By ilikeafrica in arXiv — 19 Nov 2025

Int. J. Mod. Phys. A10 (1995) 289 – 336.

1. 이 논문은 리우빌 작용과 인터세션 넘버에 관한 이론을 다룬다.

2. 리우빌 작용의 표현에서 토폴로지적 중력의 correlator가 보인다.

3. 이 논문에서는 2D 중력의 커틀라프와 배경 기하학 사이의 불등식을 유도한다.

4. 고유한 도형을 정의하고 이를 사용하여 Virasoro 대수에 대한 위상학적 해를 찾는다.

5. 리우빌 작용과 토폴로지적 중력의 이중 역할에 대해 논의하며, 그 결과를 도출한다.

6. 2D 중력에서의 커틀라프 문제와 배경 기하학 사이의 관계를 고찰하여 불등식이 존재하는 것을 증명한다.

7. 리우빌 작용을 사용하여 토폴로지적 중력을 정의하고, 그 결과에 대한 해석을 논의한다.

8. 고유 도형과 리우빌 작용 사이의 관계를 살펴보고, 고유 도형을 사용한 Virasoro 대수의 위상학적 해를 찾는다.

9. 리우빌 작용과 토폴로지적 중력 간의 직접적인 연결을 찾아낸다.

10. 커틀라프 문제와 배경 기하학 사이의 불등식이 존재한다는 것을 증명한다.

영어 요약 시작:

Int. J. Mod. Phys. A10 (1995) 289 – 336.

arXiv:hep-th/9306150v2 13 Nov 2003Int. J. Mod.

Phys. A10 (1995) 289 – 336.UNIFORMIZATION THEORY AND 2D GRAVITYI.

LIOUVILLE ACTION AND INTERSECTION NUMBERSMarco Matone∗Department of MathematicsImperial College180 Queen’s Gate, London SW7 2BZ, U.K.andDepartment of Physics “G. Galilei” - Istituto Nazionale di Fisica NucleareUniversity of PadovaVia Marzolo, 8 - 35131 Padova, Italy¶ABSTRACTThis is the ﬁrst part of an investigation concerning the formulation of 2D gravity in theframework of the uniformization theory of Riemann surfaces.

As a ﬁrst step in this directionwe show that the classical Liouville action appears in the expression of the correlators oftopological gravity. Next we derive an inequality involving the cutoﬀof 2D gravity andthe background geometry.

Another result, still related to uniformization theory, concerns arelation between the higher genus normal ordering and the Liouville action. We introduceoperators covariantized by means of the inverse map of uniformization.

These operatorshave interesting properties including holomorphicity. In particular they are crucial to showthat the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle andvanishes for deformation of the complex structure induced by the harmonic Beltrami diﬀer-entials.

By means of the inverse map we propose a realization of the Virasoro algebra onarbitrary Riemann surfaces and ﬁnd the eigenfunctions for the holomorphic covariant oper-ators deﬁning higher order cocycles and anomalies which are related to W-algebras. Finallywe face the problem of considering the positivity of eσ, with σ the Liouville ﬁeld, by propos-ing an explicit construction for the Fourier modes on compact Riemann surfaces.

Thesefunctions, whose underlying number theoretic structure seems related to Fuchsian groupsand to the eigenvalues of the Laplacian, are quite basic and may provide the building blocksto properly investigate the long-standing uniformization problem posed by Klein, Koebe andPoincar´e.∗e-mail: matone@padova.infn.it, mvxpd5::matone¶Present address

Contents1Introduction22Uniformization Theory And Liouville Equation42.1Diﬀerentials On Σ: Explicit Construction . .

. .

. .52.2Uniformization And Poincar´e Metric.

. .

.82.3The Liouville Condition. .

. .

.102.4Chiral Factorization And Polymorphicity . .

. .

. .112.5µ, χ(Σ) And The Liouville Equation.

. .

.122.6The Inverse Map And The Uniformization Equation . .

. .

. .132.7Remarks On eϕA =|A′|2(Im A)2. .

. .

. .182.8On The Standard Approach To Liouville Gravity.

. .

. .193Covariant Holomorphic Operators, Classical Liouville Action And NormalOrdering213.1Higher Order Schwarzian Operators .

. .

. .213.2Eigenfunctions Of Q(2k+1)ϕ¯zAnd Qch(2k+1)J−1H.

. .

.233.3Normal Ordering On Σ And Classical Liouville Action. .

. .

. .253.4Diﬀeomorphism Anomaly And The KN Cocycle .

. .

.294Virasoro Algebra On Σ314.1Higher Genus Analogous Of Killing Vectors. .

. .

. .324.2Realization Of The Virasoro Algebra On Σ .

. .

. .335Liouville Field And Higher Genus Fourier Analysis355.1Positivity And Fourier Analysis On Riemann Surfaces .

. .

. .355.2Real Multivaluedness And Im Ω> 0 .

. .

. .365.3Eigenfunctions .

. .

. .375.4Multivaluedness, Area And Eigenvalues .

. .

. .395.5Genus One .

. .

. .405.6Remarks .

. .

.406Liouville Action And Topological Gravity426.1Compactiﬁed Moduli Space. .

. .

.436.2⟨κd1−1 · · · κdn−1⟩. .

. .

. .441

6.3κ1 =i2π2∂∂S(h)cl. .

. .

. .457CutoﬀIn 2D Gravity And The Background Metric467.1Background Dependence In The Deﬁnition Of The Quantum Field .

. .

.467.2The CutoﬀIn z-Space. .

. .

. .477.3Univalent Functions And (∆z)2min .

. .

.481IntroductionThe last few years have witnessed remarkable progress in 2D quantum gravity [1–6]. Impor-tant connections between non-perturbative 2D gravity, generalized KdV hierarchies, topo-logical and Liouville gravity have been discovered [7–9] (see [10] for reviews).

This interplayhas provided a good opportunity for important progress both in physics and mathematics.Despite these results there are still important unsolved problems. For example a directand satisfactory proof of the equivalence of the diﬀerent models of 2D quantum gravity is stilllacking.

In particular in the continuum formulation of 2D gravity the higher genus correlatorshave not yet been worked out. One of the problems concerns the integration on the modulispace.

The Schottky problem hinders integration on the Siegel upper plane. Analogously tothe relations between intersection theory on the moduli space and KdV [7–9], the solutionof the Schottky problem is based on deep relationships between algebraic geometry andintegrable systems.In particular Shiota and Mulase [11] proved that, according to theNovikov conjecture, a matrix in the Siegel upper plane is a Riemann period matrix if andonly if the corresponding τ-function satisﬁes Hirota’s bilinear relations.

Apparently it istechnically impossible to satisfy these constraints in performing the integration on the Siegelupper plane. Similar aspects are intimately linked with 2D gravity.

In particular, resultsfrom matrix models, where the integration on the moduli space is implicitly performed,suggest that at least in some cases the integrand is a total derivative.Many of the results concerning the continuum formulation of 2D gravity have been derivedby considering its formulation in Minkowskian space whereas Liouville theory is intrinsicallyEuclidean. Furthermore, most of the results are known on the sphere and the torus.

Onthe other hand the Liouville equation is the condition of constant negative curvature. Thusto get some insight into Liouville theory it is necessary to concentrate the analysis of thecontinuum formulation of 2D gravity on surfaces with negative Euler characteristic.Acrucial step in setting the mathematical formalism for this purpose has been made by Zograf2

and Takhtajan [12, 13]. Starting with the intention of proving a conjecture by Polyakov,they showed that Liouville theory is strictly related to uniformization theory of Riemannsurfaces.In particular it turns out that the Liouville action evaluated on the classicalsolution is the K¨ahler potential for the Weil-Petersson metric on the Schottky space.

It seemsthat the results in [12, 13] have relevant implications for 2D gravity which are still largelyunexplored. In our opinion they will serve as a catalyst to formulate a “quantum geometrical”approach to 2D gravity in the framework of uniformization theory.

Some aspects concerninguniformization theory and 2D gravity have been considered in [14].The present paper is the ﬁrst part of a work whose basic aim is to attempt to bringtogether the diﬀerent branches of mathematical technology in order to investigate 2D gravityin a purely geometrical context.The organization of the paper is as follows. In section 2 we summarize basic facts aboutthe uniformization theory of Riemann surfaces and the Liouville equation.

The points we willconsider include the explicit construction of diﬀerentials in terms of theta functions. Thisis particularly useful because by means of essentially two theorems it is possible to recoverand understand, by some pedagogical “theta gymnastics”, a lot of basic facts concerningRiemann surfaces.Furthermore, we will discuss in detail the properties of the Poincar´emetric in relation to Liouville equation and uniformization theory.

One of the aims of thissection is to clarify some points concerning the properties of the Liouville ﬁeld. For examplenote that in current literature the ﬁeld eγσ is sometimes considered as a (1, 1)-diﬀerential,which is in contradiction with the fact that, since g = eγσˆg, with ˆg a background metric, eγσmust be a (0, 0)-diﬀerential.

This point is related to the diﬃculties arising in the deﬁnitionof conformal weights in Liouville theory.In section 3 we introduce a set of operators which are covariantized by means of the inverseof the uniformization map and give their chiral (polymorphic) and non chiral eigenfunctions.We will see that these operators have important properties including holomorphicity. Nextwe consider the cocycles associated to the above operators.

In particular, the cocycle asso-ciated to the covariantized third derivative is the Fuchsian form of the Krichever-Novikov(KN) cocycle. In this framework we show that the normal ordering for operators deﬁned onΣ is related to classical Liouville theory.

Another result concerns the equivalence betweenthe chirally split anomaly of CFT and the KN cocycle. In particular it turns out that thisanomaly vanishes under deformation of the complex structure induced by the harmonic Bel-trami diﬀerentials.

These results suggest to consider higher order anomalies in the framework3

of uniformization theory of vector bundles on Riemann surfaces (W-algebras).In section 4 we consider a sort of higher genus generalization of the Killing vectors whichis based on the properties of the Poincar´e metric and of the inverse map of uniformization.This analysis will suggest a realization of Virasoro algebra on arbitrary Riemann surfaces.In section 5 we introduce an inﬁnite set of regular functions which can be considered as“building-blocks” to develop the higher genus Fourier analysis. We argue that the numbertheoretic structure underlying the building-blocks is related to the uniformization problem.In particular we investigate the structure of the eigenvalues of the Laplacian.

Besides itsmathematical interest one of the aims of this investigation is to provide a suitable tool torecognize the modes of the Liouville ﬁeld σ. This is an attempt to face the problem, usuallyuntouched in the literature, of considering metric positivity in performing the quantizationof Liouville theory.In section 6 we ﬁnd a direct link between Liouville and topological gravity.

In particularwe will show that the ﬁrst tautological class, which enters in the correlators of topologicalgravity [6–8], has the classical Liouville action as potential, in particularκ1 =i2π2∂∂S(h)cl . (1.1)In the last section we discuss important aspects concerning the role of the Poincar´e metricchosen as background.

Furthermore, by means of classical results on univalent functions,we derive an interesting inequality involving the cutoﬀof 2D gravity and the backgroundgeometry whose consequences should be further investigated.2Uniformization Theory And Liouville EquationIn this section we introduce background material for later use.In particular we beginby giving a procedure to explicitly construct any meromorphic diﬀerential deﬁned on aRiemann surface. After that we introduce basic facts concerning the uniformization theory[15,16] and the Liouville equation.

Then we investigate the properties of the inverse map ofuniformization, and consider the linearized version of the Schwarzian equation {J−1H , z} =T F, with T F the Liouville stress tensor (or Fuchsian projective connection). We concludethe section with some remarks on the standard approach to Liouville gravity.4

2.1Diﬀerentials On Σ: Explicit ConstructionLet us start by recalling some basic facts about the space of the (p, q)-diﬀerentials T p,q.Let {(Uα, zα)|α ∈I} be an atlas with harmonic coordinates on a Riemann surface Σ. Adiﬀerential in T p,q is a set of functions f ≡{fα(zα, ¯zα)|α ∈I} where each fα is deﬁned onUα. These functions are related by the following transformation in Uα ∩Uβfα(zα, ¯zα)(dzα)p(d¯zα)q = fβ(zβ, ¯zβ)(dzβ)p(d¯zβ)q,f ∈T p,q,(2.1)that is fα transforms as ∂pzα∂q¯zα.In the case of the Riemann sphere bC ≡C ∪{∞}, all the possible transition functionsz−= g−+(z+) between the two patches (U±, z±) of the standard atlas are holomorphicallyequivalent to g−+(z+) = z−1+ , that is bC has one complex structure only (no moduli).

There-fore giving f+(z+, ¯z+) ﬁxes f−(z−, ¯z−) and vice versa.In the higher genus case ﬁxing a component of f in a patch is not suﬃcient to uniquelyﬁx the other functions in f ≡{fα(zα, ¯zα)|α ∈I}. As an example we consider the case ofmeromorphic n-diﬀerentials f (n) on a compact Riemann surface of genus h. The Riemann-Roch theorem guarantees that it is possible to ﬁx the points in1 Div f (n) up to (in general) hzeroes, say P1, .

. .

, Ph, whose position is ﬁxed by Ph+1, . .

. , Pp, Q1, .

. .

, Qq, q = p−2n(h−1),the conformal structure of Σ and (in general) on the choice of the local coordinates. Aninstructive way to see this is to explicitly construct f (n).

In order to do this we ﬁrst recallsome facts about theta functions. Let us denote by Ωthe β-period matrixΩjk ≡Iβjωk,(2.2)where ω1, .

. .

