Finite Conformal Maps, Gauge Symmetry, and Free Fields

The previous page introduced the operator product expansion and the idea that a conformal field theory is controlled by local operator data. We now pause to sharpen a point that will matter for the rest of the conformal field theory part of the course: how local operators transform under finite conformal maps.

In dimensions $d>2$, the connected conformal group is finite-dimensional. In two Euclidean dimensions, however, local conformal transformations are holomorphic maps. This gives a much larger local symmetry and eventually leads to the Virasoro algebra. The present page is the bridge: first we write the finite transformation law for primary fields, then we separate global Möbius transformations from general local maps, and finally we compare conformal covariance with two other local-symmetry ideas—diffeomorphism invariance and gauge invariance. The page ends by returning to the simplest dynamical fields, free oscillator modes, because the next pages will use their Wightman functions and $i\epsilon$ prescriptions as the basic analytic examples.

Required background. Lesson 15 supplies primary weights, conformal correlators, and the two-dimensional notation used here.

Helpful background. Lesson 14 develops finite conformal transformations and inversion in general dimension, while QFT I, Lesson 37 introduces covariant derivatives, gauge redundancy, and Wilson-line logic.

Primary fields under finite maps

Active finite conformal maps

We use Euclidean complex coordinates

$$z=x+iy, \qquad \bar z=x-iy, \qquad ds^2=dx^2+dy^2=dz\,d\bar z.$$

A local conformal map is

$$w=f(z), \qquad \bar w=\bar f(\bar z), \qquad f'(z)\ne0.$$

A primary field of weights $(h,\bar h)$ is transformed actively by

$$U_f O(z,\bar z)U_f^{-1} =(f'(z))^h(\bar f'(\bar z))^{\bar h}O(f(z),\bar f(\bar z)).$$

Its scaling dimension and spin are

$$\Delta=h+\bar h, \qquad s=h-\bar h.$$

Some books use the passive convention with inverse powers of $f'$. The physics is identical as long as the convention is used consistently.

A holomorphic map rescales lengths locally. Since

$$dw=f'(z)dz, \qquad d\bar w=\bar f'(\bar z)d\bar z,$$

the metric transforms as

$$ds'^2=dw\,d\bar w=|f'(z)|^2 dz\,d\bar z.$$

Thus, to linear order in its radius, a small circle near $z$ is mapped to a small circle near $w=f(z)$, multiplied by the local scale $|f'(z)|$ and rotated by the phase of $f'(z)$. Nonlinear terms in $f$ can distort a finite circle, but that distortion vanishes relative to its radius in the local limit. A primary field responds to the linear scale and rotation with weights $(h,\bar h)$.

For a pure dilation and rotation,

$$f(z)=\lambda e^{i\theta}z, \qquad \lambda>0,$$

the transformation factor is

$$(f')^h(\bar f')^{\bar h} =\lambda^{h+\bar h}e^{i\theta(h-\bar h)} =\lambda^\Delta e^{i\theta s}.$$

This is the cleanest way to remember the meanings of $\Delta$ and $s$. The sum $h+\bar h$ measures response to local scale, while the difference $h-\bar h$ measures response to local rotation. For a single-valued bosonic local field on the plane, $s$ is an integer. Fermions have half-integer spin and require a spin structure; the Ising Majorana fields have weights $(1/2,0)$ and $(0,1/2)$.

A holomorphic map $w=f(z)$ acts near an insertion by a local scale and rotation. A primary field records that local Jacobian through the factor $(f')^h(\bar f')^{\bar h}$.

