Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries

We study a class of quantum integrable systems derived from dimer graphs and also described by local toric Calabi-Yau geometries with higher genus mirror curves, generalizing some previous works on genus one mirror curves. We compute the spectra of the quantum systems both by standard perturbation method and by Bohr-Sommerfeld method with quantum periods as the phase volumes. In this way, we obtain some exact analytic results for the classical and quantum periods of the Calabi-Yau geometries. We also determine the differential operators of the quantum periods and compute the topological string free energy in Nekrasov-Shatashvili (NS) limit. The results agree with calculations from other methods such as the topological vertex.


Introduction and Summary
The developments of various prosperous topics in mathematics and physics often intersect each others. Topological string theory on Calabi-Yau manifolds has been a fruitful branch of superstring theories that encompass many recurring themes in mathematical physics, see e.g. [1]. In the seminar work [2], Nekrasov and Shatashvili (NS) proposed a connection between the partition function of Seiberg-Witten gauge theory on Ω background and certain quantum integrable systems. In the NS limit, we set one of the two Ω deformation parameters to vanish and identify the other as the Planck constant of the quantum system. This relation can be uplifted to five dimensions, where the partition functions are computed by refined topological string theory on corresponding Calabi-Yau spaces. The topological free energy in the NS limit can be viewed as a quantum deformation of the prepotential, and is computed similarly by promoting the periods of the Calabi-Yau geometries to quantum periods [3,4,5,6]. More examples in Seiberg-Witten theories can be found in [7,8,9,10,11]. The quantization conditions of the quantum system are formulated as the Bohr-Sommerfeld quantization condition where the phase volumes are computed by quantum periods. In the five dimensional case, the quantum systems are often known as relativistic models due to the exponential kinetic and potential terms in the Hamiltonians from quantizing the mirror curves of the local Calabi-Yau spaces. Inspired by earlier works [12,13,14], some novel non-perturbative contributions to the quantization conditions are conjectured in [15,16]. Various aspects of the quantization conditions, including complex value Planck constant, resurgence, wave functions, etc are further explored in e.g. [17,18,19,20,21]. The non-perturbative parts of the two types of exact quantization conditions in [15,16] are related by certain constrains on the BPS invariants known as the blowup equations [22] [23]. The blowup equations originally come from studies of Seiberg-Witten gauge theories [24] (see also [25,26]), but have now become a very effective tool for computing topological string amplitudes on various Calabi-Yau manifolds [27,28,29,30,31]. The exact quantization conditions have also been applied to related condensed matter systems, e.g. in [32,33,34,35].
Most examples of the early studies focus on geometries with mirror curves of genus one. The quantum periods and quantization conditions for quantum systems corresponding to mirror curves of higher genus were subsequently considered, in e.g. [36,37,22,38,39]. A particularly interesting class of quantum integrable systems can be constructed by dimer models on torus [40], and the quantization conditions are studied in [41,42]. The dimer models in this paper also correspond to local toric Calabi-Yau geometries and the mirror curves are encoded in the data of the bipartite dimer graphs. Some of Calabi-Yau spaces geometrically engineer 5d supersymmetric gauge theories, which are uplifts of the 4d SU (N ) Seiberg-Witten theories considered in [7]. There are a number of commuting Hamiltonians, and the multiple quantization conditions can be similarly derived from topological string free energy in the NS limit on the corresponding Calabi-Yau spaces. The studies in [41,42] mostly focus on numerical tests of the non-perturbative quantization conditions. However, in order to have a more insightful understanding of the interconnections between various subjects here, it is better to have some analytical results. In this paper we develop some analytic approaches to the problem, though mostly focusing on the perturbative aspects.
