1 Introduction

Quantization condition from exact WKB for difference equations

LPTENS 16/02


Quantization condition from exact WKB

for difference equations

Amir-Kian Kashani-Poor

Laboratoire de Physique Théorique de l’École Normale Supérieure,
Université de Recherche PSL, CNRS, Sorbonne Universités, UPMC
24 rue Lhomond, 75231 Paris Cedex 05, France


Abstract: A well-motivated conjecture states that the open topological string partition function on toric geometries in the Nekrasov-Shatashvili limit is annihilated by a difference operator called the quantum mirror curve. Recently, the complex structure variables parameterizing the curve, which play the role of eigenvalues for related operators, were conjectured to satisfy a quantization condition non-perturbative in the NS parameter . Here, we argue that this quantization condition arises from requiring single-valuedness of the partition function, combined with the requirement of smoothness in the parameter . To determine the monodromy of the partition function, we study the underlying difference equation in the framework of exact WKB.

1 Introduction

A long standing goal of topological string theory is to obtain the topological string partition function as an analytic function on the parameter space of the theory. The latter is the product of the coupling constant space – or in the case of refinement – in which the genus counting parameter – or and the coupling constant of the refinement – take values, and the appropriate moduli space of the underlying Calabi-Yau manifold . This program has been most successful in the case of toric (hence non-compact) Calabi-Yau manifolds. The topological vertex [1] and its refined variants [2, 3] permit the computation of in the large radius regime of in a power series expansion in exponentiated flat coordinates of , with coefficients that are rational functions in and , with , . The holomorphic anomaly equations [4] and their refinement [5, 6, 7, 8] can be used to compute the coefficients of in an asymptotic expansion as analytic functions on . In the compact case, some impressive all genus results for certain directions in the Kähler cone have been obtained for Calabi-Yau manifolds that are elliptically fibered, see e.g. [9]. An open question is how to define without recourse to any expansion.

In [10, 11], the open topological string partition function on a toric Calabi-Yau manifold was studied for a particular class of torically invariant branes, and the mirror curve of identified as the open string moduli space for this problem. This insight led to the computation of to leading order in . [12] proposed that to extend the computation beyond leading order in , the mirror curve had to be elevated to an operator . In fact, it is the Nekrasov-Shatashvili (NS) limit [13] of that can be determined via a quantization of the mirror curve, as lies in the kernel of [13, 14, 15],1

(1.1)

The idea to recover the closed topological string partition function from the monodromy of the open partition function was put forward in [12], and made more precise in [13, 14, 15]. In a remarkable series of papers [17, 18, 19, 20, 21, 22, 23, 24], the quantization of the mirror curve has been taken as a framework within which to define non-perturbatively. In the genus one case, the equation (1.1) can straightforwardly be rewritten as a spectral problem for the complex structure parameter of the mirror geometry,

(1.2)

Here, is put in the form via appropriate variable redefinitions [22].2 Upon specifying the function space in which is to lie, this eigenvalue problem can be solved numerically. For the higher genus case, [24] identify the mirror curve of the toric Calabi-Yau manifold with the spectral curve of a quantum integrable system determined by the toric data of . The underlying class of quantum integrable systems was introduced by Goncharov and Kenyon [28]. The complex structure parameters of the mirror curve map to the spectrum of the integrable system. [17] and follow-up works propose a quantization condition on the parameters based on a non-perturbative modification of , which roughly takes the form (see equation (3.12) below for the precise statement)

(1.3)

Here, and are the conventional perturbative and enumerative contribution to the closed topological string partition function in the NS limit. is a contribution included in the quantization condition to cancel poles of in . Condition (1.3), at real values of , has been shown to reproduce the numerical results obtained by diagonalizing the Hamiltonians of the associated Goncharov-Kenyon system numerically in a harmonic oscillator eigenbasis of to high precision.

