Conformal Gauge, Two-Dimensional Gravity, and Vacuum Polarization

The string worldsheet is a two-dimensional generally covariant system. Here we use a positive Euclidean worldsheet metric and real embedding fields in a Euclidean target, as in the geometric discussion of Lesson 31. The target signature is an explicit continuation of the site’s Lorentzian setting. Classically, local conformal coordinates reduce the matter action to that of free fields $X^\mu$, while stress-tensor constraints retain the metric equation of motion. Integrating out these fields instead produces an effective action for the background metric; its second variation is a gravitational response kernel.

This page has two intertwined goals. First, we make the conformal-gauge statement concrete: the conditions

$$(\partial_+X)^2=0, \qquad (\partial_-X)^2=0$$

say that, for an immersion, the chosen worldsheet coordinates are locally orthogonal and equally scaled on the embedded surface. Second, we compare electromagnetic and gravitational polarization operators: their nonlocal parts come from connected current and stress correlations, while local contact terms are needed for their Ward identities. We derive the source normalizations and quadratic structure here; the overall anomaly coefficient and quantum gravity measure belong to the next lesson.

Required background. Lesson 31 supplies the positive-metric Polyakov action, its boundary variation, and the distinction between local conformal gauge and global moduli.

Helpful background. Lesson 24 fixes the $T T$ normalization and central-charge convention used below.

Conformal gauge and classical constraints

Take $\mathcal T>0$, a smooth positive worldsheet metric, and real Euclidean target fields. Work locally away from a boundary, or use boundary conditions that remove the integration-by-parts terms. Define

$$z=\xi^+=\xi^1+i\xi^2, \qquad \bar z=\xi^- = \xi^1-i\xi^2,$$

and $d^2z=d\xi^1d\xi^2$. Then

$$d\xi^+d\xi^-=(d\xi^1)^2+(d\xi^2)^2, \qquad \partial_+=\partial_1-i\partial_2=2\partial_z, \qquad \partial_- =\partial_1+i\partial_2=2\partial_{\bar z}.$$

The symbols $\partial_\pm$ denote these combinations, not the coordinate derivatives $\partial/\partial\xi^\pm$. Keeping the factors of two explicit is essential when the classical quadratic operators are normalized as CFT stress tensors.

The Polyakov action is

$$S[X,g]={\mathcal T\over2}\int d^2\xi\,\sqrt g\,g^{ab}\partial_aX\cdot\partial_bX.$$

Every smooth positive two-dimensional metric admits local isothermal coordinates. In such a chart,

$$ds^2=e^{\phi(\xi)}d\xi^+d\xi^-.$$

Equivalently,

$$g_{ab}=e^{\phi}\delta_{ab}$$

in local real coordinates. In this gauge the Weyl factor drops out of the classical matter action, because

$$\sqrt g\,g^{ab}=\delta^{ab}.$$

Classical Weyl invariance permits changing $\phi$ without changing this matter action. It does not eliminate global moduli, and it need not survive quantization. The local classical action is exactly

$$S_0[X]={\mathcal T\over2}\int d^2\xi\,\partial_+X\cdot\partial_-X.$$

The equation of motion is the two-dimensional Laplace equation,

$$\partial_+\partial_-X^\mu=0.$$

Define the metric stress tensor with the inherited plus covariant-metric variation:

$$\begin{aligned} \delta S&={1\over2}\int d^2\xi\,\sqrt g\,T_m^{ab}\delta g_{ab},\\ T_{m,ab}&=\mathcal T\left[\frac12g_{ab}(\partial X)^2 -\partial_aX\cdot\partial_bX\right]. \end{aligned}$$

Varying the metric before gauge fixing gives $T_{m,ab}=0$. In conformal coordinates, define the unnormalized quadratic operators

$$U_{++}=\partial_+X\cdot\partial_+X, \qquad U_{--}=\partial_-X\cdot\partial_-X.$$

The classical Virasoro constraints are $U_{++}=U_{--}=0$. They are not the equations obtained by varying $X^\mu$. For this metric variation and local gauge reduction, see Polyakov 1987, §9.5, pp. 176–177, Eqs. 9.101–9.110, with the action normalization fixed above.

