Operators and higher genus mirror curves
We perform further tests of the correspondence between spectral theory and topological strings, focusing on mirror curves of genus greater than one with nontrivial mass parameters. In particular, we analyze the geometry relevant to the relativistic Toda lattice, and the resolved orbifold. Furthermore, we give evidence that the correspondence holds for arbitrary values of the mass parameters, where the quantization problem leads to resonant states. We also explore the relation between this correspondence and cluster integrable systems.
Université de Genève, Genève, CH-1211 Switzerland
Laboratoire de Physique Théorique de l’École Normale Supérieure
CNRS, PSL Research University, Sorbonne Universités, UPMC, 75005 Paris, France
Recently, a detailed, conjectural correspondence has been proposed between topological strings on toric Calabi–Yau (CY) manifolds, and the spectral theory of certain quantum-mechanical operators on the real line ghm (see mmrev for a review). The operators arise by the quantization of the mirror curve to the toric CY, as suggested originally in adkmv . This correspondence builds upon previous work on quantization and mirror symmetry ns ; mirmor ; acdkv ; km ; hw , and on the exact solution of the ABJM matrix model mptop ; dmp ; mp ; hmo ; hmo1 ; hmo2 ; hmmo (reviewed in hmorev ; mmabjmrev .) It leads to exact formulae for the spectral traces and the Fredholm determinants of these operators, in terms of BPS invariants of the CY. Conversely, the genus expansion of the topological string free energy arises as an asymptotic expansion of their spectral traces, in a certain ’t Hooft-like regime. In this way, the correspondence provides a non-perturbative completion of the topological string free energy. Although the general correspondence of ghm is still conjectural, it has passed many tests. In the last two years, techniques have been developed to calculate the corresponding quantities in spectral theory and shown to be in perfect agreement with the predictions of the conjecture kasmar ; mz ; kmz ; oz ; gkmr . Other aspects of the correspondence have been discussed in lst ; bgt ; grassi ; sugimoto .
The conjectural correspondence of ghm was formulated for mirror curves of genus one. The generalization to curves of higher genus was proposed in cgm . In this case, the quantization of the curve leads naturally to different operators, and one can define a generalized Fredholm determinant which encodes their spectral properties. In cgm , the higher genus version of the correspondence was verified in detail in the example of the resolved orbifold, arguably the simplest toric CY with a genus two mirror curve. It would be desirable to have more examples and tests of the conjecture in the higher genus case, which has been comparatively less explored. The first goal of this paper is to start filling this gap by analyzing in detail two important genus two geometries. After briefly reviewing the spectral theory of the operators associated to toric CY’s in Sec. 2, we study in Sec. 3 the CY geometry that leads to the relativistic periodic Toda lattice (recently studied in hm ), and the resolved orbifold. Both geometries have a mass parameter, which was absent in the example analyzed in cgm . Geometrically, they both engineer super Yang–Mills theory kkv , and they correspond to 5d, gauge theories on with different values of the 5d Chern–Simons coupling (see for example ikp ). Our focus is on the spectral traces of the operators obtained by quantization of their mirror curves. In all cases, the conjectural formulae of cgm pass the tests with flying colors. Along the way, we present in Sec. 2.3 the exact integral kernel for the inverse of a four-term operator which is relevant for the relativistic Toda case. This generalizes the results in kasmar ; kmz .
The operators arising from the quantization of mirror curves depend on the so-called mass parameters of the geometry hkp ; hkrs . They are only compact and of trace class when these parameters satisfy some positivity conditions. This is similar to what happens in ordinary quantum mechanics. A simple example is provided by the quartic oscillator, defined by the following Schrödinger operator on :
The inverse operator is trace class provided (see for example voros ). For , the potential becomes unstable and there are no longer bound states. However, there are resonant states with a discrete set of complex eigenvalues that can be calculated by using complex dilatation techniques (see for example kpbook ; reed-simon ). In section 5 of this paper, we point out that the spectral theory of the operators arising from mirror curves displays a very similar phenomenon. Namely, for general values of the mass parameters, these operators are no longer compact (this is easily seen by using semiclassical estimates). However, their spectral traces admit an analytic continuation to the complex planes of the mass parameters, with branch cuts, and they can be still exactly computed from topological string data by a natural extension of the conjecture in ghm ; cgm . In particular, we can construct an analytic continuation of the Fredholm determinant for arbitrary, complex values of the mass parameters. The vanishing locus of this analytically-continued function leads to discrete, complex values of the energy, which we interpret as resonances.
