Permutation Branes

N -fold tensor products of a rational CFT carry an action of the permutation group SN . These automorphisms can be used as gluing conditions in the study of boundary conditions for tensor product theories. We present an ansatz for such permutation boundary states and check that it satisfies the cluster condition and Cardy’s constraints. For a particularly simple case, we also investigate associativity of the boundary OPE, and find an intriguing connection with the bulk OPE. In the second part of the paper, the constructions are slightly extended for application to Gepner models. We give permutation branes for the quintic, together with some formulae for their intersections. In memory of Sonia Stanciu 1. Introductory remarks Thanks to the pioneering work by Cardy [1,2] and to further, more recent developments (see e.g. [3-8]), we have a reasonably complete understanding of a certain class of boundary conditions for rational conformal field theories. This class is distinguished by an especially simple relation between the form of the bulk partition function and the gluing automorphism Ω that connects leftand right-moving symmetry generators W (z) = Ω W (z̄) for z = z̄ : If the rational bulk theory comes with a charge-conjugate modular invariant partition function Z(q, q̄) = ∑ i∈I χi(q) χi+(q) ∗ , it is the standard gluing condition Ω = id that leads to a simple picture of boundary conditions, basically because all Ishibashi states that can be formed for standard gluing conditions are actually present in the bulk theory. The same is true for general Ω as long as, in the bulk partition function Z(q, q̄), each sector i is paired with ω(i) instead of the charge-conjugate sector i, where ω(i) labels the representation of the symmetry algebra W induced in the sector i by the action of Ω. The boundary states associated with this class of boundary conditions are often referred to as “Cardy boundary states”. Non-standard gluing conditions (for bulk theories whose partition function is not of the appropriate ω-type) are of crucial importance in string theory, starting with Dirichlet boundary conditions for free bosons. The non-trivial automorphisms Ω available for the gluing conditions of a rational CFT of course depend on the symmetry algebra W of the model. But there is a simple and natural construction that yields, from a given rational model, new ones on which universal automorphism groups act – namely taking tensor products of the original “component theory”. While this operation alone does not produce very exciting effects in the domain of bulk theories, the structure of associated boundary CFTs is much richer: The set of all boundary conditions for an N -fold tensor product, even those that preserve the full symmetry, is not simply obtained by forming tensor products of boundary conditions for the component theory, but includes boundary conditions associated with gluing automorphisms from the permutation group SN . In this paper, we present general formulae for such “permutation boundary states”, first focusing on situations where the original RCFT has a charge-conjugate (or, if all sectors are self-conjugate, a diagonal) partition function. The permutation boundary states are still “rational” in the sense that the boundary CFT is covariant under the full symmetry algebra W , but we will see that the excitation (or open string) spectra of permutation branes in general look rather different from those of tensor products of Cardy states. Our work generalises studies of permutation branes in G × G WZW models [9]. On the other hand, our boundary states are special cases of the “conformal walls” that appeared in [10]: while a general conformal wall is nothing but a symmetry-breaking conformal boundary condition for a tensor product model (not necessarily with identical factors), our permutation branes can be viewed as walls with perfect transmission (or reflection) of energy between (or within) the identical component theories.