If the vacuum is invariant under the transformation, correlation functions obey

$$\boxed{ \langle O_1(z_1,\bar z_1)\cdots O_n(z_n,\bar z_n)\rangle =\prod_{i=1}^n (f_i')^{h_i}(\bar f_i')^{\bar h_i} \langle O_1(w_1,\bar w_1)\cdots O_n(w_n,\bar w_n)\rangle, }$$

where

$$w_i=f(z_i), \qquad f_i'=f'(z_i).$$

The phrase “if the vacuum is invariant” is not a throwaway. On the plane, the vacuum of a CFT is invariant under the global conformal group. A general local holomorphic map is instead a change of conformal frame, not necessarily a unitary symmetry that leaves the state and boundary conditions fixed. Primary fields still obey the local Jacobian law when the background data are transported, while the stress tensor acquires the anomalous Schwarzian term studied later.

Global maps and Möbius transformations

The infinitesimal holomorphic transformations are

$$z\mapsto z+\epsilon(z).$$

Locally, $\epsilon(z)$ can be any holomorphic function. Globally on the Riemann sphere, however, regularity at both $z=0$ and $z=\infty$ permits only a quadratic polynomial:

$$\epsilon(z)=\alpha+\beta z+\gamma z^2.$$

These three terms generate translations, dilations plus rotations, and special conformal transformations. Exponentiating them gives the Möbius maps

$$\boxed{ f(z)={az+b\over cz+d}, \qquad ad-bc=1. }$$

The matrices

$$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$

and its negative define the same map, so the group is $PSL(2,\mathbb C)$ on the Riemann sphere. If the map is required to preserve the real line or the upper half-plane, then $a,b,c,d$ may be chosen real and the group is $PSL(2,\mathbb R)$.

The globally nonsingular holomorphic vector fields on the sphere are generated by $1$, $z$, and $z^2$. Their finite transformations combine into the Möbius map $f(z)=(az+b)/(cz+d)$.

It is useful to record the elementary identities

$$f'(z)={ad-bc\over(cz+d)^2},$$

and

$$f(z_1)-f(z_2)={(ad-bc)(z_1-z_2)\over(cz_1+d)(cz_2+d)}.$$

These identities explain why two-dimensional two-point functions are Möbius covariant. For a primary of weights $(h,\bar h)$,

$$\langle O(z_1,\bar z_1)O(z_2,\bar z_2)\rangle ={C\over z_{12}^{2h}\bar z_{12}^{2\bar h}}, \qquad z_{12}=z_1-z_2.$$

Using the two identities above,

$${(f_1'f_2')^h\over(f(z_1)-f(z_2))^{2h}} ={1\over z_{12}^{2h}},$$

and similarly for the antiholomorphic part. Thus

$$\prod_{i=1}^2(f_i')^h(\bar f_i')^{\bar h} \langle O(f(z_1),\bar f(\bar z_1))O(f(z_2),\bar f(\bar z_2))\rangle = \langle O(z_1,\bar z_1)O(z_2,\bar z_2)\rangle.$$

For a scalar primary with $h=\bar h=\Delta/2$, this becomes the familiar rotationally invariant form

$$\langle O(z_1,\bar z_1)O(z_2,\bar z_2)\rangle ={C\over |z_{12}|^{2\Delta}}.$$

For a chiral field with $\bar h=0$, the correlator is holomorphic away from coincident points:

$$\langle \psi(z_1)\psi(z_2)\rangle={C\over z_{12}^{2h}}.$$

The Ising fermion corresponds to $h=1/2$, so its chiral two-point function is proportional to $1/z_{12}$.

Domains, boundaries, and the meaning of covariance

A conformal map may act in two related but conceptually different ways. It may be an automorphism of the same background, such as a Möbius transformation of the plane vacuum. Or it may map one domain to another domain. In the second case, the correlator in the original domain $D$ is related to a correlator in the image domain $\Gamma=f(D)$ by local Jacobian factors:

$$\boxed{ \left\langle\prod_i O_i(z_i,\bar z_i)\right\rangle_D = \prod_i (f_i')^{h_i}(\bar f_i')^{\bar h_i} \left\langle\prod_i O_i(w_i,\bar w_i)\right\rangle_\Gamma. }$$