, ωh are the holomorphic diﬀerentials with the standard normalizationIαjωk = δjk,(2.3)αk, βk being the homology cycles basis. The theta function with characteristic readsΘ [ab] (z|Ω) =Xk∈Zheπi(k+a)·Ω·(k+a)+2πi(k+a)·(z+b),Θ (z|Ω) ≡Θh00i(z|Ω) ,(2.4)where z ∈Ch, a, b ∈Rh.

When ak, bk ∈{0, 1/2}, Θ [ab] (−z|Ω) = (−1)4a·bΘ [ab] (z|Ω). TheΘ-function is multivalued under a lattice shift in the z-variableΘ [ab] (z + n + Ω· m|Ω) = e−πim·Ω·m−2πim·z+2πi(a·n−b·m)Θ [ab] (z|Ω) .

(2.5)1Let {Pk} ({Qk}) be the set of zeroes (poles) of a n-diﬀerential f (n).The formal sum Div f (n) ≡Ppk=1 Pk −Pqk=1 Qk and deg f (n) ≡p −q, deﬁne the divisor and the degree of f (n) respectively. It turnsout that deg f (n) = 2n(h −1).5

Let us now introduce the prime form E(z, w). It is a holomorphic −1/2-diﬀerential both inz and w, vanishing for z = w onlyE(z, w) = Θ [ab] (I(z) −I(w)|Ω)h(z)h(w).

(2.6)Here h(z) denotes the square root of Phk=1 ωk(z)∂ukΘ [ab] (u|Ω) |uk=0; it is the holomorphic1/2-diﬀerential with non singular (i.e. ∂ukΘ [ab] (u|Ω) |uk=0 ̸= 0) odd spin structure [ab].

Thefunction I(z) in (2.6) denotes the Jacobi mapIk(z) =Z zP0ωk,z ∈Σ,(2.7)with P0 ∈Σ an arbitrary base point. This map is an embedding of Σ into the JacobianJ(Σ) = Ch/LΩ,LΩ= Zh + ΩZh.

(2.8)By (2.5) it follows that the multivaluedness of E(z, w) isE(z + n · α + m · β, z) = e−πim·Ω·m−2πim·(I(z)−I(w))E(z, w). (2.9)In terms of E(z, w) one can construct the following h/2-diﬀerential with empty divisorσ(z) = exp −hXk=1Iαkωk(w) log E(z, w)!,(2.10)whose multivaluedness isσ(z + n · α + m · β) = eπi(h−1)m·Ω·m−2πim·(∆−(h−1)I(z))σ(z),(2.11)where ∆is (essentially) the vector of Riemann constants [17].

Finally we quote two theorems:a. Abel Theorem [15]. A necessary and suﬃcient condition for D to be the divisor of ameromorphic function is thatI (D) = 0 mod (LΩ) and deg D = 0.(2.12)b.

Riemann vanishing theorem [17]. The functionΘ I(z) −hXk=1I(Pk) + ∆Ω!,z, Pk ∈Σ,(2.13)either vanishes identically or else it has h zeroes at z = P1, .

. .

, Ph.6

We are now ready to explicitly construct the diﬀerential f (n) deﬁned above. First of allnote thatef (n) = σ(z)2n−1Qpk=h+1 E(z, Pk)Qp−2n(h−1)j=1E(z, Qj),(2.14)is a multivalued n-diﬀerential with Div ef (n) = Ppk=h+1 Pk −Pp−2n(h−1)k=1Qk.

Therefore we setf (n)(z) = g(z) ef (n),(2.15)where, up to a multiplicative constant, g is ﬁxed by the requirement that f (n) be singlevalued.From the multivaluedness of the E(z, w) and σ(z) it follows that, up to a multiplicativeconstantg(z) = Θ (I(z) + D|Ω) ,(2.16)withD =pXk=h+1I(Pk) −p−2n(h−1)Xk=1I(Qk) + (1 −2n)∆. (2.17)By Riemann vanishing theorem g(z) has just h-zeroes P1, .

. .

, Ph ﬁxed by D.Thus therequirement of singlevaluedness also ﬁxes the position of the remainder h zeroes. To makemanifest the divisor in the RHS of (2.15) we ﬁrst recall that the image of the canonical linebundle K on the Jacobian of Σ coincides with 2∆[17].

On the other hand, since[Kn] =pXk=1Pk −p−2n(h−1)Xk=1Qk,(2.18)by Abel theorem we have2Div Θ (I(z) + D|Ω) = Div Θ I(z) −hXk=1I(Pk) + ∆Ω!,(2.19)and by Riemann vanishing theoremDiv Θ (I(z) + D|Ω) =hXk=1I(Pk). (2.20)Above we have considered harmonic coordinates.

If one starts with arbitrary coordinatesds2 = egabdxadxb,(2.21)2The square brackets in (2.18) denote the divisor class associated to the line bundle Kn. Two divisorsbelong to the same class if they diﬀer by a divisor of a meromorphic function.7

the harmonic ones, deﬁning the conformal form 2gz¯z|dz|2, are determined by the Beltramiequation3 eg12ǫacegcb∂bz = i∂az. Therefore we can globally choose ds2 = eφ|dz|2, eφ = 2gz¯z.This means that with respect to the new set of coordinates {(Uα, zα)|α ∈I} the metricis in the conformal gauge ds2α = eφα|dzα|2 in each patch.That is the functions in φ ≡{φα(zα, ¯zα)|α ∈I} are related by the following transformation in Uα ∩Uβφα(zα, ¯zα) = φβ(zβ, ¯zβ) + log |dzβ/dzα|2.

(2.22)By a rescaling ds2 →des2 = ρds2 it is possible to set, at least in one patch, des2α = |dzα|2.Since ρ ∈T 0,0, there is at least one patch (Uγ, zγ) where des2γ ̸= cst|dzγ|2. Finally we recallthat a property of the metric is positivity.

Thus if g = eσˆg with ˆg a well-deﬁned metric,then eσ ∈C∞+ where C∞+ denotes the subspace of positive smooth functions in T 0,0. Later,in the framework of the “Liouville condition”, we will discuss some aspects concerning thestructure of the boundary of C∞+ .2.2Uniformization And Poincar´e MetricLet us denote by D either the Riemann sphere bC = C ∪{∞}, the complex plane C, or theupper half plane H = {w ∈C|Im w > 0}.

The uniformization theorem states that everyRiemann surface Σ is conformally equivalent to the quotient D/Γ with Γ a freely actingdiscontinuous group of fractional transformations preserving D.Let us consider the case of Riemann surfaces with universal covering H and denote byJH the complex analytic covering JH : H →Σ. In this case Γ (the automorphism group ofJH) is a ﬁnitely generated Fuchsian group Γ ⊂PSL(2, R) = SL(2, R)/{I, −I} acting on Hby linear fractional transformationsw ∈H,γ · w = aw + bcw + d ∈H,γ =acbd∈Γ ⊂PSL(2, R).

(2.23)By the ﬁxed point equationw± =a −d ±q(a + d)2 −42c,(2.24)it follows that γ ̸= I can be classiﬁed according to the value of |tr γ|:3In isothermal coordinates the metric reads ds2 = eφ (dx)2 + (dy)2, where z = x + iy.Note thatconsidering x = cst as an isothermal curve, y = cst corresponds to the curve of heat ﬂow.8

1. Elliptic.

|tr γ| < 2, γ has one ﬁxed point on H (w−= w+ /∈R) and Σ has a branchedpoint z with index q−1 ∈N\{0, 1} where q−1 is the ﬁnite order of the stabilizer of z.2. Parabolic.

|tr γ| = 2, then w−= w+ ∈R and the Riemann surface has a puncture.The order of the stabilizer is now inﬁnite, that is q−1 = ∞.3. Hyperbolic.

|tr γ| > 2, the ﬁxed points are distinct and lie on the real axis, thusw± /∈H. These group elements represent handles of the Riemann surface and can berepresented in the form (γw −w+)/(γw −w−) = eλ(w −w+)/(w −w−), eλ ∈R\{0, 1}.Note that if Γ contains elliptic elements then H/Γ is an orbifold.

Furthermore, since theparabolic points do not belong to H, point JH(w+) corresponds to a deleted point of Σ. Byabuse of language we shall call both the elliptic and the parabolic points ramiﬁed punctures.A Riemann surface isomorphic to the quotient H/Γ has the Poincar´e metric ˆg as theunique metric with scalar curvature Rˆg = −1 compatible with its complex structure. Thisimplies the uniqueness of the solution of the Liouville equation on Σ.

The Poincar´e metricon H isdˆs2 =|dw|2(Im w)2. (2.25)Note that PSL(2, R) transformations are isometries of H endowed with the Poincar´e metric.An important property of Γ is that it is isomorphic to the fundamental group π1(Σ).Uniformizing groups admit the following structure.

Suppose Γ uniformizes a surface of genush with n punctures and m elliptic points with indices 2 ≤q−11≤q−12≤. .

. ≤q−1m < ∞.

Inthis case the Fuchsian group is generated by 2h hyperbolic elements H1, . .

. , H2h, m ellipticelements E1, .

. .

, Em and n parabolic elements P1, . .

. , Pn, satisfying the relationsEq−1ii= I,mYl=1ElnYk=1PkhYj=1H2j−1H2jH−12j−1H−12j= I,(2.26)where the inﬁnite cyclicity of parabolic ﬁxed point stabilizers is understood.Setting w = J−1H (z) in (2.25), where J−1H: Σ →H is the inverse of the uniformizationmap, we get the Poincar´e metric on Σdˆs2 = 2ˆgz¯z|dz|2 = eϕ(z,¯z)|dz|2,(2.27)whereeϕ(z,¯z) =|J−1H (z)′|2(Im J−1H (z))2,(2.28)9

which is invariant under SL(2, R) fractional transformations of J−1H (z). SinceRˆg = −ˆgz¯z∂z∂¯z log ˆgz¯z,ˆgz¯z = 2e−ϕ,(2.29)the condition Rˆg = −1 is equivalent to the Liouville equation∂z∂¯zϕ(z, ¯z) = 12eϕ(z,¯z),(2.30)whereas the ﬁeld eϕ = ϕ + log µ, µ > 0, deﬁnes a metric of constant curvature −µ.

Noticethat the metric gz¯z = eσ+ϕ/2, with ∂z∂¯zσ = 0, has scalar curvature Rg = −e−σ. Howeverrecall that non constant harmonic functions do not exist on compact Riemann surfaces.2.3The Liouville ConditionIf g is a (in general non singular) metric on a Riemann surface of genus h with n parabolicpoints we have4ZΣ√gRg = 2πχ(Σ),(2.31)whereχ(Σ) = 2 −2h −n.

(2.32)A peculiarity of parabolic points is that they do not belong to Σ, so that the singularitiesin the metric and in the Gaussian curvature at the punctures do not appear in g and R asfunctions on Σ.Let Σ be a n-punctured Riemann surface of genus h and with elliptic points (z1, . .

. , zm).Its Poincar´e area is [16]ZΣqˆg = 2π 2h −2 + n +mXk=1(1 −qk)!,(2.33)where q−1k∈N\{0, 1} denotes the ramiﬁcation index of zk.Let us choose the coordinates in such a way that the metric be in the conformal formds2 = 2gz¯z|dz|2.

In this case gz¯z = eσˆgz¯z (here we set γ = 1) where eσ ∈C∞+ and ˆgz¯z = eϕ/2is the Poincar´e metric. SinceRg = −2e−σ−ϕ∂z∂¯z(ϕ + σ) = −e−σ 1 + 2e−ϕ∂z∂¯zσ,(2.34)4In the following the |dz∧d¯z|2term in the surface integrals is understood.10

by (2.31) it follows thatZΣ√gRg = −ZΣ ∂z∂¯z(ϕ + σ) = −ZΣ ∂z∂¯zϕ = 2πχ(Σ).(2.35)Eq. (2.35) follows from the fact that log f ∈T 0,0 for f ∈C∞+ .

This shows that for admissiblemetric the contribution to χ(Σ) comes from the transformation property of ϕ, whereasterms such asRΣ σz¯z, eσ ∈C∞+ , are vanishing. A necessary condition in order that eσˆg be anadmissible metric on Σ is that σ satisﬁes the Liouville conditionZΣ σz¯z = 0,(2.36)which is a weaker condition than eσ ∈C∞+ .

This trivial remark is useful to understand thestructure of the boundary of space of admissible metrics eσˆg. Actually it is possible to adddelta-like singularities at the scalar curvature leaving the Euler characteristic unchanged.That is these additional singularities do not imply additional punctures on the surface.

Inparticular there are positive semideﬁnite (1, 1)-diﬀerentials gz¯z = eσˆgz¯z that, sinceRΣ√gRg =RΣ√ˆgRˆg, can be considered as degenerate metrics.An interesting case is when σ(z) =−4πG(z, w) where G is Green’s function for the scalar Laplacian with respect to the Poincar´emetric ˆgz¯z = eϕ/2. The Green function takes real values and has the behaviour −12π log |z−w|as z →w, moreover−∂z∂¯zG(z, w) = δ(2)(z −w) −√ˆg2RΣ√ˆg,(2.37)where the −√ˆg/2RΣ√ˆg = eϕ/8πχ term is due to the constant zero-mode of the Laplacian.Thus, in spite of the logarithm singularity of G, the contribution toRΣ Gz¯z coming fromthe delta function is cancelled by the contribution due to the eϕ/8πχ term.ThereforeRΣ ∂z∂¯zG(z, w) = 0 and χ(Σ) is unchanged whereas the scalar curvature becomesRˆg(z, ¯z) = −1 →Rg(z, ¯z) = −e4πG(z,w) 1 + 8πe−ϕ(z,¯z)δ(2)(z −w) +14χ(Σ)!.