Now expand the first constraint in real coordinates:

$$(\partial_+X)^2 =(\partial_1X-i\partial_2X)^2 =(\partial_1X)^2-(\partial_2X)^2-2i\,\partial_1X\cdot\partial_2X.$$

Therefore

$$(\partial_+X)^2=0$$

is equivalent to the pair of real conditions

$$(\partial_1X)^2=(\partial_2X)^2, \qquad \partial_1X\cdot\partial_2X=0.$$

So, for a nondegenerate positive-definite induced metric, the coordinate tangent vectors have equal length and are orthogonal. Geometrically, the parameter grid maps to the surface by local similarities: infinitesimal squares become infinitesimal squares up to an overall scale.

The chiral conservation equations follow directly. Since

$$U_{++}=\partial_+X\cdot\partial_+X,$$

we have

$$\partial_-U_{++} =2\partial_+X\cdot\partial_-\partial_+X=0,$$

using $\partial_+\partial_-X=0$. Similarly,

$$\partial_+U_{--}=0.$$

Thus the two quadratic components become chiral:

$$U_{++}=U_{++}(z), \qquad U_{--}=U_{--}(\bar z)$$

locally. In the quantum free-scalar theory, the normalized CFT operators are

$$\begin{aligned} T(z)&=2\pi T_{m,zz} =-2\pi\mathcal T:\partial_zX\cdot\partial_zX: =-{\pi\mathcal T\over2}:U_{++}:,\\ \bar T(\bar z)&=2\pi T_{m,\bar z\bar z} =-{\pi\mathcal T\over2}:U_{--}:. \end{aligned}$$

Normal ordering is defined by the flat vacuum contraction. For each scalar,

$$\langle\partial_zX(z)\partial_wX(w)\rangle =-{1\over4\pi\mathcal T(z-w)^2}.$$

The two double contractions therefore give $\langle T(z)T(w)\rangle=D/[2(z-w)^4]$ for $D$ independent embedding fields. This verifies $c=D$ and fixes the normalization imported from CFT; $U_{++}$ itself does not have that coefficient. See Di Francesco, Mathieu, and Sénéchal 1997, §5.3.1, pp. 128–129, Eqs. 5.73–5.83. Their normalized chiral tensor agrees with this formula; their canonical tensor has the opposite sign to $T_m$.

The Weyl factor and local curvature

In conformal gauge,

$$ds^2=e^\phi d\xi^+d\xi^-.$$

With $\Delta=\partial_1^2+\partial_2^2=\partial_+\partial_-$ and the site’s curvature convention, the exact scalar curvature is

$$R=-e^{-\phi}\Delta\phi,$$

so curvature is controlled by the Laplacian of the Weyl factor. There is no remaining derivative-normalization ambiguity after the definitions above.

Classically, the Weyl factor is pure gauge for this matter action. Quantum mechanically, the regulated matter measure is generally not Weyl invariant. This conformal anomaly produces an induced action for $\phi$, or a covariant nonlocal structure involving $R\Delta_g^{-1}R$. Its coefficient is the central topic of the next page.

For now, we only need the background-field viewpoint. Fix a metric $g_{ab}$ and define the matter partition function

$$Z[g]=\int \mathcal D X\,e^{-S[X,g]}, \qquad W[g]=-\log Z[g].$$

Use a regulated partition function and a fixed prescription for zero modes and infrared normalization; absorb any metric dependence of the measure into the regulated action when differentiating it. The functional $W[g]$ is then the induced gravitational effective action. Integrating over metrics as well leads formally to matter coupled to two-dimensional gravity:

$$Z_{\rm grav} ={1\over \operatorname{Vol}(\mathrm{Diff}\times \mathrm{Weyl})} \int \mathcal Dg\,\mathcal D X\,e^{-S[X,g]}.$$

Replacing the free fields $X^\mu$ by a minimal CFT gives the family often called minimal models coupled to two-dimensional gravity. The metric is no longer merely a background probe; it becomes part of the fluctuating geometry.

