Conformal field theory of Painlevé VI
Generic Painlevé VI tau function can be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with . Using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants, we obtain full and completely explicit expansion of near the singular points. After a check of this expansion, we discuss examples of conformal blocks arising from Riccati, Picard, Chazy and algebraic solutions of Painlevé VI.
Starting from the early seventies, Painlevé equations have been playing an increasingly important role in mathematical physics, especially in the applications to classical and quantum integrable systems and random matrix theory FS (), jmms (), Lukyanov2011 (), smj (), twairy ()–wmtb (), Zamo_PIII (). A considerable progress has been made since then in the study of various analytic, asymptotic and geometric properties of Painlevé transcendents. The interested reader is referred to clarkson (); conte (); fokas () for details and further references.
The sixth Painlevé equation (PVI)
is on the top of the classification of 2nd order ODEs without movable critical points. The latter property means that is a meromorphic function on the universal cover of . Four complex parameters and two integration constants form a six-dimensional PVI parameter space .
The most natural mathematical framework for Painlevé equations is the theory of monodromy preserving deformations. For example, equation (1) is associated to rank 2 system
with four regular singular points on . Traceless matrices () are independent of and have eigenvalues that coincide with PVI parameters. Also, .
The fundamental matrix solution is multivalued on , as its analytic continuation along non-contractible closed loops produces nontrivial monodromy. Full monodromy group is generated by three matrices which correspond to the loops in Fig. 1 (note that ). Right multiplication of by a constant matrix gives another solution, and therefore are fixed by (2) only up to overall conjugation.
As is well-known, Schlesinger equations of isomonodromic deformation of (2)
are equivalent to (1). Parameter space may be identified with and many questions on Painlevé VI can be recast in terms of monodromy. This approach, characteristic for classical integrable systems in general, turns out to be quite successful. In particular, it represents the key element of the solution of PVI connection problem jimbo (), as well as of the construction boalch (); dubrovin () and classification apvi () of algebraic solutions.
Logarithmic derivative of Painlevé VI tau function
can be expressed in terms of and (see e.g. Eq. (2.9) in dyson2f1 ()). It solves a nonlinear 2nd order ODE called -form of Painlevé VI (PVI), which may be written as
It is much more natural to work with than with for a number of reasons:
First, it is the tau function which typically shows up in the applications of Painlevé equations, representing gap probabilities in random matrix theory, correlation functions of Ising model and sine-Gordon field theory at the free-fermion point etc.
Thirdly and (arguably) most importantly, the tau function has an intimate connection with quantum field theory.
The last point was discovered by Sato, Miwa and Jimbo in the first two papers of the series smj (). There it was shown that the Riemann-Hilbert problem for rank linear systems with an arbitrary number of regular singularities on admits a formal solution in terms of correlation functions in the theory describing free massless chiral fermion copies. Besides fermions, the correlators involve local fields of another type (below we use the term “monodromy fields” instead of SMJ’s “holonomic”) which can be seen in the operator formalism as Bogoliubov transformations of the fermion algebra ensuring the required monodromy properties. General isomonodromic tau function was originally defined as the correlator of monodromy fields. In particular, for Painlevé VI it is given by a four-point correlator
We would like to put isomonodromic deformations into the context of subsequent developments in conformal field theory BPZ (); DF (); ZZ_book (). To our knowledge, no such attempt has been made so far except for a short general discussion in Moore (). The present paper mainly deals with Painlevé VI which represents the simplest nontrivial example of isomonodromy equations.
It will be argued below that the relevant chiral CFT has central charge and monodromy fields in (5) are Virasoro primaries with conformal dimensions , where . Therefore the structure of the expansion of near, say, , is strongly constrained by conformal invariance. In fact, the chiral correlator of four primary fields for any is given by
where the sum runs over conformal families appearing in the OPE of and , denotes the dimension of the corresponding intermediate primary field and is the set of external dimensions. Conformal block associated to the channel is a power series normalized as . It is completely fixed by conformal symmetry BPZ (). The structure constants , combine conformal blocks into correlation functions of specific theories and should be obtained from another source. CFT correlators usually contain contributions of holomorphic and antiholomorphic conformal blocks subject to the requirement of invariance under the action of braid group on the positions of fields. The chiral correlators (5), (6) are not invariant under this action but transform in a natural way, induced by the Hurwitz action on monodromy matrices.