This formula is the workhorse of two-dimensional boundary CFT and statistical mechanics in planar domains. It assumes that the boundary condition and state are carried from $D$ to $\Gamma$; an unnormalized partition function can additionally contain Weyl-anomaly factors. It is also a good conceptual rehearsal for quantum gravity. Coordinates by themselves are not observables; they are labels. In the active convention used here, a scalar field under a coordinate map behaves as

$$\phi(x)\mapsto \phi(f(x)),$$

so a coordinate-labeled correlator transforms covariantly:

$$\langle \phi(x_1)\phi(x_2)\rangle \mapsto \langle \phi(f(x_1))\phi(f(x_2))\rangle.$$

In an ordinary QFT on a fixed background, this is a useful symmetry statement when $f$ preserves the relevant background structure. In a gravitational theory, diffeomorphisms are gauge redundancies, so a coordinate-labeled local field is not by itself a physical observable. Physical observables must be invariant under the redundancy or relationally dressed.

A conformal map sends a domain $D$ to an image domain $\Gamma$. Primary insertions acquire local factors determined by $f'(z_i)$, while the boundary condition is transported to the image domain.

The group $PSL(2,\mathbb R)$ appears naturally when the domain is the upper half-plane. It is the subgroup of Möbius transformations preserving the boundary real axis. This is why boundary CFT and open-string worldsheets repeatedly produce $SL(2,\mathbb R)$ gauge fixing factors.

Gauge symmetry and dressed correlators

The same warning appears in gauge theory in a simpler, more familiar form. Consider a charged scalar field in Abelian gauge theory, with convention

$$\phi(x)\mapsto e^{+iq\alpha(x)}\phi(x), \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\alpha(x), \qquad D_\mu=\partial_\mu-iqA_\mu.$$

The local field $\phi(x)$ is charged. Therefore the naive two-point function

$$\langle \phi^\dagger(x_1)\phi(x_2)\rangle$$

is not a gauge-invariant observable: its phase changes by

$$e^{-iq\alpha(x_1)}e^{+iq\alpha(x_2)}.$$

There are two standard cures. The first is to make a local neutral composite, such as

$$\phi^\dagger(x)\phi(x).$$

The second is to dress the separated charged fields by a Wilson line along a path $\gamma$ from $x_1$ to $x_2$:

$$W_\gamma(x_1,x_2)=\exp\left(-iq\int_\gamma A_\mu dx^\mu\right).$$

Since

$$W_\gamma(x_1,x_2) \mapsto \exp\left(-iq\alpha(x_2)+iq\alpha(x_1)\right)W_\gamma(x_1,x_2),$$

the bilocal operator

$$\boxed{ \phi^\dagger(x_1)W_\gamma(x_1,x_2)\phi(x_2) }$$

is gauge invariant.

A separated pair of charged fields is not gauge invariant by itself. A Wilson line supplies the endpoint phases needed to make a gauge-invariant bilocal operator.

This is the gauge-theory analogue of the gravity warning above. Local charged fields are perfectly useful inside a fixed gauge or as ingredients in correlation functions, but physical questions must be expressed in gauge-invariant terms. In later pages Wilson loops will become central order parameters for confinement.

Free fields as oscillator modes

We now return to a simpler system: the free scalar field. The reason is strategic. The next page studies Wightman functions, commutators, and the $i\epsilon$ prescription. All of those analytic structures can be seen already in the free oscillator decomposition.