The paper is organized as followings. In Section 2 we review the constructions of dimer models, and derive Hamiltonians of the quantum integrable systems based on previous literatures. We shall study some examples with genus two mirror curves and correspondingly two commuting dynamical Hamiltonians. In Section 3 we study the perturbative quantum spectra of the Hamiltonians around minimal points of the phase space. A useful technical ingredient is the symplectic transformations of the quantum canonical coordinates, which are necessary to determine the energy eigenvalues of the quadratic terms. We find the symplectic transformations for the examples with simple classical minima, and further calculate the higher order spectra with standard perturbation methods in quantum mechanics. In Section 4 we systematically compute the periods and topological string free energies for the Calabi-Yau geometries, summarizing the results in previous literature. We then compute the differential operators which exactly determine quantum corrections to classical periods, generalizing earlier works [5,6] to the situation of higher genus mirror curves. Similarly, the topological free energy in the NS limit is determined by the quantum periods, and we show that this agrees with results from e.g. method of topological vertex. An interesting feature is that the differential operators are the same for differential cycles of the higher genus mirror curves. Following earlier works [14], we perform some satisfying tests of our calculations by comparing the quantum spectra from direct perturbation and from Bohr-Sommerfeld quantization using quantum periods as phase volumes. These exercise provide some exact analytic results for the classical and quantum periods of the Calabi-Yau spaces, which are difficult to directly obtain.

Dimer models and integrable systems
In [40], the authors proposed an infinite class of cluster integrable systems. 1 The most interesting ones among them are the cluster integrable systems for the dimer models on a torus. The dimer model is the study of the set of perfect matching of a graph, where the perfect matching is a subset of edges which covers each vertex exactly once. For a bipartite graph, the vertices are divided into two sets, the black set and the white set. Every edge connects a white vertex to a black vertex. For a more detail introduction to dimer models, see [45].
The dimer model can be connected to a toric diagram by Kasteleyn matrix K(X, Y ) [45], which is the weighted adjacency matrix of the graph. The determinant of the Kasteleyn matrix, happens to be the mirror curve of the corresponding Toric Calabi-Yau three-fold [46] [47]. The adjacency matrix can be computed as follows: • Multiply each edge weight of the graph a sign ±1, so that around every face, the product of the edge weights over edges bounding the face is sgn( • Construct two loops γ X , γ Y along the two cycles of the torus, we draw them as red dash lines in the diagram.
• Fix an orientation, from black to white, as the positive orientation.
• Times each edge with a factor X or Y , if the loop γ X or γ Y get through the edge with positive orientation. Times each edge with a factor 1/X or 1/Y , if the loop γ X or γ Y get through the edge with positive orientation.
v Figure 1: An illustration of δ v (ω 1 , ω 2 ). If ω 1 and ω 2 are in the counterclockwise order, and with the same direction, δ v (ω 1 , ω 2 ) = 1 2 as in (c). Any change in the clockwise order or direction gives an extra sign, e.g. (a)(b). The arrows represent the orientations of the loops ω i .
Then the Kasteleyn matrix is a matrix with row labeled by black vertices and column labeled by white vertices, with the entry as the weight between the connected black and white vertices. The entry is 0, if two points are not connected. In this paper, we are interested in Y p,q system, the determinant of the Kasteleyn matrix has the form Following [40][48], the commutation relations and the Hamiltonians of the cluster integrable systems can be read from the loops of the graph. Let ω i be the oriented loops on the graph, the Poisson bracket between cycles are defined as where Here sgn(v) = 1 for the white vertex v, and −1 for the black vertex. δ v is a skew symmetric bilinear form with Figure 1. Though more general vertex is possible, for our examples of dimer models we will only encounter cubic vertices.
To construct the basis t i of all the loops, we can first fix an arbitrary perfect matching as the reference perfect matching, then all the bases can be constructed from the difference between the reference perfect matching and another perfect matching. Then the Hamiltonians H n are the sum of all possible combinations of n of these cycles t i with the condition that they do not overlap or touch at any vertex of the tiling.

