Wilson Loops on Riemann Surfaces, Liouville Theory and Covariantization of the Conformal Group

The covariantization procedure is usually referred to the translation operator, that is the derivative. Here we introduce a general method to covariantize arbitrary differential operators, such as the ones defining the fundamental group of a given manifold. We focus on the differential operators representing the sl(2,R) generators, which in turn, generate, by exponentiation, the two-dimensional conformal transformations. A key point of our construction is the recent result on the closed forms of the Baker-Campbell-Hausdorff formula. In particular, our covariantization receipt is quite general. This has a deep consequence since it means that the covariantization of the conformal group is {\it always definite}. Our covariantization receipt is quite general and apply in general situations, including AdS/CFT. Here we focus on the projective unitary representations of the fundamental group of a Riemann surface, which may include elliptic points and punctures, introduced in the framework of noncommutative Riemann surfaces. It turns out that the covariantized conformal operators are built in terms of Wilson loops around Poincar\'e geodesics, implying a deep relationship between gauge theories on Riemann surfaces and Liouville theory.


Introduction
It is well-known that in 2D space-time a pure gauge theory is locally trivial. In particular, by ∂ µ F µν = 0, it follows that the unique nontrivial component E 1 = F 01 is a constant and the finite energy condition fixes E 1 = 0. This is a consequence of the fact that the number of degrees of freedom of a pure gauge theory in D-dimensions is D − 2. Essentially, in topologically trivial 2D space-time any gauge configuration can be absorbed by a gauge transformation.
The situation is completely different in the case one considers non-trivial topologies. In particular, there are well-known models corresponding to a topological field theory with a strict relationship with string theory. Consider the case of a SU(N) (or U(N)) Yang-Mills theory on a genus g Riemann surface Σ where the trace is on the fundamental representation. Cutting Σ along a basis of its fundamental group π 1 (Σ), one gets the relation It turns out that the partition function can be expressed in terms of path integral of the traces of matrices of the gauge group associated to each generator of π 1 (Σ) and (see [1] and references therein) where the summation is over the irreducible representations of the group, A is the area of Σ, d R the dimension of the representation R and c 2 (R) the second Casimir operator of R.
The relation between connections and the fundamental group of Σ appears also in the context of noncommutative Riemann surfaces. This corresponds to consider unitary projective representations of the uniformizing Fuchsian groups. Such representations are obtained by covariantizing the differential operators of the sl 2 (R) algebra. Roughly speaking, whereas the projective unitary representation of the (abelian) group uniformizing the torus is simply obtained by setting U k = exp(λ k (∂ k + iA k )), with A a connection 1-form, in the case of uniformizing groups one needs to covariantize the generators of sl 2 (R). In other words, first one has to find the receipe to covariantize ∂ z , z∂ z and z 2 ∂ z and then finding the unitary operators U k such that g k=1 U 2k−1 U 2k U −1 2k−1 U −1 2k = e 2πiθ I . (1.2) This case corresponds to the one of hyperbolic Riemann surfaces. However, we will consider the general case that includes Riemann surfaces with elliptic point and punctures.
We will see that finding the unitary projective representation connects several questions, such as the one of the simultaneous covariantization, originally considered in our previous work [2] [3], that now are investigated in a systematic way and solved step by step. In our investigation, we will express the Möbius transformations in terms of the differential representation of sl 2 (R). This is done by first expressing and element of PSL 2 (R) as the composition of a translation, dilatation and a special conformal transformation. However, this has a drawback for our purposes. The reason is that the order of the above composition may change according to the kind of Möbius transformation one is considering. This is related to the important question of expressing U k as the exponential of a unique operator, conjugated by a functional F k of the connection A, associated to the geodesic defining the corresponding generators of the uniformizing group, namely The problem then is to find D k and the functional F k , which is the key object to covariantize D k . As we will see, such questions are in turn related to the problem of finding the explicit relation between the normal form of π 1 (Σ) and the uniformizing group Γ of Σ. We will find such a relation, that selects the corresponding form of the generators β k of Γ. As such, these generators satisfy the relation β 4g β 4g−1 . . . β 1 = I , (1.4) which is the case of compact Riemann surfaces of genus g (our investigation extends to the case with elliptic points and punctures). As we will see, all the above questions, unanswered in [2] [3], have a solution which is essentially unique.
The operator exp(D k ) performs the Möbius transformation β −1 k of the arguments of a function, so that where β k z := a k z + b k c k z + d k . (1.6) The F k 's are directly related to the Wilson loop associated to the geodesic connecting z and β −1 k z on the upper half plane H, namely As we said, the above construction is possible once one finds D k such that any Möbius transformation can expressed in the form exp(D k ). Such a question is equivalent to the problem of finding the closed form of W such that exp(X) exp(Y ) exp(Z) = exp(W ) . (1.11) The recent solution of such a problem is a key point of our construction. In particular, it turns out that our covariantization receipt is quite general. This has a deep consequence since it means that the covariantization of the conformal group is always definite. In [4], it has been introduced an algorithm to derive the closed form of W in (1.11) for a wide class of commutator algebras, classified in [5] and applied to all semisimple complex Lie algebras in [6]. The algorithm in [4], that extends the remarkable result by Van-Brunt and Visser [7] (see also [8] for related results), exploits the associative property of the Baker-Campbell-Hausdorff formula and implementing in it the Jacobi identity. In particular, it turns out that when X, Y and Z are elements of sl 2 (R), the corresponding commutator algebras is, according to the classification in [5], a subtype of the type 4. As a result, we will see that the Möbius transformation (1.6) is represented by the unitary operator where the parameters λ (k) ± and λ (k) j , j = −1, 01, are defined in terms of the components of the matrix β k . Usually, the covariantization procedure is considered only for the translation operator, that is for the ordinary derivative. The above covariantization is the one for the conformal transformations. To derive (1.12) it has been introduced a general method to covariantize much more general operators than ∂ z . Actually, our analysis starts with the torus, whose fundamental domain are straight lines, and these are generated by derivatives. The noncommutative torus leads to a covariantization of the derivatives that apparently cannot be extended to the higher genus case. However, we reformulate the noncommutative torus in a more geometrical way. It turns out that such a geometrical formulation is the natural framework to derive, in analogy with the case of the torus, the corresponding quantities for negatively curved Riemann surfaces. Such a strategy provides the correct prescription to define the covariantized operators and leads to (1.12). The above covariantization prescription can be extended to more general cases, including AdS/CFT. In particular whenever a manifold is naturally associated to a differential representation of its fundamental group. Here we focus on the projective unitary representations of the fundamental group of a Riemann surface, which may include elliptic points and punctures, introduced in the framework of noncommutative Riemann surfaces.
The organization of the paper is as follows. In section 2, after shortly reviewing the uniformization theorem, we express the Möbius transformations in terms of the exponentiation of differential operators representing sl 2 (R). In section 3 we express the action of such operators in the form of the exponential of a linear combination of the sl 2 (R) generators. This is a basic step to formulate the covariantization of the conformal group. In section 4 we first reformulate the noncommutative torus in a more geometrical form, and find a hidden symmetry. Next, in section 5, such a geometrical analysis will be used to formulate the unitary projective representation of the group uniformizing arbitrary Riemann surfaces. In particular, we will focus on the problem of simultaneous covariantization of the differential operators representing the sl 2 (R) generatoras and of their complex conjugate. This will lead to the covariantization of the conformal operators as given in (1.12). We will also solve the problem of finding the explicit relation between the normal form of π 1 (Σ) and the uniformizing group Γ of Σ. The extension to the nonabelian case is introduced in subsection 5.7. Section 6 is devoted to the properties of the gauge length as pseudo-distance and will show that it corresponds to a Poincaré area.

