The String Coffee Table

I am glad to announce that the paper

John Baez & Urs Schreiber: Higher Gauge Theory: 2-Connections on 2-Bundles (2004)

is available now on the preprint server as

hep-th/0412325.

(This link will be generally accessible next Sunday or Monday. If it does not work yet a copy can be found here.)

Here is the abstract:

Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics — first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the concept of ‘2-bundle’ recently introduced by Bartels. A 2-bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a ‘2-connection’ on a principal 2-bundle. We describe principal 2-bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for nonabelian gerbes with connection and curving subject to a certain constraint — namely, the vanishing of the ‘fake curvature’, as defined by Breen and Messing. This constraint also turns out to guarantee the existence of ‘2-holonomies’: that is, parallel transport over both curves and surfaces, fitting together to define a 2-functor from the `path 2-groupoid’ of the base space to the structure 2-group. We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base manifold.

Readers of this blog will recall many posts concerned with these issues.

The paper builds a bridge from work on local nonabelian surface holonomy as presented in

Alvarez, Ferreira & Sánchez Guillén: A new approach to integrable theories in any dimension (1997)

H. Pfeiffer Higher gauge theory and a non-Abelian generalization of 2-form electromagnetism (2003)

F. Girelli & H. Pfeiffer Higher gauge theory - differential versus integral formulation (2003)

U. Schreiber Nonabelian 2-forms and loop space connections from 2d SCFT deformations (2004)

to the cocycle description of nonabelian gerbes as described in

L. Breen & W. Messing: Differential geometry of gerbes (2001)

P. Aschieri, L. Cantini & B. Jurčo: Nonabelian bundle gerbes, their differential geometry and gauge theory (2003)

using the categorification approach of

J. Baez: Higher Yang-Mills theory (2002)

T. Bartels: Categorified gauge theory: 2-bundles (2004)

by demonstrating that a 2-bundle with 2-connection under certain conditions defines a nonabelian gerbe with connection and curving together with a notion of globally defined nonabelian 2-holonomy (surface holonomy) for that gerbe. (This uses strict structure 2-groups which implies that the ‘fake curvature’ $dt(B) + F_A$ has to vanish.)

The basic tools here are path space differential calculus on the one hand (‘calculus of string’) and category theory (‘stringification’) on the other. Roughly, an ordinary connection $\mathcal{A}$ on path space can be used to construct a 2-functor ${hol}$ from the 2-groupoid $\mathcal{P}_2(M)$ of ‘bigons’ (surface elements, roughly) to the structure 2-group $\mathcal{G}$

Having a notion of nonabelian surface holonomy for nonabelian gerbes is necessary for writing down actions for membranes ending on stacks of 5-branes and should hence also be required for nonabelian 6-dimensional conformal field theories and their relation to Yang-Mills theory in four dimensions.

As discussed in the last section of the above paper, the generalization to coherent structure 2-groups still needs to be properly done and is not treated in that paper. But hopefully in a followup.

Happy New Year!

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?

I am not sure. Here is what I can say about this question:

To recall some notions, we need the following:

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

on any path space $\mathcal{P}_s^t(U_i)$. Here $\oint_A(\omega_1, \dots, \omega_n)$ 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 $\mathcal{A}$ 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: $F_A + dt(B) = 0$. (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 $\mathcal{A}$ can be shown to be

i.e. for vanishing fake curvature

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

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

where $T_\mathcal{A}$ denotes the parallel transport of $\mathcal{F}$ 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 $H_i = \mathbf{d}_{A_i} B_i$ over $V$ but $H_i$ here at every point is parallel transported with $A_i$ to the $\sigma$-origin and with $\mathcal{A}$ to a $\tau$-origin, where $\mathcal{A}$ 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 $\mathfr{h}$ 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 ${ker}(dt) \subset \mathfr{h}$ is trivial does the whole integral reduce to the ordinary

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.

This is a followup to Peter Woit’s recent blog entry Langlands Program and Physics.

There, Peter mentions the following

Part of this story involves the Montonen-Olive duality of N=4 supersymmetric Yang-Mills. This duality interchanges the coupling constant with its inverse, whiile taking the gauge group G to the Langlands dual group (group with dual weight lattice). The symmetry that inverts the coupling constant is actually part of a larger ${SL}(2, \mathbb{Z})$ symmetry.

