December 22, 2005

Unoriented Strings and Gerbe Holonomy

Posted by Urs Schreiber

We have a new preprint:

K. Waldorf & C. Schweigert & U. S.
Unoriented WZW Models and Holonomy of Bundle Gerbes
hep-th/0512283

Abstract:

The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models.

As a motivation, recall that an ordinary bundle $E\to M$ on a base space $M$ on which a finite orbifold group $K$ acts by diffeomorphisms $K\ni k:M\to M$ is called equivariant if there are isomorphisms ${\varphi }^{k}:{k}^{*}E\to E$ relating the bundle to any of its images under the pullback induced by the actions of the elements of the orbifold group. These isomorphism have to satisfy a certain compatibility condition. There may be different choices of such isomorphisms and hence different choices of equivariant structures on bundles.

This is relatively straightforwardly catgorified to the context of (bundle) gerbes. A 2-equivariant structure on a (bundle) gerbe $G$ over a base space $M$ is a choice of (bundle) gerbe isomorphisms ${\varphi }^{k}:{k}^{*}G\to G$ (known as ‘stable isomorphisms’ in the case of bundle gerbes) that satisfy the above compatibility condition up to coherent 2-isomorphism.

In the above paper this is not discussed in generality, but for the case where $K={ℤ}_{2}$ acts by (possibly orientation-reversing) diffeomorphisms and where the gerbe isomorphism for the nontrivial element $\sigma \in {ℤ}_{2}$ relates, in the language of bundle gerbes, not ${\sigma }^{*}G$ with $G$, but ${\sigma }^{*}G$ with ${G}^{*}$.

Here ${G}^{*}$ is the dual bundle gerbe of $G$, obtained by replacing the line bundle appearing in the definition of $G$ by its dual line bundle. Passing to the dual gerbe essentially corresponds to what in the physics literature is called the orientation involution on the worldsheet. Hence an equivariant structure

(1)${\sigma }^{*}G\to {G}^{*}$

for $\sigma \in K={ℤ}_{2}$ describes a gerbe on an orientifold, i.e. on an orbifold with ‘additional twist’.

There is a unified 2-categorical picture behind this, which will be discussed in a sequel, but the above paper just postulates such orientifold structures on gerbes and shows that this has the right properties.

An ‘orientifold structure’ on a bundle gerbe in the above sense is called a Jandl structure on a gerbe in that paper. This term was already used for certain involutive structures on Frobenius algebras in hep-th/0306164 which describe CFTs on unoriented (and possibly unorientable) worldsheets. A Jandl structure on a bundle gerbe is a geometric realization of (aspects of) an abstract Jandl structure on an algebra object in a modular tensor category.

The term derives from a rather famous example of German-language experimental poetry by the Austrian poet Ernst Jandl, who in 1995 added the following insight to mankind’s pool of wisdom:

manche meinen
lechts und rinks
kann man nicht velwechsern
werch ein illtum

I haven’t ever before tried to translate poetry, much less experimental poetry, but if the above doesn’t enlighten you the following should give you the idea:

some peopre think
light und reft
cannot be muddred up
what an ellol

Bettel use a Jandr stluctule on youl gelbe to avoid that ellol!

Introduction:

Wess-Zumino-Witten (WZW) models are one of the most important classes of (two-dimensional) rational conformal field theories. They describe physical systems with (non-abelian) current symmetries, provide gauge sectors in heterotic string compactifications and are the starting point for other constructions of conformal field theories, e.g. the coset construction. Moreover, they have played a crucial role as a bridge between Lie theory and conformal field theory. It is well-known that for the Langragian description of such a model, a Wess-Zumino term is needed to get a conformally invariant theory [Wit84]. Later, the relation of this term to Deligne hypercohomology has been realized [Gaw88] and its nature as a surface holonomy has been identified [Gaw88, Alv85]. More recently, the appropriate differential-geometric object for the holonomy has been identified as a hermitian U(1) bundle gerbe with connection and curving [CJM02]. Already the case of non-simply connected Lie groups with non-cyclic fundamental group, such as $G:=\mathrm{Spin}\left(2n\right)/{ℤ}_{2}×{ℤ}_{2}$ shows that gerbes and their holonomy are really indispensable, even when one restricts one’s attention to oriented surfaces without boundary. The original definition of the Wess- Zumino term as the integral of a three form $H$ over a suitable three-manifold cannot be applied to such groups; moreover, it could not explain the wellestablished fact that to such a group two different rational conformal field theories that differ by ‘discrete torsion’ can be associated.

