Spectral Theory and Mirror curves of Higher Genus
Recently, a correspondence has been proposed between spectral theory and topological strings on toric Calabi–Yau manifolds. In this paper we develop in detail this correspondence for mirror curves of higher genus, which display many new features as compared to the genus one case studied so far. Given a curve of genus , our quantization scheme leads to different trace class operators. Their spectral properties are encoded in a generalized spectral determinant, which is an entire function on the Calabi–Yau moduli space. We conjecture an exact expression for this spectral determinant in terms of the standard and refined topological string amplitudes. This conjecture provides a non-perturbative definition of the topological string on these geometries, in which the genus expansion emerges in a suitable ’t Hooft limit of the spectral traces of the operators. In contrast to what happens in quantum integrable systems, our quantization scheme leads to a single quantization condition, which is elegantly encoded by the vanishing of a quantum-deformed theta function on the mirror curve. We illustrate our general theory by analyzing in detail the resolved orbifold, which is the simplest toric Calabi–Yau manifold with a genus two mirror curve. By applying our conjecture to this example, we find new quantization conditions for quantum mechanical operators, in terms of genus two theta functions, as well as new number-theoretic properties for the periods of this Calabi–Yau.
Département de Physique Théorique et Section de Mathématiques,
Université de Genève, Genève, CH-1211 Switzerland \emailAddsantiago.firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
It has been conjectured in  that there is a precise correspondence between the spectral theory of certain operators and local mirror symmetry. This correspondence postulates that the Weyl quantization of mirror curves to toric Calabi–Yau (CY) threefolds leads to trace class operators on , and that the spectral determinant of these operators is captured by topological string amplitudes on the underlying CY. As a corollary, one finds an exact quantization condition for their spectrum, in terms of the vanishing of a (deformed) theta function. The correspondence unveiled in  builds upon previous work on the quantization of mirror curves [2, 3] and on the relation between supersymmetric gauge theories and quantum integrable systems . It incorporates in addition key ingredients from the study of the ABJM matrix model at large [5, 6, 7, 8, 9, 10]. These ingredients are necessary for a fully non-perturbative treatment, beyond the perturbative WKB approach of  and of other recent works on the quantization of spectral curves.
The correspondence of  between spectral theory and topological strings can be used to give a non-perturbative definition of the standard topological string. The (un-refined) topological string amplitudes appear as quantum-mechanical instanton corrections to the spectral problem, and due to their peculiar form, they can be singled out by a ’t Hooft-like limit of the so-called fermionic spectral traces of the operator. In addition, by using the integral kernel of the operator, which was determined explicitly in  in many cases, one can write down a matrix model whose expansion gives exactly the genus expansion of the topological string [12, 13]. Therefore, one can regard the correspondence of  as a large Quantum Mechanics/topological string correspondence, with many features of large gauge/string dualities. In particular, it is a strong/weak duality, since the Planck constant in the quantum-mechanical problem, , is identified as the inverse string coupling constant.
All the examples of the correspondence that have been studied so far involve local del Pezzo CYs, and their mirror curve has genus one [1, 12, 13, 14].
It was pointed out in  that the relationship between the spectral theory of trace class operators and topological string amplitudes
should hold for general toric CYs, i.e. it should hold for mirror curves of arbitrary genus. In this paper we present a compelling picture for the
spectral theory/mirror symmetry correspondence in the higher genus case. This generalization involves some new ingredients.
In the theory developed in  for the genus one case, the basic object is the spectral determinant of the trace class
operator obtained by quantization of the mirror curve. It turns out that a curve of genus leads to different operators, which are related by
The fact that we obtain a single quantization condition from a curve of genus might be counter-intutitive to readers familiar with quantum integrable systems, like for example the quantum Toda chain and its generalizations. In those systems, the quantization of the spectral curve leads to quantization conditions. This is of course due to the fact that the underlying quantum-mechanical system is -dimensional, and there are commuting Hamiltonians that can (and should) be diagonalized simultaneously. It should be noted, however, that the spectral curve by itself does not carry this additional information. In fact, in the case of quantum mechanical problems on the real line it is quite common that the quantization of a higher genus curve leads to a single quantization condition. This is what happens, for example, for the Schrödinger equation with a confining, polynomial potential of higher degree.