Differential representation of the conformal group
In this section, after shortly reviewing the uniformization theorem for Riemann surfaces, we express the PSL 2 (R) transformations, acting on the upper half-plane, in terms of the composition of three exponentiations of the generators of sl 2 (R), represented by the differential operators ℓ k = z k+1 ∂ z , k = −1, 0, 1. This will lead to consider two questions. The first is that such a prescription is not general, since the order of the composition depends on the specific Möbius transformation. The other question is related to the problem of expressing the unitary operators U k in the form of a unique exponential (2.1) As we will see, this is the natural way to define the covariantization of the conformal group. Such preliminary questions are solved in the next section, where we will derive the form of a Möbius transformation in the form of a unique exponential, whose argument is a linear combination of the ℓ k 's.

Uniformization and Liouville equation
In the following, given a 2 × 2 matrix we adopt the notation Let D be either the Riemann sphereĈ = C ∪ {∞}, the complex plane C, or the upper half-plane According to the uniformization theorem, every Riemann surface Σ is conformally equivalent to D/Γ, where Γ is a freely acting discontinuous group of fractional transformations preserving D. Let J H be the uniformizing map J H : H −→ Σ. It has the invariance property z ∈ H. By the fixed point equation γz = z, that is it follows that there are three kinds of PSL 2 (R) matrices.
The Poincaré metric on H is the metric with scalar curvature −1 This implies that the Liouville equation on Σ has the unique solution (2.4) The basic property of the Poincaré metric is that its isometry group PSL 2 (R) coincides with the automorphism group of H. The group Γ is isomorphic to the fundamental group π 1 (Σ).
If Γ uniformizes a surface of genus g with n punctures and m elliptic points with indices 2 ≤ n 1 ≤ n 2 ≤ . . . ≤ n m < ∞, then Γ is generated by 2g hyperbolic elements γ 1 , . . . , γ 2g , m elliptic elements E 1 , . . . , E m , and n parabolic elements P 1 , . . . , P n satisfying the relations The uniformizing group carries information both on the topological and the complex structures of the Riemann surface. It is easy to see that the number of parameters fixing the generators of a Fuchsian group coincides with the dimension of moduli space of complex structures of Riemann surfaces. For example, in the case of compact Riemann surfaces of genus g, the full set of generators depends on 6g real parameters which reduce to 6g − 3 upon using (2.5). On the other hand, (2.5) remains invariant under conjugation by an element of SL 2 (R) leading to 3g − 3 complex parameters.

