Quantized mirror curves and resummed WKB
Based on previous insights, we present an ansatz to obtain quantization conditions and eigenfunctions of a family of difference equations which arise from quantized mirror curves in the context of local mirror symmetry of toric Calabi-Yaus. It is a first principles construction, which yields closed expressions for the quantization condition and the eigenfunctions when . The key ingredient is the modular duality structure of the underlying quantum system. We use our ansatz to write down explicit results in some examples, which are successfully checked against purely numerical results for both spectrum and eigenfunctions. Concerning the quantization condition, we also provide evidence that, in the rational case, this method yields a resummation of a conjectured quantization condition involving enumerative invariants of the underlying toric Calabi-Yau.
Département de Physique Théorique
Université de Genève, Genève, CH-1211 Switzerland \emailAddszabolcs.firstname.lastname@example.org
Finding eigenfunctions and eigenvalues of differential or difference operators is an ubiquitous problem in physics. Finding exact expressions for them is most of the time very difficult, even in the one dimensional case where the operators act on functions on the real line (the setup often considered in standard quantum mechanics). In this paper, we will concerne ourselves with a family of operators which are polynomials of the exponentials
where the operators and satisfy the canonical commutation relation for . The operators and can be represented on functions on as multiplication and shift operators. Polynomials of and correspond to more general difference operators acting on functions. Such operators naturally arise in several areas of theoretical physics. An example would be when quantizing the spectral curves of some particular integrable systems, thus obtaining the Baxter equation which is central in the study of those systems. Another would be when quantizing mirror curves. When the genus of the underlying curve is greater or equal to one, it is often the case that the operator has a discrete spectrum. More precisely, it has been shown in many examples that the inverse operator of the quantized curve is a trace class operator on [1, 2], which implies that the spectrum of the operator is discrete.
Our main motivation for the study of such difference operators stems from local mirror symmetry of toric Calabi-Yau threefolds. It has been known for a long time that the mirror geometry of toric Calabi-Yau threefolds is encoded, through Batyrev’s construction, in a curve in called the mirror curve. Periods on that curve give the Kähler parameters in term of the complex structure parameters, as well as the derivatives of the prepotential. More recently, it has been proposed that appropriate quantizations of that same mirror curve encode interesting deformations of the prepotential, which are tightly related to topological strings and enumerative invariants of the target space [3, 4]. Building on insights coming from ABJM theory, this idea has been further developed, in particular in [5, 6]. The most complete realization of this idea is the so called “Topological String / Spectral Theory” (TS/ST) correspondence [7, 8] (for a review, see ), whose extension to the open sector has been undertaken in [10, 11].
In the TS/ST correspondence, the central object to consider is the operator given by the quantization of the mirror curve of the underlying Calabi-Yau threefold.
The main idea is that spectral quantities of that operator (such as eigenvalues, traces or eigenfunctions) can be written down explicitly using enumerative invariants of the underlying geometry
The solution presented here is an ansatz which is based on several recent developments. It has been noticed in  that the quantization conditions giving the eigenvalues which are implied by the TS/ST correspondence of  can be written in a way that is symmetric under .
The importance of this duality has also been emphasised in . Given the close relationship of our difference equations to integrable systems [19, 20], this duality is to be considered as a manifestation of the “modular double” structure  of the corresponding integrable system (see  for the case of the relativistic Toda lattice). It is then natural to assume that the symmetry under should not only be manifested in the spectrum of the difference equation, but also in its eigenfunctions. This is the point of view taken in , where an ansatz is proposed for the eigenfunctions based on this consideration.
Concretely, the ansatz of  consists in taking the resummed WKB eigenfunction and symmetrize it with respect to . Monodromy invariance of the eigenfunction thus constructed yields the quantization conditions of . Unfortunately, this idea alone does not allow one to get explicit, closed expressions for the full eigenfunction , because the resummed WKB expression is not known exactly, but only, for example, as a large expansion.
The difference equations studied in the present work are closely related to a family of integrable systems introduced in  called “cluster integral systems”, or Goncharov-Kenyon integrable systems. Given any toric Calabi-Yau threefold, one can construct a cluster integrable system, as outlined in . The archetypical example is the relativistc Toda lattice of particles, which is the cluster integrable system associated to the toric Calabi-Yau threefolds often called the resolved geometries (since they are resolutions of the singularity). For the relativistic Toda lattice, it has been shown in  using the Quantum Inverse Scattering Method, that the solutions for the spectrum and eigenfunctions of the integrable system can be built from the spectrum and eigenfunctions of the Baxter equation. Our method thus provides a solution for the relativistic Toda lattice eigenfunctions in the case where is rational. The idea that the solution of the integrable system is built in the solution of the Baxter equation may also potentially be valid for the more general family of cluster integrable systems. In that case, our method would also give a solution for them in the rational case. Let us mention that the eigenfunctions for the relativistic Toda have been constructed in [29, 28], using a different method relying on gauge theory computations of instanton partition functions in the presence of defects. That construction gives a solution for any value of , but only as an expansion in an auxiliary parameter which is related in some sense to the “size” of the system.
