Abstract
Abstract We study superconformal interfaces between $ \mathcal{N}=\left( {1,1} \right) $ supersymmetric sigma models on tori, which preserve a $ \widehat{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\left( {d,d\left| \mathbb{Q} \right.} \right) $ ), 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.
Highlights
We find that the topological interfaces between torus models form the same semi-group upon fusion
We study superconformal interfaces between N = (1, 1) supersymmetric sigma models on tori, which preserve a u(1)2d current algebra
Topological interfaces for the free boson compactified on a circle, i.e. for d = 1 have been analyzed in [8, 9]. We extend this analysis to torus models of arbitrary dimension d ≥ 1, and to theories with N = (1, 1) worldsheet supersymmetry
Summary
As explained in the above references, there are two different ways to think about interfaces: as operators mapping the states of CFT2 on the circle to those of CFT1; or as boundary conditions in the tensor-product theory CFT1⊗CFT2∗, where CFT2∗ is the parity transform of CFT2. To convert interfaces to boundary states one reflects CFT2 to CFT2∗, so that both conformal theories are defined on the half-cylinder τ ≥ 0 This exchanges the leftand right-moving modes a2n a2n. With the implicit understanding that the positive-frequency modes of CFT1 act on the left and those of CFT2 on the right of the map I10,,2bos This latter map implements the zero-mode gluing conditions on the ground states of the two u(1) Kac-Moody algebras
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