Differential representation of sl 2 (R)
Let ℓ n , n = −1, 0, 1, be the generators of sl 2 (R). Fix their normalization by ). Thus, we set Note that [ℓ n , f ] = z n+1 ∂ z f . Each element of PSL 2 (R) can be expressed as the composition of a translation, dilatation and a special conformal transformation. These PSL 2 (R) transformations are If f (z) admits a convergent series expansion, then We note that this action can be defined by considering the formal Taylor theorem. This corresponds to equating the application of a formal exponential of a formal multiple of ∂ z with a formal substitution operation. For example, In this case f (z) is an arbitrary series, including the formal ones. In particular, this series may have the form n a n z n , with n which may also take complex values. Note that the formal expansion of exp(λ −1 ℓ −1 ), should not be confused with the standard meaning of formal series expansion, referring to non-convergent expansions. In particular, note that given an element X of a Lie algebra g, equipped with a norm || · ||, exp(X) is well defined on all C whenever ||X|| is finite.
Recall that µ j (µ k z) = (µ j µ k )(z) , and set h j (z) = exp(λ j ℓ j )z. Since it follows that exp(λ k ℓ k ) exp(λ j ℓ j ) acts in reverse order with respect to the matrix representation. This implies that the representation V constructed in terms of the above differential operators acts as have the homomorphism property is preserved. This fixes the representative of µ ∈ PSL 2 (R) to be where the λ k 's are the ones corresponding to the group element µ −1 .

Parameters of the representation
In order to derive the relation between the λ k 's and µ −1 , we note that by (2.6) Since AD − BC = 1, we have and Since a global sign is irrelevant for PSL 2 (R) matrices, we can choose the + sign on the right hand side of (2.10), so that The above decomposition holds when A = 0. Group elements not attainable by the decomposition (2.9) can be reached by alternative decompositions. A decomposition which holds when D = 0 is the reversed one and Choosing the + sign on the right hand side of (2.14), we have Finally, if both A and D vanish, we can choose where the sign ambiguity reflects the fact that this operator coincides with the one associated to the inverse matrix. In the following we will need the operator version of Eq.(2.8), that is

Note that the relation (2.5) is represented by
where we restricted to the hyperbolic case and 3 Parameterization of SL 2 (R) by the closed forms of the BCH formula We have seen that in order to express an element of SL 2 (R) in the form one has to distinguish three different cases, namely On the other hand, to find the covariant form of the conformal group, we must express (3.1) in the form for any choice of i, j and k. Very recently, have been derived new closed forms for the Baker-Campbell-Hausdorff (BCH) formula that include, as a particular case, the problem (3.3). Such a solution also solves the previous question, namely the one of writing down a unique expression, in the form on the right hand side of (3.3), holding simultaneously for the three distinguished cases (3.2). The reason is that any group element can be always expressed in the form on the right hand side of (3.3). Note that the unique exception would be if the trace of the corresponding matrix is −2 in the case when such a matrix is non-diagonalizable. On the other hand, we are interested in the linear fractional transformations where there is no any problem. The expression on the right of (3.3) is quite general. This has a deep consequence since it means that the covariantization of the conformal group is always definite. Let us further illustrate such a point, by considering X, Y and Z, elements of an arbitrary Lie algebra g. Suppose that it has been found a finite linear combination W of the generators of g such that For any suitable norm, the power expansion of exp(X) exp(Y ) exp(Z) converges on all C, except in the points where one or more of the norms ||X||, ||Y || and ||Z|| are singular. It follows that if (3.4) holds in a neighborhood of the identity, then it should hold in a wider region with respect to the one related to the expansion of ln(exp(X) exp(Y ) exp(Z)). Of course, this is related to the possible singularities of the norm of W . Remarkably, as observed in [4], this never happens for PSL 2 (C). The problem of finding the closed form of W in (3.4), has been recently considered in [4] where it has been introduced an algorithm that solves the BCH problem for a wide class of cases. In [5] it has been shown that there are 13 types of commutator algebras admitting such closed forms of the BCH formula. Furthermore, it turns out that the algorithm includes all the semisimple complex Lie algebras [6], as they correspond to particular cases of commutator algebras of type 1c-i, type 4 and type 5. It turns out that sl 2 (R) corresponds to a particular subtype of the type 4 commutator algebras.
Let us shortly show the main steps of the algorithm. In [7] Van-Brunt and Visser obtained a remarkable relation providing the closed form of the BCH formula in important cases. If X, Y ∈ g satisfy the commutation relations with I a central element and u, v, c, complex parameters, then [7] exp Such a result generalizes to a wider class of cases by exploiting the associativity of the BCH formula and implementing the Jacobi identity [4]. Consider the identity then, by (3.6), This solves the BCH problem since, by (3.5), (3.6), (3.8) and (3.10), it follows that imposing [X,Ỹ ] =ũX +ṽỸ +cI , (3.13) that fixes α,ũ,ṽ andc, gives (3.14) Note that, consistently with the Jacobi identity The Jacobi identity constrains e, m, n and p by a linear system. Furthermore, note that setting Y = λ 0 Q and λ − := λ 0 α, λ + := λ 0 β, Eq.(3.8) implies, as a particular case, In [4], as a particular case of the Virasoro algebra, it has been shown that the closed form of the BCH formula in the case of sl 2 (C) is Note that, as explained above, since the ℓ k 's are the differential representation of sl 2 (R), it follows that the corresponding Möbius transformations are the ones associated to the inverse matrix. In fact, one may check that the correspondence between the λ k 's and the matrix elements given in Eqs.(2.10)(2.11), is the same of the one in [4] after the change In the next sections we will construct the projective unitary representation of the fundamental group π 1 (Σ) by means of differential operators U k acting on the Hilbert space L 2 (H). This will lead to a relation for such operators which has the form of (2.5) except that the identity on the right hand side is multiplied by a phase. In the hyperbolic case we will have As a first step, here we consider the unitarity problem. In higher genus we will see the appearance of several new structures. For example, a distinguished feature concerns the combination of differential operators that one may use to construct the unitary operators. While on the torus the exponentials e ∂x 1 and e ∂x 2 both appear separately, for g > 1 the possible operators are restricted to a specific combination of In this section we will also reconsider the formulation of the noncommutative torus and, in particular, the way the phase e 2πiθ in (1.1) is obtained. The aim is to learn from the torus as much as possible, in order to reformulate the derivation in a way which extracts the features of the construction without referring to the specifics of the torus. The fact that the fundamental group in g > 1 is nonabelian implies that, in order to determine the phase in (4.1), we cannot use the reduced BCH formula which holds when A and B commute with [A, B]. For g = 1 the associated differential generators commute, i.e. [∂ 1 , ∂ 2 ] = 0, so that it makes sense to use (4.2) to evaluate the phase coming from the (constant) commutator of the covariant derivatives. For g > 1, a computation of the phase by means of the complete BCH formula would involve quantities which are a covariantization of the already noncommuting operators such as the generators of sl 2 (R). However, this is not only a technical difficulty, rather we actually still do not know which structures the covariantization of the generators of sl 2 (R) may have. As we said, reformulating the case of the torus in a different language will suggest its natural higher genus generalization, without using the complete BCH formula. In particular, it will shed lights on the covariantization procedure. As we will see, the result is deeply related to the geometry of Riemann surfaces. In particular, constructing the unitary operators that will projectively represent π 1 (Σ), will bring us to a problem that can be seen as the one of simultaneous covariantization. Essentially, this is the problem of finding in higher genus the covariant version of a given operator O in such a way that its adjoint has the form O † A = FÕF −1 , withÕ independent of the connection A and F a functional of A.