The quotient by Weyl transformations is schematic at the quantum level. Gauge fixing introduces ghosts and moduli, while a nonvanishing total Weyl anomaly makes the conformal factor dynamical rather than removable; these effects are developed on the next page.

Metric sources and stress-tensor response

Treat the two chiral metric components as independent sources for functional differentiation. Consider the formal source slice

$$ds^2=dz\,d\bar z+h_{zz}dz^2.$$

In real coordinates its matrix is $g=I+h_{zz}B$, where

$$B=\begin{pmatrix}1&i\\i&-1\end{pmatrix},\qquad B^2=0.$$

Hence $\det g=1$ and $g^{-1}=I-h_{zz}B$. Substitution into the classical action gives

$$\delta S=-{\mathcal T\over2}\int d^2z\,h_{zz}(\partial_-X)^2 ={1\over\pi}\int d^2z\,h_{zz}\bar T.$$

This fixes both the opposite-component coupling and its coefficient. The one-component deformation is a complex source, not a real positive metric by itself. A real perturbation also includes $h_{\bar z\bar z}=h_{zz}^*$; with the mixed component held fixed, $|h_{zz}|<1/2$ ensures positivity. Quantum operator definitions and the covariant regulator can still supply local terms.

For $W[h]=-\log Z[h]$, write $S_i=\delta S/\delta h_i$ for any source component. Differentiating the normalized expectation value gives the exact regulated identities

$$W_i=\langle S_i\rangle,\qquad W_{ij}=\langle S_{ij}\rangle-\langle S_iS_j\rangle_{\mathrm{conn}}.$$

For the flat chiral source above, the one-point insertion is $\bar T/\pi$, and the Hessian has separated-point part

$${\delta^2 W\over \delta h_{zz}(x)\delta h_{zz}(y)}\bigg|_{h=0} =-{1\over\pi^2}\langle\bar T(x)\bar T(y)\rangle_{\rm conn} +\text{contact terms}.$$

For a general real linear coupling $+\kappa\int hO$, that prefactor is $-\kappa^2$. Reversing the real source sign changes the first derivative but leaves the Hessian unchanged. An imaginary coupling is a different case because its square changes sign. The real-source differentiation is also derived in Altland and Simons 2023, §7.2.1, pp. 391–392, Eqs. 7.5–7.7.

Consequently, with the momentum-conserving delta function removed from the translation-invariant correlator, the chiral contribution is

$$W_{2,\mathrm{nonlocal}}[h] =-{1\over2\pi^2}\int {d^2q\over(2\pi)^2}\, h_{zz}(q)h_{zz}(-q) \langle\bar T(q)\bar T(-q)\rangle_{\rm conn}.$$

Contact terms include action second derivatives, source-dependent operator definitions, and local counterterms. For a fixed regulated functional they are not arbitrary sign choices. The nonlocal part carries universal CFT information such as the central charge, while its local completion enforces the full Ward identity. This distinction is explicit in Polyakov 1987, pp. 185–188, Eqs. 9.156–9.172.

Vacuum polarization in QED

Consider Euclidean QED in a fixed background $A_\mu$, with a regulator preserving vector gauge invariance. To fix the source insertion, take

$$S_E=\int d^dx\,\bar\psi \left[\gamma_E^\mu(\partial_\mu-iq_{\rm el}A_\mu)+m\right]\psi, \qquad \{\gamma_E^\mu,\gamma_E^\nu\}=2\delta^{\mu\nu}.$$

The charge is $q_{\rm el}$; $q_\mu$ below denotes Euclidean momentum. Define

$$e^{-W[A]} =\int \mathcal D\psi\,\mathcal D\bar\psi\, \exp\left[-S_E[\psi,\bar\psi,A]\right].$$