Bundle gerbes will be central for the problem we address in this paper. A long series of algebraic results indicate that the WZW model can be consistently considered on unorientable surfaces. Early results include a detailed study of the abelian case [BPS92] and of SU(2) [PSS95b, PSS95a]. Sewing constraints for unoriented surfaces have been derived in [FPS94]. Already the abelian case [BPS92] shows that not every rational conformal field theory that is well-defined on oriented surfaces can be considered on unoriented surfaces. A necessary condition is that the bulk partition function is symmetric under exchange of left and right movers. This restricts, for example, the values of the Kalb-Ramond field in toroidal compactifications [BPS92]. Moreover, if the theory can be extended to unoriented surfaces, there can be different extensions that yield inequivalent correlation functions. This has been studied in detail for WZW theories based on $\mathrm{SU}\left(2\right)$ in [PSS95b, PSS95a]; later on, this has been systematically described with simple current techniques [HS00, HSS99]. Unifying general formulae have been proposed in [FHS+00]; the structure has been studied at the level of NIMreps in [SS03].

Aspects of these results have been proven in [FRS04] combining topological field theory in three-dimensions with algebra and representation theory in modular tensor categories. As a crucial ingredient, a generalization of the notion of an algebra with involution, i.e. an algebra together with an algebraisomorphism to the opposed algebra, has been identified in [FRS04]; the isomorphism is not an involution any longer, but squares to the twist on the algebra. An algebra with such an isomorphism has been called Jandl algebra. A similar structure, in a geometric setting, will be the subject of the present article.

The success of the algebraic theory leads, in the Lagrangian description, to the quest for corresponding geometric structures on the target space. From previous work [BCW01, HSS02, Bru02] it is clear that a map $k:M\to M$ on the target space with the additional property that ${k}^{*}H=-H$ will be one ingredient. Examples like the Lie group $\mathrm{SO}\left(3\right)$, for which two different extensions for the same map $k$ to unoriented surfaces are known, already show that this structure does not suffice. We are thus looking for an additional structure on a hermitian bundle gerbe which allows to define a Wess-Zumino term, i.e. which allows to define holonomy for unoriented surfaces. For a general bundle gerbe, such a structure need not exist; if it exists, it will not be unique. In the present article, we make a proposal for such a structure. It exists whenever there are sufficiently well-behaved stable isomorphisms between the pullback gerbe ${k}^{*}G$ and the dual gerbe ${G}^{*}$. If one thinks about a gerbe as a sheaf of groupoids, the formal similarity to the Jandl structures in [FRS04] becomes apparent, if one realizes that the dual gerbe plays the role of the opposed algebra. For this reason, we term the relevant structure a Jandl structure on the gerbe. We show that the Jandl structures on a gerbe on the target space M, if they exist at all, form a torsor over the group of flat equivariant hermitian line bundles on M. As explained in section 4.3, this group always contains an element ${L}_{-1}^{k}$ of order two. We show that two Jandl structures that are related by the action of ${L}_{-1}^{k}$ provide amplitudes that just differ by a sign that depends only on the topology of the worldsheet. Such Jandl structures are considered to be essentially equivalent. We finally show that a Jandl structure allows to extend the definition of the usual gerbe holonomy from oriented surfaces to unoriented surfaces. We derive formulae for these holonomies in local data that generalize the formulae of [GR02, Alv85] for oriented surfaces.

[…]

The notion of a Jandl structure naturally explains algebraic results for specific classes of rational conformal field theories. It is well-known that both the Lie group $\mathrm{SU}\left(2\right)$ and its quotient $\mathrm{SO}\left(3\right)$ admit two Jandl structures that are essentially different (i.e. that do not just differ by a sign depending on the topology of the surface). In the case of $\mathrm{SU}\left(2\right)$, this is explained by the fact that two different involutions are relevant: $g↦{g}^{-1}$ and $g↦z{g}^{-1}$, where $z$ is the non-trivial element in the center of $\mathrm{SU}\left(2\right)$. Indeed, since $\mathrm{SU}\left(2\right)$ is simply-connected, we have a single flat line bundle and hence for each involution only two Jandl structures which are essentially the same.