Direct computation of the coefficients becomes quite laborious with the growth of . However, very recently this problem was completely solved in the framework of AGT conjecture AGT () relating Liouville CFT and 4D supersymmetric gauge theories. The latter correspondence produces conjectural combinatorial evaluations of conformal blocks, subsequently proven by Alba, Fateev, Litvinov and Tarnopolsky AGT_proof (). For , which is the only case of interest for PVI, another derivation was given by Mironov, Morozov and Shakirov in MMSh1 (). In particular, AGT representation expresses the contribution of fixed level of descendants of to 4-point conformal block in terms of sums of simple explicit functions of , and over bipartitions with a fixed number of boxes.
Hence, to obtain full expansion of PVI tau function it suffices to determine the dimension spectrum of primaries present in the OPEs of monodromy fields and the associated structure constants. To formulate the final result, we introduce monodromy exponents by
Together with , these parameters define seven invariant functions on the space of monodromy data subject to a relation jimbo ()
where (). This of course agrees with the dimension of , and allows to interpret the triple as a pair of PVI integration constants. Below we assume that , are generic complex numbers verifying Jimbo-Fricke relation (7).
Let be the set of all partitions identified with Young diagrams. Given , we write and for the number of boxes in the th row and the th column of , and denote by the total number of boxes in . The quantity is called the hook length of the box .
Our main statement is the following
Complete expansion of Painlevé VI tau function near can be written as
The function is a power series in which coincides with the general conformal block and is explicitly given by
The structure constants can be written in terms of Barnes G-function,
with given by
Painlevé transcendents are generally believed to be rather complicated special functions. In our opinion, this reputation is somewhat undeserved. Painlevé VI, for instance, enjoys most of the basic properties of the Gauss hypergeometric equation. In particular, it has many elementary solutions (see apvi () and references therein), Bäcklund transformations okamoto (), quadratic Landen-type transformations kitaev_LMP (), and its connection problem is solved jimbo (). The present work adds to this list a link to representation theory of the Virasoro algebra and the series representation of PVI solutions, which can also be seen as an efficient tool for their numerical computation.
The outline of the paper is as follows. The next section starts with a brief survey of the isomonodromy problem. After exhibiting global conformal symmetry of the tau function, in Subsection 2.3 we introduce monodromy fields and explain how various mathematical objects of the theory of monodromy preserving deformations can be written in terms of correlation functions in 2D CFT. Subsection 2.4 deals more specifically with Painlevé VI. Here we present the arguments leading to (8)–(12), and discuss analytic continuation of PVI solutions and their Bäcklund transformations from CFT point of view. Section 3 is devoted to direct analytic verification of the above expansion. This task is further pursued in Section 4, where the known special PVI solutions are examined from the field-theoretic perspective. We conclude with a list of open questions and directions for future work.
2 CFT approach to isomonodromy
It is instructive to start with the general case of rank linear system with regular singular points on . Instead of (2) one has
The absence of singularity at infinity implies that constant matrices satisfy the constraint . They are assumed to be diagonalizable so that with some . The fundamental solution will be normalized by . It is useful to introduce the matrix
The coefficients of the Taylor series of around can be expressed in terms of and its derivatives. In particular,
Near the singular points, the fundamental solution has the following expansions (under additional non-resonancy assumption for ):
Here is holomorphic and invertible in a neighborhood of and satisfies . The connection matrix is independent of . Counterclockwise continuation of around leads to monodromy matrix .
