2D quantum gravity on compact Riemann surfaces with non-conformal matter

We study the gravitational action induced by coupling two-dimensional non-conformal, massive matter to gravity on a compact Riemann surface. We express this gravitational action in terms of finite and well-defined quantities for any value of the mass. A small-mass expansion gives back the Liouville action in the massless limit, the Mabuchi and Aubin-Yau actions to first order, as well as an infinite series of higher-order contributions written in terms of purely geometric quantities.


Introduction and motivation
Ever since the seminal paper by Polyakov [1] it has been known that conformal matter coupled to 2D gravity gives rise to an "effective" gravitational action that is the Liouville action S L [g 0 , g] ≡ S L [g 0 , σ] = d 2 x √ g 0 σ∆ 0 σ + R 0 σ , g = e 2σ g 0 . Being defined as the logarithm of a ratio of partition functions computed with two different metrics, any gravitational action satisfies a cocycle identity S grav [g 1 , g 2 ] + S grav [g 2 , g 3 ] = S grav [g 1 , g 3 ] . (1.4)

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The simplest example of a gravitational action satisfying this cocycle identity is the "cosmological constant action" As we will recall below, this action must be present as a counterterm to renormalize the divergences that are present in (1.2) in addition to S L . Gravitational actions other than the Liouville or cosmological constant actions can be constructed and have been studied mainly in the mathematical literature, like the Mabuchi and Aubin-Yau actions [2][3][4][5]. These latter functionals crucially involve not only the conformal factor σ but also directly the Kähler potential φ and do admit generalizations to higher-dimensional Kähler manifolds. In the mathematical literature they appear in relation with the characterization of constant scalar curvature metrics [5]. Their rôles as two-dimensional gravitational actions in the sense of (1.3) have been discussed in some detail in [6]. In particuler, ref. [6] has studied the metric dependence of the partition function of non-conformal matter like a massive scalar field and shown that a gravitational action defined by (1.3) contains these Mabuchi and Aubin-Yau actions as first-order corrections (first order in m 2 A where m is the mass and A the area of the Riemann surface) to the Liouville action. The partition function of quantum gravity at fixed area, with a gravitational action being a combination of the Liouville and Mabuchi actions, has been studied at one loop in [7] and at two and three loops in [8].
While the results of [6] concerned the first order in an expansion in m 2 A, in the present note we will derive a few results that are valid exactly at finite m. Ideally one would like to study some general matter action where non-conformal terms ∼ c i O i have been added to some conformal theory and obtain exact results in these couplings c i . We are going to be much less ambitious and simply study a single massive scalar field with action and try to establish some exact results valid for finite m. Here ∆ g is the Laplace operator for the metric g, defined with a minus sign, so that its eigenvalues are non-negative: In the remainder of this section, we discuss some basic differences between the massive and massless cases and define the matter partition functions in both cases. Then we briefly recall the Mabuchi and Aubin-Yau actions and their variations. In section 2, we summarize some technical tools involving heat kernels, zeta functions and their perturbation theory. Section 3 then gives the computation of the gravitational action for the massive scalar field, providing explicit formulae for its dependence on the metric in terms of manifestly finite and well-defined quantities. Maybe not too surprisingly, some of our massive formulae will look somewhat similar to those that can be found in [6] for the massless case. However, let us insist that our results are exact in m and valid for any finite mass. Nevertheless, we will write them in a way that immediately allows for a small mass expansion, thus JHEP01(2017)089 recovering the Liouville action in the zero-mass limit and the Mabuchi and Aubin-Yau actions as the first-order corrections. The higher-order corrections are similarly expressed in terms of purely geometric objects, but do not seem to have any known counterparts in the mathematical literature, nor is it clear to us whether they can be written as integrals over the manifold of local quantities.