As explained in cgm , since the operators arising from the mirror curve are closely related among themselves, the conjectural correspondence of cgm leads to a single quantization condition, given by the vanishing of the generalized Fredholm determinant. However, by a construction of Goncharov and Kenyon gk , it is possible to construct a quantum integrable system, called the cluster integrable system, for any toric CY, leading to commuting Hamiltonians. An exact quantization condition for this integrable system was proposed in hm ; fhm , based on the general philosophy of ns . The result of fhm generalizes the quantization condition of ghm in the form presented in wzh . Therefore, there are two different quantum problems arising from toric CY manifolds: on one hand we have the problem associated to the quantization of the mirror curve, which can be formulated in terms of non-commuting operators on . On the other hand, we have the cluster integrable system of Goncharov and Kenyon, which leads to commuting Hamiltonians on . In the genus one case, the two problems coincide, and in the higher genus case they should be closely related: as in the case of the standard Toda lattice, we expect that the quantization of the mirror curve leads to the quantum Baxter equation of the cluster integrable system. By requiring additional constraints on the solution of this equation, one should recover the quantization conditions. This program has not been pursued in detail. However, it was noticed in fhm , in a genus two example, that an appropriate rotation of the variables in the generalized spectral determinant leads to two different functions on moduli space. The intersection of the vanishing loci of these functions turns out to coincide with the spectrum of the cluster integrable system. Recently, this observation has been generalized in swh , and there is now empirical evidence that the generalized spectral determinant of cgm , after appropriate rotations of the phases of the variables, leads to at least different functions whose zero loci intersect precisely on the spectrum of the cluster integrable system111As is obvious from this discussion, the quantization condition of cgm is more general than the one in fhm , as all existing results indicate that one can recover the latter from the former.. Another goal of this paper is to further clarify the relation between the quantization conditions of cgm and of fhm . In particular, we will show in Sec. 5 that in some genus two geometries, reality and positivity conditions allow us to deduce the spectrum of the cluster integrable system from the generalized Fredholm determinant.
2 Spectral theory and topological strings
In this section, we review the construction of operators from the mirror curve of a toric CY, their spectral theory, and the connection to the topological string theory compactified on the toric CY. We mainly follow ghm ; mz ; kmz ; cgm . The material on toric CYs and mirror symmetry is standard, see for example hkt ; ckyz ; ck ; horibook .
2.1 Mirror curves and spectral theory
Toric CY threefolds can be specified by a matrix of charges , , satisfying the condition,
Their mirrors can be written in terms of complex coordinates , , which satisfy the constraint
The mirror CY manifold is then given by
defines a Riemann surface embedded in , which we will call the mirror curve to the toric CY threefold . We note that there is a group of reparametrization symmetries of the mirror curve given by akv ,
The moduli space of the mirror curve can be parametrized by the coefficients of its equation (2.5), among which three can be set to 1 by the scaling acting on and an overall scaling. Equivalently, one can use the Batyrev coordinates
which are invariant under the actions. In order to write down the mirror curves, it is also useful to introduce a two-dimensional Newton polygon . The points of this polygon are given by
in such a way that the extended vectors
satisfy the relations
This Newton polygon is nothing else but the support of the 3d toric fan of the toric Calab-Yau threefold on a hyperplane located at . The function on the l.h.s. of (2.6) is then given by the Newton polynomial of the polygon ,
Clearly, there are many sets of vectors satisfying the relations (2.11), but they lead to curves differing in a reparametrization or a global translation, which are therefore equivalent. It can be seen that the genus of , , is given by the number of inner points of . We also notice that among the coefficients of the mirror curve, of them are “true” moduli of the geometry, corresponding to the inner points of , while of them are the so-called “mass parameters”, corresponding to the points on the boundary of the polygon (this distinction has been emphasized in hkp ; hkrs ). In order to distinguish them, we will denote the former by , and the latter by , . It is obvious that we can translate the Newton polygon in such a way that the inner point associated to a given is the origin. In this way we obtain what we will call the canonical forms of the mirror curve
where is a polynomial in , . Note that, for , there is a single canonical form. Different canonical forms are related by reparametrizations of the form (2.7) and by overall translations, which lead to overall monomials, so we will write
where is of the form , . Equivalently, we can write
and we use the Weyl quantization prescription. In this way we obtain different operators, which we will denote by , . These operators are self-adjoint. The equation of the mirror curve itself is promoted to an operator
which we call the quantum mirror curve. Different canonical forms are related by the quantum version of (2.14),
where is the operator corresponding to the monomial . We will also denote by the operator associated to the function in (2.15). This can be regarded as an “unperturbed” operator, while the moduli encode different perturbations of it. We also define the inverse operators,
It is easy to see that cgm
We want to study now the spectral theory of the operators , . The appropriate object to consider turns out to be the generalized spectral determinant introduced in cgm . Let us consider the following operators,
Let us suppose that these operators are of trace class (this turns out to be the case in all known examples, provided some positivity conditions on the mass parameters are satisfied). Then, the generalized spectral determinant associated to the CY is given by
Due to the trace class property of the operators , this quantity is well-defined, and its definition does not depend on the choice of the index , due to the similarity transformation
with the convention that
This expansion defines the (generalized) fermionic spectral traces of the toric CY . Both and depend in addition on the mass parameters, gathered in a vector . When needed, we will indicate this dependence explicitly and write , . As shown in cgm , one can use classical results in Fredholm theory to obtain determinant expressions for these fermionic traces. Let us consider the kernels of the operators defined in (2.22), and let us construct the following matrix:
Then, we have that 222The determinant of the matrix is independent of the label , just like the Fredholm determinant.