As we have just mentioned, one of the main consequences of the conjecture of  is that it provides a non-perturbative definition of topological string theory. This can be also generalized to the higher genus case: as we show in this paper, the generalized spectral determinant leads to fermionic spectral traces , depending on non-negative integers . In the ’t Hooft limit
these traces have the asymptotic expansion
where are the genus free energies of the topological string, in an appropriate conifold frame. In particular, we can regard these fermionic spectral traces, which are completely well defined objects, as non-perturbative completions of the topological string partition function.
The theory of quantum mirror curves of higher genus is relatively intricate, and we develop it in full detail for what is probably the simplest genus two mirror curve, namely, the total resolution of the orbifold. We perform a detailed study of the associated spectral theory, and in particular we determine the vanishing locus of the spectral determinant on the two-dimensional moduli space, in the so-called maximally supersymmetric case . In addition, we give compelling evidence that the expansion of the topological string free energies near what we call the maximal conifold locus gives the large expansion of the fermionic spectral traces. This provides a non-perturbative completion of the topological string on this background. As a bonus, we obtain non-trivial identities for the values of the periods of this CY at the maximal conifold locus in terms of the dilogarithm, in the spirit of [15, 16].
The organization of this paper is the following. In section 2, we develop the theory of quantum operators associated to higher genus mirror curves and we construct the appropriate generalization of the spectral determinant. In section 3 we present an explicit, conjectural expression for the spectral determinant in terms of topological string amplitudes, and we explain how the large limit of the spectral traces provides a non-perturbative definition of the all-genus topological string free energy. In section 4, we test these ideas in detail in the example of the resolved orbifold. In section 5, we conclude and present some problems for future research. The Appendix summarizes information about the special geometry of the resolved orbifold which is needed in section 4.
2 Quantizing mirror curves of higher genus
2.1 Mirror curves
In this paper we will consider mirror curves to toric CY threefolds, and we will promote them to quantum operators. Let us first review some well-known facts about local mirror symmetry [17, 18] and the corresponding algebraic curves. The toric CY threefolds which we are interested in can be described as symplectic quotients,
where . The quotient is specified by a matrix of charges , , . The CY condition requires the charges to satisfy 
The mirror CY manifold is given by
The complex parameters , , give a redundant parametrization of the moduli space, and some of them can be set to one. Equivalently, we can consider instead the coordinates
The constraints (2.3) have a three-dimensional family of solutions. One of the parameters corresponds to a translation of all the coordinates:
which can be used for example to set one of the s to zero. The remaining coordinates can be expressed in terms of two variables which we will denote by , . There is still a group of symmetries left, given by transformations of the form ,
After solving for the variables in terms of , , one finds a function
defines a Riemann surface embedded in . We will call (2.10) the mirror curve to the toric CY threefold . All the information about the closed string amplitudes on is encoded in , as shown in [23, 24, 25].
The equation of the mirror curve (2.10) can be written down in detail, as follows. Given the matrix of charges , we introduce the vectors,
satisfying the relations
In terms of these vectors, the function (2.9) can be written as
Clearly, there are many sets of vectors satisfying these constraints, but they differ in reparametrizations and overall factors (as we explained above), and therefore they define the same Riemann surface. The genus of this Riemann surface, , depends on the toric data, encoded in the matrix of charges, or equivalently in the vectors . Among the parameters (2.6), there will be “true” moduli of the geometry, and in addition there will be “mass parameters”, which lead typically to rational mirror maps (this distinction has been emphasized in [26, 27].)
The quantization of mirror curves studied in , building on [2, 3], is simply based on Weyl quantization of the function (2.9), i.e. the variables , are promoted to Heisenberg operators , satisfying
In the genus one case, when the CY is the canonical bundle over a del Pezzo surface ,
the function (2.9) can be written in a canonical form, as
where is the modulus of the Riemann surface. The quantum operator associated to the toric CY threefold, , is obtained by Weyl quantization of the function , and plays the rôle of (minus) the exponentiated energy, or the fugacity.