Examples
In this subsection, we give some examples for the dimer models of 5d, N = 1 SU (3) gauge theories, with various Chern-Simons level m = 0, 1, 2, 3. The graphs of these theories were appeared during the study of 4d N = 1 quiver gauge theories, where the graphs of the dimer models are brane tiling for the quiver gauge theories. For the Y p,p system, the brane tiling is the well-known Hexagon tiling [48]. One can merge the points in the tiling for the Y p,p system to get the tiling for a Y p,q , q < p system [49]. For example, the tiling for Y 3,3 system is depicted in Figure 2a.
We choose the loops to be 2 (2.5) Only loops that are overlapped have non-vanishing Poisson brackets, they are (2.6) The Hamiltonians can be read from the graph directly from the rules in precious section: (2.7) The Poisson brackets (2.6) can be enhanced to the quantum level as the commutation relations, in terms of canonical variables {q i , p i = −i ∂ ∂q i }, we find a possible coordinates relation The R is the radius of the compactification circle from 5d to 4d, which gives a nontrivial deformation to the integrable systems. It is related to the instanton counting parameter or mass parameters in the 5d gauge theory point of view. One can get the brane tiling of Y 3,2 systems 2b (b) by merging the point 8, 11 and 2, 5 in Figure  2a. By further merging 9, 12 and 3, 6, we get Y 3,1 2c. By doing this further, we get Y 3,0 2d. In the following, we list their Poisson brackets and Hamiltonians for these models. The bases t i are the loops inherited from t i in Y 3,3 after merging the points.  The non-vanishing Poisson brackets are In terms of canonical variables, With the Hamiltonians (2.20)

Perturbative computations of quantum spectra
In this section, we consider the perturbative energy spectra of the quantum integrable systems described by genus two mirror curves, including the Y 3,m models with m = 0, 1, 2, 3, and C 3 /Z 5 model. Each model have two dynamical Hamiltonians, which are derived from dimer models. In the previous Section 2, we derived the Hamiltonians for the Y 3,m models, where the case of m = 0 was also considered in [41]. The Hamiltonians of some orbifold models including C 3 /Z 5 are available in [42]. We also note that the Y 3,3 model is equivalent to the orbifold C 3 /Z 6 model in [42]. We quantize the Hamiltonians by promoting the dynamical variables to operators with canonical commutation relations [q i , The Hamiltonians are bounded below in the phase space (q 1 , p 1 , q 2 , p 2 ). First we consider the Y 3,0 , Y 3,3 , C 3 /Z 5 models, for which the classical minima are simply located at the origin q 1 = q 2 = p 1 = p 2 = 0. We expand the Hamiltonians around the minimal point.
First we study in details the C 3 /Z 5 model, whose Hamiltonians are We expand the Hamiltonians up to quadratic order where the S 1 , S 2 are real symmetric matrices We would like to write the quadratic Hamiltonians as linear combinations of two harmonic oscillators. We consider a linear transformation where M is a 4×4 real matrix. To preserve the same canonical commutation relation, the matrix M must be a symplectic matrix M ΣM T = Σ, where Σ is the antisymmetric matrix (3.6) It turns out due to the special property that the Hamiltonians commute with each other, we can find symplectic transformation M so that the quadratic terms can be written as linear combinations of the two harmonic oscillators , There is a continuous 2-parameter family of solutions for the matrix M . Without loss of generality, we can use a particular solution with the linear coefficients Denoting the quantum levels of the harmonic oscillators (x 1 , y 1 ) and (x 2 , y 2 ) by two non-negative integers n 1 , n 2 , the quantum spectrum up to order is We can further compute the higher order corrections to energy spectra. We use the time-independent perturbation theory well-known in quantum mechanics, which separates a Hamiltonian into a zero order part and a perturbation part where the zero order part H 0 corresponds to the Hamiltonians up to quadratic order in (3.3), while the