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Generalized symmetries & entanglement
What does a quantum state reveal when its entanglement is separated into sectors of a symmetry that need not have an inverse?
The idea
What is the question?
An ordinary symmetry can often be undone by another symmetry operation. Generalized symmetries enlarge this picture: some act on extended objects, while non-invertible symmetry defects combine according to fusion rules rather than a group. These structures constrain quantum field theories even when there is no conventional conserved charge of the familiar kind.
For example, an invertible line fused with its inverse gives the identity line. An Ising-type non-invertible line can instead fuse with itself into a sum of channels, conventionally written σ × σ = 1 + ψ. This is a rule for combining defects; it is not, on its own, a prescription for assigning statistical probabilities to the outcomes.
Explore an idea
A little physics to play with.
Change the controls and watch what follows.
Interactive intuition
See how sectors share the weight.
Explore a three-sector probability model, then compare the fusion of invertible and non-invertible objects.
Sector Shannon entropy H(p)
Probabilities: 1 · ψ · σ
Distribution
pa = (1 − λ)/3 + λ da²/4 · H(p) = −Σ pa log₂ pa
Compare invertible and non-invertible fusion
σ × σ = 1 + ψ: two possible fusion channels. The σ object is non-invertible.
The fusion rule lists allowed channels, not their probabilities. It is independent of the probability slider above.
Resolving a state into symmetry sectors reveals how its weight is distributed. This sector uncertainty is one part of the information carried by a quantum state.
What this shows: a toy interpolation using the Ising quantum dimensions d = (1, 1, √2). The endpoint (¼, ¼, ½) is a dimension-weighted reference, not a universal prediction for the RCFT boundary states in the papers. The full entropy of a block-diagonal state also includes entropy within each sector: S(ρ) = H(p) + Σ paS(ρa).
Entanglement gives a way to look for their imprint. Instead of asking only for the total entropy of an interval, I study how the reduced state decomposes into generalized charge sectors. The calculation connects boundary conformal field theory, topological defects and a three-dimensional symmetry topological field theory. Boundary conditions at the entanglement cut matter: they affect the charged moments and therefore the information visible in the sector decomposition.
A related line of work studies defects in rational and meromorphic CFTs through commutant pairs. By decomposing a complicated parent theory into simpler commuting factors, one can calculate defect-modified character pairings and follow their modular transformations. This probes structure that an ordinary torus partition function can hide.
A useful distinction is between the probability of landing in a sector and the quantum entanglement within that sector. Both contribute to the full entropy, but they encode different information. The sector-resolved calculations show why simply dividing the total entropy equally between all sectors can miss physically meaningful contributions associated with quantum dimensions and boundary conditions.
This turns abstract symmetry data into quantities that distinguish quantum states and defect sectors, connecting conformal field theory with the mathematical language of anyons and topological phases.
The toolkit
How I approach it.
Boundary states, charged moments and sector projectors
Symmetry topological field theory and twisted/anyonic sectors
Modular S matrices, Verlinde eigenvalues and commutant character pairings
Research contributions
What the work shows.
- Proposed a categorical-symmetry resolution of entanglement and analysed the role of symmetry-compatible boundary states.arXiv:2402.06322 →
- Extended the construction to generalized twisted and anyonic sectors using boundary-dependent charged moments.arXiv:2409.02162 →
- Identified subleading differences between sector entropies, including violations of entanglement equipartition.arXiv:2402.06322 →arXiv:2409.02162 →
- Developed equatorial projection for defect amplitudes in commutant pairs, including explicit E₈,₁ examples beyond two characters.arXiv:2602.02700 →
Where these results apply. The published entanglement results use specified rational-CFT and boundary assumptions. Sector probabilities should not be presented as universally proportional to squared quantum dimensions without stating the relevant theory, boundary data and asymptotic regime.
Read further
Papers in this direction.
Categorical-symmetry resolved entanglement in conformal field theory
Key results4 highlights
Proposes a way to see non-invertible symmetry structure inside CFT entanglement.
- Introduces categorical-symmetry-resolved entanglement using symmetry-compatible boundary states and charge-sector projectors.
- Relates the construction to the boundary conditions placed at the entanglement cut and the assumptions needed to implement the categorical symmetry.
- Works through rational-CFT examples, including a Fibonacci subsector, where quantum dimensions enter the sector decomposition.
- Exhibits subleading violations of entanglement equipartition between different categorical charge sectors.
Scope. This is a particular boundary-state construction with compatibility and gauging assumptions; it should not be presented as an unrestricted definition for every non-invertible symmetry.
Generalized symmetry resolution of entanglement in conformal field theory for twisted and anyonic sectors
Key results4 highlights
Extends symmetry-resolved entanglement to generalized charges carried by twisted and anyonic sectors.
- Organises generalized charge sectors using the anyons of the associated three-dimensional symmetry topological field theory.
- Defines boundary-dependent charged moments and projected partition functions for resolving interval entanglement into these sectors.
- Shows how quantum dimensions and boundary data enter the sector-resolved entropies.
- Finds sector-dependent subleading terms: entanglement equipartition can fail even when the leading ultraviolet divergence is shared.
Scope. The derivation assumes a diagonal RCFT with a unique bulk vacuum; self-conjugate representations are also assumed in parts of the analysis.
Verlinde lines, anyon permutations and commutant pairs inside E₈,₁ CFT
Key results4 highlights
Uses simpler commuting CFT factors to expose topological-defect information hidden by a meromorphic parent partition function.
- Develops an equatorial-projection framework that transports defect insertions through bilinear character pairings of commutant CFTs.
- Tracks modular transformations of the insertions, relating diagonal Verlinde-line eigenvalues to fusion actions in crossed channels.
- Constructs and analyses defect amplitudes for commutant pairs inside E₈,₁, including three-character and tensor-product examples.
- Distinguishes genuinely modular-invariant partition functions from amplitudes that instead describe consistent defect or interface observables.
Scope. The explicit constructions centre on E₈,₁ and specified commutant pairs. Extensions to larger central charges are directions to develop, not a classification of all defects or meromorphic CFTs.
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