The higher genus case is much richer, due to the fact that there are different moduli for the curve. As a consequence, there will be different “canonical” forms for the curve, which we will write as
Here, is a modulus of , and in practice it is one of the s appearing in (2.13). Of course, the different canonical forms of the curves are related by reparametrizations and overall factors, so we will write
where is a monomial of the form . Equivalently, we can write
We can now perform a standard Weyl quantization of the operators . In this way we obtain different operators, which we will denote by , . These operators are Hermitian. The relation (2.18) becomes,
where is the operator corresponding to the monomial . In this relation, the “splitting” of in two square roots is due to the fact that we are using Weyl quantization, which leads to Hermitian operators. The expression (2.19) becomes, after promoting both sides to operators,
We can regard the operator as an “unperturbed” operator, while the moduli encode different perturbations of it. We will also need,
By comparing the coefficients of in the relation (2.20), we find
Amusingly, these relationships are a sort of non-commutative version of the the relations between transition functions in the theory of bundles. We will set, by convention,
We also have
Before proceeding, let us examine some examples to illustrate the considerations above.
The resolved orbifold. Let us consider the CY given by the total resolution of the orbifold , where the action has weights . This geometry has been studied in detail in various references, like for example [28, 29, 30, 31], and (refined) topological string amplitudes on this background have been recently calculated in . The vectors of charges are given by
To parametrize the moduli space, we introduce five variables , as well as the combinations
A useful choice of vectors for this example is
and the equation for the Riemann surface reads, after setting ,
However, it is easy to see that one can also choose the vectors
which leads to the equation
In Fig. 1 we show the vectors for the system (2.31) (this is sometimes called a height one slice of the fan (2.31)). Of course, although we have chosen the same notations, the variables , appearing in (2.32) are not the same ones appearing in (2.30). Rather, they are related by a canonical transformation,
In this case, the two canonical functions and are given by
and the moduli are
In the coordinates appropriate for , we have , while in the coordinates appropriate for , we have . In terms of the three-term operators introduced in ,
the unperturbed operators are
The resolved orbifold, or geometry. Let us now consider the total resolution of the orbifold , where the action has weights . This is precisely the geometry studied in the first papers on local mirror symmetry [17, 18], which engineers geometrically Seiberg–Witten theory. It has also been studied in some detail in . In this case, the charge vectors are
Like before, we can parametrize the moduli space with six coordinates , , or in terms of
The coordinates , are true moduli of the curve, while is rather a mass parameter . A useful choice of vectors is,
and after setting , we find the curve
It is easy to see that there is another realization of this curve as
Here, we can regard as a parameter, and , as the moduli. The canonical operators derived from this geometry are then given by
They can be regarded as perturbations of , and of , respectively. ∎
It was noted in , in the genus one case, that the most interesting operator was not really , but rather its inverse . The reason is that is expected to be of trace class and positive-definite, therefore it has a discrete, positive spectrum, and its Fredholm (or spectral) determinant is well-defined. It was rigorously proved in  that, in many cases, this is the case, provided the parameters appearing in the operators satisfy certain positivity conditions. In analogy with the genus one case, we expect the operators
to exist, be of trace class and positive-definite. In the concrete examples that we have considered, this actually follows from the results in . In that paper, it was shown that
exists and are of trace class. It was also shown that the inverse of
2.3 The generalized spectral determinant
According to the conjecture of , when the mirror curve has genus one, many important aspects of the spectral theory of can be encoded in the topological string amplitudes on . We would like to generalize this to mirror curves of higher genus. What are the natural questions that we would like to answer from the point of view of spectral theory? Clearly, we would like to know the spectrum of the operators in terms of enumerative data of , and in addition, as in , we would like to have precise formulae for the spectral determinants of their inverses . However, one should note that, due to (2.20), the operators are closely related, and their spectra and spectral determinants are not independent.
In a more fundamental sense, we need an appropriate multivariable generalization of the spectral determinant. In the genus one case, when is a local del Pezzo of the form (2.15), there is one single modulus , and the spectral determinant
where we denoted the eigenvalues of the positive definite, trace class operator by , . A more useful definition, advocated by Grothendieck  and Simon [32, 33], involves the fermionic spectral traces , defined as
In this expression, the operator is defined by acting on . A theorem of Fredholm  asserts that, if is the kernel of , the fermionic spectral trace can be computed as a multi-dimensional integral,
The spectral determinant is then given by the convergent series,
Another definition of the Fredholm determinant is based on the Fredholm–Plemelj’s formula,
In the higher genus case, there should exist a generalization of the spectral determinant (2.47), depending on all the moduli . We also expect to have spectral traces depending on various integers , . One motivation for this comes from the connection between fermionic spectral traces and matrix models developed in [12, 13]: in the higher genus case, we expect to have a multi-cut matrix model, and there should be as many cuts as true moduli in the model.