(2.38)Note that similar remarks extend to the positive deﬁnite metric e2πG(z,w)+ϕ. We concludethis digression by stressing that one can modify the metric by adding singularities in such away that the Euler characteristic changes.

In this case one should try to deﬁne a new surfacewith additional punctures where the (1, 1)-diﬀerential is an admissible metric.2.4Chiral Factorization And PolymorphicityBy means of chiral (in general polymorphic) functions it is possible to construct regular andnon vanishing diﬀerentials f (n,n) ∈T n,n. An important example is given by the expression11

of the Poincar´e metric in terms of the inverse map J−1H (which is a chiral polymorphic scalarfunction) or in terms of solutions of the uniformization equation (cfr. (2.70)).

Conversely, afactorized form f (n,n) = g1(z)g2(¯z) enforces us to consider chiral diﬀerentials whose degree isﬁxed by n and the topology of Σ. It is easy to see that the only change that g1 and g2 canundergo after winding around the homology cycles of Σ is to get a constant multiplicativefactor.

However these diﬀerentials have the same degree as singlevalued diﬀerentials, that isdeg g1 = deg g2 = 2n(h −1).Later we will see that similar aspects force us to consider non Abelian monodromy forthe chiral function arising in the construction of the Poincar´e metric.2.5µ, χ(Σ) And The Liouville EquationHere we consider some aspects of the Liouville equation.We start by noticing that byGauss-Bonnet it follows that ifRΣ eϕ > 0, then the equation∂z∂¯zϕ(z, ¯z) = µ2eϕ(z,¯z),(2.39)has no solutions on surfaces with sgn χ(Σ) = sgn µ. In particular, on the Riemann spherewith n ≤2 punctures5 there are no solutions of the equation∂z∂¯zϕ(z, ¯z) = 12eϕ(z,¯z).ZΣ eϕ > 0,(2.40)The metric of constant curvature on bCds2 = eϕ0|dz|2,eϕ0 =4(1 + |z|2)2,(2.41)satisﬁes the Liouville equation with the “wrong sign”, that isRϕ0 = 1−→∂z∂¯zϕ0(z, ¯z) = −12eϕ0(z,¯z).

(2.42)If one insists on ﬁnding a solution of eq. (2.40) on bC, then inevitably one obtains at leastthree delta-singularities∂z∂¯zϕ(z, ¯z) = 12eϕ(z,¯z) −2πnXk=1δ(2)(z −zk),n ≥3.

(2.43)5The 1-punctured Riemann sphere, i.e. C, has itself as universal covering.

For n = 2 we have JC :C →C\{0}, z 7→e2πiz. Furthermore, C\{0} ∼= C/ < T1 >, where < T1 > is the group generated byT1 : z 7→z + 1.12

Since σ = log(ϕ −ϕ0) does not satisfy the Liouville condition, the (1, 1)-diﬀerential eϕ isnot an admissible metric on bC.Furthermore, since the unique solution of the equationϕz¯z = eϕ/2 on the Riemann sphere is ϕ = ϕ0 + iπ, to consider the Liouville equation on bCgives the unphysical metric −eϕ0.This discussion shows that in order to ﬁnd a solution of eq. (2.40) one needs at least threepunctures, that is one must consider eq.

(2.40) on the surface Σ = bC\{z1, z2, z3} where theterm 2π P3k=1 δ(2)(z −zk) does not appear simply because zk /∈Σ, k = 1, 2, 3. In this caseχ(Σ) = −1, so that sgn χ(Σ) = −sgn µ in agreement with Gauss-Bonnet.2.6The Inverse Map And The Uniformization EquationLet us now consider some aspects of the Liouville equation (2.30).

As we have seen thePoincar´e metric on Σ iseϕ(z,¯z) =|J−1H (z)′|2(Im J−1H (z))2,(2.44)which is invariant under SL(2, R) fractional transformations of J−1H .This metric is theunique solution of the Liouville equation.An alternative expression for eϕ follows by considering as universal covering of Σ thePoincar´e disc ∆= {z||z| < 1}. Let us denote by J∆: ∆→Σ the map of uniformization.Since the map from ∆to H isw = i1 −zz + 1,z ∈∆,w ∈H,(2.45)we haveeϕ = 4|J−1∆′|2(1 −|J−1∆|2)2 = 4|J−1∆′|2∞Xk=0(k + 1)|J−1∆|2k.

(2.46)Both (2.44) and (2.46) make it evident that from the explicit expression of the inversemap we can ﬁnd the dependence of eϕ on the moduli of Σ. Conversely we can express theinverse map (to within a SL(2, C) fractional transformation) in terms of ϕ. This followsfrom the Schwarzian equation{J−1H , z} = T F(z),(2.47)whereT F(z) = ϕzz −12ϕ2z,(2.48)is the classical Liouville energy-momentum tensor (or Fuchsian projective connection) and{f, z} = f ′′′f ′ −32 f ′′f ′!2= −2(f ′)12((f ′)−12)′′,(2.49)13

is the Schwarzian derivative. Note that eq.

(2.30) implies that∂¯zT F = 0. (2.50)In the conformal gauge the metric can be written as gz¯z = eϕ+γσ/2, eγσ ∈C∞+ .

In this casethe stress tensorT γ = (ϕ + γσ)zz −12(ϕz + γσz)2,(2.51)satisﬁes the equation∂¯zT γ = −eϕ+γσ∂zRϕ+γσ. (2.52)Therefore T γ is not chiral unless Rϕ+γσ is an antiholomorphic function.

Of course the onlypossibility compatible with the fact that eγσ ∈C∞+ is Rϕ+γσ = cst. Another aspect of thestress tensor is that SL(2, C) transformations of J−1H , while changing the Poincar´e metric,leave T F invariant.On punctured surfaces there are non trivial global solutions of the equation σz¯z = 0, sothat in this case ∂¯zT γ = −γ∂¯z(ϕzσz) = −γ2eϕσz.

Furthermore, since ∂zRϕ+γσ = γβ∂zRϕ+βσ,with β an arbitrary constant, we have∂¯zT γ = −γβ eϕ+γσ∂zRϕ+βσ. (2.53)Let us deﬁne the covariant Schwarzian operatorS(2)f= 2(f ′)12∂z(f ′)−1∂z(f ′)12,(2.54)mapping −1/2- to 3/2-diﬀerentials.

SinceS(2)f· ψ =2∂2z + {f, z}ψ,(2.55)the Schwarzian derivative can be written as{f, z} = S(2)f· 1. (2.56)The operator S(2)fis invariant under SL(2, C) fractional transformations of f, that isS(2)γ·f = S(2)f ,γ ∈SL(2, C).

(2.57)Therefore, if the transition functions of Σ are linear fractional transformations, then {f, z}transforms as a quadratic diﬀerential. However, except in the case of projective coordinates,14

the Schwarzian derivative does not transform covariantly on Σ. This is evident by (2.56)since in ﬂat spaces only (e.g.

the torus) a constant can be considered as a −1/2-diﬀerential.Let us consider the equationS(2)f· ψ = 0. (2.58)To ﬁnd two independent solutions we set(f ′)12∂z(f ′)−1∂z(f ′)12ψ1 = (f ′)12∂z(f ′)−1∂zcst = 0,(2.59)and(f ′)12∂z(f ′)−1∂z(f ′)12ψ2 = (f ′)12∂zcst = 0,(2.60)so that the solutions of (2.58) areψ1 = cst (f ′)−12,ψ2 = cst f(f ′)−12.

(2.61)Since ψ2/ψ1 = cst f, to ﬁnd the solution of the Schwarzian equation {f, z} = g is equivalentto solve the linear equation2∂2z + g(z)ψ = 0. (2.62)We stress that the “constants” in the linear combination φ = aψ1+bψ2 admit a ¯z-dependenceprovided that ∂za = ∂zb = 0.The inverse map is locally univalent, that if z1 ̸= z2 then J−1H (z1) ̸= J−1H (z2).

A relatedcharacteristic of J−1His that under a winding of z around non trivial cycles of Σ the pointJ−1H (z) ∈H moves from a representative D of the fundamental domain to an equivalentpoint of another representative6 D′. On the other hand, since Γ is the automorphism groupof JH, it follows that after winding around non trivial cycles of Σ the inverse map transformsin the linear fractional wayJ−1H −→γ · J−1H = aJ−1H + bcJ−1H + d,acbd∈Γ.

(2.63)However note that (2.57) guarantees that, in spite of the polymorphicity (2.63), the classicalLiouville stress tensor T F = S(2)J−1H · 1 is singlevalued.As we have seen one of the important properties of the Schwarzian derivative is that theSchwarzian equation (2.47) can be linearized. Thus if ψ1 and ψ2 are linearly independentsolutions of the uniformization equation ∂2∂z2 + 12T F(z)!ψ(z) = 0,(2.64)6This property of J−1Hmakes evident its univalence as function on Σ.15

then ψ2/ψ1 is a solution of eq.(2.47). That is, up to a SL(2, C) linear fractional transforma-tion, we haveJ−1H = ψ2/ψ1.

(2.65)Indeed by (2.59,2.60) it follows thatψ1 = (J−1H′)−12,ψ2 = (J−1H′)−12J−1H ,(2.66)are independent solutions of (2.64). Another way to prove (2.65) is to write eq.

(2.64) in theequivalent form(f ′)12∂z(f ′)−1∂z(f ′)12ψ = 0,f ≡J−1H ,(2.67)and then to set z = JH(w). In this case (2.64) becomes the trivial equation on Hw′3/2∂2wφ = 0.

(2.68)For any choice of the two linearly independent solutions we have φ2/φ1 = w up to an SL(2, C)transformation. Going back to Σ we get J−1H = ψ2/ψ1.Note that any SL(2, R) transformationψ1ψ2−→eψ1eψ2=acbdψ1ψ2,(2.69)induces a linear fractional transformation of J−1H .Therefore the invariance of eϕ underSL(2, R) linear fractional transformations of J−1H corresponds to its invariance for SL(2, R)linear transformations of7 ψ1, ψ2.

This leads us to express e−kϕ ase−kϕ = (−4)−k ψ1ψ2 −ψ2ψ12k ,(2.70)in particular, when 2k is a non negative integer, we gete−kϕ = 4−kkXj=−k(−1)jCj+k2k ψk+j1ψk−j1ψk+j2ψk−j2,2k ∈Z+,Cjk =k!j! (k −j)!.

(2.71)On the other hand, since we can choose ψ2 = ψ1R ψ−21 , we havee−kϕ = (−4)−k|ψ|4kZψ−2 −Zψ−22k,∀k,(2.72)7Note that the Poincar´e metric is invariant under SL(2, R) fractional transformations of J−1Hwhereasthe Schwarzian derivative T F(z) = {J−1H , z} is invariant for SL(2, C) transformations of J−1H . Thus theidentiﬁcation J−1H= ψ2/ψ1 is up to a SL(2, C) transformation.16

withψ = aψ1(1 + bZψ−21 ),a ∈R\{0},b ∈R. (2.73)We note that the ambiguity in the deﬁnition ofR z ψ−2 reﬂects the polymorphicity of J−1H .This property of J−1Himplies that, under a winding around non trivial loops, a solution of(2.64) transforms in a linear combination involving itself and another (independent) solution.It is easy to check that∂2z + 12T F(z)e−ϕ/2 = 0,(2.74)which shows that the uniformization equation has the interesting property of admittingsinglevalued solutions.The reason is that the ¯z-dependence of e−ϕ/2 arises through thecoeﬃcients ψ1 and ψ2 in the linear combination of ψ1 and ψ2.Since [∂¯z, S(2)J−1H ] = 0, the singlevalued solutions of the uniformization equation are∂2z + 12T F(z)∂l¯ze−ϕ/2 = 0,l = 0, 1, .

. .

. (2.75)Thus, since e−ϕ and e−ϕϕ¯z are linearly independent solutions of eq.

(2.64), their ratio solvethe Schwarzian equation{ϕ¯z, z} = T F(z). (2.76)Higher order derivatives ∂l¯ze−ϕ/2, l ≥2, are linear combinations of e−ϕ/2 and e−ϕ/2ϕ¯z withcoeﬃcients depending on TF and its derivatives; for example∂2¯ze−ϕ/2 = −TF2 e−ϕ/2.

(2.77)In particular if ψ2(z) = TFψ1(z) then, in spite of the fact that TF is not a constant on Σ,ψ1 and ψ2 are linearly dependent solutions of eq.(2.64). A check of the linear dependence ofψ1 from ψ2 follows from the fact that{ψ2/ψ1, z} = {TF, z} = 0 ̸= T F(z).

(2.78)Let us show what happens if one sets J−1H= ψ1/ψ2 without considering the remarkmade in the previous footnote. As solutions of the uniformization equation, we can considerψ1 = e−ϕ/2 and an arbitrary solution ψ2 such that ∂z (ψ2/ψ1) = 0.

Since ∂¯ze−ϕ/2/ψ2̸= 0,in spite of the fact that {e−ϕ/2/ψ2, z} = T F, we have J−1H ̸= ψ1/ψ2.We conclude the analysis of the uniformization equation by summarizing some usefulexpressions for the Liouville stress tensorT F =nJ−1H , zo= {ϕ¯z, z} = 2J−1H′ 12 ∂z1J−1H′∂zJ−1H′ 12 · 1 = 2eϕ/2∂ze−ϕ∂zeϕ/2 · 117

= 2 e−ϕ/2ψ2!′ 12∂z e−ϕ/2ψ2!′−1∂z e−ϕ/2ψ2!′ 12· 1 = −2eϕ/2 e−ϕ/2′′ = −2ψ−1ψ′′,(2.79)with ψ given in (2.73) and ψ2 an arbitrary solution of eq. (2.64) such that ∂ze−ϕ/2/ψ2̸= 0.2.7Remarks On eϕA =|A′|2(Im A)2Sometimes in current literature it is stated that the solution of the Liouville equation iseϕA =|A′|2(Im A)2,(2.80)with A a generic holomorphic function.However the uniqueness of the solution of theLiouville equation implies that A(z) is the inverse map of the uniformization which is uniqueup to SL(2, R) fractional transformations.