One possible explanation for this ${SL}(2,\mathbb{Z})$ symmetry is the conjectured existence of a six-dimensional superconformal QFT with certain properties. Witten explains more about this in his lectures at Graeme Segal’s 60th birthday conference in 2002. His article from the proceedings volume, entitled ‘Conformal Field Theory in Four and Six Dimensions’ doesn’t seem to be available online, but his slides are, and they cover much the same material.

These slides can be found here.

The abelian case is well understood. The ${SL}(2,\mathbb{Z})$ symmetry of abelian YM follows (at least classically obviously) from realizing it as a toroidal compactification of the theory of an abelian 2-form with self-dual field strength in six dimensions, where the ${SL}(2,\mathbb{Z})$ is just the modular group of the internal torus.

It is believed that something analogous holds true for nonabelian (super)Yang-Mills (for any A-D-E gauge group), i.e. that its Montone-Olive symmetry comes from a toroidal compactification of some 6-dimensional theory involving a non-abelian 2-form.

In this set of slides, Witten calls this nonabelian 6D theory a nonabelian gerbe theory. But certainly that is just a name, to be filled with content, right?

The most glaring problem with making this concrete seems to be this:

What precisely is the duality condition in the nonabelian case and under which conditions can it be imposed?

When I talked to nonabelian gerbe people about this, one thing they said is that it is not clear that in the nonabelian case the self-duality should still be ordinary Hodge self-duality, but that it might involve in addition to the Hodge star an operation on the Lie algebra factor. But I am not quite sure what that should be.

In lack of a better idea, let me assume in the following that we want ordinary Hodge duality. Now, one sufficient condition fulfilled by an ordinary bundle to admit a self-dual field strength is that the field strength transforms covariantly.

So if $U = \{ U_i \}_{i\in I}$ is a good covering of the base space with open sets and $F_{A_i}$ is the field strength on $U_i$, then on double overlaps

obviously.

Since the covariant transformation respects Hodge self-duality, it is consistent to impose Hodge self-duality in overlapping patches $U_i$.

It is not clear at all that this remains true in general for nonabelian gerbes!

For nonabelian gerbes the general transition law for the nonabelian 3-form field strenth $H_i$ has a covariant part

plus a mess of noncovariant terms

and in particular involving this term

(The notation here is taken from equation (55) in hep-th/0409200.)

Suppose we want $H$ to be Hodge self-dual and hence $H_i$ to be Hodge-self-dual on each $U_i$. This implies that on every double overlap all these additional terms in the above transition law have to be self-dual by themselves!

So self-duality on $H$ implies further self-duality conditions on the fields $A_i$, $B_i$, $a_{ij}$, $d_{ij}$ (which are the connection 1-form, it’s 2-form cousin and two ‘transition forms’ that measure the failure of $A_i$ and $B_i$ to transform as usual.)

But these fields don’t transform covariantly themselves. So the self-duality condition on them involves still more conditions, now on triple overlaps. And so on. It is a huge mess of ever more complicated conditions that arise this way. (Unless there is some simplifying principle hidden in them, which I currently cannot see.)

It will be hard to find solutions to these conditions. One solution, though, is easy to see. Obviously, for $H$ to be self-dual it is sufficient that

(actually this seems to be easy to weaken somewhat)

and

The big question is: Are there any further restrictions on the cocycle data of a nonabelian gerbe that would allow Hodge-self-dual H? In particular, are there any with ${ad}(B_i) + F_{A_i} \neq 0$?

The above choice is curious, since it implies that, while $A_i$ and $B_i$ are nonabelian, $H_i$ takes value in an abelian subalgebra of the full nonabelian Lie algebra.

It is also the only case so far in which we know (so far) how to associate a nonabelian 2-holonomy with the nonabelian gerbe. (A paper on that is due out by end of the year. Really, I should not be blogging but be working on that…)

The existence of that nonabelian 2-holonomy seems to be, apart from the self-duality of $H$, a further important condition on whatever Witten may mean by nonabelian gerbe field theory:

We known that when lifted to M-theory these nonabelian 6-D theories come from stacks of coinciding M5s with M2s ending in them. The action of these M2s should involve the abelian volume holonomy of an abelian 2-gerbe characterized by the 4-form $dC_3$, where $C_3$ is the supergravity 3-form potential, over the world-volume of the membrane, call that suggestively but by abuse of the integral notation $\exp(i \int_V C_3)$, times a nonabelian surface holonomy of the nonabelian 2-form living on the M5s over the worldsheet of the boundary of the M2, call that ${Tr}{hol}_{\partial V}(B)$.

