Complete conformal field theory solution of a chiral six–point correlation function
Using conformal field theory, we perform a complete analysis of the chiral six-point correlation function
with the four operators at the corners of an arbitrary rectangle, and the point in the interior. We calculate this for arbitrary central charge (equivalently, SLE parameter ). is of physical interest because for percolation () and many other two-dimensional critical points, it specifies the density at of critical clusters conditioned to touch either or both vertical ends of the rectangle, with these ends ‘wired’, i.e. constrained to be in a single cluster, and the horizontal ends free.
The correlation function may be written as the product of an algebraic prefactor and a conformal block , where , with a cross-ratio specified by the corners ( determines the aspect ratio of the rectangle). By appropriate choice of and using coordinates that respect the symmetry of the problem, the conformal block is found to be independent of either or , and given by an Appell function.
The methods of conformal field theory (CFT) [1, 2] allow calculation of the correlation functions of a variety of operators, which may, in many cases, be interpreted as physical quantities in the continuum limit of two dimensional critical systems. As the number of operators grows, the calculation becomes progressively more difficult, however, and in practice there are very few results for correlators with more than four operators.
In this paper we present a full calculation of the six-point correlator
in the rectangular geometry . The conformal dimensions used through out this article and the associated central charge are
where is the Schramm-Loewner Evolution (SLE) parameter. Because of their positions the corner operators have an effective dimension , i.e. twice the usual value (see subsection II.2 for more details on this point).
is of interest because it may be used in a variety of physical models to determine the density of clusters attached to one, or both, of two distinct boundary intervals, when these intervals are each ‘wired’, i.e. constrained to belong to a single cluster. A recent paper  presents a calculation of for percolation () in a semi-infinite rectangle, and also in an arbitrary rectangle by assuming a certain –independence specific to the rectangle.
Here, we calculate completely and without assumptions, for arbitrary . This is done by solving the differential equations that satisfies. The main new step is a certain choice of coordinates. This choice allows us to derive the curious –independence mentioned, to reduce the number of variables from three to two, and write the solutions explicitly in terms of Appell functions. The application of these results to a variety of physical systems is considered in .
To begin, we express in the form
where is an appropriately chosen algebraic prefactor. With this choice it transpires that for a given rectangle, all possible solutions for can be written in the form , where is determined by . Then, using certain elliptic functions of and as intermediate variables, we find an algebraic expression for , and algebraic factors times Appell hypergeometric functions for . The full expressions for and depend on the aspect ratio of the rectangle as well as and/or .
The prefactor is independent of the details of the physical system, while changes depending on the particular observable associated with . For a conformal block of (1), only depends on or . This surprising feature, originally observed numerically (see ), indicates the presence of some unknown symmetry. It also implies that in a given rectangle appropriate ratios of two with different physical meanings are completely independent of either or , since the prefactor cancels out.
In a companion paper  we apply these results in several ways. First, to find the cluster densities for a range of critical loop models, in both dense and dilute phases and equivalently, for critical -state Potts models, probing either FK or spin clusters. Second, we extend previous results for the factorization of correlations for percolation, described in , to the critical models mentioned. Finally, for percolation, the density of horizontal crossing clusters in a rectangle with open boundary conditions on all edges is determined.
In this work, section II calculates the correlation function (1) by solving the associated PDEs, which with proper choice of co-ordinates and prefactor reduce to the Appell equations. Subsection II.1 gives the PDEs, choses and coordinates, and presents the solutions for , all of which are single conformal blocks. The interesting independence of from one coordinate mentioned (see (15)) appears here. Subsection II.2 computes the form of the correlation function prefactor in the rectangle, a not completely trivial task. Section III contains a summary of our results and some discussion.
ii.1 Solving the differential equations
In this subsection we determine and solve the differential equations for the correlation function (1). The main new step in solving the differential equations is a certain choice of coordinates, given below. This choice allows us to derive the interesting –independence mentioned, and to write the solutions in terms of Appell functions.
To begin, we consider in the upper half plane
using the methods of boundary CFT, then transform into the rectangle . In we can decompose into chiral components . Then, by conformal symmetry one may write
where . In the next section we give our reason for choosing this particular form for .
Using standard CFT methods  may be determined via the differential equations arising from the null state associated with each in (5). The presence of , for instance, means that (5) is annihilated by the operator
Next we let , which means that under the conformal map to the rectangle (8) is the cross-ratio of the image points of the corners of the rectangle. (Note that differs from the standard modular lambda parameter, which is here.) Thus one arrives at a differential equation for ,
We next transform (7) into rectangular coordinates via the conformal mapping
where , with the complete elliptic integral of the first kind, and the Jacobi elliptic function with elliptic parameter . The factor appears because our has fixed height of , which differs from the standard rectangle used to define the Jacobi elliptic functions. The aspect ratio is given by
which is the inverse of the standard elliptic aspect ratio. Conversely, the aspect ratio specifies the elliptic parameter via
The transformation (8) maps the corners of the rectangle , starting at the origin and proceeding counterclockwise, into the -plane points and , respectively.