Let us show what happens if A is consideredto be an arbitrary well-deﬁned chiral function on a compact Riemann surface.First ofall according to the Weierstrass gap theorem a meromorphic function f (0), with divisor ingeneral position, has at least h + 1 zeroes8. Thus, since deg f (0) = 0, f (0) has at least h + 1poles.

Since ∂¯zz−1 ∼πδ(2)(z), it follows that if A(z) were a well-deﬁned 0-diﬀerential thenit would induce singularities in the scalar curvature at the divisor of A, so that RϕA ̸= −1.Furthermore eϕA itself is singular when Im A(z) = 0 and will degenerate for the zeroes ofA′(z). Therefore, in order that ϕA be the solution of the Liouville equationRϕA = −1,(2.81)the ﬁeld A(z) must be a chiral and linearly polymorphic function.

In particular, under theaction of the fundamental group π1(Σ), A(z) must transform in a linear fractional way withthe coeﬃcients of the transformation in the Fuchsian group Γ whose elements are ﬁxed bythe moduli of Σ.Notice that A(z) cannot simply be a holomorphic nowhere vanishing function with con-stant multivaluedness. In this case the monodromy is Abelian.

From the analytic point ofview the commutativity of group monodromy has the eﬀect of giving a metric with singu-larities. This, for example, follows from the fact that if A →cst A, then A′/A would be awell-deﬁned one-diﬀerential so that#zeroes (A′/A) = #poles (A′/A) + 2(h −1) > 0.

(2.82)8The restriction to “points in general position” means that we are not considering Weierstrass points inDiv f (0).18

On the other hand since A must be a holomorphic nonvanishing function it follows thatDivA′ = Div(A′/A). Thus eϕA would be a degenerate metric since, if Im A(P) ̸= 0, ∀P ∈DivA′, then#zeroes (eϕA) = 4(h −1) > 0.

(2.83)Unfortunately no one has succeeded in writing down the explicit form of the inversemap in terms of the moduli of Σ. This is the uniformization problem.

In section 5 we willintroduce a new set of Fourier modes in higher genus whose properties suggest that they arestrictly related to the underlying Fuchsian group.2.8On The Standard Approach To Liouville GravityWe conclude this section by considering some aspects concerning the standard approachto Liouville gravity. Let us begin by noticing that if one parametrizes the metric in theform g = eγσˆg, where ˆg is a background metric (in particular ˆg is a positive deﬁnite (1, 1)-diﬀerential), then eγσ ∈C∞+ .

The Liouville action in harmonic coordinates readsS =ZΣ |∂zσ|2 + 1γqˆgRˆgσ + µ2γ2qˆgeγσ!. (2.84)Sometimes it is stated that at the classical level γσ transforms as in (2.22) whereas afterquantization the logarithm term is multiplied by a constant related to the central charge.Actually, to perform the surface integration ˆg must be a (1, 1)-diﬀerential and eγσ ∈T 0,0.Therefore it is unclear what the meaning of S is if one considers eγσ ∈T 1,1.

Another standardchoice is to set dˆs2 = cst|dz|2 on a patch. Once again, since it is not possible to make thischoice on the whole manifold, with this prescription the surface integral is undeﬁned.

A wayto (partly) solve these problems is to set (formally) ˆgz¯z = 1 and then consider ds2 = eγσ|dz|2,so that eγσ ∈T 1,1. In this case Rˆg is formally zero and the integrand in (2.84) reduces toF = |∂zσ|2 + µ2γ2eγσ.

(2.85)The transition from the integrand in (2.84) to F is implicitly assumed by some authors (seefor example equations (1.2) and (2.1) in the interesting paper [18]). However, since |∂zσ|2does not transform covariantly, one must add “boundary terms” toRΣ F in order to get awell-deﬁned action.

This has been done in [12]. In the case of surfaces with punctures thisboundary term corresponds to a regularization term which is crucial in ﬁxing the scalingproperties of S. This regularization procedure is related to the fact that negatively curved19

surfaces (the realm of Liouville theory) have both ultraviolet and infrared cutoﬃng prop-erties. The coupling between cutoﬀ, regularization terms in the Liouville action, modularanomaly is a highly non trivial (and interesting) aspect which is related to the propertiesof univalent functions (e.g.

Koebe 1/4-theorem) that we will discuss in the last section inthe context of 2D quantum gravity. We notice that this subject is related to classical andquantum chaos on Riemann surfaces.In the operator formulation of CFT one considers the solutions of the classical equationof motion with allowed singularities at the points where the in and out vacua are placed.This allows one to consider non trivial solutions of the equation φz¯z = 0.In Liouvilletheory it is not possible to compute the asymptotics of the stress tensor by standard CFTtechniques.

The reason is that the OPE in CFT is based on free ﬁelds techniques where< X(z)X(w) >∼−log(z −w).This explains why it is diﬃcult to recognize what thevacuum of Liouville theory is.The known results in quantum Liouville theory essentially concern the formulation onthe sphere and the torus. To get insight about the continuum formulation of 2D gravity inhigher genus is an outstanding problem.

To understand the diﬃculties that one meets withrespect to the h = 0, 1 cases we summarize few basic facts.h ≤1 A feature of bC with respect to higher genus surfaces is that its universal covering isbC itself. Therefore metrics on bC and on its universal covering coincide.

This partlyexplains why in this case computations are easier to be done. In the torus case “Li-ouville theory” is free, the reason is that the metric of constant curvature eϕ satisﬁesthe equation ϕz¯z = 0, that is ϕ = cst.

Therefore to quantize 2D gravity on the torus itis suﬃcient to use standard CFT techniques and to impose positivity on the Liouvilleﬁeld. This can easily be done because the Fourier modes on the torus are explicitlyknown9.h ≥2 In the higher genus case the metric of constant curvature on Σ has a richer geometricalstructure with respect to the Poincar´e metric on its universal covering H. Thus inquantizing the theory we must consider the geometry of the moduli space or, which isthe same, the geometry of Fuchsian groups (which is encoded in J−1H ).

To get resultsin a way similar to those derived on bC we have to shift our attention from Σ to theupper half plane whose Poincar´e metric is explicitly known (see (2.25)). In this case9In section 5 we will consider the problem of formulating higher genus Fourier analysis.20

we are not considering the underlying topology and geometry of the terms in thegenus expansion. Nevertheless non perturbative results enjoy similar properties.

Thissimilarity suggests to formulate a non perturbative approach to 2D gravity based on asort of path-integral formulation of 2D gravity on H. The reason for this is that bothH and the Poincar´e metric on it are universal (non perturbative) objects underlyingthe full genus expansion.3Covariant Holomorphic Operators, Classical Liou-ville Action And Normal OrderingHere we introduce a set of operators S(2k+1)J−1Hcorresponding to ∂(2k+1)zcovariantized by meansof J−1H . We stress that univalence of J−1H implies that these operators are holomorphic.

Next,we derive the chiral (polymorphic) and non chiral eigenfunctions for a set of operators relatedto S(2k+1)J−1H. An interesting property of these operators is that the (generalized) harmonicBeltrami diﬀerentials µ(2k+1)harm(see eq.

(3.26)) are in their kernelS(2k+1)J−1Hµ(2k+1)harm = 0. (3.1)We consider the cocycles associated to S(2k+1)J−1H.In this framework we show that thenormal ordering for operators deﬁned on Σ is related to classical Liouville theory.Theunivalence of J−1H allows us to get time-independence and locality for the cocycles.The holomorphicity of the covariantization above allows us to show that the chirally splitanomaly of CFT reduces to the Krichever-Novikov (KN) cocycle.

This suggests to introducehigher order anomalies given as surfaces integrals of (1, 1)-forms deﬁned in terms of S(2k+1)J−1H.These anomalies are related to the uniformization theory of vector bundles on Riemannsurfaces (W-algebras).Remarkably, eq. (3.1) implies that these anomalies (including thestandard chirally split anomaly) vanish in the case one considers deformation of the complexstructure induced by (generalized) harmonic Beltrami diﬀerentials.3.1Higher Order Schwarzian OperatorsLet us start by noticing that sincee−kϕ = |J−1H′|−2kJ−1H −J−1H2i2k,(3.2)21

it follows that the negative powers of the Poincar´e metric satisfy the higher order general-ization of eq. (2.74)S(2k+1)J−1H· e−kϕ = 0,k = 0, 12, 1, .

. .

,(3.3)with S(2k+1)fthe higher order covariant Schwarzian operatorS(2k+1)f= (2k + 1)(f ′)k∂z(f ′)−1∂z(f ′)−1 . .

. ∂z(f ′)−1∂z(f ′)k,(3.4)where the number of derivatives is 2k + 1.

We stress that univalence of J−1Himplies holo-morphicity of the S(2k+1)J−1Hoperators. Eq.

(3.3) is manifestly covariant and singlevalued on Σ.Furthermore it can be proved that the dependence of S(2k+1)fon f appears only throughS(2)f· 1 = {f, z} and its derivatives; for exampleS(3)J−1H = 3∂3z + 2T F∂z + T F ′,(3.5)which is the second symplectic structure of the KdV equation.A nice property of theequation S(2k+1)J−1H· eψ = 0 is that its projection on H is the trivial equation(2k + 1)w′k+1∂2k+1wψ = 0,w ∈H,(3.6)where w = J−1H (z). This makes evident why only for k > 0 it is possible to have ﬁniteexpansions of e−kϕ such as in eq.(2.71).

The reason is that the solutions of eq. (3.6) are{wj|j = 0, .

. .

, 2k} so that the best thing we can do is to consider linear combinations ofpositive powers of the non chiral solution Im w which is just the square root of inverse of thePoincar´e metric on H.The SL(2, C) invariance of the Schwarzian derivative implies thatS(2k+1)J−1H= S(2k+1)ϕ¯z,k = 0, 12, 1, . .

. .

(3.7)On the other hand by Liouville equationS(2k+1)J−1H= S(2k+1)ϕ¯z= (2k + 1)ekϕ∂ze−ϕ∂ze−ϕ . .

. ∂ze−ϕ∂zekϕ,k = 0, 12, 1, .

. .

. (3.8)In the following we will use this property of S(2k+1)J−1Hto construct the eigenfunctions for theoperatorQ(2k+1)ϕ¯z= S(2k+1)ϕ¯z2ϕ¯ze−ϕ2k+1 ,(3.9)and for its chiral analogousQch(2k+1)J−1H= S(2k+1)J−1H∂z log J−1H−2k−1 .

(3.10)22

3.2Eigenfunctions Of Q(2k+1)ϕ¯zAnd Qch(2k+1)J−1HSince[∂¯z, S(2k+1)J−1H] = 0,(3.11)it follows that besides e−kϕ other singlevalued solutions of S(2k+1)J−1H· ψ = 0 have the form∂l¯ze−kϕ. However notice that by (3.7) it follows that the set of singlevalued diﬀerentialsψl = (2ϕ¯z)l e−kϕ,l = 0, .

. .

, 2k,(3.12)is a basis of solutions of S(2k+1)J−1H· ψ = 0. To see this it is suﬃcient to substitute ψl in theRHS of (3.8) and systematically use the Liouville equatione−ϕ∂z(2ϕ¯z)l = l(2ϕ¯z)l−1.

(3.13)In the intersection of two patches (U, z) and (V, w) the ﬁeld ψl transforms as(2ϕ¯z(z, ¯z))l e−kϕ(z,¯z) =2 eϕ ¯w(w, ¯w) + 2 ¯w¯z¯z/( ¯w¯z)2l e−keϕ(w, ¯w)w−kz¯w−k+l¯z,(3.14)that is ψl decomposes into a sum of solutions for the covariant operator S(2k+1)J−1Hwritten inthe patch V .Let us consider the chiral (i.e. such that ∂¯zφ(−k) = 0) solutions ofS(2k+1)J−1H· φ(−k) = 0.

(3.15)We have φ(−k)l= (J−1H )l(J−1H′)−k l = 0, . .

. , 2k.

Note that φ(−k)land ψl have the commonproperty of generating other solutions either by changing patch (eq. (3.14) for ψl) or, in thecase of φ(−k)l, by running around cycles.

For example, in the case k = 1/2, starting fromφ(−1/2)1, to get the other linearly independent solution φ(−1/2)2it is suﬃcient to perform a nontrivial winding around Σ.SinceS(2k+1)J−1H· (2ϕ¯z)2k+l e−kϕ = λl (2ϕ¯z)l−1 e(k+1)ϕ,l ∈Z,(3.16)whereλl = (2k + 1)(2k + l)(2k + l −1) . .

. (l + 1)l,l ∈Z,(3.17)it follows that the singlevalued diﬀerentialsψl = (2ϕ¯z)l−1e(k+1)ϕ,l ∈Z,(3.18)23

are eigenfunctions of Q(2k+1)ϕ¯zQ(2k+1)ϕ¯z· ψl = λlψl,l ∈Z. (3.19)Note that ψ−2k, .

. .

, ψ0 are the zero modes of Q(2k+1)ϕ¯z. Furthermore, eq.