Now let the free scalar live in $D$-dimensional Minkowski spacetime, so there are $D-1$ spatial momentum components. With periodic boundary conditions in a finite spatial volume $V$, write

$$\phi(t,\mathbf x) =\sum_{\mathbf p} \left(a_{\mathbf p}u_{\mathbf p}(t,\mathbf x) +a_{\mathbf p}^\dagger u_{\mathbf p}^*(t,\mathbf x)\right),$$

with

$$u_{\mathbf p}(t,\mathbf x) ={1\over\sqrt V}e^{i\mathbf p\cdot\mathbf x}f_{\mathbf p}(t), \qquad f_{\mathbf p}(t)={e^{-i\omega_{\mathbf p}t}\over\sqrt{2\omega_{\mathbf p}}}, \qquad \omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

The creation and annihilation operators obey

$$[a_{\mathbf p},a_{\mathbf q}^\dagger]=\delta_{\mathbf p\mathbf q}, \qquad [a_{\mathbf p},a_{\mathbf q}]=[a_{\mathbf p}^\dagger,a_{\mathbf q}^\dagger]=0.$$

The time-dependent oscillator mode is normalized by its Wronskian:

$$\boxed{ f_{\mathbf p}\dot f_{\mathbf p}^{*}-\dot f_{\mathbf p}f_{\mathbf p}^{*}=i. }$$

Indeed, for $f=e^{-i\omega t}/\sqrt{2\omega}$,

$$f\dot f^*-\dot f f^* ={e^{-i\omega t}\over\sqrt{2\omega}} {i\omega e^{i\omega t}\over\sqrt{2\omega}} - {-i\omega e^{-i\omega t}\over\sqrt{2\omega}} {e^{i\omega t}\over\sqrt{2\omega}} =i.$$

This Wronskian is the mode-by-mode version of the canonical commutation relation

$$[\phi(t,\mathbf x),\pi(t,\mathbf y)]=i\delta^{(D-1)}(\mathbf x-\mathbf y), \qquad \pi=\dot\phi.$$

Indeed, inserting the mode expansion, relabeling $\mathbf p\mapsto-\mathbf p$ in the second term, and using $f_{-\mathbf p}=f_{\mathbf p}$ gives

$$\begin{aligned} [\phi(t,\mathbf x),\pi(t,\mathbf y)] &={1\over V}\sum_{\mathbf p}e^{i\mathbf p\cdot(\mathbf x-\mathbf y)} \left(f_{\mathbf p}\dot f_{\mathbf p}^* -\dot f_{\mathbf p}f_{\mathbf p}^*\right)\\ &={i\over V}\sum_{\mathbf p}e^{i\mathbf p\cdot(\mathbf x-\mathbf y)} =i\delta_V^{(D-1)}(\mathbf x-\mathbf y), \end{aligned}$$

where $\delta_V^{(D-1)}$ is the periodic delta function. This displays directly why the sign and normalization of the Wronskian matter.

In infinite volume,

$$\sum_{\mathbf p}{1\over V}\longrightarrow \int{d^{D-1} p\over(2\pi)^{D-1}},$$

and the expansion becomes

$$\phi(t,\mathbf x) =\int{d^{D-1} p\over(2\pi)^{D-1}}{1\over\sqrt{2\omega_{\mathbf p}}} \left( a_{\mathbf p}e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x} +a_{\mathbf p}^\dagger e^{i\omega_{\mathbf p}t-i\mathbf p\cdot\mathbf x} \right),$$

with

$$[a_{\mathbf p},a_{\mathbf q}^\dagger]=(2\pi)^{D-1}\delta^{(D-1)}(\mathbf p-\mathbf q).$$

Each momentum mode of a free scalar field is a harmonic oscillator. The Wronskian normalization of the positive-frequency solution is equivalent to the canonical commutator of the field and its conjugate momentum.

The Euclidean momentum-space two-point function of a free scalar is

$$\langle \phi(p)\phi(-p)\rangle_E={1\over p^2+m^2}.$$

At $m\ne0$, the mass introduces a scale and the theory is not conformal. In two dimensions the $m=0$ theory has a zero-mode and infrared subtlety: the scalar itself has a logarithmic correlator rather than an ordinary power-law primary correlator. The derivative sector is nevertheless conformal, and its simplest fields obey

$$\langle \partial\phi(z)\partial\phi(0)\rangle\propto {1\over z^2}, \qquad \partial\phi\text{ has weights }(1,0).$$

This small caveat is worth remembering. Not every field appearing in a Lagrangian is automatically a primary field in the CFT sense. The primary operators are the fields with definite transformation laws under conformal maps.

