Classical gravitational anomalies of Liouville theory

We show that for classical Liouville field theory, diffeomorphism invariance, Weyl invariance and locality cannot hold together. This is due to a genuine Virasoro center, present in the theory, that leads to an energy\hyp{}momentum tensor with non-tensorial conformal transformations, in flat space, and with a non-vanishing trace, in curved space. Our focus is on a field-independent term, proportional to the square of the Weyl gauge field, $W_\mu W^\mu$, that makes the action Weyl-invariant and was disregarded in previous investigations of Weyl and conformal symmetry. We show this term to be related to the classical center of the Virasoro algebra. The mechanism uncovered here is a classical version of the quantum anomalous phenomenon: the generalization to curved space only allows to keep one of the two symmetries enjoyed by the flat space theory, either Lorentz (diffeomorphism) or conformal invariance.


Liouville field theory, with flat space action
is an exactly solvable two-dimensional model that enjoys a prominent role in many fields of the theoretical and mathematical investigations.Among those, the geometry of surfaces [1], twodimensional (quantum) gravity, see, e.g., [2] and [3], string theory, see, e.g., [4], conformal field theories, such as the Wess-Zumino-Witten and the Toda models, see e.g., [5], and therefore the AdS/CFT correspondence [6].It is then of great importance to know its symmetries in all details, already at the classical level.
In particular, Liouville theory, besides being Lorentz invariant, is known to enjoy full (global) conformal symmetries in flat space, hence it belongs to the cases studied in [7].There it is assumed that Weyl and diffemorphism invariances hold together.Even though full (global) conformal symmetry is known to be in place, in [8] it was conjectured that Liouville theory might not be made both diffeomorphic and Weyl invariant, evoking a generic "classical anomaly" as the reason for that.In this letter we prove that conjecture and provide explicit formulae for such classical gravitational anomalies.A more detailed discussion on how such classical anomalies arise, in general and for Liouville, can be found in [9].
Liouville theory often emerges as an effective action, see, e.g., the recent [10].Clearly, when a theory is only effective, we can relax symmetry requirements and lack of Weyl or diffeomorphic invariance, or even lack of both, is not a big deal.On the other hand, it is entirely legitimate to study Liouville theory as a fundamental theory, as we do here.In this latter case, the request for a fully symmetric situation, both Weyl and diffeomorphic, is in order.
The concerns of [8] are not only important to probe the procedure of [7], but also in the more general investigation of the concept of anomaly, in the first place.This is our most important motivation.
As well known, see, e.g., [11,Chapter I.2], it is the process of second quantization (fields) rather than first quantization (one particle) that is responsible for the anomaly phenomenon.It must then be seen as a violation occurring while trying to enforce symmetries, valid for certain specific configurations, to setups with infinitely many more allowed configurations.
The question that we pose, and answer, here is whether something along those lines happens when we try to generalize the Lorentz and conformal symmetries of the flat space action (1) to curved space, staying entirely in a classical environment.On top of that, we also find that the classical central charge of the Virasoro algebra plays the same role as in the quantum case.Altogether, this is then clearly a classical instance of a phenomenon previously encountered only in the quantum (field theory) context, as conjectured in [8].
As we shall see, all that happens very much according to the original findings of Adler [12] and Bell and Jackiw [13] (ABJ) (see also [11; 14]): the generalization (to curved space here, to the quantum regime in ABJ) only allows to keep one of the two symmetries valid in the particular regime (flat space here, classical regime in ABJ).The nontrivial central charge of the flat theory does not spoil the full (Lorentz and conformal) invariance of the flat space action (1), but spoils either one or the other symmetry, in curved space.We must have either diffeomorphic (Lorentz [15]) anomaly or conformal (Weyl) anomaly.
Let us start by considering the diffeomorphic invariant Liouville action on a curved background routinely employed to obtain the energy-momentum tensor (EMT) that on-shell gives ensuring a zero trace in the flat limit.
No issue appears to occur in flat space, besides the unpleasant ad-hoc nature of this procedure to obtain the improved, i.e. traceless, EMT in flat space Θ µν = lim gµν →ηµν T µν L , see later Eq.(19).Nonetheless, as we shall see in the rest of the paper, nontrivial issues about conformal invariance are indeed present already in flat space, as well as in curved space.
To see that, we first notice that the lack of Weyl invariance of the action (2) and the ad-hoc nature of the improvement procedure just recalled, could be both faced at once by employing the approach of [7], based on the Weyl-gauging of the curvilinear expression for the action (1) Since under Weyl transformations, and the (Ricci gauged, in the language of [7]) action holds.Notice that, contrary to [7], we keep here the last, Φ-independent term, that is precisely the one that ensures Weyl invariance.A solution of ( 7) can be found [8] using the Green's function K(x, y), such that which is the well-known Polyakov string effective action [4].The EMT associated to the action ( 6) with (8), is traceless and covariantly conserved2 [4].The price we pay is the evident nonlocality.