(3.19) is invariantunder the substitution, ψl →Fψl, where F is an arbitrary solution of∂zF = 0. (3.20)Since the Liouville stress tensor satisﬁes the equations ∂z∂n¯z TF = 0, n = 0, 1, 2, .

. ., thegeneral solution of (3.20) depends on TF and its derivatives.

However, taking into accountpolymorphic diﬀerentials, the general solution of eq. (3.20) has the formF ≡Fψk, ψ′k, ψ′′k, .

. .,k = 1, 2,′ ≡∂¯z,(3.21)where ψ1 and ψ2 are two solutions of∂2¯z + 12TF ψ = 0,(3.22)such that ∂zψ1 = ∂zψ2 = 0 and ∂¯zψ1/ψ2̸= 0.The diﬀerentialsψchl=J−1Hl−1 J−1H′k+1 ,l ∈Z,(3.23)are the chiral analogous of ψl.

Indeed they satisfy the equationQch(2k+1)J−1H· ψchl= λlψchl ,l ∈Z,(3.24)where λl is given in (3.17). A property of S(2k+1)J−1HisS(2k+1)J−1H, ψ(n)= 0,(3.25)with ψ(n) a holomorphic n-diﬀerential.

This implies that the (generalized) harmonic Beltramidiﬀerentials satisfy the equationS(2k+1)J−1Hµ(2k+1)harm = 0,µ(2k+1)harm = ψ(2k+1)e−kϕ. (3.26)This equation will be useful in recovering the kernel of the chirally split anomaly arising inCFT.

Note that µ(3)harm is a standard harmonic Beltrami diﬀerential.The operator S(3)fBA, where∂zfBA = e−1BA,(3.27)with eBA a Baker-Akhiezer vector ﬁeld, appears in the formulation of the covariant KdVin higher genus [19,20]. The non holomorphic operators S(2k+1)fBA, k ∈Z+ deﬁne cocycles onRiemann surfaces [19].

Similar operators, not related to uniformization, have been consideredalso in [21] (see also [22]).24

3.3Normal Ordering On Σ And Classical Liouville ActionLet us now consider the meromorphic n-diﬀerentials ψ(n)jproposed in [23]. They are thehigher genus analogous of the Laurent monomials zj−n+.

By means of ψ(n)jwe can expandholomorphic diﬀerentials on Σ\{P+, P−}. Their relevance for an operator approach whichmimics the radial quantization on the Riemann sphere has been shown in [24].In terms of local coordinates z± vanishing at P± ∈Σ the basis readsψ(n)j (z±)(dz±)n = a(n)±jz±j−s(n)±(1 + O(z±)) (dz±)n ,s(n) = h2 −n(h −1),(3.28)where j ∈Z+h/2 and n ∈Z.

The (dz±)n term has been included to emphasize that ψ(n)j (z)transforms as ∂nz . By the Riemann-Roch theorem ψ(n)jis uniquely determined by ﬁxing oneof the constants a(n)±j(to choose the value of a(n)+jﬁxes a(n)−jand vice versa).

In the followingwe set a(n)+j= 1. There are few exceptions to (3.28) concerning essentially the h = 1 andn = 0, 1 cases [23].

The expression of this basis in terms of theta functions reads [24]ψ(n)j (z) = C(n)jΘI(z) + Dj;n|Ω σ(z)2n−1E(z, P+)j−s(n)E(z, P−)j+s(n),(3.29)whereDj;n = (j −s(n)) I(P+) −(j + s(n)) I(P−) + (1 −2n)∆,(3.30)and the constant C(n)jis ﬁxed by the condition a(n)+j= 1.Let us introduce the following notation for vector ﬁelds and quadratic diﬀerentialsek ≡ψ(−1)k,Ωk ≡ψ(2)−k. (3.31)Note that (3.28) furnishes a basis for the 1−2s(n) = (2n−1)(h−1) holomorphic n-diﬀerentialson ΣH(n) =nψ(n)k |s(n) ≤k ≤−s(n)o.

(3.32)In particular the quadratic holomorphic diﬀerentials areH(2) =nΩk+1−h0|k = 1, . .

. , (3h −3)o,h0 ≡32h.

(3.33)Let C be a homologically trivial contour separating P+ and P−.The dual of ψ(n)jisdeﬁned by12πiIC ψ(n)j ψk(n) = δkj ,(3.34)25

which givesψj(n) = ψ(1−n)−j. (3.35)By means of the operator S(2k+1)J−1Hwe deﬁne the quantityχ(2k+1)Fψ(−k)i, ψ(−k)j=124(2k + 1)πiIC ψ(−k)iS(2k+1)J−1Hψ(−k)j,k = 0, 1, 2, .

. .

,(3.36)that for k = 1 is the Fuchsian KN cocycleχ(3)F (ei, ej) =124πiIC12eie′′′j −e′′′i ej+ T F eie′j −e′iej,ej ≡ψ(−1)j. (3.37)Notice thatχ(2k+1)fjψ(−k)i, ψ(−k)j=124(2k + 1)πiIC ψ(−k)iS(2k+1)fjψ(−k)j= 0,∀i, j,(3.38)wherefj(z) =Z z hψ(−k)ji−1k ,for k = 1, 2, 3, .

. .

,fj(z) = ψ(0)j ,for k = 0. (3.39)An arbitrary KN cocycle has the formeχ(3) (ei, ej) = χ(3)F (ei, ej) +3h−3Xk=1akIC Ωk+1−h0 [ei, ej] ,h0 ≡32h.

(3.40)The cocycle eχ(3) (ei, ej) deﬁnes the central extension bVΣ of the h0 graded-algebra VΣ ofthe meromorphic vector ﬁelds {ej|j ∈Z + h/2}. In particular the commutator in bVΣ is[ei, ej] =h0Xs=−h0Csijei+j−s + t eχ(3) (ei, ej) ,[ei, t] = 0,(3.41)whereCsij =12πiIC Ωi+j−s[ei, ej].

(3.42)Two important properties of eχ(3) (ei, ej) are localityeχ(3) (ei, ej) = 0,for |i + j| > 3h,(3.43)and “time-independence”. Time-independence of eχ(3) means that the contribution to thecocycle is due only to the residue of the integral at the point P+ (τ = 0) or, equivalently atP−(τ = ∞) (here we are considering τ = et where t is the time parameter introduced byKrichever and Novikov which parametrizes the position of the contour C on Σ).26

Let us expand T F in terms of the 3h −3 holomorphic diﬀerentialsT F = {J−1H , z} = TΣ +3h−3Xk=1λ(F )k Ωk+1−h0,(3.44)where TΣ denotes the holomorphic projective connection on Σ obtained from the symmetricdiﬀerential of the second-kind with bi-residue 1 and zero α-periods (see [17] for the explicitexpression of TΣ).In the case of Schottky uniformization we haveT S = {J−1Ω, z} = TΣ +3h−3Xk=1λ(S)k Ωk+1−h0,(3.45)where JΩ: Ω→Σ, with Ω⊂bC the region of discontinuity of the Schottky group. Theconstants λ(F )kand λ(S)kare the (higher genus) Fuchsian and Schottkian accessory parameters.The Schottkian cocycle isχ(2k+1)Sψ(−k)i, ψ(−k)j=124(2k + 1)πiIC ψ(−k)iS(2k+1)J−1Ωψ(−k)j,k = 0, 1, 2, .

. .

,(3.46)that for k = 1 reduces to the Schottkian KN cocycleχ(3)S (ei, ej) =124πiIC12eie′′′j −e′′′i ej+ T S eie′j −e′iej. (3.47)The choice of the KN cocycle ﬁxes the normal ordering of operators in higher genus [23].In particular the normal ordering associated to χ(3)F (ei, ej) and χ(3)S (ei, ej) depends on theaccessory parameters λ(F )kand λ(S)krespectively.

On the other hand these parameters arerelated to S(h)clwhich denotes the Liouville action evaluated on the classical solution [12].To write down S(h) we must consider the Schottky covering of Σ.In this approachthe relevant group is the Schottky group G ⊂PSL(2, C). Let L1, .

. .

, Lh be a system ofgenerators for G of rank h > 1 and D a fundamental region in the region of discontinuityΩ⊂bC of G bounded by 2h disjoint Jordan curves C1, C′1, . .

. , Ch, C′h such that C′i = −Li(Ci).These curves correspond to a cutting of Σ ∼= Ω/G along the α-cycles.

The Liouville actionhas the form [12]S(h) =ZD d2z(∂zϕ∂¯zϕ + exp ϕ) −i2hXi=2ZCiϕ L′′iL′id¯z −L′′iL′idz!++ i2hXi=2ZCilog|L′i|2L′′iL′id¯z + 4πhXi=2log(1 −λi)2λi(ai −bi)2 ,(3.48)27

where ai, bi ∈bC are the attracting and repelling ﬁxed points of Li (it is possible to assumethat a1 = 0, a2 = 1 and b1 = ∞) while λi is deﬁned by the normal formLiz −aiLiz −bi= λiz −aiz −bi,0 < |λi| < 1. (3.49)We now quote the main results in [12].

The ﬁrst one concerns the quadratic holomorphicdiﬀerential Ω= T F −T S considered as a 1-form on Schottky space S. It turns out thatΩ= 12∂S(h)cl ,(3.50)where ∂is the holomorphic component of the exterior diﬀerentiation operator on S. Fur-thermoreλ(F )k−λ(S)k= 12∂S(h)cl∂zk,k = 1, . .

. , 3h −3,(3.51)where {zk} are the coordinates on S. Another result in [12] is12∂∂S(h)cl= −iωW P.(3.52)where ωW P is the Weil-Petersson 2-form on S. Furthermore, since ∂T S = 0, it follows that∂T F = −iωW P.(3.53)From the above results it follows that the diﬀerence between Fuchsian and Schottkianhigher order cocycles depends on the classical Liouville action.

In particular for the KNcocycle we haveχ(3)F (ei, ej) −χ(3)S (ei, ej) =148πi3h−3Xk=1∂S(h)cl∂zkIC Ωk+1−h0 [ej, ei] . (3.54)Similar relations hold for the Virasoro algebra on punctured Riemann spheres.

Eq. (3.54)clariﬁes how classical Liouville theory is connected with quantum aspects of operators deﬁnedon Riemann surfaces.

Let us notice that (3.54) can be generalized to the case of higher ordercocycles. Also in this case the diﬀerence between Fuchsian and Schottkian cocycles dependson the Liouville action.The investigation above solves the problem posed in [19] about time-independence andlocality of the cocycles deﬁned by covariantization.

This follows from the fact that, since thedivisor of the vector ﬁeld1J−1H′ is empty, the integrand in (3.36) has no poles outside P±.28

3.4Diﬀeomorphism Anomaly And The KN CocycleLet us now consider the chirally split form of the diﬀeomorphism anomaly [25]A(µ; e) + A(µ; e),(3.55)whereA(µ; e) =124πZΣ12e∂3zµ −µ∂3ze+ T (e∂zµ −µ∂ze)),(3.56)with e a vector ﬁeld. Here T denotes an arbitrary projective connection.

We now showthat A(µ; e) reduces to the KN cocycle. To do this we ﬁrst introduce some results on thedeformation of the complex structure of Riemann surfaces [24].

For a short introduction tothis subject and related topics see for example [26]; for more details see [27,28].To parametrize diﬀerent metrics we consider Beltrami diﬀerentials with discontinuitiesalong a closed curve. Let P+ be a distinguished point of Σ and z+ a local coordinate suchthat z+(P+) = 0.

Let us denote by Σ+ the disc deﬁned by z+ ≤1, and by A ⊂Σ+ anannulus whose centre is P+. Let Σ−be the surface deﬁned byΣ+ ∪Σ−= Σ,Σ+ ∩Σ−= A.

(3.57)We now perform a change of coordinatez+ →Z = z+ + ǫek(z+),z+ ∈A,ǫ ∈C,(3.58)with ek ≡ψ(−1)ka KN vector ﬁeld. Identifying the new annulus with the previous collar onΣ+ we get a new surface eΣ whose metric readsg(µk) = ρ(z, z)|dz + µkdz|2,(3.59)where the Beltrami diﬀerential isµk(P) =ǫ∂¯zek,if P ∈Σ+;0,otherwise.

(3.60)The KN holomorphic diﬀerentials Ωj form a dual basis with respect to µk.Indeedintegrating by parts we have1πZΣ Ωjµk = ǫδjk. (3.61)Since ek ∼z±k−h0+1±+ .

. ., it follows that for k ≥h0 we only change the coordinate z+,whereas eh0−1 (h0 ≡3h/2) changes z+ and moves P+.

For k ≤−h0 + 1, ek is holomorphic29

on Σ\{P+}, so eΣ is isomorphic to Σ because the variation induced in the annulus can bereabsorbed in a holomorphic coordinate transformation on Σ\Σ+. For |k| ≤h0−2 the vectorﬁeld ek has poles both in P+ and P−.

This change in Σ corresponds to an inﬁnitesimal modulideformation. Notice that the dimension of the space of these vector ﬁelds is just 3h −3.We are now ready to show that the anomaly A(µ; e) reduces to the KN cocycle.

Firstof all notice that by choosing (3.60) for the Beltrami diﬀerential in (3.56), the domain of thesurface integral (3.56) reduces to Σ+. Then we write A(µk; e) in the useful formA(µk; e) =ǫ24πZΣ+ev∂zv∂zv∂z∂¯zekv,(3.62)where v satisﬁes the equation12 v′v!2−v′′v = T,(3.63)that for T = T F has solutionv =1J−1H′.

