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December 3, 2004

Re: Category Theory and Physics

Posted by urs

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 τ such that

(1)T=τ 1 Sτ.

There may be subtleties with inverting τ (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 τ 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 ,

(3) f c c

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 and Sc, respectively, for any other ‘spaces’ Tc, Sc, Tc , Sc :

(4) T(f) Tc Tc

(5) S(f) Sc Sc .

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

(6) τ(c) Sc Tc

and a map τ(c )

(7) τ(c ) Sc Tc .

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 τ in such a way that the composition

(8) S(f) τ(c ) Sc Sc Tc

gives a map which equals the composition

(9) τ(c) T(f) Sc Tc Tc .

If this is true one says that there is a natural transformation τ:ST from S to T. If τ is invertible (which is the case more directly related to the concept of a similarity transformation), there is also a natural transformation τ 1 going from T to S and τ 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 objects, calls f a morphism between these objects, calls T and S functors between these morphisms and, well, calls τ 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

(10)g ik=g ijg jk

and on double overlaps U ij we have

(11)hol(A i)=g ijhol(A i)g ij 1 ,

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

(12)g ik 2 g ij 2 g jk 2


(13)hol(A i 2 )g ij 2 hol(A j 2 )(g ij 2 ) 1

where all objects appearing are stringified/categorified. So the group product g 2 g 2 is no longer a map but a functor itself, hol(A i 2 ) 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


- Lie-algebra valued 1-forms a ij on double overlaps such that the functorial version of g ik=g ijg jk becomes equivalent to

(14)g ijg jk=Ad f ijkg ik

(this is the object part of the functor) and

(15)f ikl 1 f ijk 1 g ij(f jkl)f ijl

(this is the coherence law on the natural transformation)

while the functorial version of hol(A i)=g ijhol(A j)g ij 1 is equivalent to

(16)A i+ad a ij=g ij(d+A j)g ij 1

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

(17)B i=g ij(B j)+k ij

(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

(18)a ij+g ij(a jk)f ijka ikf ijk 1 f ijkdf ijk 1 f ijk 1 A i(f ijk).

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 hol(A j 2 )(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 τ 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 1 g 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 Ω(M). Let d:Ω(M)Ω(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:Ω(M)Ω(M) such that the following equation holds

(19)d W=e Wde W

for W:Ω(M)Ω(M) any even graded operator on Ω(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 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 Wde 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.

Posted at December 3, 2004 4:44 PM UTC

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Re: Re: Category Theory and Physics

For perturbative gauge theory it is sufficient to deal with trivial bundles. Is it similarly sufficient to deal with `trivial gerbes’ (what are these?) for the purpose of writing an action, gauge tansformations etc?

Posted by: Amitabha on December 4, 2004 5:43 AM | Permalink | Reply to this

Trivial Gerbes

Yes, I have vaguely thought about this before, too. Even if there are global effects in physics we could still claim that first of all we need to understand the globally trivial case.

I see no reason not to expect something similar to be true for gerbes, too. So for physical applications it might well be that trivial gerbes are at first of the most importance, while nontrivial topologies would enter only later as sort of an afterthought.

Here by trivial gerbes I mean those where you can choose your covering to consist of just a single open set, being the entire base space.

However, I also believe that this does not make the study of these cocycle conditions (the various transitions laws for the local data) obsolete. That’s because these transition laws are formally nothing but gauge transformations (the transition from one patch to another is locally just a gauge transformation in the obvious sense), and gauge transformations need to be understood in any case.

So if you like to ignore global issues for the moment you could just as well go ahead and replace the transition laws and their categorification which I discussed above,

(1)g ik=g ijg jk
(2)hol(A i)=g ijhol(A j)g ij 1

by what is really a generalization of them, namely the gauge transformation laws

(3)g˜ ij=u ig iju j 1
(4)hol(A˜ i)=u ihol(A i)u i 1

without changing any of the conclusions.

I have a short comment on that in section 2.4.4. of my notes, though I haven’t had the time yet to work out all the details.

Right now the most important aspect of deriving the cocylce definition of the nonabelian gerbe from 2-bundles with 2-connections is to ensure that 2-bundles with 2-connection really are the same thing as nonabelian gerbes. This ensures that both languages are compatible. Since both of them have their advantages and disadvantages, this is good to know.

On the other hand, as Paolo Aschieri and Branislav Jurčo emphasize in their paper, the generic bundle appearing on a D-brane configuration is not only non-trivial but even twisted, and that the same should be true for gerbes on 5-branes. Therefore it is good to have a good undrestanding of the transition mechanism.

But yes, writing down a good Lagrangian for a 2-gauge theory is probably still the major open task. I know you and other’s have made various plausible suggestions. But would you agree that the question is still rather open? After all it could well be that no Lagrangian description for these 5-branes can exist, right?

