### 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 $K={\mathbb{Z}}_{2}$ acts by (possibly orientation-*reversing*) diffeomorphisms and where the gerbe isomorphism for the nontrivial element $\sigma \in {\mathbb{Z}}_{2}$ 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

for $\sigma \in K={\mathbb{Z}}_{2}$ 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 $G:=\mathrm{Spin}(2n)/{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ 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 $\mathrm{SU}(2)$ 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 $\mathrm{SO}(3)$, 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 $\mathrm{SU}(2)$ and its quotient $\mathrm{SO}(3)$ 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 $\mathrm{SU}(2)$, this is explained by the fact that two different involutions are relevant: $g\mapsto {g}^{-1}$ and $g\mapsto z{g}^{-1}$, where $z$ is the non-trivial element in the center of $\mathrm{SU}(2)$. Indeed, since $\mathrm{SU}(2)$ 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 $\mathrm{SU}(2)$ descend to the same involution of the quotient $\mathrm{SO}(3)$. The latter manifold, however, has fundamental group Z2 and thus twice as many equivariant flat line bundles as $\mathrm{SU}(2)$. The different Jandl structures of $\mathrm{SO}(3)$ 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.

## 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

your notation, and we talk about K-equivariant bundles, sheaves,

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

and equivariance are about

the action of K,

not about the specifics in which world

the objects acted upon live.

You can talk about K-equivariant

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

about 2-equivariant objects living

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.