A short note on dynamics and degrees of freedom in 2d classical gravity

Abstract

We comment on some peculiarities of matter with and without Weyl invariance coupled to classical 2d Einstein–Hilbert gravity for several models, in particular, related to the counting of degrees of freedom and on the dynamics. We find that theories where the matter action is Weyl invariant has generically more degrees of freedom than action without the invariance. This follows from the Weyl invariance of the metric equations of motion independently of the invariance of the action. Then, we study another set of models with scalar fields and show that solutions to the equations of motion are either trivial or inconsistent. To our knowledge, these aspects of classical 2d gravity have not been put forward and can be interesting to be remembered when using it as a toy model for 4d gravity. The goal of this note is also as a pedagogical exercise: our results follow from standard methods, but we emphasize more direct computations.

Appendices

Quantum two-dimensional gravity

In this appendix, we make contact between the classical and quantum regimes of gravity for the counting of the degrees of freedom.

The metric becomes dynamical due to quantum effects. It is convenient to write the metric in the conformal gauge using the diffeomorphisms:

$$\begin{aligned} g = \mathrm {e}^{2\phi } g_0, \end{aligned}$$

(A.1)

where $\phi$ is the Liouville mode and $g_0$ is a fixed background metric. In this gauge, the dynamics of the metric and the matter decouple and both sectors are mixed only by implementing the constraints and integrating over the moduli parameters of the surface.

An alternative interpretation is to see (A.1) as a parametrization instead of a gauge fixing. Indeed, $g_0$ is arbitrary and it is desirable to put all choices on an equal footing by introducing an invariance under diffeomorphisms of $g_{0\mu \nu }$ (background diffeomorphisms).

Since the physical metric is left invariant under the transformation

$$\begin{aligned} g_0 = \mathrm {e}^{2\omega } g'_0, \qquad \phi = \phi ' - \omega , \end{aligned}$$

(A.2)

it means that the system presents an emergent Weyl symmetry which is necessary to ensure that there is only one off-shell degree of freedom (three from $g_0$ and one from $\phi$, minus two from diffeomorphisms and one from this emergent symmetry). This emergent Weyl symmetry is not fundamental since it is very specific to the conformal gauge, at the opposite of the Weyl symmetry (2.2): this last symmetry (when it exists) can be used in any gauge and truly reduce the total number of off-shell degrees of freedom, whereas the emergent Weyl is here only not to spoil the counting due to the redundant notation. In particular, any action in the conformal gauge should have this invariance to be background-independent.

The total partition function is

$$\begin{aligned} Z = \int \mathrm {d}_g g_{\mu \nu }\, \mathrm {e}^{- S_\mu [g]} Z_m|g], \qquad Z_m[g] = \int \mathrm {d}_g \Psi \, \mathrm {e}^{- S_m[g, \Psi ]}, \end{aligned}$$

(A.3)

where $Z_m$ is the partition function of the matter and the index g on the measure indicates that it depends on the metric. Upon parametrizing the metric in the conformal gauge (A.1), one writes the partition function as:

$$\begin{aligned} Z = \int \mathrm {d}_g \phi \, \mathrm {e}^{- S_\mu [g_0, \phi ] - S_{\text {grav}}[g_0, \phi ]} \, \Delta _{\text {FP}}[g] Z_m|g_0], \end{aligned}$$

(A.4)

where $\Delta _{\text {FP}}[g]$ is the Faddeev–Popov determinant from the gauge fixing; it will be ignored in the rest of the discussion since it is not relevant. The gravitational action

$$\begin{aligned} S_{\text {grav}}[g_0, \phi ] = - \ln \frac{Z_m[\mathrm {e}^{2\phi } g_0]}{Z_m[g_0]} \end{aligned}$$

(A.5)

is the Wess–Zumino effective action¹² for the change of metric from g to $g_0$ in the matter partition function. Typically, the leading terms are the Liouville action and the Mabuchi action [1, 4, 5, 8]. The total action is then

$$\begin{aligned} S_*[g_0, \phi , \Psi ] = S_\mu [g_0, \phi ] + S_{\text {grav}}[g_0, \phi ] + S_m[g_0, \Psi ]. \end{aligned}$$

(A.6)

When the matter is Weyl invariant, the gravitational action can be computed by parametrizing the metric as $g = \mathrm {e}^{2\omega } g_0$ and by integrating the trace of the quantum energy–momentum tensor over $\omega$ from 0 to $\phi$. The latter is given by the Weyl anomaly (quantities with a subscript 0 are given in terms of the metric $g_0$)

$$\begin{aligned} \langle T^m \rangle = \frac{2\pi }{\sqrt{g}} \frac{\delta }{\delta \omega } \ln Z_m[g] = - \frac{c_m}{12}\, R. \end{aligned}$$

(A.7)

Using the relation $R = R_0 - 2 \Delta _0 \omega$ and integrating, the action (A.5) becomes the Liouville action:

$$\begin{aligned} S_{\text {grav}} = - \frac{c_m}{24\pi } \int \mathrm {d}^2 x \sqrt{g_0} \big (g_0^{\mu \nu } \partial _\mu \phi \partial _\nu \phi + R_0 \phi \big ). \end{aligned}$$