(3.64)We now use the univalence of J−1H . Indeed this guarantees that the obstruction for thereduction of (3.62) to a contour integral around ∂Σ+ (which is homologically equivalent tothe C-contour in (3.34)) comes only from possible poles of e in Σ+.

As we have seen theunivalence of J−1Himplies the holomorphicity of S(2k+1)J−1H. It has been just this property ofS(2k+1)J−1Hwhich has suggested to write the integrand in A(µk; e) in the form (3.62).Since any diﬀeomorphism can be expressed in terms of the KN vectors ej, it is suﬃcientto consider A(µk; ej) instead of A(µk; e).

By the remarks above it follows that10A(µk; ej) = ǫ2χ(3)F (ej, ek),j ≥h0 −1. (3.65)For j ≤h0 −2, the vector ﬁeld ej has poles at z = P+ and A(µk; ej) can be expressed as alinear combination of KN cocycles.Note that the Wess-Zumino condition for A(µk; ej) corresponds to the cocycle identityfor χ(3)F (ej, ek).

On the other hand writing χ(3)F (ej, ek) in terms of theta functions (the explicitform of ek ≡ψ(−1)kis given in (3.29)) one should get some constraints on the period matrixfrom the cocycle identity that presumably are connected with the Hirota bilinear relation.Thus the Wess-Zumino condition for A(µk; ej) seems to be related to the Schottky problem.We do not perform such analysis here, however we stress that the cocycle condition forχ(3)F (ej, ek) involves, besides the period matrix, the Fuchsian accessory parameters.10Note that for T ̸= T F we have a similar relation.30

Eq. (3.65) suggests to deﬁne the higher order anomaliesAµ(2k+1)j, ψ(−k)i=124(2k + 1)πZΣ ψ(−k)iR(2k+1)J−1Hψ(−k)j,k = 0, 1, 2, .

. .,(3.66)withR(2k+1)f= (2k + 1)(f ′)k∂z(f ′)−1∂z(f ′)−1 .

. .

∂z(f ′)−1∂z∂¯z(f ′)k,(3.67)where the number of derivatives is 2k + 1. Notice that the generalized Beltrami diﬀerentialsareµ(2k+1)k(P) =ǫ∂¯zψ(−k)j,if P ∈Σ+;0,otherwise.

(3.68)Here the deformation of the complex structure of vector bundles on Riemann surfaces isprovided by the space of diﬀerentialsfH(k+1) =nψ(−k)j|1 −s(−k) ≤j ≤s(−k) −1o,(3.69)which is the dual space to H(k+1) deﬁned in (3.32).By construction higher order anomalies are related to higher order cocycles in a waysimilar to eq.(3.65). In this case we must consider W-algebras and the moduli space ofvector bundles on Riemann surfaces.

We notice that the explicit expression of the KN-diﬀerentials in terms of theta functions given in (3.29) provides a useful tool to investigatethis subject.We now show that in the case one uses the generalized harmonic Beltrami diﬀerentialsµ(2k+1)harm, j = ψ(k+1)je−kϕ,j ∈[s(k + 1), −s(k + 1)], s(k) = h/2 −(k)(h −1),(3.70)the anomalies (including the standard chirally split anomaly) vanishAµ(2k+1)harm, j, ψ(−k)i=124(2k + 1)πZΣ ψ(−k)iS(2k+1)J−1Hµ(2k+1)harm, j = 0,k = 0, 1, 2, . .

..(3.71)This follows simply becauseS(2k+1)J−1Hµ(2k+1)harm, j = 0. (3.72)4Virasoro Algebra On ΣHere we consider a sort of higher genus generalization of the Killing vectors.

This generaliza-tion follows from an investigation of the kernel of the KN cocycle. This analysis will suggesttwo possible realizations of the Virasoro algebra on Σ without central extension based onthe Poincar´e metric and J−1H .

Finally a higher genus realization of the Virasoro cocycleχkj = δk,−j(j3 −j)/12 is given.31

4.1Higher Genus Analogous Of Killing VectorsIn genus zero the KN cocycle reduces to χkj which vanishes for j = −1, 0, 1, ∀k. Thisreﬂects the SL(2, C) symmetry of the Riemann sphere due to the three Killing vectors.

Forh ≥2 do not exist chiral holomorphic −k-diﬀerentials, with k = 1, 2 . .

.. The reason is thatin this casedeg ψ(−k)j= 2k(1 −h) < 0,(4.1)so that ψ(−k)jhas at least 2k(h −1) poles.

Nevertheless by eq. (3.16) the cocycles have thefollowing propertyχ(2k+1)Fψ(−k)i, (2ϕ¯z)l e−kϕ= 0,∀i,k = 0, 1, 2, .

. .

,l = 0, . .

. , 2k.

(4.2)In particularχ(3)Fei, e−ϕ= χ(3)Fei, 2ϕ¯ze−ϕ= χ(3)Fei, (2ϕ¯z)2e−ϕ= 0,∀i. (4.3)Thus, in spite of the fact that for h ≥2 Killing vectors do not exist, the non-chiral vectors(2ϕ¯z)le−ϕ, l = 0, 1, 2, can be seen as their higher genus generalization.

Let us make someremarks on this point. The Killing vectors are the solutions of the equation∂¯zv = 0.

(4.4)In the case of the Riemann sphere we can choose the standard atlas (U±, z±) with z−= z−1+in the intersection U+ ∩U−. For the component of a vector ﬁeld v ≡{v+(z+), v−(z−)}, wehave v+(z+)(dz+)−1 = v−(z−)(dz−)−1, that is v−(z−) = −z2−v+(z−1−).

Therefore ifv+l (z+) = zl+,(4.5)then v−l (z−) = −z−l+2−and the solutions of eq. (4.4) are v0, v1, v2.

To understand what happenin higher genus, we ﬁrst note that besides eq. (4.4) these vector ﬁelds are solutions of thecovariant equationS(3)f· v = 0,f(z) ≡Z zv−10 .

(4.6)Indeed in U+ ∩U−eq. (4.6) reads∂3z+v+(z+) = z2−∂z−z2−∂z−z2−∂z−z−2−v−(z−)= 0,(4.7)whose solutions coincide with the solutions of eq.(4.4).

This relationship between the zeromodes of ∂¯z and S(3)fextends to the case of −k-diﬀerentials, k = 0, 1/2, 1, . .

.. In particular32

on the Riemann sphere the 2k + 1 chiral solutions of the equation S(2k+1)f· φ(−k) = 0, whereφ(−k) are −k-diﬀerentials, coincide with the zero modes of the ∂¯z operatorS(2k+1)f· φ(−k) = 0−→∂¯zφ(−k) = 0,k = 0, 12, 1, . .

. ,(4.8)whose solutions are φ(−k)l≡{φ(−k)+l, φ(−k)−l}, l = 0, 1, 2, whereφ(−k)+l(z+) = zl+,φ(−k)−l(z−) = (−1)kz2k−l−.

(4.9)The higher genus generalization of (4.8) readsS(2k+1)J−1H· φ(−k) = 0−→∂¯zφ(−k) = 0,k = 0, 12, 1, . .

. ,(4.10)whose solutions are11φ(−k)l=J−1HlJ−1H′k ,l = 0, 1, .

. .

, 2k. (4.11)Thus a possible choice for the higher genus analogous of the Killing vectors are the polymor-phic vector ﬁeldsφ(−1)0=1J−1H′,φ(−1)1= J−1HJ−1H′,φ(−1)2=J−1H2J−1H′ .

(4.12)Similarly to the case of the Killing vectors, φ(−1)0, φ(−1)1and φ(−1)2are zero modes for χ(3)F .More generallyχ(2k+1)Fψ(−k)i, φ(−k)l= 0,∀i,k = 0, 1, 2, . .

.,l = 0, . .

. , 2k.

(4.13)However if singlevaluedness is required we must relax the chirality condition and insteadof φ(−k)lwe must consider the non chiral diﬀerentials (2ϕ¯z)le−kϕ, l = 0, 1, . .

. , 2k.4.2Realization Of The Virasoro Algebra On ΣThe previous discussion suggests a higher genus realization of the Virasoro algebra.

We ﬁrstconsider two realizations of this algebra without central extension. In the ﬁrst case we have[Lj, Lk] = (k −j)Lj+k,Lk = (2ϕ¯z)k+1e−ϕ∂z.

(4.14)11However note that ∂¯zφ(−k)l= 0, ∀l.33

Similarly we can realize the centreless Virasoro algebra on Σ considering as generatorsthe polymorphic chiral vector ﬁeldsLchk =J−1Hk+1J−1H′∂z,Lchk =J−1Hk+1J−1H′∂¯z,(4.15)so thathLchj , Lchki= (k −j)Lchj+k,hLchj , Lchki= (k −j)Lchj+k,hLchj , Lchki= 0. (4.16)Observe that the holomorphic operators S(2k+1)J−1Hcan be expressed in terms of the abovegeneratorsS(2k+1)J−1H= (2k + 1)J−1H′k+1 Lch−12k+1 J−1H′k = (2k + 1)e(k+1)ϕL2k+1−1 ekϕ.

(4.17)The structure of the generators Lchk suggests the generalizationLk = vk∂z,vk(z) = f k+1(z)f ′(z) ,(4.18)with f(z) an arbitrary meromorphic function. In this case we can deﬁne the cocycleχ(vk, vj) =124πiIC0vkS(3)f vj = j3 −j12δk,−j,(4.19)where C0 encircles a simple zero of f. Thus we have[Lj, Lk] = (k −j)Lj+k + j3 −j12δk,−j.

(4.20)To deﬁne a cocycle depending only on the homological class of the contour we considerf = eR z ω,(4.21)with ω a 1-diﬀerential. A possible choice is to consider the third-kind diﬀerential with polesat P± and with periods over all cycles imaginaryω = ∂z log E(z, P+)E(z, P−) −2πihXj,k=1 ImZ P+P−ωj!Ω(2)jk−1ωk(z),(4.22)where Ω(2) denotes the imaginary part of the Riemann period matrix.

In this case one cansubstitute the contour C0 in (4.19) with the contour C in (3.34). However we notice thatthe integrand of (4.19) with f given in (4.21) is not singlevalued for j ̸= −k.

An interestingpossibility to investigate this aspect is to set f = fn,m where the fn,m’s are deﬁned in thenext section.34

5Liouville Field And Higher Genus Fourier AnalysisIn the standard approach to 2D gravity the Liouville ﬁeld is considered as a free ﬁeld.However there is a substantial hindrance in the CFT approach to Liouville gravity. Namely,since the metric g = eσˆg must be well-deﬁned, eσ must be an element of C∞+ .

If σ wereconsidered as a free scalar ﬁeld then the metric would take non positive values as well.One of the aims of this section is to investigate uniformization theory and then provide themathematical tools to face the problem of metric positivity in considering Liouville gravityon higher genus Riemann surfaces.A long-standing problem in uniformization theory is to express the uniformizing Fuchsiangroup in terms of the Riemann period matrix Ω. A possibility is to write J−1Hin termsof Ω.In this case, after going around non trivial cycles, the coeﬃcients in the M¨obiustransformation of J−1H are given in terms of Ω.

Unfortunately to write explicitly J−1H seemsto be an outstanding problem.These aspects are related to the problem of ﬁnding the eigenfunctions of the Laplacianon Σ. Actually, there is a strict relationship between the eigenvalues of the Laplacian andgeodesic lenghts.

These lenghts are related to the trace of hyperbolic Fuchsian elements. Away to investigate this argument is by Selberg trace formula.

However one should try toinvestigate the problem of uniformization in a more direct (analytic) way. Here we introducea new set of functions on Σ whose structure seems related to the uniformizing Fuchsiangroup.5.1Positivity And Fourier Analysis On Riemann SurfacesA possible way to construct functions in C∞+ is to consider the ratio of two suitable (p, q)-diﬀerentials.

For p = q = 1, besides the Poincar´e metric, we can use any positive quadraticform likePhj,k=1 ωjAjkωk. An alternative is to attempt to deﬁne the higher genus analogousof the Fourier modes.

In the following we adopt this approach.LetG = eg−g + eg−g2= cos (2 Im g) ,(5.1)be a function on a compact Riemann surface Σ. We will see that in order that G be anon trivial regular function in T 0,0, it is necessary that after winding around the homologycycles of Σ, the function g transforms with an additive term whose imaginary part be anon-vanishing element in πZ.

Such a multivaluedness is crucial for the construction of well-35

deﬁned regular functions on Σ. We will see that there are inﬁnitely many functions, labelledby 2h integers (n, m) ∈Z2h, with the properties of g whose existence is strictly related tothe positive deﬁniteness of the imaginary part of the Riemann period matrix.5.2Real Multivaluedness And Im Ω> 0We begin by considering the holomorphic diﬀerentialω(z) =hXk=1Akωk(z),A = a + ib,(a, b) ∈R2h,(5.2)where ω1, .

. .

, ωh, are the holomorphic diﬀerentials with the standard normalization (2.3).After winding around the cyclecq,p = p · α + q · β,(q, p) ∈Z2h,(5.3)the function f(z) = eR zP0 ω transforms intof(z + cq,p) = exp hXk=1(pk +hXl=1qlΩkl)Ak!f(z). (5.4)We constrain the multivaluedness factor in (5.4) to be real for arbitrary (q, p) ∈Z2h; that iswe require that the imaginary part of the exponent in (5.4) be an integer multiple of πhXk=1bkIαjωk = πnj,j = 1, .

. .

, h,nj ∈Z,(5.5)hXk=1akΩ(2)kj +hXk=1bkΩ(1)kj = πmj,j = 1, . .