In this paper, we aim to establish that the quantization condition (1.3) arises upon imposing single-valuedness of the elements of the kernel of . To this end, we need to determine the monodromy of solutions to (1.1) as functions of the complex structure parameters on which depends. We propose to do this in the framework of exact WKB analysis applied to difference equations. The WKB analysis of difference equations has received some treatment in the literature (see e.g. [29]), however, to our knowledge, not in the form we require for our study. We thus attempt to generalize to difference equations the approach presented e.g. in [30] to the transition behavior of WKB solutions of differential equations. We find that the transition behavior in the case of linear potentials can be studied in detail. Difference equations lack the rich transformation theory required to lift the analysis rigorously to general potentials [30]. Our analysis hence relies on the conjecture that the transition behavior for potentials with simple turning points is governed only by these, and well approximated in their vicinity by the linear analysis.

To explain the non-perturbative contribution to (1.3), we will argue that by the choice of harmonic oscillator states for the numerical diagonalization, the elements of the function space are constrained to be functions in with smooth dependence on . The latter condition requires adding a non-perturbative piece in to as hitherto defined. Equation (1.3) arises from the constraint that the function thus obtained be single-valued.

We will study the monodromy problem of difference equations in section 2. In section 3, we discuss the open topological string partition function from various perspectives and explain how we expect the quantization condition (1.3) to arise. This discussion is applied to the example in section 4, in which we also present numerical evidence for the quantization condition (1.3) in the case of complex . We end with conclusions.

2 Monodromy from exact WKB

In this section, we will provide evidence linking the monodromy problem of solutions to difference equations of the form (1.1) arising upon quantization of mirror curves to the so-called quantum B-periods. We first flesh out an argument provided by Dunham [31] in the case of differential equations using exact WKB methods. We then set out to generalize these methods to difference equations.

2.1 Differential equation

We will briefly review the basics of WKB analysis in this subsection, following [30]. A somewhat more detailed review in the same spirit can be found in [32].

The starting point of the analysis is a second order differential equation

(2.1)

depending on a small parameter . is a meromorphic function of with possible dependence, which we for simplicity will take to be of the form . To solve this equation, we can make a WKB ansatz

(2.2)

with considered as a formal power series in ,

(2.3)

Plugging this ansatz into the differential equation (2.1) yields the expansion coefficients recursively,

(2.4)
(2.5)

The equation (2.4) has two solutions . The choice of sign propagates down to all expansion coefficients . We thus obtain two formal WKB solutions to (2.1), reflecting the fact that the differential equation is of second order.

Denoting

(2.6)

it is not hard to show that

(2.7)

The two formal WKB solutions can thus be expressed as

(2.8)

The two formal series will generically merely provide asymptotic expansions of two solutions to (2.1). Exact WKB analysis is concerned with recovering the functions underlying such expansions.

Borel resummation is a technique to construct a function having a given asymptotic expansion as a power series

(2.9)

It proceeds in two steps. The first is to improve the convergence behavior of (2.9) by considering the Borel transform

(2.10)

The second step is to take the Laplace transform of (2.10),

(2.11)

Here, is a half-line in the -plane, emanating from the origin at angle to the abscissa. If the sum in (2.10) and the integral in (2.11) exist, then defines a function with asymptotic expansion given by (2.9), called a Borel resummation of the formal power series (2.9). The Laplace transform (2.11) can fail to exist if exhibits a singularity on the integration path . Integrating along on either side of the singularity will then generically give rise to two functions , both with asymptotic expansion (2.9), but differing by exponentially suppressed pieces in . The position of the singularities of the Borel transform thus leads to a subdivision of the -plane into sectors. Choosing to lie in different sectors will give rise to different Borel resummations of (2.9).

When the coefficients of the formal power series (2.9) depend on a variable ,

(2.12)

the position of the poles of the Borel transform (2.10), and hence the subdivision of the -plane into sectors, will depend on . Keeping the integration path fixed, crossing certain lines in the -plane will result in poles of crossing . These lines are called Stokes lines. They divide the -plane into Stokes regions. The Borel resummation of (2.12) performed on either side of a Stokes line will yield functions whose analytic continuation to a mutual domain will differ by exponentially suppressed terms.