Summary

Finite conformal maps in two dimensions act on primary fields through local holomorphic and antiholomorphic Jacobians. The weights $(h,\bar h)$ encode scaling dimension $\Delta=h+\bar h$ and spin $s=h-\bar h$. Local holomorphic maps are infinite-dimensional, but globally regular maps on the sphere reduce to Möbius transformations.

The covariance formula for correlators is both powerful and delicate. It is exact for global conformal transformations preserving the vacuum, and it relates correlators in conformally equivalent domains. Similar conceptual care is required for local symmetries in gauge theory and gravity: coordinate-labeled or charged fields are useful, but physical observables must be invariant or properly dressed.

The free scalar field reappears as a collection of harmonic oscillators in $D-1$ spatial momentum dimensions. The positive-frequency mode $e^{-i\omega t}/\sqrt{2\omega}$ is normalized by a Wronskian, and that Wronskian is precisely what makes the equal-time canonical commutator work. This oscillator picture will now be used to derive Wightman functions, commutators, time ordering, and the $i\epsilon$ prescription.

Common pitfalls

Mixing active and passive conventions. If one writes

$$U_f O(z)U_f^{-1}=(f')^hO(f(z)),$$

then the corresponding vacuum Ward identity has positive powers of $f'$. If instead one asks for the field components in the new coordinate $w=f(z)$, inverse powers appear; neither convention is wrong, but combining them is.

Treating every holomorphic map as a global symmetry. On the sphere, only Möbius transformations are globally regular. More general holomorphic maps organize local Ward identities and Virasoro symmetry, but they need not leave the vacuum, domain, or boundary conditions unchanged.

Using an undressed charged correlator as an observable. Before gauge fixing, a separated charged two-point function is not gauge invariant. A Wilson line is not decoration; its endpoint transformation supplies exactly the missing phase.

Reusing $d$ for two different dimensions. Conformal formulas count spacetime dimensions, whereas the oscillator integral counts spatial momentum components. Here $D$ is the Minkowski spacetime dimension and the momentum measure is therefore $d^{D-1}p$.

Exercises

Exercise 1: Möbius covariance of a two-point function

Show that the two-point function

$$G(z_1,z_2)={1\over z_{12}^{2h}}$$

is covariant under the holomorphic part of the Möbius transformation

$$f(z)={az+b\over cz+d}, \qquad ad-bc=1,$$

with primary weight $h$.

Solution

First compute

$$f'(z)={ad-bc\over(cz+d)^2}={1\over(cz+d)^2}.$$

Next,

$$\begin{aligned} f(z_1)-f(z_2) &={az_1+b\over cz_1+d}-{az_2+b\over cz_2+d} \\ &={ (az_1+b)(cz_2+d)-(az_2+b)(cz_1+d) \over(cz_1+d)(cz_2+d)}. \end{aligned}$$

The numerator is

$$a d(z_1-z_2)-bc(z_1-z_2)=(ad-bc)z_{12}=z_{12}.$$

Therefore

$$f(z_1)-f(z_2)={z_{12}\over(cz_1+d)(cz_2+d)}.$$

The transformed two-point function with the primary factors is

$$(f'(z_1))^h(f'(z_2))^h{1\over(f(z_1)-f(z_2))^{2h}}.$$

Substituting the formulas gives

$${1\over(cz_1+d)^{2h}(cz_2+d)^{2h}} {(cz_1+d)^{2h}(cz_2+d)^{2h}\over z_{12}^{2h}} ={1\over z_{12}^{2h}}.$$

So the correlator is Möbius covariant.