A first screening
Mimicking the case of g = 1, where each one of the two operators U 1 and U 2 are expressed in terms of real coordinates x ≡ x 1 and y ≡ x 2 respectively, one expects that the building blocks for the solution to the quotient conditions in g > 1 have two possible forms, either , where the L n 's are some covariantized sl 2 (R) operators to be determined. Such operators should be the generalization to the case of the three generators of sl 2 (R) of the covariant derivative. Since in the case of the torus the relevant phase is expressed by means of the commutator between covariant derivatives, and considering that L n contains ∂ z = ∂ x −i∂ y , one may expect that, in the case of higher genus Riemann surfaces, both exp(L n − L † n ) and exp(i(L n + L † n )) appear. However, while the exponentials exp(∂ x ) and exp(∂ y ) generate translations, that map C, the universal covering of the torus, to itself, the operators exp(L n − L † n ) and exp(i(L n + L † n )) should generate PSL 2 (R) transformations. On the other hand, exp(i(L n + L † n )) cannot generate real Möbius transformations, so that we should discard exp(i(L n + L † n )) and restrict to exp(L n − L † n ) only. This fact is strictly related to the nonabelian nature of the group π 1 (Σ) which, in turn, is related to the condition ℑ(z) > 0 defining H. The latter reflects the fact that the translation operator along the imaginary axis exp(∂ y ) alone does not belong to the automorphisms group of H. Since among the translation operators exp(∂ x ) and exp(∂ y ) only the former is allowed, we see that comparing exp(L n − L † n ) with exp(∂ x ) = exp((∂ z + ∂z)/2) one should expect that L † n corresponds to −L n . We will see that a slightly modified version of this holds. Finding the L n 's is a problem closely related to the one of deriving the central extension for the Fuchsian group without using the BCH formula. Since the L n 's are covariant versions of the three generators of sl 2 (R), in general the nested commutators should be difficult to treat, so that, apparently, one should use the complete version of the BCH formula. Nevertheless, since we will succeed in finding the central extension of the uniformizing group, this implies that the same result should be obtained by using the complete BCH formula. Therefore, in spite of (4.4), the structure of the sl 2 (R) implies a simplification. In particular, determining the c j 1 ...jn in 1 l,m=−1 should reveal a considerable simplification of the complete BCH formula for the case at hand.

