Conformal and Topological Models

Some theories are most naturally described by local scaling operators; others by line operators, braiding, and the topology of space. This chapter compares two conformal theories in two spacetime dimensions with an Abelian topological gauge theory in three. Their shared value is that symmetry strongly organizes the observables, though it does so in different ways.

Enter this chapter

Begin with Ising Conformal Field Theory for a small set of primary fields and their scaling dimensions. Enter through Abelian Chern–Simons Theory if your question concerns anyons, linking, or the dependence of a state space on spatial topology.

Topics

1. Ising Conformal Field Theory — Identify the critical theory’s primary fields, scaling data, and its distinction from a microscopic Ising model away from criticality.
2. Compact Free Boson — Derive momentum and winding contributions to operator dimensions and check radius duality.
3. Abelian Chern–Simons Theory — Relate a quantized level to line operators, braiding, and the state count on a closed spatial surface.

How the ideas fit together

In a conformal theory, a scaling dimension describes how an operator changes under a rescaling of distances. It is not a particle mass. The Ising CFT has a discrete set of primary fields, whereas the compact boson has sectors labeled by integer momentum and winding. Changing the boson’s radius changes scaling dimensions even though its central charge stays fixed. Thus the central charge alone does not identify a CFT; see Di Francesco, Mathieu, and Sénéchal 1997, §§6.3 and 10.2–10.4.

Pure Chern–Simons theory has no propagating local bulk modes, yet it has nontrivial observables. A line can detect another line through braiding, and the quantum state space depends on the topology of a spatial surface. Neither this absence of local propagation nor a finite state count on a compact surface makes the theory empty.

Keep physical realizations separate from these ideal descriptions. A critical lattice system approaches a CFT only in a specified scaling regime. A topological theory describes selected long-distance properties of a gapped phase; it does not determine that phase’s microscopic excitation gap or response at arbitrary frequency.

Where to go next

Continue to Supersymmetric Models for further examples of symmetry constraining quantum dynamics. For a deformation away from the free-boson conformal point, compare the periodic interaction in Sine–Gordon Model.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Springer, 1997, §§6.3 and 10.2–10.4. DOI.