A local solution to (7) was found by Deser and Jackiw3 in [18] where ε 01 = +1 is the Levi-Civita symbol4 and a "conformal" parametrization of the metric gives sinh σ, and, from there, γ = ln √︁ g ++ /g −− (see Appendix B).The expression (9) includes the derivative of a generic Weyl scalar, r, to take into account the invariance of (7) for . (10) It follows that the term g µν W µ DJ W ν DJ in (6), although it keeps Weyl invariance and locality of A R [Φ], cannot be a world scalar, hence it breaks diffeomorphism invariance.
To investigate and quantify such breaking, let us start with the contribution to the EMT coming from the extra term In Appendix B it is shown that where and r µν ≡ δr/δg µν .One would like to compute ∇ µ T µν extra and compare it with known expressions of the quantum gravitational anomalies, such as the so-called consistent anomaly [14] (our notation follows [19], see also [20]) νσ or the so-called covariant anomaly [14] (our notation follows [21]) Rather than attempting a direct computation, we take a simpler road.First, we move to isothermal light-cone coordinates, that we know to always exist in two dimensions.There one has , and we indicate with a hat all quantities evaluated there5 .If we set r = 0 for a moment, we have and the computation becomes trivial, ∇ ˆµT ˆµν extra = 0. Of course, this would not guarantee general covariance, until we have a frame-independent result (see later).
On the other hand, including r in the computation gives and this expression, although it differs from the recalled anomalous quantum expressions [19; 21], it is clearly nonzero, in general.In this coordinate frame, T ˆµν extra not only guarantees Weyl invariance, through a traceless EMT, but for harmonic rs, □ ˆr ˆ= 0 As for the previous case, for r = 0, this is not enough to have general covariance.We have no guarantee that (15) holds in all frames.We have to look for how much such divergence differs from a tensor, when we move away from the isothermal frame, ∆∇ ˆµT ˆµν . This has to be, at least partially, expressible in terms of ∆W ˆµ(x) ≡ W ′µ (x ′ ) − (∂x ′µ /∂x ν )W ˆν(x).For infinitesimal diffeomorphisms, W µ transforms as (10) and, defining ∆r(x) ≡ r ′ (x ′ ) − r(x), With these that, for the choice ( 9), and for r = 0, eventually gives a compact expression This expression does not vanish for a general f µ .This proves the loss of diffeomorphism invariance in the Weyl invariant formulation of Liouville theory (6), with local solution (9).Quadratic transformations, , preserve the tensorial nature of ∇ ˆµT ˆµν extra and so do conformal transformations, obeying □f µ = 0. Therefore, for infinitesimal conformal and Poincaré transformations in flat space, the extra term, T µν extra , is covariantly conserved, regardless of the choice of r.In other words, T µν extra does not violate the symmetries of T µν in the flat limit, that is the same conclusion of [7].
To complete the proof that this lack of diffeomorphic invariance is indeed the classical version of the quantum anomaly, we need to relate it to a nontrivial center of the Virasoro algebra.To do so, let us first consider the flat limit of the EMT (3) that is traceless on-shell.The associated Noether charges, written in the light-cone frame6 through the Poisson brackets {Φ(x), Φ(y They are made of two terms, both necessary for the invariance of the flat action (1): the usual Lie derivative of the scalar field, f α ∂ α Φ, and an affine term.It is easy to verify that these charges obey an algebra with a genuine central extension where By restricting to a periodic manifold, with a periodicity P , x ± ∝ x ± + P , generators can be decomposed into and the algebra (22) can be recast into the following form that is just the Virasoro algebra with genuine central charge It is the algebra of the flat Liouville EMT components that inevitably includes the genuine center (25) and so do its transformations This center is not there in the trace of flat EMT (19), but it is proportional to the trace (4) of the curved space EMT (3).
A deeper study of this, including the general framework for classically anomalous transformations, we have done it in [9].Here we want to show that the above is indeed related to the extra term T µν extra that, in curved space, preserves Weyl invariance but breaks diffeomorphic invariance.To do so, let us first rewrite (27) as the difference between the full transformation and its tensorial part as we did earlier in the curved context.We then simply notice that, for the infinitesimal diffeomorphism, x µ → x µ − f µ (x), the non-tensorial transformation of T µν extra is where the same notation of ( 16) and ( 17) has been used.Assuming conformal diffeomorphisms and taking the flat limit we have which is exactly 7 (28) with c given by (25).This center was removed from the trace (4) but re-emerged in (30).We have then proved that, also for classical Liouville theory, lack of Weyl invariance or of diffeomorphism invariance is related to the Virasoro center, like in the quantum case.This gives a precise mathematical meaning to what we are now entitled to call "classical gravitational anomalies".Whether this is possible for more general classical systems, it is an important open question.Another direction for further research that we are considering is the connection of such anomalous transformation of the EMT with "classical Unruh and Hawking effects".which, in light-cone coordinates, can be parametrized as In this parametrization a new quantity R µ can be defined Using the following identity it can be seen that The natural way to compute the EMT for ∆A is by varying the latter with respect to γ µν To begin, we see that where The last two terms are the result of the following computation [18] δ This proves the Weyl invariance of the improved Liouville action.