Let us now vary the positions of singularities and normalization point, simultaneously evolving ’s in such a way that the monodromy is preserved. A classical result translates this requirement into a system of PDEs
It is important to note that the matrix remains invariant under isomonodromic variation of . Schlesinger deformation equations are obtained as compatibility conditions of (13), (16) and (17). Explicitly,
Lax form of the Schlesinger system (18)–(19) implies that the eigenvalues of ’s are conserved under deformation. This is of course expected due to the obvious relation between the spectra of ’s and monodromy matrices.
Isomonodromic tau function is defined by
It is a nontrivial consequence of the deformation equations that the 1-form in the r.h.s. is closed. To show that it does not depend on , one can rewrite (20) as
Here the first equality follows from (20) and the second one from the fact that is conjugate to .
Finally, let us decompose all ’s into the sum of scalar and traceless part as . It can then be easily checked that
This allows to assume without any loss of generality that ’s are traceless, but we deliberately postpone the imposition of this condition.
2.2 Global conformal symmetry
Fractional linear maps form the automorphism group of the Riemann sphere. It is clear that under the action of these transformations on , and the quantities and transform as functions and as a vector field. Our task in this subsection is to understand the effect of global conformal mappings on the tau function.
where with . One recognizes here the general expression for the three-point correlation function of (quasi)primary fields of dimensions in the two-dimensional conformal field theory.
Given the above example, it is natural to assume that for general the tau function transforms as the -point function of primaries with appropriate dimensions:
To prove the last formula, it is sufficient to consider infinitesimal transformations generated by the vector field , which amounts to checking three differential constraints:
These relations can indeed be straightforwardly demonstrated using the first equality in (21) and the condition .
2.3 Field content
The fundamental solution is completely fixed by its monodromy, normalization and singular behaviour (15). The last point is particularly important: indeed, adding an integer to any diagonal element of modifies the asymptotics of as without changing monodromy matrices. Therefore, in what follows, the notion of monodromy will include not only the set of ’s but also the choice of their logarithm branches .
Here it is assumed that , , are primary fields in a 2D CFT characterized by some central charge . Further, we want the OPEs of ’s with ’s to contain the identity operator. This forces them to have equal dimensions, to be denoted by . Normalization of these fields is fixed by the normalization of ; the leading OPE term should be equal to
Since may be represented by an entire series near , the dimensions of all other primaries appearing in this OPE should be given by strictly positive integers. Monodromy fields are defined by the condition that their complete OPEs with have the form
where are some local fields. In particular, the raw vector should be multiplied by when continued around . If one succeeds in finding a set of fields with all mentioned properties, the correlator ratio (26) will automatically give the solution of the linear system (13).
The definition (20) of the tau function arises very naturally in the CFT framework. To illustrate this, let us compute two more orders in the OPE (27). The identity field has no level 1 descendants, therefore the leading correction is given by a new primary field of dimension 1. The next-to-leading order correction comes from three sources: i) nonvanishing level 2 descendant of the identity operator given by the energy-momentum tensor , ii) level 1 current descendant and iii) new primaries of dimension 2 which can be combined into a single field . Thus
We will make a further assumption of tracelessness of , which is essentially motivated by the examples considered below. Now, substituting (28) into (26) and matching the result with (14), one finds that
Standard CFT arguments allow to rewrite the r.h.s. of the last formula as
In what follows, we will be exclusively interested in the case when
Such a condition implies, in particular, that the dimensions of monodromy fields coincide with the quantities from the previous subsection.
One possible realization of the above conditions is provided by the theory of free complex fermions. Its central charge agrees with the conformal dimension of fermionic fields , which play the role of ’s and ’s. The currents are by definition given by , while the energy-momentum tensor and the fields may be expressed as
To represent monodromy fields, recall the usual bosononization formulas
where are free complex bosonic fields with the propagator . Also note that for , monodromy matrices for the linearly transformed fermions
are obtained from ’s by conjugation by . In particular, setting , one obtains fermions , with diagonal monodromy around . Denote by bosonic fields associated to this “diagonal” fermionic basis, then monodromy field can be written as
We thus need to deal with different bosonization schemes of the same theory, each of them being adapted for representing one of the monodromy fields. The corresponding -tuples of bosons are related by complicated nonlocal transformations.