perturbation part H corresponds to the higher order terms. We denote the harmonic quantum states of the zero order Hamiltonians as |n 1 , n 2 . Then the first few order corrections to energy spectra are (3.12) To compute the next 2 order corrections, we need to expand the exponentials in the Hamiltonians (3.1) to cubic and quartic orders, and rewrite the canonical coordinates in terms of the standard creation and annihilation operators. For the the first correction n 1 , n 2 |H |n 1 , n 2 , the cubic terms have no contribution since there is an odd number of creation and annihilation operators, while the quartic terms make an order 2 contribution. The cubic terms have a 2 order contribution in the more complicated second correction term in the above equation (3.12). After some complicated calculations, we find the 2 order contributions to the quantum spectra. For the C 3 /Z 5 model, the results are 13) It is well known that the eigenvalues of a matrix do not change under a similarity transformation of the matrix. Here analogously we find that the spectra in (3.13) are independent of the choice of symplectic transformation, up to the trivial freedom of exchanging the two quantum numbers n 1 ↔ n 2 . This is easy to understand from physics point of view since the Hamiltonians are the same regardless of the choices of the canonical coordinates. Furthermore, the linear coefficients (3.9) are indeed related to the eigenvalues of certain matrices. We note that for a general evendimensional real symmetric matrix S, since det(SΣ−λI) = det(ΣS −λI) = det(SΣ+ λI) = det(ΣS + λI), the eigenvalues of SΣ and ΣS are the same and always come in pairs with opposite signs. In our context, we find that for the matrices (3.4) in the quadratic Hamiltonians, the eigenvalues of S 1 Σ and S 2 Σ are always purely imaginary and the positive imaginary parts are exactly the linear coefficients (3.9). Namely, the eigenvalues of S 1 Σ are ±ic 1 , ±ic 2 and the eigenvalues of S 2 Σ are ±ic 3 , ±ic 4 . This is also true for the Y 3,0 and Y 3,3 models discussed below. In the Appendix A we give a simple general mathematical proof of this property.
Similarly we find the symplectic transformations and the perturbative energy spectra for the Y 3,0 and Y 3,3 models in (2.20, 2.7). Again there is a continuous 2-parameter family of solutions for symplectic transformations. For the Y 3,0 model, we can use for example a solution and the perturbative energy spectrum is We see there is an apparent symmetry of the spectra. The spectra of the two Hamiltonians E 1 ↔ E 2 are exchanged if the quantum levels are exchanged n 1 ↔ n 2 . For the Y 3,3 model, the results are There is also an apparent symmetry that under a T-duality like transformation R → 1 R , the energy spectra transforms as . We need to be careful with a potential subtlety of perturbation theory here. For the first Hamiltonian of the Y 3,3 model, we see that the energy E 1 are degenerate up to order for quantum states with the same n 1 +n 2 . It turns out that this does not affect the calculations in formula (3.12), as we check that the off-diagonal elements of the perturbation in the degenerate space actually vanish, i.e. n 1 +k, n 2 −k|H |n 1 , n 2 = 0 for k = ±1, ±2. The vanishing is trivial for cubic terms in the perturbation H , while we check by explicit computation that it is also true for quartic terms.
For the remaining Y 3,1 and Y 3,2 models (2.16, 2.12), we need to determine the classical minima by solving for the critical points of Hamiltonians ∂ q i H = ∂ p i H = 0 for i = 1, 2. We find that the minima are located at the same points for the two Hamiltonians of the quantum system due to the special property that the Hamiltonians commute withe each other. In these models it is much more complicated to find the symplectic transformations that diagonalize the quadratic terms of the Hamiltonians expanded around the minima. However, we can still use the formula in Appendix A to compute the order contributions to quantum spectra in terms of the eigenvalues of certain matrices from the quadratic terms.
