1 Introduction


Exact solutions to quantum spectral curves by topological string theory

Jie Gu, Albrecht Klemm, Marcos Mariño, Jonas Reuter

Bethe Center for Theoretical Physics, Physikalisches Institut, Universität Bonn, 53115 Bonn, Germany Département de Physique Théorique et Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland


We generalize the conjectured connection between quantum spectral problems and topological strings to many local almost del Pezzo surfaces with arbitrary mass parameters. The conjecture uses perturbative information of the topological string in the unrefined and the Nekrasov–Shatashvili limit to solve non-perturbatively the quantum spectral problem. We consider the quantum spectral curves for the local almost del Pezzo surfaces of , , and a mass deformation of the del Pezzo corresponding to different deformations of the three-term operators , and . To check the conjecture, we compare the predictions for the spectrum of these operators with numerical results for the eigenvalues. We also compute the first few fermionic spectral traces from the conjectural spectral determinant, and we compare them to analytic and numerical results in spectral theory. In all these comparisons, we find that the conjecture is fully validated with high numerical precision. For local we expand the spectral determinant around the orbifold point and find intriguing relations for Jacobi theta functions. We also give an explicit map between the geometries of and as well as a systematic way to derive the operators from toric geometries.

June, 2015


1 Introduction

Topological string theory on Calabi–Yau (CY) threefolds can be regarded as a simplified model for string theory with many applications in both mathematics and physics. Topological strings come in two variants, usually called the A– and the B–models, related by mirror symmetry. When the CY is toric, the theory can be solved at all orders in perturbation theory with different techniques. The A–model can be solved via localization [1, 2] or the topological vertex [3], while the B–model can be solved with the holomorphic anomaly equations [4, 5] or with topological recursion [6, 7]. Part of the richness and mathematical beauty of the theory in the toric case stems from the interplay between these different approaches, which involve deep relations to knot theory, matrix models, and integrable systems.

In spite of these developments, there are still many open questions. Motivated by instanton counting in gauge theory [8], it was noted [9] that the topological string on toric CYs can be “refined,” and an additional coupling constant can be introduced. Although many of the standard techniques in topological string theory can be extended to the refined case [10, 11, 12, 13], this extension is not as well understood as it should (for example, it does not have a clear worldsheet interpretation). Another realm where there is much room for improvement is the question of the non-perturbative completion of the theory. Topological string theory, as any other string theory, is in principle only defined perturbatively, by a genus expansion. An important question is whether this perturbative series can be regarded as the asymptotic expansion of a well-defined quantity. In the case of superstring theories in AdS, such a non-perturbative completion is provided conjecturally by a CFT on the boundary. In the case of topological string theory on CY threefolds, there is a similar large duality with Chern–Simons theory on three-manifolds, but this duality only applies to very special CY backgrounds [14, 15]2.

One attractive possibility is that the topological string emerges from a simple quantum system in low dimensions, as it happens with non-critical (super)strings. Since the classical or genus zero limit of topological string theory on a toric CY is encoded in a simple algebraic mirror curve, it has been hoped that the relevant quantum system can be obtained by a suitable “quantization” of the mirror curve [16]. In [17], it was shown that a formal WKB quantization of the mirror curve makes it possible to recover the refined topological string, but in a special limit –the Nekrasov–Shatashvili (NS) limit– first discussed in the context of gauge theory in [18]. The quantization scheme in [17] is purely perturbative, and the Planck constant associated to the quantum curve is the coupling constant appearing in the NS limit.