In the case , this formula becomes
One finds, for example
as well as
The generalized spectral determinant encodes the spectral properties of all the operators in a single strike. Indeed, one has cgm ,
i.e. the generalized spectral determinant specializes to the spectral determinants of the unperturbed operators appearing in the different canonical forms of the mirror curve.
The standard spectral determinant of a single trace-class operator determines the spectrum of eigenvalues, through its zeros. The generalized spectral determinant (2.23) vanishes in a codimension one submanifold of the moduli space. It follows from (2.33) that this submanifold contains all the information about the spectrum of the operators appearing in the quantization of the mirror curve, as a function of the moduli , .
2.2 Spectral determinants and topological strings
The main conjectural result of ghm ; cgm is an explicit formula expressing the generalized spectral determinant and spectral traces in terms of of enumerative invariants of the CY . To state this result, we need some basic geometric ingredients. As discussed in the previous section, the CY has moduli denoted by , . We will introduce the associated “chemical potentials” by
The CY also has mass parameters, , . Let . The Batyrev coordinates introduced in (2.8) can be written as
The coefficients determine a matrix which can be read off from the toric data of . One can choose the Batyrev coordinates in such a way that, for , the ’s correspond to true moduli, while for , they correspond to mass parameters. For such a choice, the non-vanishing coefficients
form an invertible matrix, which agrees (up to an overall sign) with the charge matrix appearing in kpsw . We also recall that the standard mirror map expresses the Kähler moduli of the CY in terms of the Batyrev coordinates
where is a power series in with finite radius of convergence. Together with (2.36), this implies that
This quantum mirror map will play an important rôle in our construction. In addition to the quantum mirror map, we need the following enumerative ingredients. First of all, we need the conventional genus free energies of , , in the so-called large radius frame (LRF), which encode the information about the Gromov–Witten invariants of . They have the structure333The formula of differs from the one in the topological string literature by the linear term, which usually doesn’t play a role in non-compact CY models. The addition of this term makes the formulae in the rest of the paper more compact.
In these formulae, are the Gromov–Witten invariants of at genus and multi-degree . The coefficients , are cubic and linear couplings characterizing the perturbative genus zero and genus one free energies, while is the so-called constant map contribution bcov . The constants , which can be obtained from the refined holomorphic anomaly equation hk ; krewal , usually appear in the linear term of (see below, (2.48)). The total free energy of the topological string is the formal series,
and is the topological string coupling constant.
As found in gv , the sum over Gromov–Witten invariants in (2.42) can be resummed order by order in , at all orders in . This resummation involves the Gopakumar–Vafa (GV) invariants of , and it has the structure
Note that, as formal power series, we have
In the case of toric CYs, the Gopakumar–Vafa invariants are special cases of the refined BPS invariants ikv ; ckk ; no . These refined invariants depend on the degrees and on two non-negative half-integers or “spins”, , . We will denote them by . We now define the NS free energy as
The first term in this series, , is equal to , the standard genus zero free energy.