Due to global issues (completely analogous to how the coupling of the string to an abelian 2-form involves abelian gerbe holonomy) the product

has a couple of subtleties. (For the case of 1-dimension lower these, and their solution, are nicely discussed in the above mentioned paper by Aschieri& Jurčo).

Therefore, in order to understand nonabelian theories in 6D (and, incidentally, the general configuration of the fundamental objects of M-theory) it would be very helpful to have a notion of nonabelian surface holonomy ${hol}_{\partial V}(B)$ that makes the above expression well-defined.

I do have a (global!) nonabelian surface holonomy for nonabelian 2-bundles and nonabelian gerbes for the case ${ad}(B_i) + F_{A_i} = 0$, i.e. for the only known case in which the existence of a self-dual 3-form field strength is known. But I have not yet checked if it makes the above action for the M2 brane globally well defined.

I was on the road again and then had some teaching to do, which kept me from replying to the comments to my last entry, which appeared on Luboš Motl’s weblog.

I currently find myself applying some category theory to string theory and made some comments on how I found the notion of categorization to harmonize very much with what one might call stringification, which should be some ‘section’ over the ‘space of theories’ in the bundle defined by the projection map given by taking the point particle limit of string dynamics - if you wish ;-). (Stringification is not a standard term at all, though something along these lines seems to have been discussed by A. Andrianov and A. Dynin at this conference in September this year, though I haven’t read (or in fact found) their articles.)

More concretely, the fact that boundaries of membranes attached to stacks of 5-branes conceptually roughly appear as a higher-dimensional generalization of how boundaries of strings (points) give rise to ordinary gauge theory by replacing these points with strings (and the strings with membranes) suggests a stringification of gauge theory, much like, I believe, replacing point particles in supersymmetric quantum mechanics with loops gives RNS strings themselves. And it turns out that if this process is regarded from the point of view of categorification which replaces points with ‘arrows’ (morphisms) it produces naturally the structure that is expected to describe these ‘gauge strings’ namely nonabelian gerbes.

As far as I understand Luboš does not doubt that this might be true, but he emphasizes (and has emphasized in previous discussions before) that one should not get lost in abstract formalism and lose sight of the physics.

I pointed out that first of all I believe that categories are not at all as detached to the physicists way of thinking as they may sometimes appear. On the contrary, the concept of a category is there to capture the essence of the concept of gauge/duality transformation, which is something very close to every physicist’s heart.

So here is the deal: We know that while all things which are the same are equal, some are less equal than others. The most prominent example for this is a similarity transformation, where $S$ and $T$ are regarded as equivalent whenever there is an invertible $\tau$ such that

There may be subtleties with inverting $\tau$ (i.e. in the cases in which there is not quite a similarity transformation but still an intertwining relation) so that a more safe way to express the same idea is to write

For this equation to be interesting, all the symbols $S$, $T$ and $\tau$ should denote maps of one sort or other. Since the maps $S$ and $T$ are equivalent if the above is true they should be thought of as the images $T(f)$, $S(f)$ of some archetypical map $f$, their equivalence class. But if $f$ is a map from $c$ to $c^\prime$,

the usefulness of the above concept of similarity transformation does not require $T(f)$ and $S(f)$ to go between the same ‘spaces’, but they could instead map from (what is conventionally written as) $Tc$ and $Sc$ to $Tc^\prime$ and $S c \prime$, respectively, for any other ‘spaces’ $Tc$, $Sc$, $Tc^\prime$, $Sc^\prime$:

$\,$

But this means that in general the map $\tau$ should depend on the ‘space’ it is coming from, so that we have a map $\tau(c)$

and a map $\tau(c^\prime)$

There is nothing deep or artificial here, one is just trying to allow for the most general situation in which a similarity transformation makes good sense. The notation might look like overkill for such a simple task, but it turns out that there are situations where keeping track of all these sources and targets is required and useful. I’ll give an example below.