The formulas (22)–(24) are a signature of the well-known decomposition of fermionic CFT into the direct sum of two WZW theories. Fermion and monodromy fields are given by products of fields from the two summands:
Bosonic field in the factors is expressed in terms of fields introduced before as . The fields , and live in the WZW theory and can be formally written as ordered exponentials of integrated linear combinations of -currents. It should be emphasized that they are Virasoro primaries but not necessarily WZW primaries. The fields and have the same dimension , in accordance with the central charge KZ (). The dimension of is equal to , where as above, stands for the traceless part of .
Now it becomes clear that imposing the tracelessness of corresponds to factoring out the piece from the fermionic theory. This innocently looking procedure is in fact crucial, as it drastically reduces the number of primary fields in the OPEs and thus makes the computation of correlation functions much more efficient as compared to fermionic realization. Therefore, in what follows we set , remove the hats from ’s, ’s, ’s and ’s to lighten the notation, and interpret the isomonodromic tau function as a correlation function of primaries with dimensions in a CFT with .
We close this subsection with an example of application of field-theoretic machinery in the case . It is somewhat distinguished from CFT point of view, since for the dimension of ’s and ’s corresponds to level 2 degenerate states, and the dimension 1 of is degenerate at level 3. Hence the correlation functions
have to satisfy linear PDEs of order 2 and 3, fixed by Virasoro symmetry. This results into the following statement (cf observations made in novikov ()):
Under assumption , the matrices
satisfy the differential equations
2.4 Painlevé VI
Recall that global conformal symmetry allows to fix the positions of three singular points. Painlevé VI equation corresponds to setting , and sending these three points to , and . The remaining singular point, , represents the cross-ratio of singularities, which is preserved by Möbius transformations.
For , let us denote by the eigenvalues of . Preceding arguments show that PVI tau function defined by (3) is nothing but the four-point correlator of monodromy fields,
and that these fields are Virasoro primaries with dimensions in a conformal field theory. The field at infinity should be understood according to the usual CFT prescription
It is clear that auxiliary fields should have monodromy around all fields in the OPE of and . Let denote the eigenvalues of and be its diagonalizing transformation. Since is defined only up to an integer, it is natural to expect that the set of primaries present in the OPE of and consists of an infinite number of monodromy fields with and
i.e. of all possible monodromy fields associated to the monodromy matrix . Taking into account that conformal dimension of is equal to , the first part of our main statement (formulas (8)–(10)) now follows from the general formula (6) and AGT combinatorial representations of conformal blocks AGT ().
The structure constants of the expansion (8) can be determined from the so-called Jimbo asymptotic formula jimbo (), expressing the asymptotics of PVI tau function as in terms of monodromy. In fact we have already obtained the “easier half” of this formula. E.g. if , then (8) implies that the leading behaviour of is given by
Subleading asymptotics, fixing the second PVI integration constant, can be rewritten in the form of a recursion relation on the coefficients . Namely,
where is defined by (12). This relation can be easily solved in terms of Barnes functions, with the answer given by (11). It is interesting to note that, up to a common multiplier and appropriately symmetrized factors, ’s essentially coincide with the chiral parts SW () of the corresponding structure constants in the time-like Liouville theory HMW (); Zamo_Liouville ().
The structure constants (11) can not be completely factorized into the products of three-point functions due to the presence of the parameter . This is an artifact of non-trivial braid group action on the correlation functions of monodromy fields.
To illustrate what we have in mind, consider the analytic continuation of along a closed counterclockwise contour around the branch point . In general, such a continuation induces an action of the 3-braid group (more precisely, of the modular group ) on monodromy dubrovin (). In the case at hand, new monodromy matrices are given by
so that and
Therefore, the change of the branch of is encoded in the change of the structure constants. On the other hand, one can perform analytic continuation directly in the expansion (8). Up to an irrelevant overall factor, this amounts to multiplication of by . Since both results should coincide, the structure constants have to satisfy the functional relation