Compact Free Boson

The compact free boson is a two-dimensional conformal field theory whose oscillator modes are free but whose global spectrum remembers a circle-valued field. Momentum and winding distinguish it from a noncompact scalar. The radius is physical data: changing it changes operator dimensions even though the central charge stays one.

Circle-valued scalar and normalization

Use Euclidean real coordinates $(x,y)$, with

$$S_E=\frac1{8\pi}\int\mathrm dx\,\mathrm dy\, \bigl[(\partial_xX)^2+(\partial_yX)^2\bigr], \qquad X\sim X+2\pi R,\quad R>0 .$$

For $z=x+iy$, this normalization gives $X_L(z)X_L(0)\sim-\log z$ and the corresponding antiholomorphic contraction. The stress tensor is $T=-\tfrac12:(\partial X_L)^2:$, with $c=1$. Both $X$ and $R$ are dimensionless in this convention.

On a spatial circle, the target zero mode has integer momentum label $n$, while a configuration can wind $w$ times around the target circle. The integers $n,w$ label different sectors; winding is not an oscillator excitation.

Momentum, winding, and operator dimensions

The left and right momenta are

$$p_L=\frac nR+\frac{wR}{2}, \qquad p_R=\frac nR-\frac{wR}{2}.$$

A normal-ordered vertex with these momenta has weights $h=p_L^2/2$, $\bar h=p_R^2/2$. Hence

$$\Delta=\frac{n^2}{R^2}+\frac{w^2R^2}{4}, \qquad s=h-\bar h=nw.$$

The integer spin is a useful check of full-field locality. Cocycles, which are zero-mode phase operators, must also be chosen consistently when multiplying charged vertex operators. The spectrum and operator construction are developed in Di Francesco, Mathieu, and Sénéchal 1997, §§6.3 and 10.2–10.4.

For example, the lowest pure-momentum pair $n=\pm1,w=0$ has $\Delta=1/R^2$, whereas the lowest pure-winding pair has $\Delta=R^2/4$. The radius determines which pair is lighter in scaling dimension.

A direct T-duality check

Under $R\mapsto2/R$ and $n\leftrightarrow w$, substitution gives $p_L\mapsto p_L$ and $p_R\mapsto-p_R$. Both weights are unchanged. Momentum and winding therefore describe the same spectrum after relabeling, with the self-dual radius $R=\sqrt2$ in this normalization.

Correlation functions, OPE coefficients, and finite-size levels are natural observables. The noncompact limit changes the zero-mode measure and turns discrete momentum into a continuum; it is not just a radius substitution inside every normalized correlator. Adding a vertex perturbation can break conformal invariance and lead toward models such as sine–Gordon.

Develop the calculation

Free Bosons and Vertex Operators gives neutrality conditions, cocycles, and the torus partition function in the same momentum normalization. Compare Sine–Gordon for a periodic perturbation and Ising CFT for a discrete primary spectrum.

References

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