Free Bosons and Vertex Operators

The compact free boson is the simplest CFT in which local oscillator algebra, a compact zero mode, momentum and winding, cocycle phases, and modular completion all matter simultaneously. Once one normalization is fixed, every vertex-operator dimension, OPE exponent, neutrality rule, and T-duality map follows. The discussion is restricted to a compact target circle; the noncompact limit changes the Hilbert-space measure and makes correlators distribution-valued in momentum labels.

Required background. The Virasoro algebra and the stress tensor fix the central charge and cylinder shift. Conformal OPE data distinguish primary normalization, OPE coefficients, and conformal-family sums.

Helpful background. Coincident products and contact terms explain normal ordering and why a regulator-dependent self-contraction is removed before defining a vertex operator.

Normalization, oscillators, and the compact zero mode

Use the Euclidean action

$$S=\frac{1}{4\pi}\int d^2z\, \partial X\,\bar\partial X, \qquad X\sim X+2\pi R.$$

This is the $\alpha'=2$ convention. Splitting $X=X_L(z)+X_R(\bar z)$ gives

$$X_L(z)X_L(0)\sim-\log z, \qquad X_R(\bar z)X_R(0)\sim-\log\bar z,$$

with no singular mixed contraction. The currents and stress tensors are

$$J(z)=i\partial X_L(z), \qquad J(z)J(0)\sim\frac1{z^2}, \qquad T(z)=-\frac12:\!\partial X_L\partial X_L\!:,$$

and similarly in the barred sector. Wick contraction gives the standard $TT$ OPE with $c=1$. These normalizations and the compact spectrum are derived in Di Francesco, Mathieu, and Sénéchal 1997, §§6.3, 9.1, and 10.2–10.4.

On a spatial circle, configurations can wind:

$$X(\sigma+2\pi)=X(\sigma)+2\pi wR, \qquad w\in\mathbb Z.$$

Single-valued wavefunctions on the target circle give momentum $n/R$, $n\in\mathbb Z$. The left- and right-moving momenta are

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

The compact zero mode is a quantum-mechanical coordinate on a circle. Integrating it enforces Kronecker momentum conservation, rather than the Dirac delta and target-volume factor of a noncompact scalar. That distinction is developed on Noncompact Continuous Spectra and Plancherel Measures.

Vertex operators, dimensions, and cocycles

Define the normal-ordered vertex operator

$$V_{n,w}(z,\bar z) =\kappa_{n,w} :\!e^{ip_LX_L(z)+ip_RX_R(\bar z)}\!:,$$

where $\kappa_{n,w}$ is a zero-mode cocycle operator. Its weights are

$$h_{n,w}=\frac{p_L^2}{2}, \qquad \bar h_{n,w}=\frac{p_R^2}{2}.$$

Consequently,

$$\Delta_{n,w}=\frac{n^2}{R^2}+\frac{w^2R^2}{4}, \qquad s_{n,w}=h_{n,w}-\bar h_{n,w}=nw\in\mathbb Z.$$

The integer spin is an immediate locality check for the full momentum–winding lattice. Wick’s theorem gives the leading OPE

$$V_{n,w}(z,\bar z)V_{n',w'}(0) \sim \varepsilon\bigl((n,w),(n',w')\bigr) z^{p_Lp_L'}\bar z^{p_Rp_R'} V_{n+n',w+w'}(0)+\cdots.$$

The monodromy exponent is

$$p_Lp_L'-p_Rp_R'=nw'+wn'\in\mathbb Z.$$

This verifies full-field mutual locality, but it does not remove operator-ordering phases. A bimultiplicative cocycle $\varepsilon$ must be chosen so that exchange phases, Hermitian conjugation, and OPE associativity agree. Different cocycle representatives related by rephasing vertex operators describe the same physics; omitting cocycles altogether can give inconsistent signs even when every conformal weight is correct.

On the sphere, a correlator is nonzero only if

$$\sum_i p_{L,i}=0, \qquad \sum_i p_{R,i}=0,$$

equivalently $\sum_i n_i=\sum_i w_i=0$. With an ordering and cocycle convention fixed,

$$\left\langle\prod_iV_{n_i,w_i}(z_i,\bar z_i)\right\rangle =\mathcal E \prod_{i<j}z_{ij}^{p_{L,i}p_{L,j}} \bar z_{ij}^{p_{R,i}p_{R,j}},$$

where $\mathcal E$ is the product of cocycle phases. The compact zero-mode integral enforces momentum neutrality; regularity and separate chiral charge conservation enforce the equivalent winding condition on the sphere. Branches are fixed first on a radial-ordering domain, and integer mutual monodromy makes the completed full correlator single-valued.

T-duality and the torus spectrum

The momentum–winding construction and its duality are reviewed in Ginsparg 1990, §§6 and 8. In the present convention, the transformation

$$R\longmapsto\frac2R, \qquad n\longleftrightarrow w$$

leaves $p_L$ invariant and sends $p_R\mapsto-p_R$. It therefore preserves $h$, $\bar h$, every oscillator degeneracy, and the lattice OPE after the corresponding cocycle isomorphism. This is T-duality in the present $\alpha'=2$ convention; a source using $\alpha'=1$ instead writes $R\leftrightarrow1/R$.