In fact, I am beginning to appreciate a comment made a while ago by Pfeiffer and others, that maybe one should look for a categorization of the concept of ‘action’ and ‘Lagrangian’. But that seems to be really a bold and problematic step, because ultimately it would mean to find a categorization of the notion of path integral itself. Without any hints or guidance from physics this does not appear too promising.

Posted by: Urs Schreiber on December 5, 2004 3:38 PM | Permalink | PGP Sig | Reply to this

Re: Trivial Gerbes

One possibility that I have (vaguely :-) ) thought about is to have an action – a single functional – for an open string with particles at the endpoints. This is probably not a fundamental open string, but could be something like a qcd string. Then consider an action which is a sum of terms – one for the dynamics of the string, and two for the endpoints – with constraints which set the endpoints of the string on the solutions of the equations of motion of the particles. This looks promising, because the action is now an integral over a surface plus an integral over its boundary – a `natural’ arena for 2-groups.

But even if you could write a sensible action, which is invariant under a 2-group, the physics seems to be trivial, at least for qcd strings, which are singlets. Which means there is no overall phase or `transformation’ when you parallel transport such a string. But maybe that is the content of B+F=0.

Does this make any sense to you?

Posted by: Amitabha on December 5, 2004 4:25 PM | Permalink | Reply to this

Categorized actions

Yes, something like this looks plausible, but there seem to be a couple of devils in the details :-)

Sure, we’d have one part of the action associated with the bulk of the string and the other associated with the boundary. But if, as you vaguely suggest, think of the action as some functor, why would you multiply the arrow part and the object part of the image of that functor?

The abelian case illustrates the problem:

An open string on a D-brane carrying a gauge field A and with an abelian Kalb-Ramond field switched on in the bulk is described by a path integral, which, apart from the NG action contribution, has the abelian gerbe holonomy of B over the bulk of the string times the nonabelian A-holonomy over the boundary. We can think of the ablian gerbe as coming from a 2-bundle with crossed module (G,H,α,t) with G trivial, H abelian and t:h1 , so that dt(B i)+F A i=0 trivially. But then the A-field of that 2-bundle in not that living on the D-brane.

And in a sense it is good that it is not, because if it were there’d be no good reason to multiply the arrow part of the 2-holonomy with its object part. For more general groups, it would not even be clear what notion of multiplication to use.

To me any generalization to actions taking values in categories (as opposed to ordinary actions which take values in sets, namely in the set of real numbers) seems to get us into very deep water, since it would directly mess with the probabilistic interpretation of QM, which, after all, is based on the absolute square of some amplitude to be interpretable as a probability density and hence take values in the real numbers.

Seems like you’d have to understand 2-probability amplitudes in order to make something like this work. And even if you can come up with that - will it describe real physics?

Posted by: Urs Schreiber on December 5, 2004 5:54 PM | Permalink | PGP Sig | Reply to this

Re: Categorized actions

I was not thinking of the action as a functor – I can’t see any need to think of the action as anything but a number. You are probably thinking of action as a map from paths to complex numbers (via e iS), but to call that a functor seems to create confusion. Think of the action as a number. For any classical system which can be broken into pieces, the action is the sum of the actions for the pieces. So for a string with particles at the ends, the action is that for the surface plus that for the boundary.

If the background gauge fields are Abelian, their contributions to the action will be their holonomies, but if they are non-Abelian, you get something else – basically you have to saturate the group indices so that you have a number for the action. So you add numbers, or multiply phases, not anything more complicated, whatever be the gauge group.

Posted by: Amitabha on December 6, 2004 10:20 AM | Permalink | Reply to this

Re: Categorized actions

Sorry, I should have been more specific:

An open string in an abelian B and non-abelian A field background is described by a path integral over

(1)exp(i ΣL NG)exp(i ΣB)Trhol Σ(A)

where Σ is the worldsheet, Σ its boundary, L NG the kinetic Nambu-Goto action (or the Polyakov action if you like), ΣB is shorthand for the correct surface integral of B over the worldsheet (involving ablian gerbe holonomy) and the last term is the trace over the holonomy of A over the boundary of the worldsheet.

My point was the following: If you want to have nonabelian B here this will involve a surface holonomy in a 2-bundle. This takes values not in some group, but in a 2-group. For closed surfaces and strict 2-groups this 2-group element will be specified by an ordinary abelian group element together with something possibly non-abelian, and one can naturally pick out the abelian label and use that in the path integral.

Posted by: Urs Schreiber on December 6, 2004 10:48 AM | Permalink | PGP Sig | Reply to this

Re: action for string with charged ends

Oh, ok. I was saying something slightly different, as you must have figured out. I was thinking of not tracing the holonomy, but saturating it with the charge vector, as you would do for a classical charged particle. That way you get Wong’s equation for both ends. But I don’t know how to write the action for the string that connects them.