The paper is structured as follows: in section 2, we present the family of difference equations, and study them using the WKB ansatz. In section 3, we derive the main result, and present the different formulas to compute its components. Section 4 is devoted to examples and tests of the formula. In section 5, we investigate the relationship between our quantization condition and the conjectural one of [17, 19, 20]. In the last section, we give some concluding remarks.
2 About the difference equation
We introduce the family of difference equations. They are then studied using the WKB ansatz, yielding the WKB eigenfunction, which is central in our construction. In this section, we do not yet assume that is rational. For simplicity, we will focus on the cases where the difference equation is of second order, which we sometimes call “hyperelliptic cases” because the underlying Riemann surface is a two-sheeted cover of the plane. But the method is in principle also applicable to higher order cases.
2.1 The difference equation and its dual
Since our main motivation is the TS/ST correspondence, our starting point is a mirror curve of genus , defined by a curve
where takes the form
The and are integer numbers. The parameter is a true modulus of the curve, whereas the parameters can be mass parameters as well as other true moduli, depending on the curve
This quantization procedure is the one used in the TS/ST correspondence. It gives the operator , which can be expressed as a polynomial of and using the BCH formula to split exponentials. In the representation, the operator acts as a derivative . The operator acts as a shift operator on a function, since its action amounts to Taylor expanding it. The operator given by
is then a difference operator acting on functions. Its inverse
has nice properties: it has been shown in many examples in [1, 2] that it is a trace class operator on . As such, it admits a discrete set of eigenvalues and eigenfunctions . An eigenfunction is in the kernel of the operator given by states such that
but not necessarily all functions in this kernel are eigenfunctions of since they may not be in the image of . Indeed, the operator should really be considered as the inverse of , so its domain (and thus the domain of ) should be restricted to the image . We will sometimes call the eigenfunctions “on-shell” eigenfunctions, and other functions in the kernel of will be called “off-shell” eigenfunctions by abuse of language (since they are not truly eigenfunctions of ). All these functions satisfy the difference equation
but only for the specific values do we find on-shell eigenfunctions. We will focus on hyperelliptic curves, where the difference equation is of order 2.
In the case where the underlying mirror curve is of genus greater then 1, we have true moduli, of which are among the . In principle, they could act as spectator parameters (like the other ) and take arbitrary values. As we will see in the examples, our method only gives the spectrum and eigenfunctions for some specific values of those moduli which, in this sense, become also quantized. Their values turn out to correspond to the energy sets of the corresponding integrable system (made out of commuting hamiltonians).
The spectrum and eigenfunctions of the operator can be numerically well approximated using the difference equation (2.7) and a hamiltonian truncation method on a basis of , which we take to be the harmonic oscillator basis. As we will see, all the numerical results used to test against exact expressions are generated using this technique.
with given by (2.4). The point is that there exist two other operators which commute with this operator, and so all the algebra spanned by them commutes with . Indeed, for any integers , we have
For example, the following operator , which we call the dual operator,
commutes with , and so it can be diagonalized simultaneously by the set of on-shell eigenfunctions of the operator . The eigenvalue of will be denoted . The operator has essentially the same form as , but a priori the different moduli and parameters can take arbitrary values. In the following, we will see some examples of relations between and (and and ), and how they arise concretely. The difference equation given by the dual operator is
By rescaling the eigenfunction , and renaming
which is exactly of the same form as the initial difference equation (2.7), but using the dual variables. We call this the dual difference equation.
Let us comment on the role of the dual difference equation. If we consider only the difference equation given by , we remark that any -periodic function can be multiplied to a solution in order to get another solution. By requiring that a solution is simultaneously a solution for the dual difference equation given by , we should obtain a unique eigenfunction for each level, since this constraint should fix the multiplicative ambiguity. Since the dual difference equation has the same form as the original difference equation, the small expansion and the small of the eigenfunction should be closely related. We will use this argument when we will want to construct the eigenfunction from the resummed WKB.
2.2 WKB eigenfunction at small
One possible approach to study the difference equation (2.7) is to consider the small regime, using the well known WKB ansatz:
Using this ansatz, for any we can write
Inserting this into the difference equation and expanding everything at small , we can recursively solve for order by order in . Then, we can integrate to obtain . Let us define to be the solution of . We find for the first orders
The natural domain of is not the -plane, but the spectral curve itself, which is a multi-sheeted cover of the plane. Since we consider the hyperelliptic case only, we have two sheets.
At large , we have
where is an order 2 polynomial, is an order 1 polynomial, and is equal to minus the polynomial part in which appears in the large expansion. Both and are independent of . It can be verified that the higher are only functions of : by this we mean that there is no polynomial in in the large expansion. We use this remark and build the truncated WKB function
which only depends on through .