An alternative to the BCH formula on the torus
Here we revisit the covariantization of the translation operators. These are naturally associated to the tessellation of the plane, so that their covariantization leads to consider the algebra of the noncommutative torus. For sake of simplicity, we consider the covariantization of translations operators along the Cartesian coordinates of R 2 , denoted by x 1 and x 2 . These operators are naturally associated to orthogonal tori. The investigation will lead to the computation of the phase in Eq.(1.1) without using the BCH formula. This alternative will indicate the way to covariantize the PSL 2 (R) operators, therefore providing the extension of our analysis to g > 1, where the direct use of the BCH formula is, apparently, inextricable. Let us consider the connection Given two operators A and B, and a function f (B) satisfying suitable properties, we have We define the functions F k (x 1 , x 2 ), k = 1, 2, by that compared with (4.6) yields The solution of this equation is where the contour integral is with x 2 fixed. Similarly where the contour integral is with x 1 fixed. In the following we will use the notation where in the integrand one has A 1 (a 1 , x 2 ) if k = 1 and A 2 (x 1 , a 2 ) if k = 2. There is an observation that simplifies considerably the construction. The key point is that since in the two contour integrals either x 1 or x 2 are fixed, so that either dx 1 = 0 or dx 2 = 0, it follows that both the integrands, da 1 A 1 and da 2 A 2 , can be replaced by the full connection A = A 1 dx 1 + A 2 dx 2 . Therefore, Since exp(λ k ∂ k ) is the translation operator, we have This allows for a very geometric derivation of the phase in (1.1).
Since we are investigating the covariantization of ∂ x 1 and ∂ x 2 , we consider tori whose fundamental domain F is a rectangle. Denote by λ 1 and λ 2 its base and height respectively. By (4.7) and Stokes' theorem where F is the curvature of A The above shows that the phase does not equal the curvature F . The fact that our derivation and the one made with the reduced BCH formula (4.2) coincide, is due to the fact that a possible solution is given by a constant F 12 . A possible choice to get a constant phase is to set F 12 = 2πθ/λ 1 λ 2 that corresponds to that by (4.2) and (4.6) gives that is (1.1). However, only in the case in which F 12 is constant does one have Let us now show why apparently the constant curvature connection is the unique solution.
Let us add the suffix x 1 x 2 to F in order to indicate its dependence on the base-point. Also, note that F is univocally determined by x 1 and x 2 . We need to show that the integral of F on F x 1 x 2 is independent of the point (x 1 , x 2 ). That is for any (x ′ 1 , x ′ 2 ) ∈ R 2 . Any point in R 2 can be obtained by a translation Let us apply the translation µ to the entire fundamental domain and denote it by µF x 1 x 2 . Since

so that Eq.(4.11) is satisfied only if
This fixes F to be a constant two-form, up to a non-constant contribution with vanishing surface integral. This implies a hidden invariance that we consider below.

The hidden invariance
Let us still consider tori with rectangular fundamental domain. We now show that the above investigation also allows to find an additional invariance of the operators under a suitable transformation of the connection A k . Such an invariance, which is not evident by analyzing the U k 's expressed in the form exp(λ k (∂ k + iA k )), is a consequence of the fact that exp(λ k ∂ k ) is a translation operator. By (4.7) we see that also U k is a translation operator Therefore, the U k 's have the invariance property whenever the h k 's satisfy To preserve unitarity of U k , we require |h k | = 1. Thus, the transformation of the connection leaves U k invariant.
In general (4.14) is not a gauge transformation, as each component transforms according to a different h k . Consequently, under (4.14) the curvature F transforms tõ Of course, this is consistent with the fact that the transformation (4.14) is not a gauge transformation unless h 1 = cnst h 2 . Above we saw how to remove the gauge connection A from the covariant derivative in U k . The result is that the gauge connection appears integrated along straight lines, that is geodesics with respect to the flat metric, the right language to extend the construction to higher genus. Upon factorization, the connection is acted on by the inverse of the differential operator (integration). This shows a close relation between integrals of the connection along geodesics, i.e.
x k +λ k x k A, and covariantized operators. More precisely, inverting (4.7) yields exp i