Returning to the WKB analysis of a second order differential equation, the Stokes phenomenon implies that a Borel resummation of the formal WKB solutions (2.8) will yield a different basis of the solution space depending on the Stokes region in which the Borel resummation is performed. This behavior can be studied by first considering the case of a linear potential , then using the transformation theory of differential equations to reduce the analysis of more general potentials to this case. For potentials with only simple zeros, the results are as follows: the Stokes lines emanate from zeros of , called turning points. Simple turning points have three Stokes lines and a branch cut emanating from them. The trajectory of Stokes lines depends on the choice of integration path for the Laplace transform and is determined by the equation

(2.13)

with the position of the turning point. As the Borel resummation of the formal WKB solutions in any Stokes region yields a basis of solutions to the differential equation (2.1), each such pair can be expressed as a linear combination of any other such pair. Neighboring Stokes regions are assigned transition matrices which enact the linear transformation relating the associated two pairs of solutions. The form of the transition matrices depends on the normalization of the WKB solutions, determined by the lower bound on the integration in the exponential of the WKB ansatz (2.2). Choosing this lower bound to be the turning point from which the Stokes line separating the two Stokes regions emanates yields independent transition matrices.

The exact form of the transformation matrices can be determined, as mentioned above, by solving the differential equation with linear potential explicitly, and then mapping the general situation to this case. The space of solutions to the linear problem is spanned by Airy functions. For our purposes, we will only need the product of the three transition matrices which arise when we circumnavigate a turning point in counter clockwise order, crossing three Stokes lines consecutively, but without crossing the branch cut. To compute , it suffices to know that the Airy functions are single-valued in the vicinity of the turning point. It follows that

(2.14)

where the matrix relates the Borel resummation of in the same Stokes region, but on either side of the branch cut. Crossing the branch cut interchanges and , and leads to a factor of due to the square root in the denominator of (2.8). This reasoning yields

(2.15)

The sign depends on conventions that we will not bother to fix, as it will cancel in our considerations.

Let us now consider a potential with two simple turning points, giving rise to a Stokes pattern as depicted in figure 1.

Figure 1: A generic potential with two turning points; the branch cut is chosen to connect the two turning points.

We want to consider the monodromy of a pair of WKB solution as we encircle the two turning points once. If we begin with a pair of WKB solutions normalized at the turning point , we must change their normalization by multiplying by the matrix

(2.16)

in order for their transition behavior upon circumnavigating the turning point to be governed by (2.15). The superscript is to denote the Stokes region in which the integration path from to lies. The exponential entries in the normalization matrix are called Voros multipliers. They are to be understood as the Borel resummation of the indicated formal power series. Such Borel resummations exhibit interesting jumping behavior with regard to the choice of integration path for the Laplace transform, as e.g. recently discussed in [32].

In total, we obtain the monodromy matrix

(2.17)

with the superscripts and indicating that the integration path connecting and is to be taken above/below the branch cut. The integration cycle is accordingly a path encircling the branch cut.

Requiring a pair of WKB solutions in the situation depicted in figure 1 to be single-valued is hence equivalent to demanding

(2.18)

2.2 Difference equation

Unlike differential equations, difference equations have no obvious transformation theory. Under variable transformation other than linear, their form changes drastically. We will perform an exact WKB analysis in the case of linear potential in the following, but not be able to offer an intrinsic criterion determining for which potentials the linear approximation is justified.

Generalities

Consider a difference equation in the form

(2.19)

for a potential which we take to be independent for simplicity. With the WKB ansatz

(2.20)

being a normalization factor which we shall fix below, we obtain

(2.21)

Expanding

(2.22)

we obtain

(2.23)

and furthermore

(2.24)

Hence,

(2.25)

where we have chosen a branch in (2.23).

