Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP

Abstract

To any solution of a linear system of differential equations, we associate a kernel, correlators satisfying a set of loop equations, and in presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the q-th reductions of KP -which contain the pp, qq models as a specialization.x k q, indexed by integer a 1 , . . . , a k P v1, dw, called k-points correlators, or shortly, correlators. and we show that the k-point correlators satisfy a set of linear equations (Theorem 2.1) and a set of quadratic equations (Theorem 2.2). We use the name loop equations to refer collectively to those set of equations. We also introduce a notion of "insertion operator" (Definition 2.5) allowing the derivation of k-linear loop equations for k ď d (the size of the differential system) from the master ones. These results are of purely algebraic nature and hold for any system (1-1). When L depends on a set of parameters t preserving the monodromy of the solutions, we can also associate to Ψpx, tq a Tau function T p tq, defined up to a constant prefactor.Secondly, in Section 3, we study the semiclassical expansion in powers of and describe in detail the monodromy of its coefficients (Section 3.2-3.4). We introduce in Definition 3.3 the notion of "expansion of topological type" -also referred to as the TT property -and show that the expansion can be computed by the topological recursion of [EO07] when the TT property holds. In practice, the main consequence of our theory is Theorem 3.1, and in presence of isomonodromic times, this also allows the computation of the expansion of ln T p tq (Corollary 4.2).Finally, in Section 5, we apply our theory to the linear system associated to the q-th reduction of KP, and illustrate it more specifically with examples of the pp, qq models (Section 6). As a motivation, those hierarchies are believed to describe the algebraic critical edge behavior that can be reached in the two hermitian matrix model, and universality classes of 2d quantum gravity coupled to conformal field theories [Moo90, DS90, GM90, dFGZJ94]. In any q-th reduction of KP, we show ( § 5.6-5.8) that the TT property holds, and that our Theorem 3.1 can be applied. CommentsThe earlier work [BE09a] described the construction of Section 2 for general 2ˆ2 rational systems, but implicitly assumed the TT property. It was illustrated for p2m`1, 2q systems in [BE09b], and entails a rigorous proof -modulo checking the TT property, wh...

Correlation functions of the KdV hierarchy and applications to intersection numbers over M¯g,n

“…We are grateful to Erik Carlsson and Fernando Rodriguez Villegas for pointing out that L λµ in Theorem 1.14 are transition matrices from monomial basis to homogeneous basis of symmetric functions. We thank Bertrand Eynard for bringing our attention to the papers [6], [2]. D. Y. is grateful to Youjin Zhang for his advising.…”

Lax matrix solution of c = 1 conformal field theory

“…The reason why we focus on the c = 1 case is the availability of powerful techniques [8] which will enable us to solve the theory with the help of a Lax matrix -a more powerful object than the spectral curve, to which it is nevertheless directly related. Let us announce our plan.…”

“…Let us announce our plan. We begin with reviewing the Ward identities in Conformal Field Theory in section 2.1, before using the ansatz of [8] for solving them in the case c = 1 in section 2.2. We use this for solving c = 1 Conformal Field Theory, by computing fourpoint conformal blocks in section 3 and the three-point correlation function in section 4 (which is independent from section 3).…”

“…By a solution we mean functions Z N , W 1 , W 2 , · · · such that W m solve the Ward identities (2.13), while Z N solves the global Ward identities (2.17). To find solutions, we will use a determinantal ansatz [8]. If N ≥ 3, this ansatz leads to the construction of one or more solutions.…”

“…Inserting these W m s in the left-hand side of the Ward identities (2.13) yields values P m+1 (I, y) of the right-hand side which were computed in [8], and which we now require to agree with our definition (2.14) of P m+1 (I, y). This requirement will result in constraints on the Lax matrix L(y).…”