Write $j_{\rm bil}^\mu=\bar\psi\gamma_E^\mu\psi$. The source-conjugate insertion is $J_{\rm src}^\mu=-iq_{\rm el}j_{\rm bil}^\mu$, so

$$\langle J_{\rm src}^\mu(q)\rangle_A ={\delta W[A]\over \delta A_\mu(-q)}.$$

This Euclidean insertion must be continued before identifying a real Lorentzian electric current. In particular, its separated-point Hessian contains $+q_{\rm el}^2\langle j_{\rm bil}^\mu j_{\rm bil}^\nu\rangle_{\rm conn}$, since $-(-i)^2=+1$. Around the zero-current vacuum, linear response is

$$\delta\langle J_{\rm src}^\mu(q)\rangle=\Pi_{\mu\nu}(q)\delta A_\nu(q),$$

where $\Pi_{\mu\nu}=\delta^2W/\delta A_\mu\delta A_\nu$ includes the required local terms. It is this declared Euclidean Hessian that we call the polarization tensor. A Lorentzian diagram’s insertion $i\Pi$ is a different convention; the insertion and dressed propagator are distinguished in Schwartz 2014, p. 309, Eq. 16.47 and §16.3, Eqs. 16.48–16.51.

Gauge invariance implies current conservation. In momentum space this gives the Ward identity

$$q_\mu\Pi_{\mu\nu}(q)=0.$$

For the symmetric polarization tensor in a parity-even rotationally invariant vacuum, and for $q\ne0$, transversality fixes the tensor structure:

$$\boxed{ \Pi_{\mu\nu}(q) =q^2\Pi(q^2)\delta^\perp_{\mu\nu}(q), \qquad \delta^\perp_{\mu\nu}=\delta_{\mu\nu}-{q_\mu q_\nu\over q^2}. }$$

The quadratic effective action is

$$W_2[A]={1\over2}\int {d^dq\over(2\pi)^d}\, A_\mu(q)\Pi_{\mu\nu}(q)A_\nu(-q).$$

Using transversality,

$$A_\mu(q)q^2\delta^\perp_{\mu\nu}A_\nu(-q) ={1\over2}F_{\mu\nu}(q)F_{\mu\nu}(-q),$$

where the site-wide forward transform gives $\partial_\mu\mapsto-iq_\mu$ and hence

$$F_{\mu\nu}(q)=-i(q_\mu A_\nu(q)-q_\nu A_\mu(q)).$$

Therefore, if $\Pi(q^2)$ is regular near $q^2=0$, the low-momentum effective action begins as a local Maxwell term,

$$W_2[A]\to {\Pi(0)\over4}\int {d^dq\over(2\pi)^d}\, F_{\mu\nu}(q)F_{\mu\nu}(-q)+\cdots.$$

A pole in the scalar coefficient changes the infrared structure. If

$$\Pi(q^2)\sim {\mu^2\over q^2},$$

then

$$W_2[A]\sim {\mu^2\over2}\int A_\mu\delta^\perp_{\mu\nu}A_\nu ={\mu^2\over4}\int F_{\mu\nu}{1\over q^2}F_{\mu\nu}.$$

This expression is nonlocal when written in terms of $F_{\mu\nu}$ and is gauge invariant. It can arise when massless matter or a collective mode has been integrated out. The coefficient $\mu^2$ is not yet a normalized photon mass. With a positive local Maxwell term $Z_A F^2/4$, the transverse inverse propagator in this approximation is $Z_Aq^2+\mu^2$. If its analytic continuation describes a stable physical pole, then $m_\gamma^2=\mu^2/Z_A$.

In a superconductor the relevant limit is static and transverse. For example, a local phase action $\rho_s(\nabla\theta-q_*\mathbf A)^2/2$ with $\rho_s>0$ gives $q_*^2\rho_s\mathbf A_T^2/2$ after eliminating $\theta$. Combined with magnetic energy $Z_B\mathbf B^2/2$, this yields $\lambda^{-2}=q_*^2\rho_s/Z_B$. Zero frequency is taken before the long-wavelength limit. Longitudinal charge screening or optical Drude weight alone does not imply this Meissner response; see Altland and Simons 2023, §7.4, pp. 407–410, Eqs. 7.41–7.47 and the London-response exercise, and the static versus optical response comparison.

