The String Coffee Table

Re: Category Theory and Physics

Posted by Urs Schreiber

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.

Posted at December 3, 2004 4:44 PM UTC