2.3 Resummed WKB from recursion
By adapting the manipulations done in , we can resum the small WKB expansion order by order in another expansion parameter, here . Let us define
Shifts of in in the eigenfunction correspond to multiplying by in the truncated function . The difference equation can be rewritten in terms of only, by using the explicit forms of . In our hyperelliptic cases, it can be put in the form
where and are rational functions of , which also may depend on the moduli and parameters, as well as on . It is useful to change variables and use
Using this, we find that (for the appropriate parametrization) the difference equation takes the rather general form
where is a strictly positive integer, and is the remaining part coming from the difference equation. This form suggests that we can solve this -equation in a large expansion. The ansatz we use is
The leading part is universal, and is essentially a quantum dilogarithm:
To perform the recursion at large , we divide everything by and use that
At each order in large , we get a linear equation determining recursively. The recursion can be solved for , which are rational functions of . The functions are given by
which can be formally solved in the following way:
This is especially useful if we work with an -expanded , since
Also, we find that the larger is, the larger is the leading power of in the large expansion of . In the end, after going back to the original variable , we find the following structure:
and so, formally,
In the above, we have collected the into the vector . The are polynomials in the variables and . If we expand this expression at small , we retrieve the large expansion of all the WKB corrections. So this expression is effectively a resummation of the small WKB expansion. To illustrate this procedure, we give here as examples the resummed WKB eigenfunctions for some cases which are associated to mirror curves of toric Calabi-Yau threefolds.
For the geometry called local , we have only one true modulus which is . The mirror curve and the WKB eigenfunction are
For the geometry called local , we have one true modulus which is , and one extra parameter which is a mass parameter. We rename it . The mirror curve and the WKB eigenfunction are
For the geometry called the resolved , we have two true moduli, and an extra one which is . We rename this second one . The mirror curve and the WKB eigenfunction are
In each of these three cases, the polynomial part of the large WKB expansion is given by
3 The rational case
In the following, we focus on the rational case, where is given by times a rational number. Using pole cancellation and modularity, we manage to write down an exact formula for the on-shell eigenfunction . The truncated WKB eigenfunction is the only ingredient, but it comes with its modular dual which is invisible in the small WKB expansion. As we will see, the resummation of the large expansion of can be down explicitly in the rational case.
3.1 Pole cancellation and modular duality
When is of the form
for positive coprime integers and , the quantity is a root of unity:
The formal solution (2.32) is ill-defined since it has poles when is a multiple of . We introduce a regulating parameter and consider the small expansion by setting
We expand (2.32) in small by using
As it is, the naive resummation of the WKB expansion given by is singular at rational . We conclude that it has to be corrected by something which
1) is non pertubative at small so that it is invisible in the small WKB expansion,
2) cancels the poles in the rational case.
Also, we have not taken into account the modular duality structure outlined in the previous section. Indeed, our point of view was to start with the small WKB resummation of the eigenfunction. However, we could have equally well started from the dual equation (2.13) also satisfied by the on-shell eigenfunction, and consider its small WKB expansion. By doing the same recursive procedure, we would end up with a very similar expression for the truncated dual-WKB eigenfunction
where , and . The are precisely the same polynomials as in (2.32), since the dual equation is of the same form as the initial equation. So we would expect an eigenfunction which is symmetric under the exchanges and . Following what is suggested in , let us add its dual to the resummed WKB, which is a non-perturbative contribution at small :
The dual part also has poles when for integer . Using (3.3) and expanding at small , we obtain
for all positive integers . This defines relations
Since the are polynomials in and , these relations are algebraic at fixed . Of course the system seems strongly overdetermined, but nevertheless, we find that there actually are solutions as a consequence of the form of the . Some examples can be found below. The on-shell values of depend on , so we should write and . Once these relations are fixed, our claim is that (3.7) is the full non-pertrubatively complete truncated WKB eigenfunction in the large expansion.
Let us remark quickly that, in contrast with , we did not make use of the so called quantum mirror map, or quantum A-periods. In  (and also in the setup of the TS/ST correspondence of [7, 8, 10]), the quantum mirror map fixes the relations between the moduli/parameters and and their duals and . Here, we impose these relations in the rational case using pole cancellation. This is less general but in some sense more natural and straightforward from the point of view of the difference equation.
3.2 Finite contribution
Requiring cancellation of poles and modular duality is what motivated us to write expression (3.7). Let us now work out the finite terms at . We insist that when varying , we should also vary the modulus (corresponding to the the eigenvalue), since the proposal is expected to be only valid on-shell. By using that , the first part gives
where . We have denoted the derivatives of . Every term in the sum above can be written using the function defined in (2.31), which is finite for any value of . Also, as we will see below, it can be obtained exactly for any integers . For integers we have that . We define for ,
The instance with , which appears often, can also be written
The function also appears in the following combination:
So (3.11) becomes
The function can be obtained through
whereas the functions and can be obtained by direct differentiation of .
Let us now look at the expansion of the dual part. We find
Explicitly, . Also, and have several sources of dependance, and we have denoted their total derivative w.r.t. by and . As before, we can write all the contributions in terms of a unique function , which is defined as
This is basically where we replaced all the variables by their duals. As before, we define for
Using (3.9), we see that the special case is related to ,
Now that we have performed the variation with respect to , it is considered to be fixed in what follows. The relation between parameters and their duals are the algebraic ones (3.10) at fixed . So we can write
and similarly for the term with the derivative w.r.t. . These are exactly the kind of terms appearing in the expansion of the first part. So (3.17) becomes