For the Y 3,1 model, the minima are at where r is the only positive root of the polynomial equation, with numerical value e.g. r = 0.921599 for R = 1. The quantum spectra are For the Y 3,2 model, the minima are at where r is now the only positive root of the equation with numerical value e.g. r = 0.665055 for R = 1. The quantum spectra are Without solving the symplectic transformations for these two Y 3,1 and Y 3,2 models, there is an ambiguity of exchanging the quantum numbers n 1 ↔ n 2 in the spectra. This can be fixed by comparing with the derivatives of periods of the corresponding Calabi-Yau geometries.

From topological strings to energy spectra
In this section we will show that the spectrum problem can be solved by utilizing the well-known method in topological string theory. More precisely, we use the method developed in [4], and calculate the energy spectra by imposing the Bohr-Sommerfeld quantization condition on the quantum B-periods of the mirror curves. First we summarize some basic facts about the classical/quantum mirror curves, and the general relations between topological strings and the energy spectra. After that, we will demonstrate how we calculate the energy spectra from the quantum periods in some concrete models.

General aspects of classical/quantum curves
We consider topological string theory on the toric Calabi-Yau three-fold, where the topological information in the B model are captured by a mirror curve. A genus-g mirror curve is defined by the algebraic equation for x, y ∈ C, W (e x , e y ; z) = 0, (4.1) where z = (z 1 , z 2 , ..., z s ) are the complex structure moduli parameters with s := b 1,1 ≥ g. Generally, there are g dynamical moduli corresponding to g compact Aand B-cycles of the Riemann surface, and the s − g remaining ones are known as non-dynamical mass parameters. We can define two kinds of classical periods called as A-and B-periods by integrating y = y(x; z) around compact A-cycles and their dual B-cycles, where y(x; z) is the solution of (4.1). The mirror maps connecting the Kähler moduli with the complex moduli can be written as linear combinations of A-periods and mass parameters where the mass terms depend only on logarithm of mass parameters and will not appear in quantum corrections. Here C ij is the intersection matrix of compact divisors and the base curves we have chosen. With a suitable choice of base curves, parts of the g × s matrix C ij happens to be the Cartan matrix of the gauge group in the context of geometric realizations of gauge theories.
In the similar way, the dual B-periods give the derivatives of the genus-zero topological string amplitude, so-called prepotential F 0 (t), From the prepotential, we define the Bohr-Sommerfeld volumes as the derivatives of prepotential with appropriate shift This shift can be derived from the S-dual like invariance of the classical volumes [50]. It can be absorbed into the genus zero free energy by adding a t i linear term [22]. In For a toric Calabi-Yau three-fold, an efficient way to calculate A-and B-period is to solve the Picard-Fuchs equations defined by where Q α i is the charge vector and x i is the homogeneous coordinate of the toric variety. The differential operator L α 's are known as the Picard-Fuchs operators. The variables x i relate to z through the Batyrev coordinates (4.7) The A-and B-periods correspond to logarithmic and double-logarithmic solutions. Now we promote the classical variables x, y to the quantum operators x, y with the canonical commutation relation, where Ψ(x) is the wave function. We can solve the difference equation by utilizing the WKB analysis, Then, we can define quantum version of two periods, called as quantum A-and Bperiod, where we expand w(x; ) in , (4.12) In our example, w (2n−1) , n ∈ Z >0 can be expressed as the total derivative of simple functions with no monodromy. Thus, its contour integral vanishes, and only 2ncorrections survive.