Parallel developments [19, 20, 21, 22, 23, 24] in the study of the matrix model for ABJM theory [25] shed additional light on the quantization problem. It was noted in [24] that the quantization of the mirror curve leads to a quantum-mechanical operator with a computable, discrete spectrum. The solution to this spectral problem involves, in addition to the NS limit of the refined topological string, a non-perturbative sector, beyond the perturbative WKB sector studied in [17]. Surprisingly, this sector involves the standard topological string. The insights obtained in [24] thanks to the ABJM matrix model apply in principle only to one particular CY geometry, but they were extended to other CYs in [26], which generalized the method of [24] for solving the spectral problem. A complete picture was developed in [27], which made two general conjectures valid in principle for arbitrary toric CYs based on del Pezzo surfaces: first, the quantization of the mirror curve to a local del Pezzo leads to a positive-definite, trace class operator on . Second, the spectral or Fredholm determinant of this operator can be computed in closed form from the standard and NS topological string free energies. The vanishing locus of this spectral determinant gives an exact quantization condition which determines the spectrum of the corresponding operator. The first conjecture was proved, to a large extent, in [28], where it was also shown that the integral kernel of the corresponding operators can be expressed in many cases in terms of the quantum dilogarithm. The second conjecture has been tested in [27] in various examples.

The conjecture of [27] establishes a precise link between the spectral theory of trace class operators and the enumerative geometry of CY threefolds. From the point of view of spectral theory, it leads to a new family of trace class operators whose spectral determinant can be written in closed form — a relatively rare commodity. From the point of view of topological string theory, the spectral problem provides a non-perturbative definition of topological string theory. For example, one can show that, as a consequence of [27], the genus expansion of the topological string free energy emerges as the asymptotic expansion of a ’t Hooft-like limit of the spectral traces of the operators [38].

The conjecture of [27] concerning the spectral determinant has not been proved, but some evidence was given for some simple CY geometries in [27]. Since the conjecture holds in principle for any local del Pezzo CY, it is important to test this expectation in some detail. In addition, working out the consequences of the conjecture in particular geometries leads to many new, concrete results for both, spectral theory and topological string theory. The goal of this paper is to test the conjecture in detail for many different del Pezzo geometries, in particular for general values of the mass parameters, and to explore its consequences. In order to do this, we use information on the refined topological string amplitudes to high genus, which lead for example to precision tests of the formulae for the spectral traces of the corresponding operators.

In more detail, the content of this paper is organized as follows. In Section 2 we explain in detail how to obtain the geometries appropriate for operator analysis from mirror symmetry of global orbifolds. As an example, we work out the mass-deformed del Pezzo, which realizes a perturbation of the three-term operator considered in [28]. In Section 3, we review and expand the conjecture of [27], as well as some of the results on the spectral theory of quantum curves obtained in [28, 39]. In Section 4, we apply these general ideas and techniques to four different geometries: local , local , local and the mass deformed del Pezzo surface. In all these cases we compute the spectrum as it follows from the conjectural correspondence, and we compare it to the numerical results obtained by direct diagonalization of the operators. We also compute the first few fermionic spectral traces, as they follow from the conjectural expressions for the spectral determinants, and we compare them with both analytic and numerical results. In the case of local , we work out the explicit expansion at the orbifold point. This leads to analytic expressions for the spectral traces, in terms of Jacobi theta functions and their derivatives. In the case of the operator, we also compare the large limit of its fermionic spectral traces, obtained in [38], to topological string theory at the conifold point. The conjecture turns out to pass all these tests with flying colors. In the Appendices, we collect information on the Weierstrass and Fricke data of local CY manifolds, and we explain the geometric equivalence between local and local .

2 Orbifolds, spectral curves and operators

As we mentioned in the introduction, the conjecture of [27] associates a trace class operator to mirror spectral curves. Let us denote the variables appearing in the mirror curve by , . The corresponding Heisenberg operators, which we will denote by , , satisfy the canonical commutation relation


Since the spectral curves involve the exponentiated variables , after quantization one finds the Weyl operators


As shown in [28], the simplest trace class operator built out of exponentiated Heisenberg operators is




For example, the operator arises in the quantization of the mirror curve to the local geometry. Since these operators can be regarded as building blocks for the spectral theory/topological string correspondence studied in this paper, it is natural to ask how to construct local toric geometries which lead to operators after quantization.

It turns out that, to do this, one has to consider an orbifold with a crepant resolution. This means that the resolution space is a non-compact Calabi-Yau manifold, i.e. it has to have trivial canonical bundle. The section of the latter on has to be invariant and it is not hard to see that this condition is also sufficient. For abelian groups, is the most general choice in the geometrical context3, and has a toric description. In fact all local toric Calabi-Yau spaces can be obtained by elementary transformations, i.e. blow ups and blow downs in codimension two, from .