Exercise 2: Global holomorphic vector fields

Why are the globally regular infinitesimal holomorphic conformal transformations on the Riemann sphere generated only by

$$\epsilon(z)=\alpha+\beta z+\gamma z^2?$$

Solution

Near $z=0$, regularity allows a Taylor expansion

$$\epsilon(z)=\sum_{n\ge0}a_n z^n.$$

Now examine the same vector field near $z=\infty$ using $u=1/z$. Since

$$\epsilon(z)\partial_z =\epsilon(1/u){d u\over dz}\partial_u =-u^2\epsilon(1/u)\partial_u,$$

regularity at $u=0$ requires

$$-u^2\epsilon(1/u)$$

to have no negative powers of $u$. A term $a_n z^n$ becomes

$$-u^2 a_n u^{-n}=-a_n u^{2-n}.$$

This is regular at $u=0$ only if $2-n\ge0$, or $n\le2$. Hence only $n=0,1,2$ survive:

$$\epsilon(z)=\alpha+\beta z+\gamma z^2.$$

These are the infinitesimal generators of $PSL(2,\mathbb C)$.

Exercise 3: Wilson-line endpoint phases

Using the gauge transformations

$$\phi(x)\mapsto e^{+iq\alpha(x)}\phi(x), \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\alpha(x),$$

show that

$$\phi^\dagger(x_1)\exp\left(-iq\int_{x_1}^{x_2}A_\mu dx^\mu\right)\phi(x_2)$$

is gauge invariant.

Solution

The Wilson-line factor transforms as

$$\begin{aligned} \exp\left(-iq\int_{x_1}^{x_2}(A_\mu+\partial_\mu\alpha)dx^\mu\right) &= \exp\left(-iq\int_{x_1}^{x_2}A_\mu dx^\mu\right) \\ &\quad\times \exp\left(-iq[\alpha(x_2)-\alpha(x_1)]\right). \end{aligned}$$

The charged fields transform as

$$\phi^\dagger(x_1)\mapsto e^{-iq\alpha(x_1)}\phi^\dagger(x_1), \qquad \phi(x_2)\mapsto e^{+iq\alpha(x_2)}\phi(x_2).$$

Multiplying all factors gives the phase

$$e^{-iq\alpha(x_1)}e^{-iq[\alpha(x_2)-\alpha(x_1)]}e^{+iq\alpha(x_2)}=1.$$

Therefore the dressed bilocal operator is gauge invariant.

Exercise 4: Wronskian and canonical normalization

Let

$$f(t)={e^{-i\omega t}\over\sqrt{2\omega}}.$$

Verify the Wronskian condition

$$f\dot f^*-\dot f f^*=i,$$

and explain why this normalization is needed in the free-field expansion.

Solution

We have

$$\dot f=-i\omega f, \qquad \dot f^*=i\omega f^*.$$

Therefore

$$f\dot f^*-\dot f f^* =i\omega |f|^2+i\omega |f|^2 =2i\omega {1\over2\omega}=i.$$

In the field expansion, the equal-time commutator $[\phi,\pi]$ receives one contribution from the positive-frequency mode and one from the negative-frequency mode. The Wronskian is exactly the coefficient that remains after subtracting these two pieces. Setting it equal to $i$ ensures

$$[\phi(t,\mathbf x),\pi(t,\mathbf y)]=i\delta^{(D-1)}(\mathbf x-\mathbf y).$$

A different normalization of $f$ would require a compensating change in the normalization of the creation and annihilation operators.

References

• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer (1997), chapters 4–6.
• P. Ginsparg, “Applied Conformal Field Theory,” in É. Brézin and J. Zinn-Justin, eds., Fields, Strings and Critical Phenomena, Les Houches Session XLIX, North-Holland (1990), 1–168.
• A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics 3, Harwood Academic Publishers (1987).
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), chapters 8, 14, and 25.
• M. Srednicki, Quantum Field Theory, Cambridge University Press (2007), chapters 3, 22, 57, and 67.