Massive versus massless matter
One should keep in mind that adding the mass term is not just a perturbation by some operator that has a non-zero conformal weight. This is due to the (would-be) zero-mode of the scalar field that is absent from the action for zero mass but obviously plays an important rôle for non-zero mass. In particular, this means that the relevant quantities of the massive theory are not simply given by those of the massless theory plus order m 2 corrections. This is most clearly examplified by the Green's function G(x, y) of the operator ∆ g + m 2 . The latter is hermitian and has a complete set of eigenfunctions ϕ n with eigenvalues λ n ≥ m 2 : (∆ g + m 2 )ϕ n (x) = λ n ϕ n (x) , n = 0, 1, 2, . . . . n + m 2 . The eigenfunctions which may be chosen to be real, are orthonormalized as As is clear from (1.7), ∆ g always has a zero mode and, hence, (1.10) We always refer to ϕ 0 as the zero-mode, even in the massive case. The Green's function for m = 0 is given by In general, if B is any quantity defined for m = 0, we will denote by B (0) the corresponding quantity for m = 0. But, for m = 0, the Green's function is not simply G (0) since λ (0) 0 = 0 and, obviously, the zero-mode must be excluded from the sum. Then The subtraction of 1 A on the r.h.s. ensures that, when integrated d 2 x √ g . . ., one correctly gets zero. We will consistently put a tilde over the various quantities we will encounter if the zero-mode is excluded from the sum. 1 In particular, using (1.10), we can write for the massive Green's function

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where G has a smooth limit as m → 0. Moreover, the smallest eigenvalue contributing in Clearly, the massive Green's function does not equal the massless one plus order-m 2 corrections: their is a crucial 1 m 2 A term in (1.13) and (1.14). The matter partition function is defined with respect to the decomposition X = n≥0 c n ϕ n as In the massless case, since λ (0) 0 = 0, the integration over c 0 would be divergent and instead one replaces it by a factor √ A. Thus Of course, the determinants det and det are ill-defined and need to be regularized. We will use the very convenient regularization-renormalization in terms of the spectral ζ-functions: and similarly for ζ (0) (s). By Weil's law (see e.g. [9]), the asymptotic behaviour of the eigenvalues for large n is λ n ∼ n A and, hence the spectral ζ-functions are defined by converging sums for Re s > 1, and by analytic continuations for all other values. In particular, they are well-defined meromorphic functions for all s with a single pole at s = 1 with residue 1 4π (see e.g. [9]). A straightforward formal manipulation shows that There is a slight subtlety one should take into account, see e.g. [9]. While the field X is dimensionless, the ϕ n scale as A −1/2 ∼ µ where µ is some arbitrary mass scale (even if m = 0), and the c n as µ −1 . It follows that one should write D g X = n µdcn 2π . This results in Z mat = n λn µ 2 −1/2 , so that every ζ (0) is changed into (1.20)

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The regularization-renormalization of determinants in terms of the ζ-function may appear as rather ad hoc, but it can be rigorously justified by introducing the spectral regularization [9]. The regularized logarithm of the determinant then equals ζ (0) + ζ(0) ln µ 2 plus a diverging piece ∼ AΛ 2 (ln Λ 2 µ 2 + const), where Λ is some cutoff. This diverging piece just contributes to the cosmological constant action (1.5), and this is why the latter must be present as a counterterm, to cancel this divergence.
Thus, finally where the first line refers to the massive case and the second line to the massless one.

Mabuchi and Aubin-Yau actions
Let us briefly recall the basic properties of the known gravitational actions. While the Liouville action (1.1) can be written in terms of g 0 and the conformal factor σ, the Mabuchi and Aubin-Yau actions are formulated using also the Kähler potential φ. They are related by where ∆ 0 denotes the Laplacian for the metric g 0 with area action on a Riemann surface of genus h can then be written as [6] S while the Aubin-Yau action takes the form As already mentioned, they both satisfy a cocycle identity analogous to (1.4) and were shown [6] to appear in S grav in the term of first order in an expansion in m 2 A. Note that (1.22) relates the variations δσ and δφ as It is then straightforward to show that the variations of the Liouville, Mabuchi and Aubin-Yau actions are given by

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Thus the Liouville and Mabuchi actions obviously admit the constant scalar curvature metrics as saddle-points at fixed area. Although not obvious from the previous equation, the variation of the Aubin-Yau action when restricted to the space of Bergmann metrics is similarly related to metrics of constant scalar curvature [5].
2 Some technical tools