2.1 Toric description of the resolution of abelian orbifolds

Let be the order of . Invariance of implies that the exponents of the orbifold action of the group factor on the coordinates defined by


add up to for . The resolution leading to the A–model geometry with abelian is described by standard toric techniques [53], while the procedure that leads to the B–model curve is an adaptation of Batyrev’s construction to the local toric geometries [54][29]. The toric description of the resolution, see [53], is given by a non-complete three dimensional fan in , whose trace at distance one from the origin is given by an integral simplicial two dimensional lattice polyhedron . Let , , , be the set of exponents of all elements of , then the two dimensional polyhedron is simplicial and is the convex hull of


in the smallest lattice generated by the points . Let us give the two fundamental types of examples of this construction.

Consider as type (a) generated by (2.5) , with and , is the convex hull of . The point is by (2.6) an inner point of , which we choose to be the origin of , while is spanned by , . Choosing the canonical basis and for and dropping the redundant first entry in the coordinates of , we find that


We will argue below that the mirror curve seen as the Hamiltonian always contains an operator of type .

Consider as type (b) with generated by (2.5), where with , and with . We require and and either4 or . The point is by (2.6) an inner point of , which we choose to be the origin of , while we can span by and . Choosing the canonical basis and for we find similarly as before


Let be the number of all lattice points of that lie only inside faces of dimension and not inside faces of dimension , and all points on dimension faces. , i.e. the number of lattice points inside , counts compact (exceptional) divisors of the smooth non-compact Calabi-Yau 3-fold , while , i.e. the number of lattice points inside edges, counts non-compact (exceptional) divisors of , which are line bundles over exceptional ’s. Their structure can be understood as follows. If with is a subgroup of that leaves a coordinate in invariant, then it acts as on the remaining and its local resolution contains an type Hirzebruch sphere tree of ’s whose intersection in is the negative Cartan matrix of the Lie algebra . These ’s are represented in the toric diagram as lattice points on the edge of that is dual to the invariant coordinate.

In the mirror geometry described below, is identified with the genus and the number of complex structure parameters deformations , , of the family of mirror curves , while counts independent residua , , of the meromorphic differential on that curve. In the field theory, correspond to vevs of dynamical fields while the are mass parameters5. In the resolution , the parameters are associated by the mirror map to the volumes of the curves determining the volume of the compact (exceptional) divisors, while the parameters are associated by the mirror map to the volumes of the of the sphere trees in the resolution of the singularities. The curve classes that bound the Kähler cone are linear combinations of these curves classes. The precise curve classes with that property are encoded in the generators of the Mori cone.

For orbifolds is simplicial. Thus it is elementary to count




Let us give a short overview over local Calabi-Yau geometries that arise as resolved orbifolds. We have seen that has always an inner point which we called and by (2.9, 2.10) it is easy to see that in the case (a) the orbifold with , the orbifold with , and the orbifold with are the only orbifolds whose mirrors are related to elliptic curves, i.e. . It is easy to see that is respectively. For one has several choices of the exponents, e.g. for the choice leads to a genus two mirror curve. In the case (b) orbifolds with genus one mirror curves are the orbifold with and , the orbifold with and , and the with and .

2.2 The mirror construction of the spectral curves

Above we described toric local Calabi–Yau threefolds that arise as resolved abelian orbifolds and can serve as A–model geometries for topological string. Let be, a bit more general, an arbitrary non-complete toric fan in , not necessarily a simplicial trace, and . The Calabi–Yau condition is equivalent to the statement that the 1–cone generators , end on a hyperplane one unit distance away from the origin of , and . We choose the coordinate system of such that the first coordinate of is always 1. The 1–cone generators satisfy linear relations. If the Mori cone is simplicial, we can choose them to be the Mori cone generators6 with , such that


Due to their interpretation in 2d supersymmetric gauged linear sigma models, are also called the charge vectors. The triviality of the canonical bundle is ensured if


