On Generalised Discrete Torsion

Abstract: For a 2d gauged sigma model with target space $M$ and discrete gauge group $G$, we consider a generalisation of Vafa's discrete torsion $H^2(BG; U(1))$ that assigns different local discrete torsion phases to different singular loci of the orbifold $M/G$. Our generalised discrete torsion lives in $H^2_G(M; U(1))$, and gives a consistent implementation of Gaberdiel and Kaste's prescription for inserting such local discrete torsion phases by hand at higher genus. We revisit the original application to $T^6/\mathbb{Z}_2^2$ and $T^7/\mathbb{Z}_2^3$ orbifold CFTs, and determine what smooth Calabi-Yau and $G_2$ geometries result from different choices of the generalised discrete torsion. We find that the local discrete torsion phases can be different from each other, but are not completely independent either; in the $T^7/\mathbb{Z}_2^3$ case for example, the orbifold CFTs only realise 3 out of the 9 possible Betti numbers of $G_2$ resolutions constructed by Joyce.

• The paper extends discrete torsion classification from group cohomology H²(BG; U(1)) to equivariant cohomology H²_G(M; U(1)).
• The study uses explicit computations in torus orbifolds like T⁶/Z₂² and T⁷/Z₂³ to demonstrate that local phase assignments are globally constrained.
• The results reconcile modular invariance in CFT with classical geometric resolutions, delineating allowed Hodge and Betti numbers in orbifold models.

Generalised Discrete Torsion in 2d Orbifold Sigma Models

Introduction and Motivation

The paper "On Generalised Discrete Torsion" (2604.01225) systematically addresses the higher-genus consistency and effective classification of discrete torsion phases in two-dimensional $\sigma$-models with finite group orbifolds. Traditional discrete torsion, as elucidated by Vafa, is controlled by $H^2(BG; U(1))$ when orbifolding by a finite group $G$. However, when the orbifold singular locus of $M/G$ decomposes into various strata — e.g., multiple fixed tori — the question arises whether one can independently assign local discrete torsion phases to each singular stratum, as is possible in the corresponding classical geometric resolutions.

The paper's principal contribution is a rigorous extension of discrete torsion from group cohomology $H^2(BG; U(1))$ to equivariant cohomology $H^2_G(M; U(1))$. This refinement reveals that while local torsion phases can differ at distinct singular loci, their possible assignments exhibit nontrivial global constraints: not all local choices are allowed independently. Explicit computations for orbifolds of tori, such as $T^6/Z_2^2$ and $T^7/Z_2^3$, establish the spectrum of allowed topologies and demonstrate the limitations of the CFT construction relative to geometrically smooth resolutions.

Discrete Torsion: From Group Cohomology to Equivariant Cohomology

The distinction between ordinary and generalised discrete torsion hinges on the cohomological invariants used to classify allowable projective phases in orbifold CFTs. For a finite abelian $G$ acting on $M$, the relevant symmetry-protected topological (SPT) phases — and, correspondingly, the topological terms in the gauged sigma model — are parameterised not by $H^2(BG; U(1))$0 but by the $H^2(BG; U(1))$1-equivariant cohomology $H^2(BG; U(1))$2. This group accounts for both group-theoretic torsion and topology-sensitive $H^2(BG; U(1))$3-field holonomies that are invariant under the group action.

The formalism naturally reduces to:

• $H^2(BG; U(1))$4 in the case $H^2(BG; U(1))$5 is a point (pure group cohomology).
• $H^2(BG; U(1))$6 when $H^2(BG; U(1))$7 is trivial (ordinary flat $H^2(BG; U(1))$8-fields).

In the context of tori orbifolds, the entire structure of equivariant cohomology is concretely computed by central extensions associated with the group $H^2(BG; U(1))$9 arising as suitable extensions of $G$0 by lattice translations, reflecting the correspondence between orbifold data and commutator relations in the covering group.

Local and Global Constraints on Discrete Torsion Assignments

A central outcome is the identification of the obstruction to independently assigning discrete torsion phases at distinct singular loci. While classical geometric resolutions (e.g., Joyce’s desingularisations for toroidal orbifolds) allow for fully local choices, the CFT construction via generalised discrete torsion intrinsically enforces cohomological constraints among the local assignments.