The quantum corrected prepotential F (t; ), so-called NS free energy, is defined by refined topological string free energy in the NS limit It satisfies a similar equation as the prepotential where t i (z; ) are the quantum mirror maps. Comparing both sides of (4.14), we can obtain the recursion relations which enable us to fix F i (t) completely, up to irrelevant constants and mass parameters. The Bohr-Sommerfeld volumes (4.5) also have quantum corrections, In quantum mechanics, the phase volume should be quantized. In our case, the B-periods are quantized, The dynamical complex structure moduli will correspond to Hamiltonians of the quantum systems as we will see in concrete examples. As in the case of NS free energy, by expanding the quantum B-periods in , we can determine the quantum corrections to the energy eigenvalues recursively. The B-periods have to vanish in the classical limit at the minimal energy points, which correspond to the conifold points in the topological string moduli space. Thus, to solve the spectral problem from the topological strings, we have to calculate the phase volumes at the conifold point. It turns out that there is no logarithmic cut for the classical volumes (B-periods) at the conifold points, so they are the same as the mirror maps up to numerical factors, where Coni denotes the conifold point, and t i,c (z c ; ) is quantum mirror map expanded around the conifold point. The numerical factor in the coefficients of t i,c (z c ; ) can be determined by comparing with the derivatives of the classical volumes at conifold point or the perturbative computation as we have done in the previous section.
Therefore, we can calculate the eigenvalues only by using the quantum mirror maps near the conifold points. Now we move to the computation of the quantum periods. It is straightforward to calculate the quantum A-periods from the definition by taking residues, whereas the direct computations of B-periods are usually not so easy. Here we utilize the differential operator method proposed in [3], and developed in [5].
The important fact is that the quantum A-periods can be given by the classical periods with differential operators as follows, where and coefficients of θ z i are given by rational functions of z i . This means that we can obtain the differential operators in the conifold frame by transforming from large radius frame to the conifold frame z i → z c,i . Then, by acting the operators on the classical A-periods expanded near the conifold point, we can obtain the quantum corrections in the conifold frame. Since the mass parameters are annihilated by the differential operators, they do not receive quantum corrections. According to (4.3), the quantum mirror maps are determined by the same differential operators as 2n D 2n t i (z), i = 1, 2, · · · , s. (4.21) Interestingly, the differential operators that we will treat in our study do not depend on the choice of the cycles 3 . Also, the classical mirror maps can be calculated from the Picard-Fuchs operators. Therefore, it is enough to calculate one of the quantum Aperiods to derive the differential operators and determine the quantum mirror maps. By combining (4.18) with (4.21), the quantum corrections to the volumes and their derivatives with respect to the eigenvalues are given by where p i ∈ Z ≥0 , n ∈ Z >0 , and j = 1, 2, ..., s. To calculate the right hand side, we use Remarkably, this structure is the same as the quantum B-period; the quantum corrections can be calculated by acting above operators on the classical B-periods, (4.23) This means that once we derive the differential operators D 2n from the quantum Aperiod that we know how to calculate systematically, we can obtain the quantum B-period which is difficult to obtain by the direct computation of the cycle integral. Similar to previous paper [5], we can derive recursion relations for the NS free energy by expanding the equations (4.21, 4.14, 4.13). We can explicitly do this for the first and second correction terms F 1,2 (t), which are determined by the differential operators D 2 , D 4 . In our examples, the differential operators will be a linear combinations of first and second derivatives of the complex structure moduli. Suppose where the coefficients s i , s i,j are rational functions of complex structure moduli z i 's. Denote the classical mirror maps as t i , then it is straightforward to compute Combining the 2 equations of (4.13, 4.14, 4.21), we find the linear coefficients s (2) i cancel out. The equation for first order NS free energy is then l,m θ l (t j )θ m (t k )(∂ t i ∂ t j ∂ t k F 0 )] = 0, n = 1, 2, · · · , g. (4.27) If s = g and the matrix C i,j is invertible, it cancels out in the above equation.
Otherwise, in general we need to solve the equations including the C matrix. Similarly, repeating the same computation to the next order, we have (4.28) Again, the linear coefficients s (4) i cancel out. By using (4.27), if the matrix C is invertible, we can eliminate F 1 , and obtain the relation between F 2 and F 0 .

Examples
We shall demonstrate the previous computations in some concrete models. In our examples, we focus on the genus-2 mirror curves: C 3 /Z 5 and Y 3,m with m = 0, 1, 2, 3. Most of classical computations have already done in e.g. [36,51,41,42,22], and we gather the results to make the paper self-contained. In the following computations, we may omit some arguments in the functions for short notation.