So in conclusion, one finds that the essence of a ‘similarity transformation’ is that there are two maps $S(f)$ and $T(f)$ (which represent the same archetype $f$) and that they can be composed with maps $\tau$ in such a way that the composition

gives a map which equals the composition

If this is true one says that there is a natural transformation $\tau: S \Rightarrow T$ from $S$ to $T$. If $\tau$ is invertible (which is the case more directly related to the concept of a similarity transformation), there is also a natural transformation $\tau^{-1}$ going from $T$ to $S$ and $\tau$ is called a natural isomorphism.

(With respect to Luboš’s comments it should be noted that just because one draws arrows here does not make any of this ‘discrete’ in a general sense. The natural isomorphisms are just as discrete or non-discrete as any equation $A = B$ would be, with something appearing on the left and something appearing on the right. Even though below I want to identify strings with morphisms, this does not mean at all that one has to think about discretized ‘bits’ of string in any way. )

This is nothing but the careful analysis of the boring old concept of a similarity (or intertwining) transformation.

Category theory is really not more nor less than some nomenclature to describe what I tried to describe above. In that nomenclature one calls the ‘spaces’ $c$ and $c^\prime$ objects, calls $f$ a morphism between these objects, calls $T$ and $S$ functors between these morphisms and, well, calls $\tau$ a natural transformation (or a natural isomorphism if invertible). But this are nothing but fancy words. All that has happened is that the concept of similarity transformation has been formulated in a general way such that all of its essence is captured.

So for instance consider a state of some Yang-Mills gauge theory. It can be defined by specifying for every path in spacetime an associated element of the gauge group (its holonomy). But several such states are to be considered equivalent if there is a gauge transformation relating them. One can easily convince oneself that this situation is equivalently described in terms of the above fancy category theoretic nomenclature by saying that

- a state in the gauge theory is a functor from the groupoid of paths in spacetime to the gauge group (regarded as a group of morphisms on a single object)

- a gauge transformation between two states is a natural isomorphism between two such functors .

This is just a different and maybe somewhat fancy way to describe precisely the ordinary concept of a gauge transformation. Category theory is all about gauge transformations.

The point is just that it so happens that by reformulating the concept of gauge transformation in terms of more abstract sounding concepts like ‘functors’ and ‘natural transformations’ people were able to usefully recognize and understand the presence of gauge equivalences in cases less trivial than that of the above example, as for instance briefly summarized by John Baez here.

Instead of talking about these major applications, I would like to sketch again (what I did in entries (I) and (II) before) how by categorifying ordinary gauge theory one rather easily finds nonabelian gerbes, and how this is precisely stringification of gauge theory.

So the deal is this: In order to categorify something we take all elements of that something which are not maps yet and think of them as maps (morphisms) from something to something else, i.e. we take all the points that are there and now think of them as strings. Analogously, all maps must become natural transformations (‘strings become membranes’) and so on. This way one lifts up everything one dimension and thinks of what was structureless before (a point) as now having internal structure (‘oscillations’) coming from its linear extension (its stringiness). Pointlike things are either equal or not, but linear things can be equal up to similarity transformations, i.e. up to natural isomorphisms. So what was an equation between points in the original theory becomes a natural isomorphism between morphisms in the categorified theory.

I want to categorify gauge theory. A state of gauge theory is a principal fiber bundle with connection. This again is defined by

- a good cover $U$ of the base space $M$ of the bundle $E$

- group $G$-avlued 0-forms $g_{ij}$ on double overlaps $U_{ij}$

- ${Lie}(G)$-valued 1-orms $A_i$ on single overlaps

such that the following equations hold:

On triple overlaps $U_{ijk}$ we have

and on double overlaps $U_{ij}$ we have

where ${hol}(A)$ denotes the holonomy of $A$.

Now categorify. Points must become strings. So now base space $M$ becomes a 2-space, that of based loops (strings), which we should think of as morphisms from their basepoint to itself. The above maps $g_{ij}$ and $A_i$ now become functors $g_{ij}^2$ and $A_i^2$ from double overlaps of our stringy base space into a categorified version of the gauge group, which is called a 2-group. Finally, the transition equations become natural isomorphisms

and

where all objects appearing are stringified/categorified. So the group product $g^2 \cdot g^2$ is no longer a map but a functor itself, ${hol}(A^2_i)$ is also a functor which now computes holonomy of surfaces instead of of lines.

