A worldsheet extension of $ O\left( {d,d\left| \mathbb{Z} \right.} \right) $

Abstract

Abstract:We study superconformal interfaces between N = (1, 1) supersymmetric sigma models on tori, which preserve a u(1) 2d current algebra. Their fusion is non-singular and, using parallel transport on CFT deformation space, it can be reduced to fusion of defect lines in a single torus model. We show that the latter is described by a semi-group extension of O(d, d|Q), and that (on the level of Ramond charges) fusion of interfaces agrees with composition of associated geometric integral transformations. This generalizes the well-known fact that T-duality can be geometrically represented by Fourier-Mukai transformations.Interestingly, we find that the topological interfaces between torus models form the same semi-group upon fusion. We argue that this semi-group of orbifold equivalences can be regarded as the α deformation of the continuous O(d, d) symmetry of classical supergravity.

Entanglement entropy through conformal interfaces in the 2D Ising model

“…Here, contributions from the oscillators of the bosonic and fermionic part of the system cancel out in the limit δ → 0, such that only the term s/2 remains. This is similar to the computations in [25], where the limit of two parallel interfaces approaching each other was considered. Note 4 One can also consider the GSO projection of the supersymmetric model.…”

“…As was shown in [25] the index of the sublattice is a useful quantity to characterize topological information of an interface, in other words, the information that does not change under deformations of the interface or bulk theories. In a similar way, T naturally exists for any interface and characterizes the transmissivity, in other words, how far away the interface is from being topological.…”

“…Having an interface between two 2D free fermion conformal field theories as on the left of figure 1, its interface operator has the general form [24,25] …”

“…The matrix O ∈ O(2) specifies the interface and can be given in terms of a boost matrix Λ ∈ O(1, 1) which guarantees that the interface preserves conformal invariance, see [24,25] for more details. Their exact relation is given by…”

“…The signs η i S specify the preserved SUSY of the two theories and the GSO projection requires to sum over both choices for η in the final step. As explained in [25] the choices of sign can be absorbed in the gluing matrix Λ ∈ O(1, 1) for the fermions, such that specific entries in the gluing matrix for bosons and fermions can differ by signs. Since the entanglement entropy does not depend on these choices, we simply assume that bosons and fermions are glued by the same matrix Λ (or equivalently O) that we used…”