Following hmmo , we now define the modified grand potential of the CY . It is the sum of two functions. The first one is
We note that the function is only known in a closed form in some simple geometries. The second function is the “worldsheet” modified grand potential, which is obtained from the generating functional (2.44),
It involves a constant integer vector (or “B-field”) which depends on the geometry under consideration. This vector satisfies the following requirement: for all , and such that is non-vanishing, we must have
We note that the characterization above only defines up to . A difference choice of does not change . The total, modified grand potential is the sum of the above two functions,
and it was introduced in hmmo . Note that if we define
which differs from by the terms proportional to , can be written in a slightly more compact form, i.e.444Note that is with the contributions from the constant maps removed. The latter can be understood as absorbed in .
where all but the constant are hidden in (refined) topological string free energies. Here we introduce the notation , meaning that is shifted by in the terms of order (the instanton contributions). In particular,
The right hand side of (2.57) defines a quantum-deformed (or generalized) Riemann theta function by
The r.h.s. of (2.57) can be computed as an expansion around the large radius point of moduli space. In the so-called “maximally supersymmetric case” , it can be written down in closed form in terms of a conventional theta function. There is an equivalent form of the conjecture which gives an integral formula for the fermionic spectral traces:
In practice, the contour integration along the imaginary axis can be deformed to a contour where the integral is convergent. For example, in the genus one case the integration contour is the one defining the Airy function (see hmo2 ; ghm ).
An important consequence of the representation (2.59) is the existence of a ’t Hooft-like limit in which one can extract the genus expansion of the conventional topological string. The ’t Hooft limit is defined by
In this ’t Hooft limit, the integral in (2.59) can be evaluated in the saddle-point approximation, and in order to have a non-trivial result, we have to consider the modified grand potential in the limit
In this limit, the quantum mirror map appearing in the modified grand potential becomes trivial. We will assume that the mass parameters scale in such a way that
are fixed as . In the regime (2.61), the modified grand potential has an asymptotic genus expansion of the form,
In these equations, we have introduced the rescaled Kähler parameter
The arguments and of the modified grand potential are related to the rescaled Kähler parameters by
We have assumed that the function has the expansion
The saddle point of the integral (2.59) as is then given by
It follows from this equation that the ’t Hooft parameters are flat coordinates on the moduli space. The frame defined by these coordinates will be called the maximal conifold frame (MCF). The submanifold in moduli space defined by
has dimension (the number of mass parameters of the toric CY), and we will call it the maximal conifold locus (MCL). It is a submanifold of the conifold locus of the CY . By evaluating the integral (2.59) in the saddle-point approximation, we find that the fermionic spectral traces have the following asymptotic expansion in the ’t Hooft limit:
The leading function in this expansion is given by a Legendre transform,
If we choose the Batyrev coordinates in such a way that the first correspond to true moduli, and the remaining correspond to mass parameters, we find
where denotes the inverse of the truncated matrix (2.37). In view of the results of abk , the higher order corrections can be computed in a very simple way: the integral (2.59) implements a symplectic transformation from the LRF to the MCF. The functions are precisely the genus free energies of the topological string in the MCF.
2.3 Perturbed operators
In order to test the conjectures of ghm ; cgm , it is very useful to have explicit results on the spectral theory side. In particular, since we have a precise conjecture for the values of the fermionic spectral traces of the relevant operators in terms of enumerative invariants, we would like to have independent, analytic computations of these traces.
In many cases, the operators which appear in the quantization of mirror curves are perturbations of three-term operators of the form
These operators were introduced and studied in kasmar . It turns out that the integral kernel of their inverses can be explicitly computed in terms of Faddeev’s quantum dilogarithm. This makes it possible to calculate the standard traces
in terms of integrals over the real line. These integrals can be computed by using the techniques of garkas , or by using the recursive methods developed in tw ; py ; hmo ; oz . Once the “bosonic” spectral traces (2.74) have been computed, the fermionic spectral traces follow by simple combinatorics. Mixed traces, as those appearing in (2.31), (2.32), can be also obtained in terms of integrals.
In this section, we will study a four-term operator which is a perturbation of (2.73). This operator reads,
Let us introduce the parameter by555Do not confuse the introduced here with the ’t Hooft variables defined in (2.61).
We will now obtain an explicit expression for the integral kernel of
They are related to the operators , appearing in (2.75) by
We then have,
We now use Faddeev’s quantum dilogarithm fk ; faddeev-penta ; faddeev (our conventions for this function are as in kasmar ). It satisfies the following identity (a similar identity was already used in kasmar ):
Its behaviour under complex conjugation is given by,
Let us denote
We now recall that Faddeev’s quantum dilogarithm satisfies the following difference equation,