A worldsheet extension of O(d,d;Z)

TL;DR

The paper develops a worldsheet framework for quasi-symmetries of toroidally compactified string theories via superconformal interfaces that preserve $\widehat{u}(1)^{2d}$. By unfolding and fusing these interfaces, the authors reveal a semi-group structure extending $O(d,d;\mathbb{Q})$ whose Ramond-sector fusion corresponds to geometric integral transformations (Fourier–Mukai-type) on D-branes, generalizing T-duality. Topological interfaces form a parallel semi-group, interpreted as orbifold equivalences that rescale the effective string coupling by the index of the charge sublattice, while leaving masses invariant. The analysis extends from $d=1$ circles to general torus models, establishing a defect monoid that mirrors the geometric transformation properties of D-branes and linking CFT defects to algebraic and geometric symmetries. The results provide a coherent picture in which worldsheet defects encode non-invertible transformations as quasi-symmetries of classical supergravity, with potential implications for the arithmetic structure of string theory and for applications in condensed-mMatter realizations of defects.

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.

Figures (1)

• Figure 1: The fusion of two interfaces corresponds to taking the size, $\delta$, of the middle region to zero. Only the $\tau$ axis is drawn in the figure. The $\sigma$ coordinate parametrizes either a circular space, or a periodic Euclidean time.