We now introduce the real coordinates
This choice of co-ordinates is a key step, as we will see. It simplifies the equations, leading to a solution of the PDEs. In addition, our results are either algebraic or hypergeometric when written in terms of and .
The conformal transformation now becomes
The coordinate , to within a factor , is the half-plane image of the projection onto the bottom edge of the rectangle, i.e. so that . The coordinate is slightly more complicated and is determined by first taking the projection on the left end of the rectangle. The half-plane image of this point is in the interval , and we define via so that . We will see in the following paragraphs that this choice of coordinates introduces a useful symmetry and allows us to compare directly with results from .
Transforming (7) into these coordinates (with the help of Mathematica) yields
Three additional equations are derived by cyclicly permuting the indices on the variables in (6) and following the steps above. These three additional equations can also be obtained from (13) by the conformal symmetries of rectangles with arbitrary and fixed height : reflection about , reflection about , and reflection over with a concurrent scaling by in order to preserve the height. The last of these may be implemented by a change of aspect ratio (i.e. ), exchanging and , and then scaling by a factor of , so that, for example, . The third symmetry operation introduces a conformal covariance factor due to the scaling, but this is absorbed into the prefactor and does not effect . The three symmetry operations translate, respectively, into
Now comes a central mathematical result. There is a linear combination of the four differential equations giving
(Since is a variable here we have altered the usual Jacobian notation and write e.g. in place of .) Thus all solutions must be of the form
(An depending on alone is not possible, as follows from (13)).
Now, for a given aspect ratio, depends only on and depends only on . As is spelled out in detail in , (15) then implies that the ratio of correlation functions investigated in  for percolation is independent of (the horizontal coordinate) in the rectangle. This peculiar symmetry was, as noted in , first observed via simulations. Using (15) shows that this symmetry is exact in the continuum limit, and holds for many critical systems.
Inserting into the differential equations and taking linear combinations allows us to cancel out all dependence and arrive at the three equations
In  we calculated (1), but only for the case of percolation (), and in the limit where goes to the bottom edge of the rectangle. Here, because the dependence is entirely contained in the prefactor of (5) we expect that the solution space of (17)–(19) for should contain the functions calculated in , up to differences in the prefactor. Guided by this, and with a little algebra, we find that making the substitution
and taking an appropriate linear combination of the resulting equations gives the standard form of Appell’s hypergeometric differential equations
with the Pochhammer symbol , is he first of the Appell hypergeometric functions.
This set of Appell functions allows us to write five solutions for , valid for all , as
For our ranges of and values, (22)–(26) exhaust the convergent Frobenius series solutions to the differential equations that can be expressed with a single . We can also find five other convergent Frobenius series solutions that can be expressed with a single Appell function of the second type,
With the definition
(note that is the parameter of the loop models) we have
These Frobenius series solutions are useful, since each one is a single conformal block. It is possible to identify the particular block in each case by examining the leading terms, but this can be done more elegantly using the Coulomb gas formalism (see ).
The physical content for loop and -state Potts models is examined in . The normalizations of these solutions are chosen in part for consistency with the vertex operator formulation used there. We also show that the particular functions , , and form a natural basis of the three dimensional solution space for the critical models mentioned.
The hypergeometric functions in the conformal blocks simplify for certain values, corresponding to various physical models. This is explored in .
Finally, recall that a second set of conformal block solutions that depend on and follows from .
ii.2 Common functional factor and corner operators
We next complete the transformation of the correlation function (5) from the upper half plane into the rectangle using (8), by computing the common functional prefactor in the rectangle, as a function of (alternatively ) and the parameter .
We’ve chosen to write the upper half plane prefactor as
This particular form was motivated by the derivation of the analogous correlation function for percolation in the semi-infinite strip, . We found that it simplified the analysis to define the prefactor so that in it takes the form , where is the strip one point function. For percolation (), the exponent is the ratio of leading exponents in the bulk-boundary fusion rules when the bulk operator approaches a free versus fixed boundary i.e. on a free boundary and on a fixed boundary (see  or  for an explanation of the boundary conditions). Similarly, we have chosen (36) so that when mapped into the rectangle the prefactor satisfies
The successful analysis of in the last subsection demonstrates the merit of this choice. In the remainder of this subsection we explicitly determine .
Because the mapping between the upper half plane and rectangle is singular at the corners the definition of the boundary condition changing operators placed there needs to be adjusted. Following  we use the convention
where defines the rectangular corner operator as a boundary operator approaches a corner . It follows that the conformal weight of the rectangular corner operator is related to the weight of the associated boundary operator by .
Thus we may write the rectangular geometry correlation functions as