(4.32) We further define the derivatives of ω 0 (ρ i ), Then, the mirror maps are given by, (4.34) The derivatives of the prepotential are ∂F 0 ∂t 1 = 2ω 1,1 + 2ω 1,2 + 3ω 2,2 , ∂F 0 ∂t 2 = ω 1,1 + 6ω 1,2 + 9ω 2,2 . (4.35) The classical B-periods Π d,i (i = 1, 2) are given by the formula (4.4), where the matrix C ij of this model is, From the prepotential, the Bohr-Sommerfeld volumes are vol (0) where the complex structure moduli z 1 , z 2 are related to the quantum systems of the dimer model by The classical volumes vanish at conifold point, z 1 = −1/25, z 2 = 1/5, or E 1 = E 2 = 5, which is checked numerically. Now let us consider the quantum periods. Correspondingly, the classical mirror curve is replaced by the difference equation, According to [4], the quantum A-periods are given by taking the residue, We note that as familiar from literature, the logarithmic term is not captured by the residue calculations and is added by hand. We express the coefficients Π (n) by the differential operator method. Since the differential operators giving Π (n≥4) is tedious long expression, here we provide the differential operator giving the leading correction to the classical periods as an example 4 , By using the operator, we can obtain the leading correction to the quantum mirror maps t To check the consistency, let us calculate the NS free energy near the large radius point. By solving the recursion relations (4.27) and (4.28) with the matrix (4.36) invertible and cancelled out, we find the NS free energy whose instanton parts [F n ] inst. are given by (4.43) They agree with the topological vertex computations. Now we are ready to calculate the quantum corrections to the energy spectra. The all-order Bohr-Sommerfeld quantization condition in this case is given by where vol i (E 1 , E 2 ; ) are the quantum corrected phase volumes. To obtain the quantum corrected spectrum, we define E i and vol i (E 1 , E 2 ; ) as series of , (4.45) The classical Bohr-Sommerfeld volumes have to vanish in the classical limit = 0 of (4.68) at the minimum which corresponds to the conifold point. By expanding (4.68) in , we can obtain E (n) i as a function of vol where we omit the arguments (E m 1 , E m 2 ) of vol (n) i 5 . By comparing (4.47) with perturbative calculations (3.10), we obtain exact values of the E 1,2 -derivatives of phase 5 We will use this expression for other models, where the arguments of vol (n) i in these models are (E m1 , E m2 , R).
volumes at the classical minimum, With the change of variables (4.38), we find We check this is indeed true numerically. The classical mirror maps near the conifold point are The coefficients of z c,1 , z c,2 in the classical mirror map are fixed by the relation (4.22). We can calculate the next leading order of the energy spectrum E 1,2 by looking at 2 -order term of (4.44). To obtain them, we need to calculate the second derivatives of the volumes and first quantum correction which can be calculated from the formula (4.22) with (4.41). After some computations, we find
(4.56) Then, the classical mirror maps and the derivatives of the prepotential are given by

57) and
∂F 0 ∂t 1 = ω 11 + ω 12 + 1 2 where The classical B-periods Π d,i (i = 1, 2) are given by formula (4.4) with where the first 2 × 2 block is the Cartan matrix of SU (3). From the prepotential, the Bohr-Sommerfeld volumes are where the complex structure moduli z 1 , z 2 , z 3 are related to the quantum systems by The classical phase volumes should vanish at the classical minimum, z 1 = z 2 = 1 3(1+R 2 ) . We check numerically this is indeed true for e.g. R = 1. Now let us consider the quantum mirror curve, Ψ(x − i ) + z 1 z 2 2 z 3 e 3x e 3i 2 Ψ(x + i ) + z 1 z 2 2 e 3x + z2e 2x + e x + 1 Ψ(x) = 0 (4.63) By taking the residue of w(x; ), we find a quantum A-period, (4.64) The differential operator giving the first quantum correction is (4.65) Then, we can obtain the 2 correction to the quantum mirror map and quantum B-period by acting above differential operator on the classical periods, We note that in this model, the t 3 depends only on mass parameter R and receives no quantum correction.