Unitary projective representations of the Fuchsian group 5.1 The problem of simultaneous covariantization
In the following we will see that, as expected, in higher genus it is convenient to use complex coordinates z andz rather than real ones x and y. Some of the operators we will work with have the structure exp(∇ z − ∇ † z ) , which is unitary by construction. The operators ∇ z and ∇ † z will be some appropriate covariantization of ∂ z and ∂z. Since we will also consider the covariantization of the sl 2 (R) differential operators, the noncommutativity of the latter will naturally lead to the method described in the previous subsections, rather than to the reduced BCH formula (4.2). The alternative derivation of the phase considered above may be applied to other cases if one can set 3 exp for some suitable function F (z,z). The fact that we need such simultaneous covariantization, i.e., that we need to express both the covariant derivative and its adjoint as standard derivatives conjugate by the same function, has already been used in the case of the torus. In that case we considered the action of unitary operators on the F k rather than the reduced BCH formula. It is clear that if we had, for example, ∇ z ∝ F ∂ z F −1 and ∇ † z ∝ F −1 ∂zF , then it would not be possible to extend the above method to g > 1. In that case Eq.(5.1) would not hold, and it would not be possible to get the phase on the right hand side of (4.1) by using the action of differential operators exp(∂ z + ∂z)F −1 (z,z) = F −1 (z + 1,z + 1) exp(∂ z + ∂z) .
As an example, one can consider differential operators acting on automorphic functions on the upper half-plane, or, what is equivalent, covariant operators acting on sections of line bundles on a Riemann surface Σ 4 . These operators can be seen as a sort of covariantization of ∂ z and ∂z. More precisely, consider the metric tensor g zz , so that ds 2 = 2g zz dzdz. The covariant derivative acting on K λ , where K is the canonical line bundle on Σ, is Formally, one can consider this as a suitable covariantization of ∂ z . The scalar product on It follows that the adjoint of ∇ λ In the literature there are also other examples of covariant differential operators on higher genus Riemann surfaces. For example one can consider covariant operators acting on sections of K µ ⊗K ν , and then take the dual space to be sections of K ρ ⊗K σ for some ρ and σ. In particular, for a suitable choice of the weights µ, ν, ρ and σ, one can obtain covariant operators of the form g λ zz ∂ z g −λ zz , with adjoints −g −λ zz ∂zg λ zz which, in a certain sense, exhibit more symmetry than (5.2) and (5.3). Nevertheless, also in this case we have ∂ z conjugate by g λ zz and g −λ zz , whereas ∂z is conjugate in the reverse order, that is by g −λ zz and g λ zz . Thus, an apparently unavoidable feature of covariant operators is that they never admit the simultaneous covariantization for some F . On the other hand, we have just seen that we need precisely the property that, given a covariant differential operator (in particular, an sl 2 (R) differential operator) and its adjoint, both of them should be expressed as a non covariant operator conjugate by F and F −1 in the same order. Let us further illustrate this point. In evaluating the adjoint operator, one performs an integration by parts and simultaneously takes the complex conjugate. Complex conjugation is the crucial point. In fact, if we now construct covariantized operators by conjugating them by F and F −1 , with F a phase, then complex conjugation corresponds to the inversion of F . Thus, the unique solution is to choosē and define Note that the condition |F | = 1 essentially follows also by requiring the unitarity of the operator in (5.1). Let us consider the scalar product integrating by parts, by (5.5) we have showing that the adjoint ∇ † z is constructed by conjugating ∂z in the same way as −∂ z is conjugate to get ∇ z ∇ † z = −F ∂zF −1 . Therefore, we found the way of covariantizing the derivatives as in (5.4). Later on we will apply the above method to the case of the sl 2 (R) differential operators. The above integration domain is H rather than the Riemann surface itself, as was the case previously. Our equations will be defined on the upper half-plane, as we are interested in constructing a projective unitary representation of the group π 1 (Σ) by means of operators acting on L 2 (H). In particular, the action of our operators will not be restricted to automorphic forms, which is the case when the equations are to be projected onto the Riemann surface. In this respect we now show that trying to perform a similar trick in the case of scalar products defined on a Riemann surface would lead to imaginary powers of the metric. Denoting by ψ (µ,σ) a section of K µ ⊗K σ , we see that where ρ, κ are real numbers. 5 The integrand in (5.6) is a (1, 1)-form, and the action of the derivatives is covariant, that is, they act on 0-differentials (e.g. ∂ z g −iκ zz ψ (iκ,1−ρ) ). Also, the complex conjugate of a (µ, ν) differential is a (ν,μ) differential. 6 Eq.(5.6) implies that the adjoint of As a consequence of the fact that both ∇ z and ∇ † z are obtained by conjugating ∂ z and −∂z by g iκ zz and g −iκ zz , we have that a function of any linear combination of ∇ z and of its 5 The construction can be generalized to the case in which the weight ρ is a complex number. The only difference consists in replacing ψ (iκ,1−ρ) by ψ (iκ,1−ρ) . One may also consider replacing iλ by a complex number µ, however in such a case g µ zz would not be a phase and simultaneous covariantization would not be possible. 6 Differentials have been studied in the literature with real [9][10] and complex weights [11].
adjoint has the property The appearance of the phase ties together several mathematical aspects which have a physical meaning. In particular, considers the operators (5.7) and (5.8) on the upper half-plane endowed with the Poincaré metric ds 2 = y −2 |dz| 2 = 2g zz |dz| 2 = e ϕ |dz| 2 , then one would obtain ∇ z = y −2iκ ∂ z y 2iκ and ∇ † z = −y −2iκ ∂zy 2iκ . In this respect, it is interesting to observe that the Poincaré Laplacian ∆ = −4y 2 ∂z∂ z , satisfies the equation ∆y where the eigenvalues are The problem of simultaneous covariantization led us to introduce imaginary powers of the metric. In turn, the associated Laplacian has eigenfuctions corresponding to complex powers of such a metric. On the other hand, the appearance of this complex power lies at the heart of the mass gap 1/4 in (5.9) which never appears in flat spaces such as in the case of the torus. Thus, there is a strict relationship between the noncommutativity of the fundamental group π 1 (Σ), the structures derived from imposing the simultaneous covariantization, and the structure of the Laplacian eigenvalues themselves. The mass gap 1/4 is in fact a sort of regularization induced by the negative curvature (which in turn is related to the nonabelian nature of π 1 (Σ)). In this context, it is worth mentioning that in [12] curvature brings the infrared and ultraviolet behavior of QCD under analytic control without any conflict with gauge invariance.

The unitary covariantized operators
Set e n (z) = z n+1 , n = −1, 0, 1, and define the operators L n = e 1/2 n ∂ z e 1/2 n = e −1/2 n ℓ n e 1/2 n = e n (∂ z + ∂ z ln e 1/2 n ) , that is This deformation of the ℓ n has no effect on the algebra, that is Let us define the covariantized operators where F (z,z) is an arbitrary function of unitary modulo Since the L with (5.10) one obtains ℓ † n = −ē −1 nl nēn . In the case n = −1 we have ∂ † z = −∂z, so that, since ∂ z = (∂ x − i∂ y )/2, we see that the construction reproduces the usual adjoint operation in the case of ∂ x and ∂ y . The basic property of the adjoint L (F ) † n is that it is obtained by conjugating −L n with F and F −1 . This means that L n and −L n are covariantized in the same way. This solves the aforementioned conjugation problem. Actually, the operator is the sum of L n andL n covariantized by means of the same conjugation A rigorous proof of unitarity goes as follows. Set where 7 w = e ℓn ze −ℓn = (a n z + b n )/(c n z + d n ), and note that that is ∂ z w = e n (w)/e n (z), so that g n /|∂ z w| 2 = 1/g n .
We then see that the operators e Λ (F ) n are unitary where in the first equality we used the fact that PSL 2 (R) is the automorphism group of H.