The analytic structure of the inverse function is best understood by expressing it as

(2.26)

The function has two branch points at respectively due to the square root functions, and one at due to the logarithm. Choosing the branch cuts for the square roots and the logarithm in the negative real direction, the branch cuts of the two square roots cancel beyond , at which point the branch cut of the logarithm begins.3 The sheet structure of is hence such that the branch cut along connects two sheets related via a sign flip, as

(2.28)

whereas the branch cut beyond connects sheets related via a shift of the imaginary part by .

The zero of (2.26) lies at . The expansion of around this point has as leading term. We thus identify the points as the turning points of the difference equation. As long as , we can approximate the behavior of the difference equation in the vicinity of such a turning point by a linear potential.

WKB for a linear potential

The difference equation with a linear potential is

(2.29)

Setting in (2.23) and (2.24), the WKB coefficients can be integrated, yielding

(2.30)

By (2.25) and using (2.26), the leading order behavior of the WKB solution is thus

(2.31)

In fact, we can check explicitly that this WKB solution provides an asymptotic expansion for a solution of the difference equation (2.29), as we can construct a solution to this equation based on the Bessel function [33].4 Recall that the Bessel function satisfies the recursion relation

(2.32)

The function thus solves the difference equation (2.29).5

The Bessel function is known to have asymptotic behavior, for along the real axis and constant positive , given by (see e.g. [35])

(2.33)

Upon the identification

(2.34)

this coincides with the WKB result (2.31), with the normalization fixed at

(2.35)

We have hence matched the leading behavior of the WKB solution (2.31) to the asymptotic expansion of the Bessel function for positive real and small and positive. To study the Stokes phenomenon, we will now take advantage of the fact that the Bessel function is also the solution to a differential equation, to which we can apply the exact WKB methods reviewed in the previous section. Indeed, solves the differential equation

(2.36)

We can eliminate the linear term and cast the equation in the form (2.1) by considering , which satisfies

(2.37)

The conventional theory of exact WKB analysis of differential equations allows us to determine the Stokes behavior of the WKB expansion of the solutions to this differential equation. By relating this expansion to the WKB solution of the difference equation, we can derive the Stokes behavior of the latter.

Making a WKB ansatz6 , we obtain the leading terms

(2.38)
(2.39)

By comparing to the asymptotic expansion (2.33), we can fix the normalization of the WKB expansion to

(2.40)

Comparing to

(2.41)

the uniqueness of asymptotic expansions in power series implies

(2.42)

We have here assumed that the WKB expansions (2.41) and (2.40) yield asymptotic expansions to the indicated solutions of the difference and differential equation respectively. In the case of the differential equation, this is guaranteed by general theory. We have verified (2.42) to high order in .

We next address the question of how the Stokes lines of the two asymptotic expansions

(2.43)

are related. The Borel transforms of the two expansions are given by

(2.44)

and

(2.45)
(2.46)
(2.47)

Hence,

(2.48)
(2.49)
(2.50)

The Borel sum of the WKB series of the difference equation hence indeed equals, for real and small positive , . From the theory of exact WKB for differential equations, we know that the Borel transform has a branch point in the -plane at . The Laplace transform performed along the real axis, i.e. with in the notation of (2.11), will hence be ill-defined for , identifying this condition as determining the location of the Stokes line. By (2.45), hence exhibits a branch point at , i.e. . The condition determining the location of the Stokes line is therefore . We conclude that the location of the Stokes lines of the difference equation is determined by the phase of , just as a naive generalization of the conventional WKB results would have suggested. By

(2.51)

the Stokes line structure close to the turning point at is the same as around a simple turning point in the case of a differential equation of the form (2.1), see figure 2.

Figure 2: Flow lines for .

The behavior of the Borel resummed WKB solution upon crossing Stokes lines emanating from the turning point is governed by the general theory. In particular, the transition behavior of upon circumnavigating a turning point is given by the matrix (2.15), and inherits this behavior.