The torus partition function is

$$Z_R(\tau,\bar\tau) =\frac1{|\eta(\tau)|^2} \sum_{n,w\in\mathbb Z} q^{p_L^2/2}\bar q^{p_R^2/2}, \qquad q=e^{2\pi i\tau}.$$

The $\eta$ factors contain the oscillator trace and the vacuum shifts $q^{-1/24}\bar q^{-1/24}$. Under $T:\tau\mapsto\tau+1$, the phase is $e^{2\pi i(h-\bar h)}=e^{2\pi inw}=1$. Under $S:\tau\mapsto-1/\tau$, Poisson resummation exchanges momentum and winding, proving modular invariance only when the complete lattice sum and its normalization are retained. At special radii the chiral algebra extends and the same sum reorganizes into finitely many extended-algebra characters; generic radii are compact and unitary but not rational with respect to a finite set of such sectors.

Construction data and what each path must add

The compact-boson branch of the figure should be read as a sequence, not as a conclusion from $c=1$: the charge lattice and cocycles determine local fields, and the full lattice sum supplies modular completion.

Schematic construction map. For the compact boson, $c=1$ fixes neither the radius nor the momentum–winding lattice; locality additionally needs cocycles, and modular invariance needs the complete lattice sum. The other branches impose different selection and completion rules.

The compact-boson branch can be checked semantically as follows:

Layer	Declared data	Independent check	Failure if omitted
Chiral algebra	$J=i\partial X_L$, $J(z)J(0)\sim z^{-2}$	Sugawara tensor has $c=1$	Dimensions float with an unstated normalization
Spectrum	$(n,w)\in\mathbb Z^2$ and the displayed $p_L,p_R$	$h-\bar h=nw\in\mathbb Z$	Full fields can have untracked monodromy
Local product	Normal ordering and cocycle $\varepsilon$	Exchange and associativity phases agree	Bare exponentials can have inconsistent signs
Selection rule	Separate left/right neutrality	$\sum n_i=\sum w_i=0$ on the sphere	Zero-mode integral or winding conservation fails
Modular completion	Full momentum–winding lattice	$T$ uses integer spin; $S$ follows by Poisson resummation	A momentum-only truncation is not modular invariant
Normalization	$X_LX_L\sim-\log z$, $X\sim X+2\pi R$	T-duality is $R\leftrightarrow2/R$	Radius and weight formulas differ by factors of two

Exactly marginal radius deformation

The spinless operator $J(z)\bar J(\bar z)$ has weights $(1,1)$. In the compact free boson it changes the target-space radius while leaving $c=\bar c=1$ fixed. The statement is coordinate-dependent at the level of the coupling: a reparametrization of $R$ changes the numerical component of the Zamolodchikov metric, although its positive-definite line element in a unitary theory is invariant.

At every radius, $z=e^w$ maps the plane vacuum to

$$\langle T_{\rm cyl}\rangle=-\frac1{24}, \qquad \langle\bar T_{\rm cyl}\rangle=-\frac1{24}.$$

With $W=-\log Z$ and this volume’s curvature convention, the same $c=1$ anomaly is written $\langle T^\mu{}_{\mu}\rangle=+R^{(2)}/(24\pi)$. The positive trace coefficient and negative cylinder Casimir shift are therefore two convention-consistent manifestations of the anomaly, not opposite choices of $c$.

A reproducible calculation should cover the $c=1$ plane–cylinder shift and the $J\bar J$ radius baseline at $R=0.8,1,1.25$. Local-renormalization-group, curved-background, and relevant-flow stages belong to a later chapter; they are not part of this compact-boson derivation.

Common pitfalls

Taking the noncompact limit inside Kronecker-normalized correlators. As $R\to\infty$, sums become integrals and Kronecker deltas become Dirac deltas with volume factors. The limiting normalization must be changed before interpreting correlators.

Using $R\leftrightarrow1/R$ with these momenta. That formula belongs to a different $\alpha'$ convention. Here $p_{L,R}=n/R\pm wR/2$, so the dual radius is $2/R$.

Dropping winding sectors. A momentum-only trace is not closed under modular $S$. Projection or truncation cannot substitute for the complete lattice.

Exercises

Show that the T-duality map preserves the full torus summand.

Solution

Under $R'=2/R$, $n'=w$, and $w'=n$,

$$p_L'=\frac{w}{2/R}+\frac{n(2/R)}2 =\frac{wR}{2}+\frac nR=p_L,$$

while

$$p_R'=\frac{wR}{2}-\frac nR=-p_R.$$

Therefore $q^{p_L'^2/2}\bar q^{p_R'^2/2}$ equals the original summand, and the integer relabeling is bijective.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.
• Ginsparg, Paul. “Applied Conformal Field Theory.” In Fields, Strings and Critical Phenomena, Les Houches Session XLIX, edited by Édouard Brézin and Jean Zinn-Justin, 1–168. Amsterdam: North-Holland, 1990. arXiv.