For $G$1, the 19-dimensional discrete part of $G$2 (modulo continuous $G$3-field parameters) encodes local assignments at the various fixed tori and points. However, achieving a fully smooth geometry — i.e., a total desingularisation with maximal Hodge numbers — is only possible for global assignments corresponding to ordinary discrete torsion. This is because singular loci are interconnected, and resolving intersecting strata in a non-coherent way is forbidden by CFT modular invariance at higher genus.

In the $G$4 setting, the analysis is refined. Even though the fixed loci (here, various $G$5) do not intersect, group cohomology computations show that only three out of nine possible Betti numbers (appearing in Joyce's $G$6 resolutions) are realised by orbifold CFTs with generalised discrete torsion. The non-independence of local torsion phases arises from relations among the group extensions governing the model, manifest in explicit combinatorial constraints on the phases.

Impact on Hodge and Betti Numbers: Calculation and Interpretation

The paper provides explicit calculations for the dependence of Hodge and Betti numbers on discrete torsion phases in the orbifold CFT. For each model, the possible values are determined by the allowed set of torsion phases, with local assignments projected to fixed loci and further to fixed points encoding the phase picked up by twisted sectors:

• In $G$7, contributions to $G$8 and $G$9 from each singular stratum depend on the corresponding local discrete torsion phase, but the full set of possible CFT Hodge numbers forms only a strict subset of what one can obtain by classical geometric desingularisation.
• In $M/G$0, the set of allowed $M/G$1 is further reduced. While the geometric construction allows for any $M/G$2 for $M/G$3 by independent resolutions, the orbifold CFT only permits those with $M/G$4, corresponding to constrained assignments dictated by the analysis of $M/G$5.

These results contradict claims in the earlier literature (e.g., Gaberdiel-Kaste) that arbitrary local discrete torsion assignments are consistent for arbitrary resolutions in orbifold CFT, demonstrating the necessity of taking higher genus consistency and global orbifold structure into account.

Formal and Mathematical Aspects

The computations leverage both explicit central extension calculations and cell-complex analysis. The matching of the CFT-based discrete torsion classification with Sharpe’s equivariant-gerbe formalism and with the structure of equivariant differential cohomology is demonstrated, particularly in the context of flat $M/G$6-fields and their holonomies. The relation to group abelianisations and the associated surjectivity properties of the mapping from discrete torsion for the abelianised group to the relevant cohomology classes are discussed in detail.

Moreover, the nontrivial structure of obstructions is connected to the redundancy in parameterising discrete torsion via extensions, with the cell-complex computations providing explicit bases for the relevant group generators and their images.

Implications and Future Directions

The results concretely specify the moduli of allowed CFT resolutions in toroidal orbifolds, informing the construction of string compactifications where singularities must be resolved in a manner compatible with worldsheet consistency. The mismatch between geometric and CFT-resolvable topologies underscores inherent limitations of symmetric orbifold CFTs in realising the entire classical moduli space of smooth $M/G$7 and Calabi–Yau manifolds derived from orbifold desingularisations.

On the mathematical side, the analysis unifies earlier approaches based on group cohomology, gerbes, and SPT phases, and clarifies the precise role played by equivariant cohomology in the classification of discrete torsion. The explicit characterisation opens avenues for systematic study of more general finite-group orbifolds and their resolutions, as well as for the systematic study of the effect of discrete torsion in orientifold compactifications and models with nontrivial higher-form symmetries.

Conclusion

This paper provides a definitive classification of generalised discrete torsion in 2d orbifold sigma models, resolved through the precise computation of $M/G$8 and its local restrictions. The results sharpen the understanding of how discrete torsion phases modulate the topology of orbifold CFTs, and the global constraints they must satisfy. The analysis demonstrates that, albeit more flexible than ordinary discrete torsion, generalised discrete torsion does not allow for complete independence of local phases except in geometrically trivial situations, reconciling CFT modular invariance and higher-genus consistency with the topological possibilities of classical geometry. This places bounds on the topologies accessible via orbifold CFTs and informs future work on string compactifications and related moduli problems.