Selecting the Fuchsian generators
We now digress on the possible realizations of the fundamental relation (2.5) for a Fuchsian group in the hyperbolic case. Being differential operators, the V k in (2.17), have the property of acting in the reverse order with respect to the matrix product. This aspect raises a subtlety in considering the relationship between π 1 (Σ) and the uniformizing group Γ. Namely, let us consider the normal form for Σ. This is a polygon whose symbol is where {a k , b k } is a basis for π 1 (Σ). Cutting the surface along these cycles one obtains a simply connected domain whose vertices are connected by elements of the covering group.
If one considers this domain as sitting on the upper half-plane, then one can consider it as a fundamental domain with the transformations connecting the vertices given by elements of Γ. Let us order the vertices of the polygon in the counterclockwise direction and denote them by z = z 0 , z 1 , . . . z 4g−1 , z 4g = z. We denote this fundamental domain for Γ by with the vertices joined by geodesics. Note that since the geodesics are univocally determined it follows that the 4g-gon fundamental domain itself is univocally determined by the action of the Fuchsian generators on the base-point z.
A consequence of (5.13) is that the elements of Γ satisfy a similar fundamental relation. Among these, the one we wrote for the γ k in (2.5), is the canonical one, that is the one in which the generators appear in the sequence γ 2j−1 γ 2j γ −1 2j−1 γ −1 2j . This version is obtained using the identification Other representations can be found e.g. by Dehn twisting. 8 Note that these transformations leave Γ invariant and should not be confused with the ones obtained by conjugating Γ in 9 PSL 2 (R). Anyway, the usual representation for the fundamental relation satisfied by the generators of Γ given in (2.5) does not fit with the aim of our construction. Actually, what we essentially need is to provide a central extension of the Fuchsian group. In particular, we are looking for operators provinding a projective representation of Γ, such 8 These correspond to in general non simultaneous conjugation of Γ's generators by suitable strings of the generators themselves (see for example [13] [14]). 9 While the conjugation in PSL 2 (R) can be used to fix three real parameters, so that the number of real independent moduli reduces to 3 × ♯ generators −3 (due to the fundamental relation) −3 (due to conjugation)=6g − 6, the Dehn twists correspond to a discrete set of transformations whose existence implies the nontrivial orbifold structure of the moduli space of Riemann surfaces.
that the fundamental relation is modified by a phase. To discuss this aspect we first need to introduce the Fuchsian matrices β k defined by Since z 4g = z, we have that the β k satisfy the fundamental relation The associated operators providing a differential representation of Γ are 1 are defined in such a way that with [·] denoting the integer part. Note that where j = 0, 1, and k ∈ [1, g]. We can use these relationships between the β k to select 2g elements, say β 4k−3 , β 4k−2 , k ∈ [1, g] which can be seen as a complete set of generators for Γ. In particular, by (5.20) we can express β 4k−1 , β 4k , k ∈ [1, g] in the form

The Wilson-Fuchs operators
In (2.17) we introduced the operators V k which are defined in terms of the ℓ n . However, in order to construct unitary operators, we should use the L n 's rather than the ℓ n 's. Nevertheless, since the algebras of the ℓ n 's and L n 's coincide, we have that the commutation properties between the V k , and therefore fundamental relation (2.16), would remain invariant if the ℓ n in V k are replaced by the L n 's. Similarly the T k ≡ T (β k ) would satisfy the same fundamental relation under the replacement L n → ℓ n ,L n →l n . Let us now consider the operators T k . Since their action is in the reverse order with respect to the one of the matrix product, by (5.16) and (5.18) we have Let A be a U(1) connection. We set where the contour of integration is the Poincaré geodesic connecting z and w. Recall that Poincaré geodesics are semi-circles centered on the real axis. Semi-circles through ∞ correspond to straight lines parallel to the imaginary axis. In the following we will mainly be interested in the case in which w is dependent on z, in particular we will consider the function Let b be an arbitrary real number. Consider the Wilson loop and define the Wilson-Fuchs operators where the T k have been defined in (5.17). Each U k defines the function F k (z,z) solution of the equation Since by (5.18) we have We also note that since Eqs.(5.11) and (5.27) show that the U k 's are unitary operators

Covariantizing the generators of the uniformizing group
We now have all the ingredients to define the covariantization of the generators of the uniformizing group. We have seen how to covariantize a given operator of sl 2 (R). The problem now is the following. Let us write T k in the form Knowing D k allows one to define the covariant operator in the natural way, that is It follows that in order to find D k , and therefore D k , one needs to solve the BCH problem (5.31). On the other hand, as reviewed in section 3, the problem (5.31) belongs to the class of the new closed forms for the BCH formula derived in [4]. In particular, by (3.18), (5.27), (5.30), and the fact that, for any, the k Λ j,k 's satisfy the sl 2 (R) commutation relations, with the same normalization of the ones satisfied by the ℓ n 's (and L n 's), it follows that Let us explicitly rewrite D k in terms of differentials operators. We have 34) This is a basic result. Starting with the analogy with the case of the torus, we considered several issues, such as the problem of the simultaneous covariantization, that led to consider various differential representations of sl 2 (R), and then the problem of unitarity. This culminated with the use of the recent results on the BCH formula, that, finally, implied (5.34).

