Topological defects and fusion rules

In D equal to two, a line has codimension one and can implement an ordinary zero-form symmetry. Its topological property permits shrinking a thin strip between two parallel lines into a fused line, subject to their junction data. A group-like line has an oppositely oriented inverse whose fusion gives only the identity line. Topological lines can occur inside a gapless conformal theory; their existence does not require the whole bulk theory to be metric independent.

For a finite set of simple defects, record nonnegative integer multiplicities for their fusion outcomes. A simple line cannot itself be decomposed as a direct sum of smaller allowed lines. Associativity constrains the multiplicities, but different ways of joining and reassociating lines can involve nontrivial linear maps on junction spaces. Knowing only the multiplication table therefore does not reconstruct the full fusion category or its anomalies.

Consider the three invertible lines of an anomaly-free Z₃ symmetry, generated by g with its cube equal to the identity. Their fusion is addition modulo three. Now form the reducible direct sum of all three lines. Multiplying this sum by itself produces every group element three times, because for each target label there are three ordered pairs of summands. The resulting object is not a new group element with a group inverse.

In any Hilbert-space representation of this ordinary unitary group action, dividing the corresponding summed operator by three gives a projector onto invariant states. The division is an operator normalization, not a physical direct sum with fractional sector multiplicities. A projection can have a kernel even in a perfectly unitary theory because it is not time evolution. This elementary reducible example prepares the algebraic idea of noninvertibility without claiming that every projection defines a new simple symmetry defect.

For a unitary finite fusion setting, positive quantum dimensions multiply under fusion and add under direct sum. Every invertible simple group defect has dimension one, while the reducible sum above has dimension three. A noninteger quantum dimension can appear for a genuinely simple noninvertible line, but that value alone does not specify the action on every state or the complete junction theory. Quantum dimension is not ordinary spatial dimension or the number of local field components.

The same fusion ring can admit different associator data, so a numerical multiplication check is necessary but incomplete. Physical questions also depend on which defects exist in the chosen theory, how they end on boundaries, and how they pass local or extended operators. A toy representation can test an exact algebraic consequence, but it cannot establish a microscopic symmetry, a protected phase, or a categorical anomaly from its matrix entries alone.