In order to construct this generalization, we consider the following operators,
The operators were defined in (2.22), while the operators are defined by (2.20). We will assume that the are of trace class (this can be verified in concrete examples). We now define the generalized spectral determinant as
Different choices of the index lead to operators related by a similarity transformation, and their determinants are equal. The generalized spectral determinant (2.54) can be of course regarded as the conventional spectral determinant of the operator
As shown in , if the operators are of trace class, as we are assuming here, (2.54) is an entire function on the moduli space parametrized by . This function can be expanded around the origin , as follows,
with the convention that
This expansion defines the (generalized) fermionic spectral traces , as promised. These are crucial in our construction, since they will provide a non-perturbative definition of the topological string partition function on . Fredholm’s formula (2.50) can be now used to give an explicit expression for these traces. Let us consider the kernels of the operators defined in (2.53), and let us construct the following matrix:
Then, we have that
As we showed above, the definition does not depend on the choice of . Note that the expansion (2.57) has detailed information about the traces of all the operators and their products.
Let us write some of the above formula in the case , since we will use them later in the paper. In this case, the fermionic spectral traces can be written as
One finds, for example
as well as
As we mentioned above, the integral (2.60) should be regarded as a generalized multi-cut matrix model integral.
What is the motivation for the definition (2.54)? We should expect the generalized spectral determinant to contain information about the operators (2.44). To see that this is the case, let us consider the spectral determinant
By using Fredholm–Plemelj’s formula (2.52), the log of this function can be computed as
We first note that,
By expanding each denominator in a geometric power series, we find that (2.66) is given by
In this equation,
except that all the terms have a strictly positive power of . It follows that
In addition, a simple inductive argument shows that
In this derivation, we have taken as our starting point the operator and its spectral determinant, but it is clear that we could have used any other operator , . In particular, we have
If we set all moduli to zero in (2.74), except for , we find
Therefore, the generalized spectral determinant specializes to the spectral determinant of the unperturbed operators appearing in the different canonical forms of the curve. We will see for example that the generalized spectral determinant associated to the resolved geometry gives, after suitable specializations, the spectral determinants of the operators and .
The attentive reader has probably noticed that the operators defined in (2.53) are not Hermitian, in general. However, the generalized spectral traces defined by (2.57) are real (for real ). This follows immediately from (2.73), which expresses (2.54) as a product of spectral determinants of Hermitian operators.
The generalized spectral determinant (2.54) vanishes in a codimension one submanifold of the moduli space. This submanifold is a global analytic set, since it is determined by the vanishing of an entire function (see ). It contains all the required information about the spectrum of the operators appearing in the quantization of the mirror curve. For example, it follows from (2.74) that it gives the spectrum of eigenvalues of a given operator , as a function of the other moduli , . Since this holds for the different operators , , it follows that their spectra are closely related. Heuristically, this can be already seen from (2.20). Let us suppose that is an eigenstate of , with eigenvalue , and for given values of the , . Then,
is an eigenstate of with eigenvalue , where the parameters appearing in , , are the same ones that appear in , for , while . Of course, since is not bounded, the relation (2.76) only holds if the square integrability of the wavefunction is not jeopardized. This is the case in the examples that we have looked at, like the resolved orbifold.
2.4 Comparison to quantum integrable systems
In the theory that we have developed in the previous sections, the quantization process leads to different operators. However, these operators are related by reparametrizations of the coordinates and the relation (2.20). In particular, there is a single quantization condition for all of them, given by the vanishing of the generalized spectral determinant (2.54), as in . This vanishing condition selects a discrete family of codimension one submanifolds in the moduli space parametrized by . We will determine this family in some detail in the case of the orbifold.
As we mentioned in the Introduction, our quantization scheme might be counter-intuitive for readers familiar with quantum integrable systems, in which the quantization of a genus spectral curve leads typically to quantization conditions. In order to appreciate the difference between the two quantization schemes, let us review in some detail the case of the periodic Toda chain of sites. This system is classically integrable, with Hamiltonians in involution (see  for an excellent exposition of the classical chain). In the quantum theory, the Hamiltonians can be diagonalized simultaneously and one obtains in this way quantization conditions that determine their spectrum completely . An elegant way to obtain the spectrum is by using Baxter’s equation [40, 41], which in the case of the Toda chain is given by,
The can be interpreted as the Hamiltonians of the Toda chain. It was shown in  that, by requiring to be an entire function which decays sufficiently fast at infinity, one recovers the quantization conditions of .