Static response and dielectric screening

In a medium, Lorentz invariance need not be present and a single scalar function no longer describes the polarization tensor. This subsection uses a homogeneous isotropic linear medium in three spatial dimensions and Gaussian units. Let $\phi$ denote the total physical static potential; this is a physical electrostatic response convention, not an identification with the Euclidean source insertion above. Define

$$\rho_{\rm ind}(k)=\chi_{00}(k,0)\phi(k).$$

Here $\chi_{00}$ responds to the total potential, rather than just the externally applied potential. Define $\Pi_{00}\equiv-\chi_{00}$. A positive $\Pi_{00}$ then increases the dielectric function, and Gauss’s law becomes

$$k^2\phi(k)=4\pi\rho_{\rm ext}(k)-4\pi\Pi_{00}(k,0)\phi(k).$$

One then introduces the displacement field by

$$\mathbf D=\varepsilon\mathbf E, \qquad \nabla\cdot\mathbf D=4\pi\rho_{\rm ext}.$$

In momentum space this gives

$$\boxed{ \varepsilon(k)=1+{4\pi\over k^2}\Pi_{00}(k,0). }$$

The screened potential sourced by $\rho_{\rm ext}$ is exactly

$$\phi(k)={4\pi\rho_{\rm ext}(k)\over k^2\varepsilon(k)}.$$

For example, a positive constant $\Pi_{00}$ replaces the Coulomb denominator by $k^2+k_s^2$, where $k_s^2=4\pi\Pi_{00}$, giving an exponentially screened potential. More generally, dominant isolated poles in complex spatial momentum give exponential scales; branch cuts can produce other tails. Dynamical collective modes are poles of the retarded longitudinal response. Zeros of $\varepsilon_R(\omega,k)$ can produce such poles when the residue is nonzero, including damped poles at complex frequency. Poles of $\varepsilon_R$ itself are not generally poles of the screened potential.

The full response to the external potential is different: since $\phi=\phi_{\rm ext}/\varepsilon$, it is $\delta\rho_{\rm ind}/\delta\phi_{\rm ext}=-\Pi_{00}/\varepsilon$. This explains why formulas written with a full density correlator naturally determine $\varepsilon^{-1}$. Compare Altland and Simons 2023, §7.3, p. 405, Eqs. 7.34–7.36.

For spatial currents one similarly decomposes

$$\Pi_{ij}(\omega,\mathbf k) =\Pi_T(\omega,k)\left(\delta_{ij}-{k_ik_j\over k^2}\right) +\Pi_L(\omega,k){k_ik_j\over k^2}.$$

The transverse and longitudinal channels are distinct in a rest frame; charge conservation relates the longitudinal current channel to density response. The relativistic vacuum is more restrictive because Lorentz invariance ties the channels together.

Gravitational polarization

The metric response is the gravitational analog of vacuum polarization. Perturb flat space by

$$g_{ab}=\delta_{ab}+h_{ab}.$$

The metric stress tensor defined above is the source conjugate to $g_{ab}$:

$$\delta W[g] ={1\over2}\int d^2x\,\sqrt g\,\langle T_m^{ab}\rangle\delta g_{ab}.$$

Expand $W=W_0+W_1+W_2+\cdots$ around flat space. Its quadratic part is

$$W_2[h] ={1\over2}\int {d^2q\over(2\pi)^2}\, h_{ab}(q)\Pi^{ab,cd}(q)h_{cd}(-q).$$

The symmetric source derivative is understood so that $\delta W=\int(\delta W/\delta h_{ab})\delta h_{ab}$. The flat-space Hessian has separated-point part

$$\Pi^{ab,cd}(q) \equiv {\delta^2W\over\delta h_{ab}(q)\delta h_{cd}(-q)}\bigg|_{h=0} =-{1\over4}\langle T_m^{ab}(q)T_m^{cd}(-q)\rangle_{\rm conn} +\text{contact terms}.$$