. , h,mj ∈Z,(5.6)whereΩ(1)kj ≡Re Ωkj,Ω(2)kj ≡Im Ωkj.

(5.7)Thus after winding around αj we havef(z + αj) = exp(aj + iπnj)f(z),(5.8)whereas around βjf(z + βj) = exp" hXk=1akΩ(1)kj −πnkΩ(2)kj+ iπmj#f(z). (5.9)36

Eqs. (5.5,5.6) show an interesting connection between a fundamental property of Riemannsurfaces and the existence of regular functions with real multivaluedness.

Namely, for eachﬁxed set of integers (n, m) ̸= (0, 0), positivity of Ω(2) guarantees the existence of a non trivialsolution of eqs.(5.5,5.6). We haveak = det Ω(2;k)det Ω(2) ,bk = πnk,nk ∈Z,(5.10)where Ω(2;k) is obtained by the matrix Ω(2) after the substitutionsΩ(2)kj →π mj −hXl=1nlΩ(1)lj!,j = 1, .

. .

, h.(5.11)For practical reasons we change the notation of f and ωfn,m(z) = expZ zP0ωn,m,ωn,m(z) =hXk=1 det Ω(2;k)det Ω(2) + iπnk!ωk(z). (5.12)We now illustrate some interesting properties of the functions fn,m.5.3EigenfunctionsLet (n, m) ∈Z2h be ﬁxed and consider the scalar Laplacian ∆g,0 = −gz¯z∂z∂¯z with respectto the degenerate metricds2 = 2gz¯z|dz|2,gz¯z = |ωn,m|22A,(5.13)where A normalizes the area of Σ to 1 (see eq.(5.26)).

The functionsψk(z, ¯z) = 1√2 fn,m(z)f n,m(z)!k+ f n,m(z)fn,m(z)!k,k = 0, 1, 2, . .

.,(5.14)with (n, m) ﬁxed, are eigenfunctions of ∆g,0 with eigenvalues∆g,0ψk(z, ¯z) = λkψk(z, ¯z),λk = 2Ak2,k = 0, 1, 2, . .

. .

(5.15)Note that2A∞Xk=11λk= ζ(1) = π26 ,4A2∞Xk=11λ2k= ζ(2) = π490. (5.16)The orthonormality of the eigenfunctionsZΣ√gψjψk = δjk,(5.17)37

follows from the fact that |ωn,m(z)|2 exp kR z ωn,m −R z ωn,mis a total derivative.Let us notice that for k /∈Z the functionsf n,m/fn,mk are in general (see below)not well-deﬁned. This shows that arbitrary powers of well-deﬁned scalar functions can bemultivalued around the homology cycles.

In this sense the possible values of k in (5.14) areﬁxed by “boundary conditions”. This aspect should be taken into account in consideringoperators such as eαφ in Liouville and conformal ﬁeld theories.We stress that in general there are other eigenfunctions besides ψk, k ∈N.

For examplewhen all the 2mj’s and 2nj’s are integer multiple of an integer N then the eigenfunctionsinclude ψk/N, k ∈N whose eigenvalue is 2Ak2/N2. More generally one should investigatewhether the period matrix has some non trivial number theoretic structure.

For examplethe problem of ﬁnding the possible solutions of the equationωn′,m′(z) = c · ωn,m(z),(5.18)with both (n, m) and (n′, m′) in Z2h and c a (in general complex) constant, is strictly relatedto the numerical properties of Ω.On the other hand if eq. (5.18) has non trivial solutions (by trivial we mean c ∈Q) thenfn′,m′(z)f n′,m′(z) = ecR z ωn,m−cR z ωn,m ̸= fn,m(z)f n,m(z)!kc ∈C\Q,k ∈Q.

(5.19)Therefore∆g,0φc(z, ¯z) = 2A|c|2φc(z, ¯z),∆g,0 = −2A|ωn,m|−2∂z∂¯z,(5.20)whereφc(z, ¯z) = 1√2 fn′,m′(z)f n′,m′(z) + f n′,m′(z)fn′,m′(z)!. (5.21)The relevance of these functions resides in the non trivial number theoretic (chaotic) structureof the eigenvaluesλc = 2A|c|2.

(5.22)To understand this it is suﬃcient to write eq. (5.18) in the more transparent formm′j −hXk=1Ωjkn′k = c mj −hXk=1Ωjknk!,j = 1, .

. .

, h.(5.23)To each (n, m) and (n′, m′) satisfying (5.23) corresponds a possible value of c.38

Since (5.18) (or equivalently (5.23)) are equations on Ωone should think that they arerelated to the Hirota bilinear relations or equivalently to the Fay trisecant identity12. Aswell-known these must be satisﬁed by Ω(Schottky’s problem).

These remarks indicate thatthe aspects of number theory underlying the structure of Fuchsian groups should be relatedto integrable systems such as the KP hierarchy.The construction of the functions hn,m = fn,m/f n,m indicates that the solutions of theequation∂zψ(z, ¯z) = µ(z)ψ(z, ¯z),(5.24)with µ a holomorphic diﬀerential and ψ a smooth function are µ = ωn,m and ψ = hn,m. Itseems that equations such as (5.18) and (5.24) provide analytic tools to investigate aspectsof number theory, structure of moduli space, chaotic spectrum etc.5.4Multivaluedness, Area And EigenvaluesTo evaluate A we can use the Riemann bilinear relations2A = i2ZΣ ωn,m ∧ωn,m = −ImhXj=1Iαjωn,mIβjωn,m =hXl,k=1(akal + bkbl)Ω(2)lk ,(5.25)and by eqs.

(5.5,5.6)A = π2hXl=1(al ml −hXk=1Ω(1)lk nk!+ πhXk=1nlΩ(2)lk nk). (5.26)The multivaluedness of fn,m is related to A (the area of the metric |ωn,m(z)|2).

In particular,after winding around the cycle cn,−m = −m · α + n · β, we havePn,−mfn,m(z) = e−2Aπ fn,m(z),(5.27)where Pq,p is the winding operatorPq,pg(z) = g(z + cq,p). (5.28)12Recall that the Fay trisecant identity is the higher genus generalization of the Cauchy (= bosonization)formulaQi

Comparing (5.27) with (5.15) we get the following relationship connecting multivalued-ness, area and eigenvaluesλk = 1π logfn,m(z)fn,m(z + k2cn,−m). (5.29)Thus we can express the action of the Laplacian in terms of the winding operator.

This rela-tionship between eigenvalues and multivaluedness is reminiscent of a similar relation arisingbetween geodesic lenghts (Fuchsian dilatation) and eigenvalues of the Poincar´e Laplacian(Selberg trace formula).5.5Genus OneOne of the properties of the fn,m’s is that in the case of the torus the functionsφn,m(z, ¯z) = 1√2 fn,m(z)f n,m(z) + f n,m(z)fn,m(z)!,(n, m) ∈Z2,(5.30)coincide with the well-known eigenfunctions for the Laplacian −2∂z∂¯z. To prove this wechoose the coordinate z = x + τy with τ = τ (1) + iτ (2) the torus period matrix.

Eq. (5.10)givesa = π(m −nτ (1))τ (2),b = πn,(5.31)thus, choosing P0 = 0, we getφn,m(z, ¯z) =√2cos 2π(nx + my),(n, m) ∈Z2,(5.32)andλn,m = 2π2(m −τn)(m −τn)/τ (2)2,(n, m) ∈Z2.

(5.33)5.6RemarksLet us make further remarks about hn,m = fn,m/f n,m in (5.14). First of all note that inconsidering hkn,m (= hkn,km) as eigenfunctions of the scalar Laplacian, the indices (n, m) areﬁxed whereas in the case of the torus the eigenfunctions are hn,m with (n, m) running in Z2.Thus if we insist on using hkn,m with ﬁxed (n, m) also on the torus, then we will lose inﬁnitelymany eigenfunctions of the Laplacian −2∂z∂¯z.

Therefore, for analogy with the torus case, acomplete set of eigenfunctions in higher genus should be labelled by (n, m) ∈Z2h. Howeverthe structure of eq.

(5.23) suggests that there is a constraint on the values of (n, m) whichshould reduce the number of indeces to h.40

Unfortunately it is very diﬃcult to recognize a complete set of eigenfunctions in highergenus. This question is related to the problem of ﬁnding the explicit dependence of thePoincar´e metric eϕ on the moduli13.

The reason is that if in the torus case the complete setof eigenfunctions should reduce to {φn,m} then for analogy the Laplacian on higher genussurfaces must be deﬁnite with respect to the constant curvature metric, that is the Poincar´emetric. Really, each metric in the form g(p)z¯z = Phj,k=1 ωjA(p)jk ωk, with A(p) a positive deﬁnitematrix, reduces to the constant curvature metric on the torus.However det′ ∆g(p),0 anddet′ ∆g(q),0 are related by the Liouville action for the Liouville ﬁeld σ = log g(p)z¯z /g(q)z¯z in thebackground metric g(q).Notice that starting from the requirement of real multivaluedness, that is the “Diraccondition” (5.5,5.6), we end in a natural way with a set of functions with important prop-erties.

In particular, since this condition is the basic feature underlying the construction ofeigenfunctions in the case of the torus, it seems that the Dirac condition is a guidance toformulate Fourier analysis on higher genus Riemann surfaces as well. Thus the propertiesof the fn,m’s suggest that they are a sort of “building-blocks” to construct a complete set ofeigenfunctions for the scalar Laplacian of a well-deﬁned metric.

In particular the set of realfunctionsF =cos (2Imgn,m(z)) , sin (2Imgn,m(z))(n, m) ∈Z2h,gn,m(z) ≡Z zP0ωn,m,(5.34)resemble higher genus Fourier modes. Furthermore, by the analogy with the torus case, oneshould investigate whetherλn,m = 2hXl,k=1(akal + bkbl),(n, m) ∈Z2h,(5.35)are eigenvalues of the Laplacian with respect to some metrics.

Note that the term akal +bkblin (5.35) appears in the expression for the area of the metric |ωn,m(z)|2 (see (5.25)). On theother hand Ω(2)/ det Ω(2) reduces to 1 in genus 1, therefore still by analogy with the toruscase one should consider possible candidates for eigenvalues also the following quantitiesµn,m =2det Ω(2)hXl,k=1(akal + bkbl)Ω(2)lk = 4An,mdet Ω(2) ,(n, m) ∈Z2h,(5.36)13As we have seen this would be equivalent to ﬁnding the explicit dependence of J−1Hon the moduli of Σand then to solving long-standing problems in the theory of uniformization, Fuchsian groups etc..41

with An,m ≡A where the dependence of the area A on (n, m) is given in (5.26). Otherquantities that should be evaluated areZh(Ω) =Y(n,m)∈Z2h\(0,0)λn,m,(5.37)andeZh(Ω) =Y(n,m)∈Z2h\(0,0)µn,m,(5.38)that on the torus reduce to the determinant of the LaplacianZ1(τ) = eZ1(τ) = τ (2)2|η(τ)|4.

(5.39)It is possible to get some insight on Zh(Ω) and eZh(Ω) by investigating the behaviour of theλn,m’s and µn,m’s under pinching of the separating and non-separating cycles of Σ. Thiscan be done because the behaviour of the period matrix near the boundary of the modulispace is well-known. In particular the structure of the λn,m’s and µn,m’s seems to be suitableto recover the eigenvalues of the Laplacian on the torus in the “ﬁrst” component of theboundary of the compactiﬁed moduli space.

However we do not perform such analysis here.Another possible investigation concerning the results in this section is the analysis ofthe subspace of the diﬀerentials in T p,q made up of the scalar functions hn,m, hn,m suitablycombined with products of the KN diﬀerentials ψ(p)kand ψ(q)l .Going back to the construction of functions in C∞+ we notice that, considering the func-tions in the set F as Fourier modes on higher genus Riemann surfaces, the Liouville ﬁeldcan be expanded asσ(z, ¯z) =X(n,m)∈Z2han,mfn,mf n,m,an,m = a−n,−m. (5.40)6Liouville Action And Topological GravityIn this section we show that the classical Liouville action appears in the intersection numberson moduli space.

These numbers are the correlators of topological gravity as formulated byWitten [6, 7]. This result provides an explicit relation between topological and Liouvillegravity.42

6.1Compactiﬁed Moduli SpaceWe now introduce the moduli space of stable curves Mh, that is the Deligne-Mumfordcompactiﬁcation of moduli space. Mh is a projective variety and its boundary D = Mh\Mh,called the compactiﬁcation divisor, decomposes into a union of divisors D0, .

. .

, D[h/2] whichare subvarieties of complex codimension one.A Riemann surface Σ belongs to Dk>0 ∼= Mh−k,1 × Mk,1 if it has one node separating itinto two components of genus k and h −k. The locus in D0 ∼= Mh−1,2 consists of surfacesthat become, on removal of the node, genus h −1 double punctured surfaces.

Surfaces withmultiple nodes lie in the intersections of the Dk.The compactiﬁed moduli space Mh,n of Riemann surfaces with n-punctures z1, . .

. , znis deﬁned in an analogous way to Mh.

The important point now is that the puncturesnever collide with the node. Actually the conﬁgurations with (zi −zj) →0 are stabilized byconsidering them as the limit in which the n-punctured surface degenerates into a (n −1)-punctured surface and the three punctured sphere.Let us go back to the space Mh.The divisors Dk deﬁne cycles and thus classes inH6h−8(Mh, Q).

