Riemann-Hilbert treatment of Liouville theory on the torus: The general case
Dipartimento di Fisica, Università di Pisa and
INFN, Sezione di Pisa, Largo B. Pontecorvo 3, I-56127
We extend the previous treatment of Liouville theory on the torus, to the general case in which the distribution of charges is not necessarily symmetric. This requires the concept of Fuchsian differential equation on Riemann surfaces. We show through a group theoretic argument that the Heun parameter and a weight constant are sufficient to satisfy all monodromy conditions. We then apply the technique of differential equation on a Riemann surface to the two point function on the torus in which one source is arbitrary and the other small. As a byproduct we give in terms of quadratures the exact Green function on the square and on the rhombus with opening angle in the background of the field generated by an arbitrary charge.
Liouville theory plays a pivotal role in several fields both at the classical [1, 2, 3, 4] and quantum level [5, 6, 7, 8, 9, 10, 11]. Liouville action provides the Faddeev-Popov determinant in two dimensional gravity  and it gives the complete hamiltonian structure of -dimensional gravity coupled with particles . It appears also in the soliton solutions to the gauged non linear Schroedinger equation on the plane [14, 15, 16]. Liouville field has been suggested to play the role of a fifth dimension in the AdS-CFT treatment of QCD .
Recently a renewed interest has developed due to a conjecture  that Liouville theory on a Riemann surface of genus is related to a certain class of , -dimensional gauge theories and the conjecture has been supported by extensive tests on genus and [19, 20, 21] and proven in a class of cases .
In the topology of the sphere important results regarding the four point conformal correlation functions have been obtained in  and their relation to the one point function on the torus suggested in  and proven in .
We recall that the classical solution provides the starting point for the semiclassical expansion and that such semiclassical expansion was used to confirm the first few terms of the bootstrap solution on the sphere, pseudosphere and disk topologies and also to give some results on higher point functions when one source is weak [25, 26, 27].
The simplest situation is given by the topology of the sphere for which many results are available both at the classical and quantum level. The torus topology is intrinsically more complicated that the sphere. For example the one point function on the torus appears to be of comparable complexity as the four point function on the sphere .
In a previous paper  Liouville theory on the torus was examined in the simpler situation in which the distribution of charges is invariant under reflection . As the map given by the Weierstrass function is invariant under reflection it turns out that one can map the problem on the Riemann sphere, where well developed techniques exist along with several results. In the symmetric situation, periodicity, which is the fundamental constraint, is obtained by imposing monodromicity in . In fact monodromic behavior in at the singular points points (see  and section 3 of the present paper) combined with reflection symmetry is equivalent to the periodicity constraint along the non collapsible cycles which in the general case is much more difficult requirement to implement.
In  the exact solution for the one point function on the square was given and several problems which can be dealt with perturbation technique solved.
In the present paper we deal with the problem in full generality i.e. when the distribution of charges is not necessarily invariant under reflections; to do this the shall need the full description of the torus as a double sheet cut-plane and we shall need the notion [29, 30, 31] of differential equation on a Riemann surface. As discussed in  the differential equation which solves the Liouville equation in the case of the torus contains even in the simplest case of the one point function a parameter, the Heun parameter which does not appear in the case of the sphere topology with three sources. Such parameter, along with an other weight parameter has to be determined by imposing monodromicity on the two sheet cut plane or equivalently by imposing the periodic boundary conditions. In the case of the square and of the rhombus with opening angle such parameter turns out to be zero and the problem is reduced to an hypergeometric equation (see Appendix B); on the other hand the exact non perturbative determination of the Heun parameter in the general case to our knowledge in not an accomplished task [32, 33, 34, 35].
The structure of the paper is the following: In section 2 we display the mathematical framework of the formulation of differential equations on a Riemann surface.
In section 3 we show with a group theory argument how the Heun parameter along with the weight parameter give the necessary and sufficient degrees of freedom to satisfy all the monodromy requirements.
Then in section 4 we solve the problem of the addition of a small charge when the solution for a single (non necessarily small) charge is given, as it is the case of the square and of the rhombus with opening angle . The treatment however is completely general.
As a by product we obtain is section 6 the expression by quadratures of the exact Green function on the background generated by an arbitrary charge. The expression holds also for the general torus; the only difference is that while for the square and the rhombus of opening angle we know the expression of the unperturbed solution, for the generic torus such unperturbed solutions are not known in terms of usual functions.
