The String Coffee Table

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 $L^k_{-1}$ of order two. We show that two Jandl structures that are related by the action of $L^k_{-1}$ 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.

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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.