To construct the Calabi–Yau threefold on which the mirror B–model topological string lives [54][29], one introduces variables in satisfying the conditions


Then the mirror manifold is given by




Due to the three independent actions on the subject to the constraints (2.13), only the following combinations


are invariant deformations of the B–model geometry. If are the Mori cone generators, the locus is the large complex structure point, which corresponds to the large volume limit of the A–model geometry. The parametrize the deformations of . It is equivalent and often more convenient to replace (2.13) and (2.15) by




respectively. Using (2.17) one eliminates of the variables. One extra variable can be set to using the overall action. Renaming the remaining two variables and the mirror geometry (2.14) becomes


which describes a hypersurface in . Note that all deformations of are encoded in . In fact the parameter dependence of all relevant amplitudes of the B–model on can be studied from the non-compact Riemann surface given by the vanishing locus of the Newton–Laurent polynomial in


and the canonical meromorphic one form on , a differential of the third kind with non-vanishing residues, given as


Because of its rôle in mirror symmetry and the matrix model reformulation of the B–model, is called the mirror curve or the spectral curve respectively, while is the local limit of the holomorphic Calabi–Yau form on the B–model geometry.

The coefficients , , of the monomials that correspond to inner points parametrize the complex structure of the family of mirror curves. To see this, note that all other coefficients can be set to one by automorphisms of a compactification of the mirror curve (2.20), e.g. of , which do not change the complex structure. However the other datum of the B–model, the meromorphic one form , is only invariant under the three actions on the coordinates of . Therefore depends on coefficients of the monomials on the boundary. We will set the coefficients of three points on the boundary to one, e.g. , in Figure 2.1. The coefficients of the other points on the boundary are then the mass parameters , . In this way the can be seen as functions of the complex structure variables and the independent mass parameters .

Let us consider the orbifold geometry with the trace given in (2.7). To get the desired operator from the mirror curve, we associate to the point , to the point , and scale the coordinate that corresponds to the point to , while we denote the coefficient of the coordinate by . This choice guarantees that the coordinate associated to the point is expressed by solving (2.17) as . Let us set all the other for , then the mirror curve has the shape


where are monomials of mass parameters. Note that the function can be regarded as a “perturbation” of the function


and will be identified with the energy of the quantum system discussed below. (2.23) is the function which, upon quantization, leads to the operator (2.4). If , then the limit , , corresponds to a partial blow up of the orbifold . Recall that all points on the trace and the corresponding bounding fans as coordinate patches have to be included to define as a smooth variety.

In the rest of the paper we will only be concerned with the cases where . This corresponds to smooth toric local Calabi–Yau threefolds whose spectral curves are elliptic curves. In particular, we consider the anti-canonical bundles of almost del Pezzo surfaces


which have toric descriptions in terms of traces , which are one of the 16 2-d reflexive polyhedra7. All of these except one, which involves a blow up, can be obtained by blow downs from the orbifold geometries discussed in the last section. In order to treat the toric cases in one go, we consider the largest polyhedra for abelian group quotients with depicted in Figure 2.1.

Figure 2.1: Toric traces for with and . They correspond to case (b) in Section 2.1 with and respectively.

We compactify the corresponding mirror curves (2.20) in , but do not use the automorphism to eliminate the . Rather we bring the corresponding mirror curves to the Weierstrass form


using Nagell’s algorithm, see Appendix 6.1. In particular in that appendix we give in (6.3) and (6.4) the and for the mirror geometries of and . They can be specialized to the corresponding data of all examples discussed in detail in the paper, by setting parameters in these formulae to zero or one according to the embedding of the smaller traces into the traces depicted in Figure 2.1.

Let us introduce some conventions, which are usefull latter on. After gauging three coefficients of the boundary monomials to one by the action, (2.16) becomes The charge is the intersection number of the anti canonical class and the curve in the curve class that bound the corresponding Mori cone generator on . Any such curve has a finite volume and lies entirely in . Since is almost del Pezzo


We define


and the reduced curve degrees


as well as


Then (2.16) implies


In [33, 35] is used as the default elliptic modulus instead of , because is the large complex structure point (LCP) in the moduli space of , and therefore convenient for computations around the LCP. In the following we will use the two variables interchangeably, preferring for the formal discussions related to the spectral problems, and for computations around the LCP.