The two involutions of $\mathrm{SU}\left(2\right)$ descend to the same involution of the quotient $\mathrm{SO}\left(3\right)$. The latter manifold, however, has fundamental group Z2 and thus twice as many equivariant flat line bundles as $\mathrm{SU}\left(2\right)$. The different Jandl structures of $\mathrm{SO}\left(3\right)$ are therefore not explained by different involutions on the target space but rather by the fact that one involution admits two essentially different Jandl structures. Needless to say, there remain many open questions. A discussion of surfaces with boundaries is beyond the scope of this article. The results of [FRS04] suggest, however, that a Jandl structure leads to an involution on gerbe modules. Most importantly, it remains to be shown that, in the Wess-Zumino-Witten path integral for a surface $\Sigma$, the holonomy we introduced yields amplitudes that take their values in the space of conformal blocks associated to the complex double of $\Sigma$, which ensures that the relevant chiral Ward identities are obeyed. To this end, it will be important to have a suitable reformulation of Jandl structures at our disposal. Indeed, the holonomy we propose in this article also arises as the surface holonomy of a 2-vector bundle with a certain 2-group; these issues will be the subject of a separate publication.

Posted at December 22, 2005 10:27 AM UTC

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Re: Unoriented Strings and Gerbe Holonomy

I object on the terminology
“2-equivariant structure”. In the
case of ordinary bundles and sheaves,
we are given a group, say K, to follow
and in general we talk about
K-equivariant objects in any fibered category over a space with K-action.

Now you categorified the space, but you
are still, if i get your setup right,
working with ordinary group K acting.
So it can be space, 2-space, n-space,
whatever but it is just equivariant
appropriately for that 1-categorical
object. So K-equivariance is still
1-equivariance. The prefix
the action of K,
not about the specifics in which world
the objects acted upon live.
tigers and you do not care if tigers
are categorified or not; equivariance
is the property which can be abstractly said in fibered setup without need to understand the nature of
objects which are equivariant,
as long as they live in a fiber of
a fibered category over an object
which has genuine action of K.

You can however categorify the group K.
If group K is a categorical group
or a bigroupoid then you will also need
K-equivariance. Thus, one can talk
in a fiber of some fibered 2-category
of objects above something
on which a categorical group acts.
The 2-equivariant objects will then.
make a 2-category.

This kind of setup has to my knowledge
never been written down in full detail
and I am myself working on a rigorous
development of that theory; the
applications may be 2-Galois theory
and alike.

Let me say few words about one approach.
Let p: F->Cat(C) be a fibered 2-category
whose base 2-category is a 2-category
of inner categories in some category
C with finite limits. Let G be a group
in Cat(C), that is a coherent categorical
group and M just an object in Cat(C)
and let G acts on M coherently.

Then you can use Yoneda lemma
for 2-categories and associate to
G a pseudofunctor G’ from Cat(C) to
categorical groups. Composing
this pseudofunctor with the projection
F->Cat(C) you get a pseudofunctor G’p
from F to cat groups. Now, you want to
say that object P in F has a
2-equivariant structure. You
can also take the pseudofunctor P’
corresponding to P by 2-Yoneda
– this one is from F to Cat(Sets)
and also M’ which is from Cat(C) to Cat(Sets).

Now, it is not difficult
to say what an action
of G’p on P’ – it is simply
a 2-natural transformation of
2-functors from G’p times P to P
which appears to be an action
for each object you evaluate at
the 2-functors;
now for equivariance you say
also that action is compatible
with the action
downstairs – that of G’ on M’ and
you are done. Of course, one has
to write down properly all the data of
such natural transformations as
there are coherences involved
all around, but the scheme of the
definition is clear.

Now a different thing is how the natural
2-fibered categories like F->Cat(C)
appear. I had an idea that
for a 1-fibered category
H->C there should be an induced
2-fibered 2-category F(H)->Cat(C)
however F(H) is NOT (as I learned
from experience and got the final blow
of truth from Prof. C. Hermida with
a counterexample) CartCat(H) as you get
something what is not 2-fibered
(though some properties work).
CartCat(H) was to be inner cats in
H whose structure morphisms
are 1-cartesian. There seem to be
a more sensible candidate for which
a hiint is in an old paper by MacLane
and Pare’ but needs some additional work
and I am not yet sure if the construction
is the appropriate one. But this
question is just one possible
source of 2-fibered cats over Cat(C),
of course also you can consider actions
of 2-groups on gerbes, stacks etc.
and look at objects over those as in
your paper but with interest in
the first modifying the construction
to organize those guys in 2-fibered
2-category (after some modifications)
and then look at 2-equivariance
of those. This is a bit longer
plan though than the 1-equivariance
of objects over 2-bundles and gerbes
I think.