The heat kernel
The heat kernel and integrated heat kernel for the operator ∆ g + m 2 are defined in terms of the eigenvalues and eigenfunctions (1.8) as The corresponding K, K (0) and K (0) are defined similarly. The heat kernel K is the solution of d dt Note that it immediately follows from either (2.1) or (2.2) that the massless and massive heat kernels are simply related by As is also clear from (2.1) (and Weil's law λ n ∼ n A ), for t > 0, K(t, x, y) is given by a converging sum and is finite, even as x → y. For t → 0 one recovers various divergences, in particular exhibits the short distance singularity of the Green's function which is well-known to be logarithmic. The behaviour of K for small t is related to the asymptotics of the eigenvalues λ n for large n, which in turn is related to the short-distance properties of the Riemann surface. It is thus not surprising that the small-t asymptotics is given in terms of local expressions of the curvature and its derivatives. Indeed, one has the well-known small t-expansion: y) is the geodesic distance squared between x and y as measured in the metric g. For small t, the exponential forces 2 to be small (of order √ t) and one can use normal coordinates around y. This allows one to obtain quite easily explicit expressions for the a r (x, y) in terms of the curvature tensor and its derivatives. They can be found e.g. in [9]. Here, we will only need them at coinciding points y = x, where Let us note that in the massless case and if the zero-mode is excluded one has instead and similarly for ζ(s, x, y), etc. Note that ζ(1, x, y) = G(x, y), while for s = r = 2, 3, . . . these local ζ-functions coincide with the G r (x, y) defined above in (1.15). They are related to the heat kernel by (2.8) For s = 0, −1, −2, . . ., Γ(s) has poles and the value of ζ(s, x, y) is entirely determined by the singularities of the integral over t that arise from the small-t asymptotics of K. As shown above, the latter is given by local quantities on the Riemann surface. In particular, On the other hand, the values for s = 1, 2, 3, . . . or the derivative at s = 0 cannot be determined just from the small-t asymptotics and require the knowledge of the full spectrum of ∆ g + m 2 , i.e. contain global information about the Riemann surface. Clearly, ζ(1, x, y) = G(x, y) is singular as x → y. For s = 1, ζ(s, x, y) provides a regularization of the propagator. More precisely, it follows from (2.8) that ζ(s, x, x) is a meromorphic function with a pole at s = 1 and that the residue of this pole is a 0 (x,x) 4π = 1 4π . Thus [9] G ζ (x) = lim s→1 µ 2(s−1) ζ(s, x, x) − 1 4π(s − 1) (2.10) is well-defined. (Here µ is an arbitrary scale.) This is an important quantity, called the "Green's function at coinciding points". One can give an alternative definition of G ζ by subtracting the short distance singularity from G(x, y) and taking x → y, see e.g. [10,11]. More precisely [9] the relation is One can show [9] that both definitions of G ζ are equivalent and define the same quantity. Again, the same relations hold between G ζ (y), G(x, y) and ζ(s, x, x). Note that G ζ (y) contains global information about the Riemann surface and cannot be expressed in terms of local quantities only.

Perturbation theory
We want to study how the eigenvalues λ n and eigenfunctions ϕ n change under an infinitesimal change of the metric. Since g = e 2σ g 0 , the Laplace operator ∆ g and hence also ∆ g +m 2

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As recalled above, ζ(s, x, x) has a pole at s = 1 with residue 1 4π , hence (3.4) From (2.10) one sees that ζ reg (1, x, Using also (2.9) we find 3 Since G ζ = G ζ + 1 m 2 A , we arrive at two equivalent expressions for δζ (0) + δζ(0) ln µ 2 : As it stands, this result is exact in m and holds whether m 2 A is small or not. Let us insists that the G ζ and G ζ appearing on the right-hand side are the massive ones. The first line is the appropriate way of writing to study the small m 2 asymptotics, as G ζ has a smooth limit for m → 0.

The massless case
Let us quickly show, how in the massless case one recovers the well-known Liouville action. For m = 0 one has δ ζ (0) (0) = 0 and (3.6) immediately gives Note that the last term precisely cancels the variation of the − 1 2 ln A A 0 in S (0) grav [g 0 , g], cf (1.21), and one gets the well-known result cf. eq. (1.26). Of course, this is just the contribution of one conformal scalar field with c = 1. The determinant that arises from fixing the diffeomorphism invariance gives the same contribution but with a coefficient + 26 24π , so that overall one gets