Now all one has to do is apply the definition of a natural isomorphism mentioned at the beginning of this post, to see what these categorified transition laws amount to in terms of local $p$-forms. Note how the appearance of natural isomorphisms where before only equations have been inserts a new level of gauge transformation. Gauge strings have 1-gauge transformations and also 2-gauge transformations. Category theory is the language that describes all sorts of gauge transformations.

So if you work it out (for details and proofs see my notes) you get (after having clarified some basic issues of path space differential geometry) rather easily that the existence of the above two natural transformations encodes

- group-valued 0-forms $f_{ijk}$ on triple overlaps

and

- Lie-algebra valued 1-forms $a_{ij}$ on double overlaps such that the functorial version of $g_{ik} = g_{ij}g_{jk}$ becomes equivalent to

(this is the object part of the functor) and

(this is the coherence law on the natural transformation)

while the functorial version of ${hol}(A_i) = g_{ij}{hol}(A_j)g_{ij}^{-1}$ is equivalent to

(this is again the object part of the functor) and

(which is the morphism part of that functor, where $B$ is a 2-form describing the surface holonomy and $k_{ij}$ is the curvature of $a_{ij}$ with respect to $A$)

whose coherence law says that

These are precisely the transition laws of a nonabelian gerbe.

All that distinguishes a gerbe from a bundle comes through the higher-dimensional nature of the categorification process. In particular, the general notion of gauge transformation manifested in the concept of a natural isomorphism enters crucially:

It is easy to convince oneself that the maps ${hol}(A_i^2)$ and $g_{ij}^2 \cdot {hol}(A^2_j)(g_{ij}^2)^{-1}$ don’t go between the same source and target objects! Hence without the general notion of gauge transformation embodied in the concept of a natural isomorphism we wouldn’t even know how these two maps could be equivalent. But it turns out that there is a $\tau$ going between them as described as the beginning, and it is encoded in that 1-form $a_{ij}$.

One could talk about many more details here, like how by turning other equations implicit in the ordinary idea of a bundle into natural transformations allows to get twisted nonabelian gerbes (which ‘are’ actually abelian 2-gerbes) or how turning the equation between $(g_1(g_2 g_3))$ and $(g_1g_2) g_3)$ into a natural isomorphisms yields degrees of freedom carrying three ‘group indices’ - all nice examples of how the general concept of gauge transformation appearing in category theory is physically very useful, but I would rather like to conclude with something else:

To me, the above procedure by which categorifying gauge theory yields stringified gauge physics suggests that something more general should hold.

I believe I am beginning to see how the following conjecture can be made precise and be proven:

Perturbative RNS string theory is a categorification of supersymmetric quantum mechanics and natural isomorphisms in this context describe gauge and duality transformations of superstring backgrounds.

I have mentioned related ideas many times before and am guaranteed to bore any half-way regular reader of this weblog, but the idea is just so appealing that I cannot resist doing it once again (adding a little more detail):

Let $M$ be the bosonic configuration space of a relativistic ($N=2$) superparticle such that the full super config space is the exterior bundle $\Omega(M)$. Let $d : \Omega(M) \to \Omega(M)$ be the deRham operator on that bundle. Now a given dynamics of (i.e. background fields for) that superparticle is described by another operator $d^W : \Omega(M) \to \Omega(M)$ such that the following equation holds

for $W : \Omega(M) \to \Omega(M)$ any even graded operator on $\Omega(M)$.

Now categorify this. $M$ will become a 2-space as in the above example of categorified gauge theory, with its arrow space being the configuration space of a closed string. There should be a notion of the exterior 2-bundle over that 2-space (it is actually pretty obvious, though there is one detail of this thing, if it exits, which is still puzzling me). In any case, there is a family of 2-maps $\mathbf{d}$ on that exterior 2-bundle which are odd graded and nilpotent on rep-invariant sections, and the categorified version of specifying a dynamics (a background field configuration) is given by specifying a natural transformation between $e^{-W}\mathbf{d}e^{W}$ one and another such map. It seems that this way one ends up with 2D SCFT and with natural isomoprhisms giving gauge and duality transformations of superstring backgrounds. Then the above gauge string theory can be regarded as just a special case of this, in some sense which would be needed to be made precise.