Computing the phase
Let us consider the following string of operators Moving the operators T −1 k on the right we obtain This imply that dA must be PSL 2 (R)-invariant. It is well known that, up to an overall constant factor, the Poincaré form is the unique PSL 2 (R)-invariant (1,1)-form. We choose the proportionality factor to be 1, that is has constant negative curvature that is, ϕ satisfies the Liouville equation This implies the following properties for A From √ gR = − 1 2 e ϕ and the Gauss-Bonnet theorem we have

Nonabelian extension
Up to now we considered the case in which the connection is Abelian. However, it is easy to extend our construction to the nonabelian case in which the gauge group U(1) is replaced by U(N). The operators U k now become the path-ordered exponentials where the T k are the same as before, times the U(N) identity matrix. Eq.(5.35) is replaced by A . (5.44) Given an integral along a closed contour σ z with base-point z, the path-ordered exponentials for a connection A and its gauge transform A U = U −1 AU + U −1 dU are related by [15] P exp i This implies that the only possibility to get a coordinate-independent phase is for the curvature (1,1)-form F = dA + [A, A]/2 to be the identity matrix in the gauge indices times a (1,1)-form η, that is F = ηI .
It follows that P exp −ib where, we recall, the contour integral is along the Poincaré geodesic connecting z and w. Let us denote by x 0 the center of this geodesic and ρ its radius. In polar coordinates we have z − x 0 = ρe iαz , w − x 0 = ρe iαw and dx/y = ρd cos α/ρ sin α = −dα, so that where α zw ≡ α z − α w , we have Note that in terms of z and w, we have which shows the dependence of α z (and α w ) on z and w. Also note that both α z and α w range between 0 and π, with the extremes corresponding to points on the extended real axis R ∪ {∞} = ∂H.

The gauge length as pseudo-distance
We now show that ℓ A (z, w) = |d A (z, w)| , (6.3) is in fact a pseudo-distance that we call gauge length. The symmetry property follows from the antisymmetry of d A (z, w) while the triangle inequality follows from the fact that ℓ A (z, w) = |α zw | or, equivalently, from the observation that ℓ A (z 1 , z 2 ) + ℓ A (z 2 , z 3 ) − ℓ A (z 1 , z 3 ) is the Poincaré area of the geodesic triangle with vertices z 1 , z 2 and z 3 . Since ℓ A (z, w) = 0 iff ℜ(z) = ℜ(w) , it follows that ℓ A (z, w) cannot be a distance. Note that Eq.(6.4), seen as an inequality involving angles, is similar to the one satisfied by the angles of triangles in Euclidean geometry. While in Euclidean geometry the Schwarz inequality is satisfied both by the angles and by the lenghts of the sides of triangles, in the case of hyperbolic geometry, the gauge length coincides with the angles themselves. Another property of this pseudo-distance is that, as we said, it has π as upper bound corresponding to the case in which the two points are on the real axis, so that ℓ A (z, w) < π , ∀(z, w) ∈ H 2 .

The gauge length as Poincaré area
The gauge length has some interesting properties which are worth mentioning. For example, while the geodesic distance between a point in H and one on the real axis measured with respect to the Poincaré metric is divergent, the corresponding gauge distance is finite. As a consequence, measuring the gauge distance between one point on a Riemann surface and a puncture on it gives a finite result. In particular, the greatest gauge length Denote by z k , k = 1, . . . , n, n ≥ 3, the vertices of an n-gon with angles α 1 , . . . , α n . Its gauge length is given by where z n+1 ≡ z 1 . Eq.(6.5) implies that the gauge length of an n-gon is PSL 2 (C)-invariant, that is ℓ This invariance can be also seen by observing that ℓ (n) A can be expressed in terms of cross ratios. In particular, for the geodesic triangle we have The invariance (6.6) is then a consequence of the PSL 2 (C)-invariance (µx 1 , µx 2 , µx 3 , µx 4 ) = (x 1 , x 2 , x 3 , x 4 ) , which together with the fact that any cyclic permutation corresponds to an involution (x 4 , x 1 , x 2 , x 3 ) = (x 1 , x 2 , x 3 , x 4 ) (x 1 , x 2 , x 3 , x 4 ) − 1 , constitute the main properties of the cross ratio.

Möbius transformations as gauge transformations
While on one hand Eq.(5.36) implies the equality on the other hand the Poincaré metric is PSL 2 (R)-invariant while A is not. As a consequence the variation of A under a PSL 2 (R) transformation can only be a total derivative. This is in fact the case, as A(µz, µz) = A(z,z) − i∂ z ln(cz + d)dz + i∂z ln(cz + d)dz . (6.7)