If we assume that the monodromy of the WKB solutions of difference equations is governed by Stokes lines emanating from turning points , and that the behavior upon crossing such lines is captured by the analysis for linear potential just presented, then the analysis of section 2.1 applies, leading to the single-valuedness condition (2.18) in the case of potentials with two turning points.

3 The open topological string and the mirror curve

3.1 The conjectured quantization condition

We begin this section by reviewing the quantization condition discussed in the introduction as presented in [24]. In this form, it applies to the topological string on an arbitrary toric Calabi-Yau manifold . The mirror to such a space is given by a pair , consisting of a complex curve together with a meromorphic 1-form , the 5d analogue of the Seiberg-Witten differential [36]. is given as the zero locus of a polynomial

(3.1)

which can be constructed, up to linear redefinitions of the variables and , from the toric data of [37]. The latter can be presented as a grid diagram, given by the intersection of the three dimensional fan of with the plane. The number of interior points of the grid diagram corresponds to the genus of . Each such point gives rise to a modulus which enters as a parameter in . Each of the boundary points of the grid diagram beyond the first three gives rise to an additional parameter or in , referred to as a mass parameter in [38].7 The moduli , , coincide in the large radius limit with , the exponentials of the flat coordinates on the complexified Kähler moduli space of . These are chosen among the simply logarithmic solutions of the underlying Picard-Fuchs equations governing the periods of the meromorphic 1-form on the curve . They are paired with doubly logarithmic solutions corresponding to -periods. In contrast, the mass parameters or correspond to residues of the 1-form and do not have dual partners. They are given as algebraic functions of the exponentiated logarithmic solutions to the Picard-Fuchs system.

The polynomial can be promoted to an operator by setting

(3.2)

This operator is conjectured to have the open topological string wave function on in the NS limit, , in its kernel,

(3.3)

with identified as the open string modulus. [24] identifies the equation

(3.4)

as the quantum Baxter equation for the Goncharov-Kenyon integrable system determined by the toric data of . The eigenvalues of the Hamiltonians of this system map to the complex structure parameters , , of . Solving the quantum Baxter equation (3.4) with appropriate boundary conditions on is equivalent to solving the spectral problem. Numerical evidence for this beyond the genus one case is reported in [23].

The conjectured quantization condition [24] is a set of equations, indexed by integers , whose solution set of -tuples is to coincide with the Goncharov-Kenyon spectrum. The ingredients that enter into the quantization condition are the Nekrasov-Shatashvili limit of the refined topological string free energy [2, 13], , and the quantum mirror map [15]. encodes integer invariants associated to the Calabi-Yau [39]. These appear most naturally when it is expressed in terms of the flat coordinates on the complexified Kähler moduli space of . We can distinguish between two contributions to . First, there is a perturbative contribution which depends on the triple intersection numbers of the compact toric divisors of (suitably generalized to the non-compact setting) and integers , which have not been given a geometric interpretation yet,

(3.5)

The second contribution depends on integer invariants of the geometry, with a -tuple mapping to a class in via the choice of coordinates , and the half-integers indicating a representation of , and has the form

(3.6)

with as introduced above, and . Following [24], we have indexed this contribution with due to its enumerative interpretation [40, 2, 6, 39]. In the spirit of [13], one would then like to impose a quantization condition on the parameters via

(3.7)

is the intersection matrix between a basis of curve classes in , corresponding to a basis of the Mori cone of the toric geometry and the coordinates , and the torically invariant divisors of . It arises in [41] to relate the derivatives of the prepotential to these divisors.