(A.8)

Studying the action (A.6) as a classical action with background metric $g_0$ gives information about the semi-classical properties of the theory. In Sect. 3, we have argued that classical 2d gravity with unitary matter is trivial for a certain class of models. But, this does not imply that the quantum behaviour is also trivial and that forbidden classical systems have no quantum dynamics. As a specific example of this fact, consider the Liouville theory: classically, the cosmological constant is forbidden, but it is necessary to include it to compute the path integral (in particular, to absorb divergences and ambiguities [6]). Surprisingly, the semi-classical limit does not resemble the classical gravity one started from: the quantum effects are not negligible in this limit and render the dynamics non-trivial.

Invariance under the emergent Weyl symmetry (A.2) links the trace of the energy–momentum tensor $T^*_{0\mu \nu }$ to the variation of the full action with respect to the Liouville mode:

$$\begin{aligned} T_0^* = - \frac{1}{2}\, \frac{\delta S_*}{\delta \phi }. \end{aligned}$$

(A.9)

Another requirement is the one of background independence: the full action (A.6) should be independent of the gauge choice (A.1), i.e. it should not depend on the background metric $g_0$ or on the specific decomposition. This is true, in particular, if the variation of the total action with respect to $g_0$ vanishes, implying that the full stress–energy tensor is zero:

$$\begin{aligned} T^*_{0\mu \nu } = 0. \end{aligned}$$

(A.10)

From (A.9) the trace equation is automatically satisfied if $\phi$ satisfies its equation of motion: as a consequence, (A.10) provides two constraints on the matter and this number does not depend on whether the matter is Weyl invariant or not.

Finally, the variations of (A.6) give decoupled equations of motion for $\phi$ and $\Psi$. As a conclusion, one sees that quantum effects have given dynamics to the Liouville field and there is a total of $N - 1$ degrees of freedom (matter plus gravity) since the matter equations of motion are not expected to decrease further this number.

Analogy: four-dimensional gauge anomaly

There exists a full analogy between the description of 2d gravity and 4d chiral gauge theory. For example, one finds the emergence of a gauge symmetry (respectively Weyl and $\mathrm {U}(1)$) through the choice of a convenient parametrization and the appearance of a Wess–Zumino action.

1.1 Massive vector field

Let’s consider a massive vector field $\mathcal {A}_\mu$ (playing the role of $g_{\mu \nu }$) with Proca action in d dimensions:

$$\begin{aligned} S_A = - \int \mathrm {d}^d x \left( \frac{1}{4}\, \mathcal {F}_{\mu \nu } \mathcal {F}^{\mu \nu } + \frac{m^2}{2} \mathcal {A}_\mu \mathcal {A}^\mu \right) . \end{aligned}$$

(B.1)

If $m^2 = 0$, then it enjoys a $\mathrm {U}(1)_g$ gauge symmetry

$$\begin{aligned} \mathcal {A}_\mu = \mathcal {A}'_\mu + \partial _\mu \alpha , \end{aligned}$$

(B.2)

which reduces the d components to $(d - 1)$ off-shell dofs, and furthermore to $(d - 2)$ on-shell dofs. If $m^2 \ne 0$, then there is one on-shell constraint, such that there are only $(d-1)$ on-shell degrees of freedom.

In full similarity with 2d gravity, after coupling to matter fields, the matter action may or may not be invariant under the $\mathrm {U}(1)_g$, even if $m^2 \ne 0$.

It is convenient to adopt another parametrization (called Stückelberg) for the vector field, where the spin 0 component is separated from the spin 1 component:

$$\begin{aligned} \mathcal {A}_\mu = A_\mu + \partial _\mu a, \end{aligned}$$

(B.3)

where $A_\mu$ and a (called the axion) play respectively the roles of $g_{0\mu \nu }$ and $\phi$ (from the point of view of Lorentz representations, the trace of $g_{\mu \nu }$ is similar to the divergence of $\mathcal {A}_\mu$), the only difference being that $A_\mu$ is dynamical. In these variables, the action reads:

$$\begin{aligned} S_A = - \int \mathrm {d}^d x \left( \frac{1}{4}\, F_{\mu \nu } F^{\mu \nu } + \frac{m^2}{2} (A_\mu + \partial _\mu a) (A^\mu + \partial ^\mu a) \right) . \end{aligned}$$

(B.4)

In order to avoid introducing an additional degree of freedom ($A_\mu$ has d components and a has 1), this system should be invariant under an emergent $\mathrm {U}(1)_e$

$$\begin{aligned} A_\mu = A'_\mu + \partial _\mu \alpha , \qquad a = a' - \alpha . \end{aligned}$$

(B.5)

Note that this symmetry is not fundamental and is just a consequence of the parametrization (B.3) that has been chosen.

