Orbifolds from Modular Orbits

TL;DR

The paper develops a modular-orbits framework to construct orbifolds of 2D CFTs by quotienting with global symmetries, ensuring modular invariance from the start and reducing reliance on explicit twisted Hilbert spaces. It shows the method works cleanly for cyclic groups and extends to continuous flavor symmetries via flavored partition functions, enabling explicit spectra in twisted sectors and revealing a modified projection structure in asymmetric cases. Through concrete examples with free bosons and the Ising model, the authors illustrate how orbifolds can reproduce standard radius changes, detect anomalies, and realize fibered CFTs where noncompact directions control twisted sectors. The approach offers a unifying, modular-invariance-driven procedure that can accommodate discrete torsion, anomalies, and higher-genus generalizations, with potential applications to more complex group extensions and OPE analyses. Overall, the work provides a practical, symmetry-driven path to consistent orbifold theories and deepens the connection between modular properties, defect lines, and orbifold spectra.

Abstract

Given a two-dimensional conformal field theory with a global symmetry, we propose a method to implement an orbifold construction by taking orbits of the modular group. For the case of cyclic symmetries we find that this approach always seems to be consistent, even in asymmetric orbifold cases where the usual construction does not yield a modular invariant theory; our approach keeps modular invariance manifest but may give a result that is equivalent to the original theory. For the case that the symmetry is a subgroup of a continuous flavor symmetry, we can give explicit constructions of the spectrum, with twisted sectors corresponding to a non-standard group projection on an enlarged twisted sector Hilbert space.

Figures (4)

• Figure 1: The untwisted sector partial traces $Z_{1,g}$ are given by inserting a group element in the trace. If we think of the action of the group element as being implemented by a topological defect wrapping the spatial circle at a fixed time, then Figure (a) represents $Z_{1,g}$. To get $Z_{g,1}$, we perform a $\tau\rightarrow\tau'=-1/\tau$ modular transformation, resulting in a defect wrapping the time circle, as in Figure (b).
• Figure 2: In figure (a) we have a representation of $Z_{g,1}(\tau,\bar{\tau})$, while the dashed lines indicate an alternative fundamental domain corresponding to $\tau'=\tau-2$. In (b) we focus on this new fundamental domain, paying attention to where the topological defect labeled by $g$ is located. By deforming this picture we arrive at figure (c), which we recognize as a representation of $Z_{g,g^2}$. For the group $\mathbb{Z}$ generated by an element $g$, any partial trace can be represented uniquely as such a diagram with $g$ lines that never cross. For example, $Z_{2,3}$ is illustrated.
• Figure 3: Three TDLs can be joined at a trivalent junction. In full generality, an ordering should be specified, with the incoming lines listed in clockwork order; we mark the last line listed with a $\times$, following Chang:2018iay. There are two ways of joining four TDLs (with $g_1g_2g_3g_4=1$) using trivalent junctions, and these can be related by a phase $\theta(g_1,g_2,g_3)$. Inequivalent phases are classified by $H^3(G,\operatorname{U}(1))$. If there is a representative for which the phase is trivial, then we say that the symmetry is non-anomalous. Otherwise, the anomaly is given by the class in $H^3(G,\operatorname{U}(1))$.
• Figure 4: We start with the $Z_{g,g^n}$ partial trace in (a). In (b) we draw a dashed identity line between two $g$ lines. Going to (c), we use a crossing relation, which results in a $g^2$ line connecting the two junctions, and which multiplies the evaluation of the diagram by a phase $e^{i\theta(g^{-1},g,g)}$, where $\theta\in H^3(G,\operatorname{U}(1))$ represents the anomaly. From (c) to (d) we simply deform the picture, sliding the junction on the right around the cycle on the torus. From (d) to (e) we perform another crossing, introducing an additional factor $e^{i\theta(g^{-1},g^2,g)}$. Repeating the sliding and crossing steps until all the horizontal $g$ lines have been absorbed we get to (f). The total phase from the crossings is $\gamma=\exp(i\sum_{j=1}^{n-1}\theta(g^{-1},g^j,g))$.