Assume no diffeomorphism anomaly and transformations compatible with the boundary prescription. Write $\delta h=\delta_0h+\delta_1h$, where $\delta_1h=\mathcal L_\epsilon h$ and the linearized coordinate variation is

$$\delta_0h_{ab}(q)=-i\bigl(q_a\epsilon_b(q)+q_b\epsilon_a(q)\bigr).$$

At order $h$, diffeomorphism invariance says

$$\delta_0W_2+\delta_1W_1=0.$$

If the renormalized flat one-point function vanishes, then $W_1=0$. The full Hessian, including its local contact terms, is therefore transverse:

$$q_a\Pi^{ab,cd}(q)=0, \qquad q_c\Pi^{ab,cd}(q)=0.$$

Translation invariance alone does not set $W_1$ to zero: a cosmological term $\lambda\int\sqrt g$ has a constant nonzero first derivative, and $\delta_1W_1$ participates in its Ward identity. Also, discarding local momentum polynomials from a transverse Hessian need not leave an exactly transverse distribution. Its remaining divergence may be local and must be canceled by the contact completion. Polyakov derives precisely this cancellation in Polyakov 1987, p. 188, Eqs. 9.167–9.172.

For a normalized plane CFT vacuum with $\langle T\rangle=0$, the holomorphic stress tensor has the universal separated-point two-point function

$$\langle T(z)T(0)\rangle ={c/2\over z^4}$$

in the standard normalization of the Virasoro OPE; see Di Francesco, Mathieu, and Sénéchal 1997, §5.4, p. 135, Eqs. 5.121–5.122. Thus $c$ measures the stress polarization after the source conversion is fixed. Holomorphy away from $z=0$ does not remove distributional contacts. With $z^{-4}=-(1/6)\partial_z^3(z^{-1})$ and $\bar\partial(z^{-1})=\pi\delta^{(2)}(z)$,

$$\bar\partial\,{c\over2z^4} =-{\pi c\over12}\partial_z^3\delta^{(2)}(z).$$

This local divergence illustrates why the full metric Hessian needs more than its separated-point $TT$ correlator.

In momentum space the correlator supplies the nonlocal quadratic part of the induced action. Its covariant form is the Polyakov structure

$$W_{\rm ind}[g]\supset C_{\rm ind}\,c\int d^2x\sqrt g\,R\Delta_g^{-1}R,$$

where $\Delta_g=\nabla^a\nabla_a$ and $C_{\rm ind}$ is the coefficient to be derived from the trace anomaly in the next lesson. The inverse requires boundary conditions and, on a closed compact surface, projection off the constant zero mode; global area and zero-mode terms are then separate. See Polyakov 1987, pp. 188–189, Eqs. 9.170–9.177. Polyakov’s linear curvature has the opposite sign to the site’s, which leaves the displayed quadratic curvature structure unchanged. For this lesson, the essential analogy is

$$\text{current response }\langle JJ\rangle \quad\longleftrightarrow\quad \text{stress response }\langle TT\rangle.$$

The former determines induced electrodynamics; the latter determines induced two-dimensional gravity.

The quadratic curvature structure becomes local in conformal gauge. Around flat space, $R^{(1)}=-\Delta\phi$. With the zero-mode prescription fixed and boundary terms vanishing,

$$\int R^{(1)}\Delta^{-1}R^{(1)} =\int\phi\Delta\phi =-\int(\partial\phi)^2.$$

This fixes the integration-by-parts sign; the coefficient $C_{\rm ind}$ still determines the induced kinetic coefficient. The conformal factor therefore acquires dynamics when the total anomaly, including ghosts and other sectors, does not cancel. Its full Liouville action, including potential and global terms, goes beyond this quadratic calculation.

Summary

Conformal gauge turns the Polyakov worldsheet into a two-dimensional field theory with chiral stress-tensor constraints. The conditions $(\partial_\pm X)^2=0$ say geometrically that the worldsheet coordinates are locally conformal: the tangent vectors are orthogonal and equally scaled. The classical Weyl factor decouples from the free scalar action, but the quantum matter measure remembers it through stress-tensor correlations.

The background-field viewpoint makes the next step systematic. For $W=-\log Z$, the Hessian is $\langle S_{ij}\rangle-\langle S_iS_j\rangle_{\rm conn}$. A background gauge field couples to a current; a background metric couples to the metric stress tensor. Their complete response kernels satisfy the corresponding Ward identities, with the gravitational quadratic identity requiring a zero flat tadpole. Local contact terms cannot generally be dropped while retaining exact transversality. In two-dimensional CFT, the normalized stress correlator is controlled by the central charge and supplies the nonlocal induced gravitational action.