1.2 Effective action

For the rest of this section, we focus on $d = 4$. The action for the system is given by

$$\begin{aligned} S[\mathcal {A}, \psi ] = S_A[\mathcal {A}] + S_f[\mathcal {A}, \psi ] \end{aligned}$$

(B.6)

where $S_A$ is Proca action (B.1) and $S_f$ is the action for a chiral fermion $\psi$ (the dependence on the conjugate ${{\bar{\psi }}}$ is implicit everywhere)

(B.7)

This model is discussed for example in [17]. Since the fermion is chiral, it obeys the relations $\psi = \gamma _5 \psi = P_L \psi$ where $P_L$ is the projector on left-handed chirality. The partition function of the system is

$$\begin{aligned} Z = \int \mathrm {d}\mathcal {A}_\mu \, \mathrm {e}^{- S_A[\mathcal {A}]}\, Z_f[\mathcal {A}], \qquad Z_f[\mathcal {A}] = \int \mathrm {d}\psi \, \mathrm {e}^{- S_f[\mathcal {A}, \psi ]}, \end{aligned}$$

(B.8)

where $Z_f$ is the fermion partition function. The terms $S_A$ and $Z_f$ play respectively the roles of $S_\mu$ and $Z_m$ in (A.3).

In terms of the parametrization (B.3), the partition function becomes (we ignore the ghost action since it decouples)

$$\begin{aligned} Z = \int \mathrm {d}A_\mu \mathrm {d}a \, \mathrm {e}^{- S_A[A, a] - S_{\text {WZ}}[A, a]}\, Z_f[A], \end{aligned}$$

(B.9)

where the Wess–Zumino action is:

$$\begin{aligned} S_{\text {WZ}}[A, a] = - \ln \frac{Z_f[A + \partial a]}{Z_f[A]}. \end{aligned}$$

(B.10)

The main difference with 2d gravity is that $A_\mu$ is dynamical and it does not drop from the path integral, but this does not modify the general argument. Note that the gauge invariance (B.5) ensures that the same number of degrees of freedom is described as by (B.8).

While the full action is not invariant $\mathrm {U}(1)_g$ if $m^2 \ne 0$, the fermion action is invariant under the transformation $\psi = \mathrm {e}^{i g \alpha (x)} \psi '$ together with (B.2), and the associated current $J^\mu = {{\bar{\psi }}} \gamma ^\mu \psi$ is conserved classically. On the other hand, there is a gauge anomaly due to the chirality of the theory and the current is not conserved quantum mechanically:

(B.11)

This can be used to determine the $S_{\text {WZ}}$ in full similarity with the derivation of the Liouville action (A.8). The quantum current is the variation of the effective action (B.8) with respect to the gauge field:

$$\begin{aligned} \langle J^\mu \rangle = - \frac{\delta }{\delta \mathcal {A}_\mu } \ln Z_f[\mathcal {A}]. \end{aligned}$$

(B.12)

Then, one can vary from $\mathcal {A}_\mu$ to $A_\mu$ continuously by parametrizing $\mathcal {A}_\mu = A_\mu + \partial _\mu \alpha$. From the variation $\delta \mathcal {A}_\mu = \partial _\mu \alpha$, the relation (B.12) can be integrated from $\alpha = 0$ to $\alpha = a$ which results in (after an integration by part)

$$\begin{aligned} S_{\text {WZ}}[A, a] = - \int \mathrm {d}^4 x\, a\, \partial _\mu \langle J^\mu \rangle = - \frac{g^3}{48 \pi ^2}\ \int \mathrm {d}^4 x\, a \, F_{\mu \nu } {\widetilde{F}}^{\mu \nu }, \end{aligned}$$

(B.13)

where the expression (B.11) and the identity $\mathcal {F}_{\mu \nu } = F_{\mu \nu }$ have been used. This computation is possible because the mass term is outside the matter path integral which is effectively gauge invariant (classically). In particular, the anomalous contribution arises from the fermion measure. This mirrors 2d gravity where the cosmological constant lies outside the matter path integral.

The total action in the parametrization (B.3) reads

$$\begin{aligned} S[A, a, \psi ] = S_A[A, a] + S_{\text {WZ}}[A, a] + S_f[A, \psi ]. \end{aligned}$$

(B.14)

This action should be invariant under the emergent $\mathrm {U}(1)_e$; in particular this implies that the variation of the Wess–Zumino term $S_{\text {WZ}}$ is exactly the one necessary for cancelling the gauge anomaly of the fermion action in terms of the field$A_\mu$ (the anomaly related to $\mathrm {U}(1)_g$ is still present). Note also that the fermions and the axion are not coupled to each other, in the same way, that the matter and the Liouville mode do not couple in 2d gravity.

Another point where the analysis differs from 2d gravity: the axion does not get its dynamics from the anomaly-generated term, but from the mass of $\mathcal {A}_\mu$ in (B.1). One could then think that the axion is not dynamical if one starts with $m^2 = 0$. However, such a mass term would be generated at 1-loop from the cubic WZ vertex and the tree mass term is necessary to remove the divergence. A similar story holds for the cosmological constant in 2d gravity.