In order to display the workings of the technique discussed in sections 2 and 3, in Appendix A we give the perturbative determination of the conformal factor for a weak source in the general case i.e. when the source is not symmetric wrt to the position of the kinematical singularities which arise in the transition from the global covering variable to the coordinates which provide a one-to-one description of the torus.
In Appendix B for completeness we report the solution for the square given in  and we also add the solution in terms of hypergeometric functions for the rhombus of opening angle . The deformation technique developed in  can be applied to both of these cases.
2 Differential equations on a Riemann surface
We give here the elements of the theory of differential equations on a Riemann surface. For more details see e.g. [29, 30, 31]. A surface with the topology of the sphere can be mapped on the compactified plane where the usual theory of linear differential equations apply (see e.g. [36, 37, 38, 39, 40]). To extend the concept of differential equation to a Riemann surface it is useful to rewrite the second order differential equation
in the form
and we have denoted in eq.(1) by a prime the differentiation wrt . For well known topological reasons when the manifold has genus or higher we have no global coordinate at our disposal and thus we must use more than one chart with analytic transition functions.
It is useful to consider as a differential of the fractionary order , i.e. in the change from the variable to the variable we have
One easily finds, denoting with a dot the derivative wrt
The usual transformation of a connection gives
The term is the Schwarzian derivative of the transformation
It is of interest to notice that the assumed nature (5) of the solutions of eq.(1), which leaves the Wronskian of two solutions unchanged, makes the well known expression  for the conformal factor in terms of two independent solutions of the differential equation (1)
a differential of order as required for an area element
One can extend the concept of Fuchsian differential equation to differential equations on a Riemann surface provided the term or better the component of the connection is a meromorphic function on the Riemann surface i.e.  the ratio of two polynomials of the same order in the homogeneous coordinates describing the surface.
3 The torus
The torus is described by the cubic curve in euclidean coordinates 
or in homogeneous coordinates
To the Liouville equation
there corresponds the differential equation in given by
with a meromorphic function on the torus with second order poles at due to the presence of the Schwarz derivative in (10) and a second order pole at
where the appearing in the residue of the double pole at is given by . Due to the factor the pole is present only on the first sheet. The term is necessary to assure that does not show an irregular singularity at infinity. The are the accessory parameters. The above structure can be trivially generalized to any finite number of sources. Obviously we could place the source at simplifying the structure of as was done in  but in the perspective of the general non symmetric situation (see e.g. section 4) we work here in full generality.
The first two lines in the expression of is simply the contribution of the Schwarzian derivative for the transition from the covering variable to the variable . The asymptotic behavior of in the local covering variable at infinity , with is given by
at i.e. absence of charges at implies
and thus at the end we have one free accessory parameter, say which has to be used to satisfy the monodromies at and along the two fundamental non contractible cycles. We want to show at the non perturbative level how one free accessory parameter is sufficient for the purpose.
Let and be two independent solutions of
of Wronskian . We must find combinations
is monodromic. As the Wronskian is fixed the transformation must be of type, which corresponds to six real degrees of freedom. In addition we notice that the monodromy of (27) is unchanged under an transformation which corresponds to three real degrees of freedom. Thus the provide effectively only three real degrees of freedom to which we have to add the real and imaginary part of , i.e. real degrees of freedom. We shall have to satisfy the monodromy condition at all singularities , and along the two fundamental non contractible cycles and .
With regard to the cycle we can use three degrees of freedom (e.g. those given by the ) to obtain
which is sufficient to give the nature of the transformation . Finally we use the remaining two real degrees of freedom (the real and imaginary part of ) to impose
This is not enough to give the nature of the transformation . However from the contour shown in Fig.1, obtained by deforming a loop around , the following relation holds
being the elliptic transformation around . The reason is that the monodromy transformations around each of the singular points and is simply , because the indices at those points are in and at and a turn in the local covering variable corresponds to a double turn in and in at infinity. Now the following simple theorem holds: If in eq.(30) and eq.(29) is satisfied and is elliptic, then and as a consequence also .
The positive number is fixed by the elliptic nature of the product (30) i.e.
Eq.(33) has the discrete solution . In general is not the only solution of eq.(33). The results of Picard [1, 3] which apply also to topologies other than the sphere, assure us that the values of the free parameters which realizes the monodromy are unique and thus is the only solution which realizes all the monodromies. In the next section we shall apply a perturbative version of this theorem to the addition of weak sources.