The massive case
Our starting point is obtained by combining eqs (1.21) and (3.6), using also (3.8): Our task then is to rewrite the second term on the r.h.s. as the variation of some local functional. Let us compute δG ζ (x). In order to do so, we first establish a formula for δG(x, y) under a variation δg = 2δσ g of the metric and thus under a corresponding variation δ∆ g = −2δσ∆ g of the Laplace operator. One can then either use the definition (1.11) as an infinite sum and the perturbation theory formula (2.13) and (2.14), or directly use the defining differential equation (1.11). In any case one finds To obtain the variation of G ζ , according to (2.11) one needs to subtract the variation of the short-distance singularity. Now, the geodesic distance g (x, y) transforms as (see e.g. appendix A1 of [9]) It follows that lim x→y δ ln µ 2 2 g (x, y) = 2 δσ(x) . (3.14) Upon integrating this over x one encounters One can then rewrite (3.10) as (3.17)

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Note that in the second term, we can replace G ζ by G ζ since the difference is m 2 2 d 2 x √ g 1 m 2 A = 1 2 , whose variation vanishes. Next, we use (2.8) to rewrite the last term as m 4 ∞ 0 dt t d 2 x √ g K(t, x, x)δσ(x), and establish a formula for the variation of the integrated heat kernel K(t). Since δλ 0 = 0 we have where we used (2.13) and (2.3). It then follows that where we integrated by parts and used (2.8). The boundary terms do not contribute 4 since K vanishes at t = ∞ as e −λ 1 t and λ 1 − m 2 > 0. Upon inserting this into (3.17) we finally get (3.20) Note that in the last term the t-integral is convergent both at t = 0 and at t = ∞. This is immediately integrated as Thus we have expressed the gravitational action S grav [g 0 , g] in terms of (local) functionals of g and g 0 that are all perfectly well-defined without any need of analytical continuation (contrary to the initial ζ (0)). Recall also from our remark at the end of section 1.2 that, if we define S grav [g 0 , g] through the spectral cutoff regularization of the logarithm of the 4 Had we started with δK rather than δ K and written this equation for K and ζ(2, x, x), the − 1 A √ gδσ would have appeared as the boundary term.

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determinant, the r.h.s. also involves the variation of the cosmological constant action µ 0 A with a coefficient µ 0 ∼ Λ 2 (ln λ 2 µ 2 + const), to be cancelled by a corresponding counterterm. With view on the small mass expansion studied below, it will be useful to rewrite (3.14) to obtain the variation of G ζ in terms of quantities that all have well-defined limits as m → 0. Recall that G = 1 m 2 A + G and G ζ = 1 m 2 A + G ζ . Thus (3.14) can be rewritten as Using (1.25), we can express δσ in the second term as −A∆δφ/4 (the δA/2A piece doesn't contribute since G has no zero-mode). Using the differential equation (1.13) satisfied by G, one finds for the second term with δS AY given in (1.26). Thus (3.24) In exactly the same way we also get (3.25) While these are exact relations valid for all m, they are written in a way that makes the small mass expansions obvious.

Small mass expansion
Equations (3.20) and (3.21) are non-perturbative in m. However, they are also written in a way that immediately allows for a perturbative expansion in m, since G ζ and K have smooth limits as m → 0. The m 2 → 0 limits of (3.20) and (3.21) also exhibit an extra term Order m 2 A contributions. Let us first give the order m 2 A-correction S Next, it follows from (1.13) and (1.14), upon letting x → y and subtracting the shortdistance singularity, that G ζ (x) = G ζ (x)+O(m 2 A). Thus the term of order m 2 A of the gravitational action can be read from (3.21) as (3.28) where, following [6], we have introduced ζ (x; g) . (3.29) The variation of G ζ was given in (3.24) and that of G This relation has been derived before in [6]. Using (1.22), it it then straightforward to obtain ([6]) ζ (x; g 0 ) . (3.32) For genus h = 0 and choosing g 0 to be the round metric on the sphere, G ζ (x; g 0 ) is a constant, and one directly gets the Mabuchi action for h = 0. More generally, for arbitrary genus, [6] show that where g c is the canonical metric on the Riemann surface. Finally, (3.28) becomes (3.34) While the last term contributes to the cosmological constant action, the other terms are to be considered as the genuine order m 2 A correction to the gravitational action, and it involves the Mabuchi and Aubin-Yau actions.
Higher-order contributions S (r) grav for r ≥ 2. It is straighforward to obtain the expansion in powers of m 2 of the terms in (3.20) or (3.21). Denoting the term ∼ m 2r by S