Common pitfalls

A conformal-gauge metric is not the same thing as a flat metric. Locally

$$g_{ab}=e^\phi\delta_{ab},$$

but the scalar curvature can be nonzero when $\phi$ varies.

Do not vary the gauge-fixed action and forget the constraints. The equations $U_{++}=U_{--}=0$ come from varying the metric before fixing conformal gauge. The normalized quantum $T$ additionally includes the specified tension, $2\pi$ factor, and normal ordering.

The coupling $h_{zz}\bar T/\pi$ is an index statement, not a contradiction. In complex coordinates the unperturbed metric has only mixed components, so raising indices interchanges the chiral components.

A transverse quadratic term need not be a local field-strength action. A structure such as

$$A_\mu\delta^\perp_{\mu\nu}A_\nu$$

is gauge invariant for nonzero momentum but contains the nonlocal projector; equivalently it is proportional to $F_{\mu\nu}(q)q^{-2}F_{\mu\nu}(-q)$.

For a real linear source $\kappa hO$, either sign of $\kappa$ gives the prefactor $-\kappa^2$ in the Hessian. Distinguish this from an imaginary Euclidean current coupling, and retain the local terms fixed by the regulated source prescription.

Do not infer the unique QED projector form from transversality alone when parity-odd structures are allowed. Nor does a chiral stress correlator by itself provide a diffeomorphism-invariant gravitational Hessian: the anomaly and contact-completion conditions matter.

Exercises

Exercise 1: Geometry of conformal coordinates

For a real immersion in a Euclidean target, start from

$$\partial_+=\partial_1-i\partial_2.$$

Show that $(\partial_+X)^2=0$ is equivalent to

$$(\partial_1X)^2=(\partial_2X)^2, \qquad \partial_1X\cdot\partial_2X=0.$$

Solution

Expand:

$$(\partial_+X)^2 =(\partial_1X-i\partial_2X)\cdot(\partial_1X-i\partial_2X).$$

This gives

$$(\partial_+X)^2 =(\partial_1X)^2-(\partial_2X)^2 -2i\partial_1X\cdot\partial_2X.$$

For this complex number to vanish, its real and imaginary parts must vanish separately. Therefore

$$(\partial_1X)^2-(\partial_2X)^2=0,$$

and

$$\partial_1X\cdot\partial_2X=0.$$

These are exactly the equal-length and orthogonality conditions.

Exercise 2: Field strength and the transverse projector

For nonzero Euclidean momentum, let

$$\delta^\perp_{\mu\nu}=\delta_{\mu\nu}-{q_\mu q_\nu\over q^2}.$$

Show that

$$A_\mu(q)q^2\delta^\perp_{\mu\nu}A_\nu(-q) ={1\over2}F_{\mu\nu}(q)F_{\mu\nu}(-q),$$

where again $\partial_\mu\mapsto-iq_\mu$ in the site-wide forward transform,

$$F_{\mu\nu}(q)=-i(q_\mu A_\nu(q)-q_\nu A_\mu(q)).$$

Solution

Here $F(q)=-i(q\wedge A(q))$ and $F(-q)=+i(q\wedge A(-q))$, so the derivative factors multiply to $(-i)(+i)=+1$. Compute the remaining contraction

$$(q_\mu A_\nu-q_\nu A_\mu)(q_\mu A_\nu-q_\nu A_\mu) =2q^2A_\mu A_\mu-2(q_\mu A_\mu)^2.$$

In momentum-space bilinear notation this means

$$F_{\mu\nu}(q)F_{\mu\nu}(-q) =2A_\mu(q)(q^2\delta_{\mu\nu}-q_\mu q_\nu)A_\nu(-q).$$