Both data (2.20,2.21) are only fixed up to symplectic transformations


which preserve . In the rest of the paper, we will often call (2.20) the spectral curve of as well.

2.3 Weierstrass data, Klein and Fricke theory and the B–model solution

According to the theory of Klein and Fricke we get all the information about the periods and the Picard–Fuchs equations for the holomorphic differential, which reads

in the Weierstrass coordinates of an elliptic curve, from properly normalized and and the -invariant of the elliptic curve


A key observation in the treatment of Klein and Fricke is that any modular form of weight , w.r.t. (or a finite index subgroup ), fulfills as a function of the corresponding total modular invariant (or ) a linear differential equation of order , see for an elementary proof [58]. In particular can be meromorphic and the basic example [59] is that can be written as the solution to the standard hypergeometric differential equation as


While solutions to the hypergeometric equation transform like weight one forms, other such objects such as in particular the periods can be obtained by multiplying them with (meromorphic) functions of the total invariant (or , which is a finite Galois cover of ). For example the unnormalized period is a weight one form that fulfills the second order differential equation


which is simply to be interpreted as the Picard–Fuchs equation for . It is easy to see that another way to write a solution to (2.34) is . These or dependent meromorphic factors can be fixed by global and boundary properties of the periods. In particular one can get the normalized solutions of the vanishing periods of at a given cusp as


for properly normalized . Note that the mass parameters appear in this theory as deformation parameters, which are generically isomonodronic8. Similarly the normalized dual period to (2.35) is for and





as readily obtained from the Frobenius method for hypergeometric functions.

The monodromy group for loops on the -plane acts on (2.35), (2.36) as a subgroup of index inside , where is the branching index of the Galois cover of to defined by (2.32) and , where is the Galois group of the covering (2.32).

In (2.35), (2.36) is the flat coordinate and the derivative of the prepotential w.r.t. the former near the corresponding cusp9. These structures exist due to rigid special geometry and the fact that near the large complex structure point is a generating function for geometric invariants of holomorphic curves of genus zero in the Calabi-Yau .

The refined amplitudes and are given in (4.17) and (4.21) respectively. The refined higher amplitudes can be defined recursively by the refined holomorphic anomaly equation [10, 60]


where is the almost holomorphic second Eisenstein series, which is a weight two form under , and the prime on the sum means that and are omitted. is a model dependent constant. It is convenient to define the an-holomorphic generator , as well as and , so that by virtue of the Ramanujan relations


the r.h.s. of (2.38) becomes a polynomial in , while the derivatives w.r.t. can be converted to derivatives w.r.t.


It follows that


in other words, is a polynomial of degree in , where is determined by (2.40), while is determined from the regularity conditions on and the gap behaviour at the conifold divisor [10]. The refined BPS states can be ontained from the large radius expansion of the .

2.4 The mass deformed geometry

Let us exemplify this construction with the function , leading to the operator . The polyhedron is depicted below.

Figure 2.2: The polyhedron 10 with the choice of the mass parameters and the modulus .

The Mori cone vectors, which correspond to the depicted triangulation, are given below


Following the procedure described in (2.17), one obtains the standard form of the Newton–Laurent polynomial as


The monomials are ordered as the points in the figure and we rescaled and and multiplied by .

With the indicated three mass parameters and the parameter , the Mori vectors determine the following large volume B–model coordinates


The anti-canonical class of the del Pezzo corresponds to an elliptic curve, which in turn has the following Mori vector


This equation implies that is the correct large volume modulus for this curve independent of the masses. By specializing the expression in Appendix 6.1 as and scaling with we get the following coefficients of the Weierstrass form:


Note there is a freedom of rescaling by an arbitrary function

without changing the Weierstrass form, if the coordinates of the Weierstrass form are also rescaled accordingly. Our particular choice of scaling makes sure that and becomes the logarithmic solution at the large complex structure point at , which corresponds to . We get as the transcendental mirror map , with . The non-transcendental rational mirror maps are