this a discrepancy?

No, both take value in the same subalgebra. Locally the path space curvature is

(1)${ℱ}_{𝒜}\left[X\right]={\oint }_{A}\left(B\right)\left[X\right]=\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{W}_{A}\left(\sigma ,1\right)\left({H}_{\mu \nu \lambda }\right)\left(X\left(\sigma \right)\right){X}^{\prime \mu }\left(\sigma \right){\mathrm{dX}}^{\nu }\left(\sigma \right)\wedge {\mathrm{dX}}^{\lambda }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$

Here the integral is over a given path $X:\left[0,1\right]\to M$, ${W}_{A}$ denotes the parallel transport with respect to $A$ along this path, $H$ is the ‘curvature 3-form’ and ${\mathrm{dX}}^{\mu }\left(\sigma \right)$ are the path space differential forms.

$H$ takes values in an abelian ideal and hence so does ${W}_{A}\left(H\right)$.

You can find that explained in detail in section 2.3.2 of our new preprint hep-th/0412325.

the problem is difficult

Yes, probably. More thinking is certainly necessary.

Posted by: Urs Schreiber on January 3, 2005 11:48 AM | Permalink | PGP Sig | Reply to this

### Re: Re: Categorified Gauge Theory

I should add that I did not want to stop the discussion on sci.physics.research. The reason why I wrote the above entry was that I thought some formulas would be helpful.

Posted by: Urs Schreiber on January 3, 2005 11:57 AM | Permalink | PGP Sig | Reply to this

Post a New Comment