Since

$$q^2\delta^\perp_{\mu\nu}=q^2\delta_{\mu\nu}-q_\mu q_\nu,$$

we obtain

$$A_\mu q^2\delta^\perp_{\mu\nu}A_\nu ={1\over2}F_{\mu\nu}F_{\mu\nu}.$$

Exercise 3: Dielectric function from Gauss’s law

In a homogeneous isotropic linear medium in three spatial dimensions and Gaussian units, let $\phi$ be the total static potential. Assume the screening convention in which Gauss’s law is

$$k^2\phi(k)=4\pi\rho_{\rm ext}(k)-4\pi\Pi_{00}(k,0)\phi(k).$$

Derive

$$\varepsilon(k)=1+{4\pi\over k^2}\Pi_{00}(k,0).$$

Solution

Move the induced term to the left-hand side:

$$\left[k^2+4\pi\Pi_{00}(k,0)\right]\phi(k)=4\pi\rho_{\rm ext}(k).$$

Define the dielectric function by

$$k^2\varepsilon(k)\phi(k)=4\pi\rho_{\rm ext}(k).$$

Comparing the two equations gives

$$\varepsilon(k)=1+{4\pi\over k^2}\Pi_{00}(k,0).$$

If one defines the density-density response with the opposite sign, the displayed $\Pi_{00}$ is replaced by $-\Pi_{00}$. The physical screened denominator is the invariant content.

Exercise 4: Diffeomorphism Ward identity

In a diffeomorphism-anomaly-free theory with zero renormalized flat metric one-point function, let the complete quadratic action, including local contact terms, be

$$W_2[h]={1\over2}\int h_{ab}(q)\Pi^{ab,cd}(q)h_{cd}(-q).$$

Absorb the Fourier factor $-i$ into the parameter $\epsilon_b$ in this exercise. For transformations compatible with the boundary prescription, show that invariance under

$$\delta h_{ab}(q)=q_a\epsilon_b(q)+q_b\epsilon_a(q)$$

implies

$$q_a\Pi^{ab,cd}(q)=0$$

for the full Hessian. Explain what remains of this statement after removing local momentum polynomials.

Solution

Vary $W_2$ under $\delta h_{ab}$:

$$\delta W_2 =\int \delta h_{ab}(q)\Pi^{ab,cd}(q)h_{cd}(-q).$$

Substitute

$$\delta h_{ab}=q_a\epsilon_b+q_b\epsilon_a.$$

Because $\Pi^{ab,cd}$ is symmetric in $a,b$, the two terms give the same contribution:

$$\delta W_2 =2\int \epsilon_b(q)q_a\Pi^{ab,cd}(q)h_{cd}(-q).$$

The functions $\epsilon_b(q)$ and $h_{cd}(-q)$ are arbitrary. Diffeomorphism invariance therefore requires

$$q_a\Pi^{ab,cd}(q)=0.$$

This is exact for the full contact-completed Hessian under the stated zero-tadpole hypothesis. After discarding local momentum polynomials, the remaining representative can have a local polynomial divergence, canceled by the contact completion; it need not be exactly transverse by itself.

Exercise 5: Scaling of the stress-tensor kernel

In a two-dimensional CFT with a unit-normalized plane vacuum and the normalized stress tensor satisfying

$$\langle T(z)T(0)\rangle={c/2\over z^4},$$

explain by dimensional analysis why the corresponding momentum-space kernel has engineering dimension two.

Solution

The stress tensor has scaling dimension $2$ in two dimensions, so its chiral two-point function scales as

$$\langle T(z)T(0)\rangle\sim {1\over z^4}.$$

The Fourier transform in two dimensions is schematically

$$\Pi(q)\sim \int d^2x\,e^{iq\cdot x}{1\over z^4}.$$

Under $x\mapsto \lambda x$, the measure contributes $\lambda^2$ and the correlator contributes $\lambda^{-4}$. Equivalently, rescaling momentum gives

$$\Pi(\lambda q)=\lambda^2\Pi(q)$$

for the nonlocal homogeneous part, up to contact terms and possible logarithms from the renormalized distribution. This fixes engineering dimension two; it does not say that the chiral kernel is the rotational scalar $q^2$. Its spin and Ward identities require nontrivial light-cone momentum dependence.

References

• Altland, A., and B. Simons. Condensed Matter Field Theory. Third edition. Cambridge University Press, 2023. DOI: 10.1017/9781108781244.
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