Baxter’s equation can be obtained by formally “quantizing” the spectral curve of the Toda chain, which can be written as
The conserved Hamiltonians are the moduli of the curve. The variables and can be regarded as canonically conjugate variables, in which plays the rôle of the momentum. In order to quantize (2.79), we promote , to Heisenberg operators. In the position representation we have
If we now regard (2.79) as an operator equation, acting on a wavefunction of the form
we recover Baxter’s equation (2.77). As already noted by Gutzwiller , this procedure is purely formal, since the spectral curve (2.79) does not determine by itself the conditions that has to satisfy, and one needs additional information. A more detailed analysis [39, 42, 43] shows that this information is provided by the standard integrability of the original many-body problem, which forces to be entire and to decay at infinity in a prescribed way.
Analyticity of leads to all-orders Bohr–Sommerfeld quantization conditions,
where are appropriate cycles on the curve (2.79). It was conjectured in  that these conditions can be derived from the Nekrasov–Shatashvili (NS) limit of the instanton partition function of , Yang–Mills theory. This limit leads to a quantum-deformed prepotential , where are flat coordinates parametrizing the Coulomb branch and is the genus of the Seiberg–Witten curve. The conjecture of  states that the periods appearing in (2.85) are related to this prepotential by
In addition, the flat coordinates are related to the through a “quantum” mirror map,
where are appropriate cycles on the spectral curve. This conjecture was verified, in the very first orders of the perturbative WKB expansion, in [44, 45]. Additional evidence for this claim has been also provided in for example .
Therefore, in the case of quantum integrable systems of the Toda type, one has quantization conditions, which in the all-orders WKB quantization can be written as in (2.85). The solution to these conditions on the -dimensional moduli space parametrized by the Hamiltonians is a set of points, i.e. a submanifold of codimension . In Fig. 2 we show a cartoon of how the quantization conditions, in the case of , lead to such a discrete spectrum. This cartoon should be compared to Figure 4 of , which shows the result of the actual calculation.
As we already noted, the quantum version of the Toda spectral curve does not lead by itself to a well-defined spectral problem: one needs additional conditions that follow from a detailed analysis of the original integrable system, which has Hamiltonians in involution and requires quantization conditions. However, this does not mean that a curve of genus should always lead, after quantization, to quantization conditions. The simplest example showing that this is not the case is a one-dimensional particle in an (even) confining potential, with a classical Hamiltonian
has genus . The “quantization” of this curve leads to a standard eigenvalue problem for a Schrödinger equation with potential . For real , , the spectrum is real and discrete, and there should be a single quantization condition, expressing the energy as a function of the parameter and the quantum number . Semiclassically, and for sufficiently large, the quantization condition is simply given by the Bohr–Sommerfeld rule,
where is the cycle associated to the turning points of the classical motion. Therefore, although the curve (2.88) has genus , when interpreted as describing a particle in an even, confining potential, its quantum version should lead to a single quantization condition, associated to the preferred cycle . One could think that the other cycles of the higher genus curve do not play a rôle. However, this is not so. The reason is that, in the exact WKB method, one should consider complex trajectories around all possible cycles of the underlying curve, and these trajectories will lead to complex instanton corrections to (2.90), as first pointed out in the seminal paper .
The quantization of higher genus mirror curves considered in this paper is in fact very similar to the quantization of the curve (2.89): there is in principle no need to specify quantization conditions, since (at least in the cases we have considered) the relevant operators have a well-defined discrete, positive spectrum which is determined by a single quantization condition. This condition determines a discrete family of codimension one submanifolds in moduli space. A cartoon for what we expect when is shown in Fig. 3. At the same time, our quantization scheme leads to a genuine -dimensional problem, as reflected in the fact that we have different operators and our generalized fermionic spectral traces depend on different integers. Our goal will be to determine the quantization condition, as well as the generalized spectral determinant (2.54) and spectral traces, from the topological string amplitudes on . The cartoon in Fig. 3 can be compared to the actual calculation of such a family in the example of the resolved geometry, and for in Fig. 5.