4 Addition of a weak source
In the present section we shall apply perturbation theory to give in terms of quadratures the two point conformal factor when one source is arbitrary and the second small. A similar problem was solved in  in the simpler situation of the perturbation provided by two weak symmetrical sources and which could be treated with methods of the sphere topology. Here instead being the situation non symmetrical we have to exploit in full the two sheet representation of the torus given by the cubic (14). We shall keep the treatment at full generality; we recall that in two instances ( the square, and the rhombus of opening angle as given in Appendix B) the exact solution is known and thus we shall for a perturbation of these situations have a solution in terms of quadratures. The same formulas apply starting from the general non perturbative one point function with the difference that in this case the unperturbed functions are given by the solution of the Heun equation, at a special value of the Heun parameter for which we do not possess an explicit formula.
The equation to be solved is
where , describing a source of arbitrary strength at the origin i.e. , is given by 
with the non perturbative Heun parameter fixed to the value which provides the monodromic one point solution on the torus. We know the exact value of such a parameter only for the special cases of the square and the rhombus with opening angle where for symmetry reasons. We shall denote by and the two unperturbed solutions, with real Wronskian , which realize the monodromic conformal factor
The perturbation is given by
and it describes the additional weak source at . We had to allow in (38) for a new accessory parameter at and the are the changes of the accessory parameters at of eq.(4). Again they are subject to two Fuchs relations which are imposed by the condition that the behavior of at remains unchanged, i.e. the source at the origin () remains unchanged. They are
The analogue of eq.(24) here is absent as now the origin is a singular regular point. Thus we are left with two free accessory parameters, say and . The perturbed solutions are given by
and the constants correspond to the addition of the unperturbed solutions and they satisfy to leave the Wronskian unchanged. As we have already discussed in section 3 if one factors the transformations the provide only three real degrees of freedom. We shall first examine the monodromy at the new singularity . Taking into account that the integrands of contain a first and second order pole at whose residues are
we have that the change of such integrals under a tour around is
The monodromy matrix of and at is given by
Monodromy at requires
It is worth noticing that the is fixed independently of all the other parameters. As the transformation at is elliptic i.e. real, the condition (47) is sufficient to assure that the transformation belongs to . On the other hand can be explicitly proven using the expression of as follows. Eq.(47) tells us that
which, being and real gives
i.e. . We add now a remark which will be essential in the following development. If we consider a contour , see Fig.2, which embraces both the origin and the weak singularity at the resulting monodromy is elliptic. More precisely the product of the perturbation of the original unperturbed elliptic transformation at and the monodromy near the identity at is elliptic. In fact
is still elliptic because the source at the origin is unchanged and ellipticity tells us that and the elliptic monodromy near the identity can be written as
with and , and all of order . Their product has the diagonal elements
with trace real and by continuity, of modulus less than 2. Thus is elliptic.
We can now deform the contour which embraces the origin and as shown in Fig.2.
Around the singular points we have
due to the cancellation of the contributions given by the integrals . In the local covering variable we have
so that for a tour around the singularity i.e. we have and and thus the monodromy matrices at are all . Thus we have
We can now repeat the argument given in section 3 to conclude that the imposition of the nature of and the relation is sufficient to assure the nature of transformations . As a consequence and being already we have also .
5 Determination of and
We come now to the determination of the parameters and . The perturbed is
We denote with and the solutions computed at the same point on the torus but reached through a cycle
We have for the unperturbed problem
and for the perturbed one
giving the constancy of .
Writing , for the cycle the relation gives
The are of the form
We see that only the combinations
appear in such relation. This, as discussed in section 3, is due to the remnant invariance of the monodromy conditions. Thus the previous is a system of two linear non homogeneous equation in , , and , .
The condition gives
which is one inhomogeneous linear equation.
To this we must add the relation which reads
which is an other system of two linear inhomogeneous equations.
The equality of the real parts of (64) and (70) and the vanishing of the imaginary part of (64) and (70) and (69) provide a linear inhomogeneous system of four equations in the unknown , , , . The outcome substituted into (64) provides .
6 The Green function on the one-point background
Taking the derivative wrt of the equation
satisfied by the we computed in the previous section we obtain the exact Green function on the one-point background. In fact we have
can be simply computed by taking the derivative of obtained in the previous section from and , as the logarithm of the Jacobian appearing in the transition from to does not depend on . We have adopting the choice (we recall that the previous difference never can vanish)