It turns out that the components of D together with the divisor associatedto [ωW P]/2π2 provide a basis for H6h−8(Mh, Q).The main steps to prove this are thefollowing. First of all recall that the Weil-Petersson K¨ahler form ωW P extends as a closedform to Mh [29], in particular [30][ωW P]2π2∈H2(Mh, Q),(6.1)which by Poincar´e duality deﬁnes a cycle DW P/2π2 in H6h−8(Mh, Q).

The next step is dueto Harer [31] who proved that H2(Mh, Q) = Q so that by Mayer-VietorisH2(Mh, Q) = Q[h/2]+2. (6.2)In [30] Wolpert constructed a basis of 2-cycles Ck, k = 0, .

. .

, [h/2] + 1 for H2(Mh, Q) andcomputed the intersection matrixAjk = Cj · Dk,j, k = 0, . .

. , [h/2] + 1,(6.3)where D[h/2]+1 ≡DW P/2π2.

The crucial result in [30] is that Ajk is not singular so that theclasses associated to Dk, k = 0, . .

. , [h/2] + 1 are a basis for H6h−8(Mh, Q).Let us now deﬁne the universal curve CMh,n over Mh,n.It is built by placing overeach point of Mh,n the Riemann surface which that point denotes.

Of course Mh,1 can be43

identiﬁed with CMh. More generally Mh,n can be identiﬁed with CnMh\{sing} whereCnMhdenotes the n-fold ﬁber product of the n-copies C(1)Mh, .

. .

, C(n)Mh of the universalcurve over Mh and {sing} is the locus of CnMhwhere the punctures come together.Finally we deﬁne KC/M as the cotangent bundle to the ﬁbers of CMh,n →Mh,n, it isbuilt by taking all the spaces of (1, 0)-forms on the various Σ and pasting them together intoa bundle over CMh,n.6.2⟨κd1−1 · · · κdn−1⟩Let Σ be a Riemann surface in Mh,n. The cotangent space T ∗Σ|zi varies holomorphicallywith zi giving a holomorphic line bundle L(i) on Mh,n.

Considering the zi as sections of theuniversal curve CMh,n we have L(i) = z∗iKC/M.Let us consider the intersection numbers [6,7]⟨τd1 · · · τdn⟩=ZMh,nc1L(1)d1 ∧· · · ∧c1L(n)dn ,(6.4)where the power di denotes the di-fold wedge product. Notice that, since c1L(i)is a two-form, ⟨τd1 · · · τdn⟩does not depend on the ordering and, by dimensional arguments, it maybe nonvanishing only if the charge conservation condition P di = 3h −3 + n is satisﬁed.Moreover, due to the orbifold nature of Mh,n, the intersection numbers will generally berational.Related to the τ’s there are the so-called Mumford tautological classes [32].

Let π :Mh,1 →Mh be the projection forgetting the puncture. The tautological classes areκl = π∗c1 (L)l+1=Zπ−1(p) c1 (L)l+1 ,l ∈Z+, p ∈Mh,(6.5)where L is the line bundle whose ﬁber is the cotangent space to the one marked point ofMh,1.

The κ’s correlation functions are ⟨κs1 · · · κsn⟩= ⟨∧ni=1 κsi, Mh⟩. To get the chargeconservation condition we must take into account the fact that integration on the ﬁbre in (6.5)decreases one (complex) dimension so that κl is a (l, l)-form on the moduli space.

It followsthat the nonvanishing condition for the intersection numbers ⟨κs1 · · · κsn⟩is Pi si = 3h −3.There are relationships between the κ’s and τ’s correlators. For example performing theintegral over the ﬁber of π : Mh,1 →Mh, we have⟨τ3h−2⟩=ZMh,1c1(L)3h−2 =ZMhκ3h−3 = ⟨κ3h−3⟩.

(6.6)44

To ﬁnd the general relationships between the κ’s and τ’s correlators it is useful to write⟨κs1 · · · κsn⟩in the following way [7]⟨κd1−1 · · · κdn−1⟩=ZCn(Mh) c1 ˆL(1)d1 ∧· · · ∧c1 ˆL(n)dn ,(6.7)where ˆL(i) = π∗iKC(i)/Mand πi : CnMh→C(i)Mh is the natural projection. Then noticethat CnMhand Mh,n diﬀer for a divisor at inﬁnity only.

This is the unique diﬀerencebetween ⟨κd1−1 · · · κdn−1⟩and ⟨τd1 · · · τdn⟩as deﬁned in (6.4) and (6.7). Thus it is possible toget relations for arbitrary correlators.6.3κ1 =i2π2∂∂S(h)clWe now show how the scalar Laplacian deﬁned with respect to the Poincar´e metric entersin the expression of the ﬁrst tautological class on the moduli space.

In order to do this weﬁrst introduce the determinant line bundles on the moduli space Mhλn = det ind ∂n. (6.8)They are the maximum wedge powers of the space of holomorphic n-diﬀerentials.

The linebundles λ1(≡λH) and λ2 are the Hodge and the canonical line bundles respectively.In [33] it has been shown that κ1 = ωW P/π2 thus, by standard results on ωW P, we haveκ1 = 6iπ ∂∂log det Ω(2)det′ ∆ˆg,0,(6.9)where ∂, ∂denote the holomorphic and antiholomorphic components of the external deriva-tive d = ∂+ ∂on the moduli space. Therefore the ﬁrst tautological class can be seen as thecurvature form (that is κ1 = 12c1(λH) in Mh) of the Hodge line bundle (λH; ⟨, ⟩Q) endowedwith the Quillen norm⟨ω , ω⟩Q = det Ω(2)det′ ∆ˆg,0,ω = ω1 ∧.

. .

∧ωh. (6.10)As we have seen the Liouville action (3.48) evaluated on the classical solution is a potentialof ωW P projected onto the Schottky space.

Thus in this spaceκ1 =i2π2∂∂S(h)cl ,(6.11)which provides a direct link between Liouville and topological gravity.45

7CutoﬀIn 2D Gravity And The Background MetricHere we apply classical results on univalent (schlicht) functions in order to derive an inequal-ity involving the cutoﬀof 2D gravity and the background geometry.7.1Background Dependence In The Deﬁnition Of The QuantumFieldAn important aspect arising in quantum gravity is the problem of the choice of the cutoﬀ. In2D quantum gravity a related problem appears when we consider the norm of the Liouvilleﬁeld σ deﬁned byg = eσˆg,(7.1)with ˆg a background metric that we suppose to be in the conformal form ds2 = 2ˆgz¯z|dz|2.The choice of the background is an important step as it deﬁnes the classical solution.

Toexplain this point more thoroughly, we ﬁrst consider the relationship between the scalarcurvaturesqˆg∆ˆg,0σ = √gRg −qˆgRˆg,(7.2)where∆ˆg,0 = −ˆgz¯z∂z∂¯z,(7.3)is the scalar Laplacian for the conformal metric. When Rg = cst < 0, σ = σcl of eq.

(7.2)is the solution of the classical equation of motion deﬁned by the Liouville action in thebackground metric ˆg. Thus both the solution of the equation of motion and the splittingσ = σcl[ˆg] + σqu[ˆg],(7.4)are background dependent.

The background dependence appears in the path-integral for-mulation of Liouville gravity where the measure Dˆgσ, deﬁned by the scalar product||δσ||2ˆg =ZΣqˆgeσ|δσ|2,(7.5)is not translationally invariant.Let us now choose the Poincar´e metric as backgroundˆgz¯z = eϕ2 ,(7.6)46

where ϕ is given in (2.44). Before investigating its role in deﬁning the quantum cutoﬀlet usnotice that sinceσcl[ˆg = eϕ] = 0,(7.7)the σ ﬁeld in (7.4) reduces to a full quantum ﬁeld and the Liouville action for σ, writtenwith respect to the background metric ˆg and evaluated on the classical solution, reduces tothe area of ˆg which is just the topological number −2πχ(Σ).7.2The CutoﬀIn z-SpaceIn [3] it was conjectured that the Jacobian that arises in using the translation invariantmeasure||δσ||2 =ZΣqˆg|δσ|2,(7.8)is given by the exponential of the Liouville action with modiﬁed coeﬃcients.

Argumentsin support of this conjecture may be found in [34]. Some aspects of this conjecture arerelated to the choice of the regulator.

We now show how the choice of the Poincar´e metricdˆs2 = eϕ|dz|2 as background makes it possible to ﬁnd an inequality involving the quantumcutoﬀand classical geometry. Including the Liouville ﬁeld σ we haveds2 = eφ|dz|2,φ = σ + ϕ,σ = σcl[eϕ] + σqu[eϕ] = σqu[eϕ].

(7.9)It is well-known that the cutoﬀin z space (∆z)2min is z-dependent(∆s)2min = eφ(∆z)2min = ǫ,(7.10)that is(∆z)2 ≥ǫe−σ−ϕ. (7.11)We stress that the cutoﬀarises already at the classical level.

As an example we considera Riemann sphere with n ≥3 punctures (we choose the standard normalization zn−2 =0, zn−1 = 1 and zn = ∞)Σ = C\{z1, . .

. , zn−3, 0, 1}.

(7.12)Near a puncture the Poincar´e metric has the following behaviourϕ(z) ∼−2log|z −z′| −2log|log|z −z′||. (7.13)Note that eϕ is well-deﬁned on the punctured surface: to deleting the point provides asort of “topological” cutoﬀfor ϕ which is related to the univalence of the inverse map ofuniformization.47

The topological cutoﬀis related to the covariance of the Poincar´e metric. To understandthis it is instructive to write down the Liouville action on the Riemann spheres with n-punctures [12]S(0,n) = limr→0 S(0,n)r= limr→0ZΣr∂zφ∂¯zφ + eφ+ 2π(nlogr + 2(n −2)log|logr|),(7.14)Σr = Σ\ n−1[i=1{z||z −zi| < r} ∪{z||z| > r−1}!,where the ﬁeld φ is in the class of smooth functions on Σ with the boundary condition givenby the asymptotic behaviour (7.13).

Eq. (7.14) shows that already at the classical level theLiouville action needs a regularization whose eﬀect is to cancel the contributions comingfrom the non covariance of |φz|2 /∈T 1,1 and provides a modular anomaly for the Liouvilleaction which is strictly related to the geometry of the moduli space [13].

This classical geo-metric context is the natural framework to understand the relationships between covariance,regularization and modular anomaly. In particular the relation between regularization andconformal weight in this framework is similar to the analogous relation which arises in CFTwhere the scaling behaviour is ﬁxed by normal ordering and regularization.

The fact thatclassical Liouville theory encodes a quantum feature such as regularization may be connectedto the fact that for the canonical transformation relating a particle moving in a Liouvillepotential to a free particle, the eﬀective quantum generating function is identical to its clas-sical counterpart [35] (no normal ordering problems). Furthermore, as we have seen, thelink between classical Liouville theory and normal ordering appears also in the analysis ofcocycles.7.3Univalent Functions And (∆z)2minThe correlators of the one dimensional string have the structure (see [36] for notation)G(p1, .

. .

, pN) = F(p1, . .

. , pN) + A(p1, .

. .

, pN)P ǫ(pi) + 2b ,(7.15)where the reason for the denominator, instead of the usual delta-function, is that the Liouvillemode represents a positively deﬁned metric. It seems that the boundaries of the space ofthe “half-inﬁnite” conﬁguration space are related to the inequalities that naturally appear inthe theory of univalent functions and in particular to their role in the uniformization theorywhich, as it has been shown in [12], is strictly related to classical Liouville theory.48

Let us consider a simply connected domain D of bC with more than one boundary point.The Poincar´e metric on D readseϕD(z,¯z) = 4|f ′D(z)|2(1 −|fD(z)|2)2,(7.16)where fD : D →∆is a conformal mapping. We are interested in the bounds of eϕD.

By anapplication of the Schwarz lemma it can be proved thateϕD(z,¯z)(∆Dz)2 ≤4,(7.17)where ∆Dz denotes the Euclidean distance between z and the boundary ∂D. The lowerbound iseϕD(z,¯z)(∆Dz)2 ≥1,(7.18)where now it is assumed that ∞/∈D.

By eq. (2.72) we can express the metric in terms ofthe Schr¨odinger wave functions satisfying (2.64) so that (7.18) reads∆Dz ≥ψψR ψ−2 −R ψ−22i.(7.19)Eq.

(7.18) follows from the Koebe one-quarter theorem [37] stating that the boundary of themap of |z| < 1 by any univalent and holomorphic function f is always at an Euclideandistance not less than 1/4 from f = 0. Thus if D is the unit disc and f(0) = 0 we have|f(z)| ≥1/4.

We stress that (7.18) resembles a sort of geometric uncertainty relation. Inparticular (7.17) and (7.18) can be considered as infrared and ultraviolet cutoﬀrespectively.Let us now consider the cutoﬀon D. By (7.11) and (7.17) it follows that(∆z)2 ≥ǫ4e−σD(∆Dz)2,(7.20)which relates the quantum cutoﬀto the background geometry.A related result concerns the Nehari theorem [38].

It states that a suﬃcient conditionfor the univalence of a function g ise−ψ|{g, z}| ≤2,|z| < 1,(7.21)whereas the necessary condition ise−ψ|{g, z}| ≤6,|z| < 1,(7.22)49

where eψ = (1−|z|2)−2. It can be shown that the constant 2 in (7.21) cannot be replaced byany larger one.

Eqs. (7.21,7.22) are inequalities between the Poincar´e metric and the modulusof the Schwarzian derivative of a univalent function which is related to the stress tensor.Acknowledgements: I would like to thank G. Bonelli, J. Gibbons and G. Wilson forstimulating discussions.

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출처: arXiv:9306.150 • 원문 보기

Int. J. Mod. Phys. A10 (1995) 289 – 336.

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