The crucial ingredient in the quantization condition of [24], inspired by the so-called pole cancellation mechanism in [42], is to consider a third contribution to the quantization condition based on , but evaluated at (up to a detail to which we return presently)

(3.8)

The inspiration behind including this term stems from the observation that the contribution (3.6) to the free energy has poles, due to the sum over , at for all integer values of and . As a function of , it hence necessarily exhibits at best a natural boundary of analyticity on the unit circle (in fact, we will see in the example of local in section 4.5 that even away from the unit circle, the expansion (3.6) is not convergent). The quantization condition (3.7) as it stands is hence ill-defined, at least for such values of . The evaluation point (3.8) is chosen to precisely cancel the contribution from each of these poles: for , the pole which arises in (3.7) at , is canceled by the contribution of the corresponding derivative of the NS free energy evaluated at (3.8) at , . This almost works as is: the residues evaluate to

(3.9)

and

(3.10)

The sign factors in (3.9) and (3.10) can be adjusted such that the two terms cancel if the Kähler parameters can be shifted by a -field that satisfies

(3.11)

for all pairs for which [43]. The existence of such a -field has been shown for many classes of examples, but a proof of its existence for all toric geometries is still lacking. Combining these elements yields the conjectured quantization condition

(3.12)

The equations (3.12) can be solved to express the Kähler parameters in terms of the integers . The so-called quantum mirror map, discussed further in section 4, then maps these solutions to the eigenvalues of the Goncharov-Kenyon spectral problem.8

3.2 The open topological string partition function

The open topological string partition function, as defined in [44], serves as a generating function for open Gromov-Witten invariants, counting maps, in an appropriate sense, from Riemann surfaces with boundary to a Calabi-Yau manifold with branes on which these boundaries are constrained to lie. We will call this partition function . When is a toric Calabi-Yau manifold, this notion can be refined [45, 46], and leads to a formal series in two expansion parameters and .

Beginning with [12], it has been gradually understood [14, 15] that the monodromy of the open topological string partition function is intimately related to the corresponding closed topological string partition function. The mirror curve to the toric Calabi-Yau is identified as the open string moduli space [10, 11], such that becomes a function on . The leading order contribution to in an expansion is then given by

(3.13)

where is the meromorphic 1-form introduced in section 3.1. Thus, the monodromy of this leading contribution around the - and -cycles of the mirror curve coincide with the periods of . These determine the prepotential of via the special geometry relations

(3.14)

In refined topological string theory, is the leading contribution in the formal expansion of in and . It was argued in [15], based on insights from [12, 13, 14], that the higher order corrections to in the NS limit9 , should arise as the monodromy of , given by

(3.15)

with identified with the exponent of the WKB ansatz discussed in section 2. The special geometry relation (3.14) now takes the form

(3.16)

This proposal was checked explicitly in [14] for pure 4d gauge theory. In the framework of the AGT correspondence [47], the necessity to take the NS limit to relate to becomes particularly transparent, see [48]. (3.16) was further checked in both the 4d and 5d setting in [8, 49, 38]. It was shown to follow from the AGT correspondence in [50] for gauge theory.

The refined open topological string partition function on toric geometries can also be defined as an index. In this incarnation, it takes the form [45, 46]

(3.17)

in an expansion in the appropriate exponentiated coordinates on the open and closed string moduli space. denotes the open string modulus encoding the position of the brane and the value of a Wilson line along the boundary of the topological string worldsheet. are the exponentials of the flat closed string moduli appearing in (3.16), and , . The over the open string modulus is to indicate that a naive choice of this coordinate must be modified by factors of closed string moduli in order to obtain integer open string invariants . The need for such so-called flat open coordinates was first exposed in [10, 11] (see also [51], where an alternate algorithm was proposed to compute these coordinates). In the Nekrasov-Shatashvili limit , we set

(3.18)

The expansion of in open and closed string moduli coincides with that of in and , giving rise to the conjecture that an underlying function should exist from which both descend. We hence ask to what extent the relation between the monodromy of the open topological string and persists in the expansion .

For simplicity, let us restrict the discussion to the case of genus 1 mirror curves . The -cycle in these geometries is given by the phase of the open string modulus . The -cycle is encoded in the branch cut structure of as a function of . clearly does not contribute to the -monodromy of under . Indeed, this monodromy is due to a contribution to which is linear in , and which can easily be computed from the quantum curve [15, 38]. Thus, in an expansion,