Posted by: zoran skoda on December 23, 2005 2:12 AM | Permalink | Reply to this

Re: Unoriented Strings and Gerbe Holonomy

I do not know if I was clear
that I implied that the P lives
in the fiber over M.
Equivariant objects live
in the fiber over the object which
is recipient (space) of the action;
I assumed this was clear.
in your picture where you work
with elements of K you do not need
to look at the whole fibered category
over variable spaces as it is in
the approach which I use here.

Posted by: zoran skoda on December 23, 2005 2:41 AM | Permalink | Reply to this

2-equivariance

Dear Zoran,

thank you very much indeed for this detailed comment.

I guess you are right that my use of the term 2-equivariance in the above is sub-optimal.

If one takes the standpoint that it is obvious that a 2-gadget has more freedom of being equivariant (namely up to coherent 2-isomorphism) than a 1-gadget, then certainly what I am mentioning above is indeed just ordinary equivariance, in that the base space, the orbifold group as well as its action are all at the 1-level.

At least so it seems. There is a subtlety here, which should eventually be sorted out. Namely, in the more 2-categorical treatment of the notion of Jandl structures on gerbes which we are working on, the gerbe really comes from a 2-functor with domain a 2-category of 2-paths. Hence in this description the base space really is a 2-space, namely a smooth category whose morphisms are paths and whose objects are points. The action of the ordinary group $K$ of course immediately lifts to paths by ordinary pushforward. But I guess that, more generally, there are genuine 2-group-like structures which may act on a space of paths, regarded as a category in this sense.

So what I am trying to say is that it might still be that a Jandl structure on a bundle gerbe is a special case of proper 2-equivariance even in your more sophisticated sense, and even though this is not manifest in the above paper (and we didn’t even remotely try to address this).

I will definitely be thinking more about this in the near future. I will take a close look at what you wrote in detail as soon as I find the time. Right now I have to run as I have to supervise an exercise group.

Posted by: Urs on December 23, 2005 9:09 AM | Permalink | Reply to this

Re: Unoriented Strings and Gerbe Holonomy

My remark was not only about the
possibility and need to look
at more general case,
but also on the way the language
works.

We instead of precisely saying which
beast we deal with we assign to it a short
name. So instead of calling it the
12-tuple G = (G_1,G_2,m,p,d,i,…) where
G_1 is an object of objects etc….you
say “2-gadget”. Here “2” modify the
together give a shorthand description of
the thing whose true name is G.

If you take K a group and G a beast
and P bundle on beast G,
and say “K-equivariant
smooth a bundle P on group G”
then you can shorten and say more
vague description of each name in
the term and then you KEEP the position
of the nickname/abbreviation
at the same spot!

K modifies word “equivariant”
smooth modifies word “bundle”
and group modifies name G.

Each of the 3 can be 1-gadget, or

So you have combinations like

2-equivariant 1-object over 2-object
and
1-equivariant 2-object over 1-object
and so on.

Number 2 can be placed
in 7 ways on a nonzero combination of
3 places:

K-equivariant object P over object G is

is a sample of the descriptive statement

p-equivariant r-object over s-object

where p,r, and s, allude to some
features of K,P and G respectively.

It is more precisely, without
doubt, not to shuffle p, r, and s
when you get impressed with r or s.

But each combination
of the 3 numbers
have its best-defined place within
the vaguest term
“equivariant object over object”.

It seems to me that you are alluding to
have in your work in progress
a 2-gerbe (which is a 3-object)
as recipient of the action. It is not
clear to me if you will be intersested in
equivariance of bundles over it or
2-bundles over it.

It may be that you will eventually want
2-equivariance of 2-objects
over 3-gadgets in which the coherences
of the 3-gadget are nontrivial only in
the 2-sense (coherence does not involve
nontrivial 3-cells).

Posted by: zoran skoda on December 25, 2005 12:03 AM | Permalink | Reply to this

Re: Unoriented Strings and Gerbe Holonomy

Hi Zoran,

in the end, I am interested in

a 2-functor $\mathrm{tra}:{P}_{2}\to C$, where ${P}_{2}$ and $C$ are some 2-categories.

There will be some action of some (1- or 2-)group on ${P}_{2}$ and I want the 2-functor $\mathrm{tra}$ to be ‘compatible’ with this, in a suitable sense.

In suitable circumstances that 2-functor would be called a “2-bundle with 2-connection” and hence I would call its compatibility condition a ‘2-equivariance’ condition.

Since everything in sight is 2-categorical, I could just say ‘equivariant’ and it should be clear from the context what it would mean.

Posted by: Urs on December 27, 2005 12:51 PM | Permalink | Reply to this
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