Basically I was thinking of the q¯q singlet state in qcd, which looks like

ψ¯(x)U(Γ(x,y))ψ(y)0 ,

for q¯q connected by a string Γ.

The action could go into a path integral that describes the evolution of this state. Then roughly speaking, the A field governs the evolution of ψ and ψ¯, while the B field governs the evolution of the string from Γ to some later configuration Γ.

Posted by: Amitabha on December 6, 2004 11:46 AM | Permalink | Reply to this

Re: action for string with charged ends

Hm, now I am not sure. The path integral integrand that I wrote down is that describing open (fundamental, though) strings attached to a D-brane with that A-field turned on and with a B-field in the bulk. Since the endpoints of the fundamental string are essentially formal ‘quarks’ on the brane this seems to be essentially the same setup as you are envisioning. Are you saying there is another action describing this situation than the one I wrote down?

Posted by: Urs Schreiber on December 6, 2004 12:12 PM | Permalink | PGP Sig | Reply to this

Re: action for string with charged ends

The qcd strings describe q¯q bound states, while quarks are open string states in string theory … so a qcd string would be some sort of effective thing living on the brane, but I have no idea what the `effective action’ should be.

Anyway, I put things badly in the earlier post. We have no disagreement over the Abelian B case – the action you wrote down is the correct one. The question is how to construct an action involving the non-Abelian B field, an action which should be invariant under a 2-group `transformation’. We don’t know how to get such an action from string theory. Even if it is possible to derive such an action, we currently don’t know how to proceed.

Perhaps we could guess something about that action from some general arguments. That was what I was trying to do. Not with a whole lot of success, clearly. :-)

Posted by: Amitabha on December 6, 2004 5:54 PM | Permalink | Reply to this

Nonabelian surface action

Maybe one can do the following, though:

We know that the strings that we are interested in are boundaries of membranes. Their worldvolume couples to an abelian 3-form C 3 whose field strength dC 3 defines an abelian 2-gerbe. Their boundary, which is the worldsheet of the boundary string, must couple to something nonabelian.

Regardless of whether a worldsheet theory for nonabelian strings makes sense as a means to compute S-matrix elements for these strings (in general it will not since these strings are not weakly coupled) the worlvolume action of these membranes must at least be well defined.

But this implies that the higher-dimensional analogue of the path integral integrand that I wrote down before must be well defined. Let V be the worldvolume of the membrane, then

(1)exp(i VC)hol V(B)

must be a well-defined expression. Here the first factor is again shorthand for the proper surface holonomy computed from abelian 2-gerbes and the right factor is some notion of 2-holonomy for the nonabelian 2-form B over the boundary worldsheet of the membrane.

We know (in principle at least) how the left factor transforms. Hence we can check our notion of surface holonomy (and hence of nonabelian surface action) by checking that the above product is well defined.

As I said, I think this is the ultimate condition on any worldsheet theory of nonabelian strings. Whether or not from this worldsheet theory one can extract a perturbation theory for interacting such strings is a different question.

Posted by: Urs Schreiber on December 6, 2004 6:09 PM | Permalink | PGP Sig | Reply to this

Re: Trivial Gerbes

I should have asked this in the previous reply:

For the trivial gerbe at least, the coherence laws are just the gauge transformations for A and B which leave F+B invariant, right? That is

A=gAg 1 dgg 1 +a

B=gBg 1 d Aa[a,a]

and the last coherence law (you are missing an = sign in the original post) says that a transforms in the adjoint?

Posted by: Amitabha on December 5, 2004 4:50 PM | Permalink | Reply to this

Re: Trivial Gerbes

This is the transition law for the 2-holonomy, and, yes, it leaves dt(B i)+F i invariant. The more general transition law for a nonabelian gerbe would allow B to pick up an additional contribution d ij in this transformation. For 2-bundles with strict structure 2-group this d ij appears as the 2-map part of the transition functor g ij 2 but as long as the 2-space that is used is that of based loops (instead of open paths) it must take values in an abelian subalgebra and does not appear in the B transition law. This of course is due to the fact that for 2-bundles this law comes from surface holonomies, which only exist for dt(B)+F=0 .

Note that the coherence law (in the category theoretic sense) gives the transition law for the a ij.

BTW, setting dt(B i)+F A i=0 the cocycle description of a nonabelian gerbe allows this gerbe to have self-dual field strength 3-form H, globally. In general the requirement to have such a global self-dual field strength leads to a horribly complicated condition on all the cocycle data. I would not be too surprised if one could show that dt(B i)+F A i=0 (or to be in the center) is the necessary and sufficient condition for a nonabelian gerbe to admit a globally defined self-dual 3-form field strength H. The calculation is straightforward but very tedious…

Posted by: Urs Schreiber on December 5, 2004 5:40 PM | Permalink | PGP Sig | Reply to this

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