To check the consistency, let us calculate the NS free energy near the large radius point. In the similar way as in the C 3 /Z 5 case, we can obtain the NS free energy by solving the recursion relations (4.27) and (4.28) whose instanton parts [F n ] inst. are given by (4.67) They agree with the topological vertex computations. Accidentally, it turns out that the derivatives with mass parameter ∂ t 3 F 1 also satisfies a similar equation although it does not formally appear in (4.27) for this model. Now we are ready to calculate the quantum corrections to the energy spectra. The all-order Bohr-Sommerfeld quantization conditions are given by where vol i (E 1 , E 2 , R; ) are the quantum corrected phase volume. To obtain the quantum corrected spectrum, we define E i and vol i (E 1 , E 2 , R; ) as series of , (4.69) The classical Bohr-Sommerfeld volumes have to vanish in the classical limit = 0 of (4.68) at the minimum E which corresponds to the conifold point. In the following, we demonstrate the computation for R = 1. The leading corrections to the spectra are given by (4.47). By comparing them with direct perturbative calculations (3.14), we obtain the exact value of E 1,2 -derivatives of the volumes, With the changes of variables, we find They agree with the direct computation numerically.
To obtain the derivatives of the volumes, we use the classical periods near the conifold point, The coefficients of z c,1 and z c,2 are fixed by the relation (4.22). From them, we can obtain the next leading order of the quantum corrections to the energy spectra by looking at 2 -order of (4.68). After some computations, we have 2 =E 1 | n1↔n2 . (4.75) They agree with the perturbative computation given in (3.14).
(4.80) The classical B-periods Π d,i (i = 1, 2) are given by formula (4.4) with the noninvertible matrix (4.81) (4.87) Then, we can obtain the quantum mirror map and quantum B-period by acting the differential operator on the classical periods, The NS free energy near the large radius point can be calculated from the general formulae (4.27) and (4.28) whose instanton parts [F n ] inst. are given by They agree with the topological vertex computations. Now we are ready to calculate the quantum corrections to the energy spectra. The Bohr-Sommerfeld quantization condition is given by (4.68) with the quantum corrected spectra and volumes defined in (4.69). The classical Bohr-Sommerfeld volumes have to vanish in the classical limit = 0 of (4.68) at the minimum which corresponds to the conifold point. In the followings, we do the computations for a particular case r = 2 −1/9 . The leading corrections to the spectra are given by (4.47). By comparing with the perturbative computation (3.18), we find the exact values of E 1,2 -derivatives of the volumes at the conifold point, (4.91) which agree with the numerical computation.
In this case we do not calculate the classical mirror map around the conifold point, but when one wants to calculate higher corrections to the energy spectra as in the case of Y 3,0 , the classical mirror map is needed to obtain the higher order quantum corrections to the (derivatives) of the volumes via the formulae (4.18), (4.22).
(4.103) Then, we can obtain the quantum mirror maps and quantum B-periods by acting above operator on the classical periods, as in (4.88).
We do not write down the NS free energy in this case since the computation process is completely the same as the case of Y 3,1 , but one can show that the NS free energy calculated from the differential operators agree with the topological vertex computations. Now we are ready to calculate the quantum corrections to the energy spectra. The Bohr-Sommerfeld quantization condition is given by (4.68) with the quantum corrected spectra and volumes defined in (4.69). The classical Bohr-Sommerfeld volumes have to vanish in the classical limit = 0 of (4.68) at the classical minimum which corresponds to the conifold point. For simplicity, we do the computation for r = 2 −4/9 . The leading corrections to the spectra are given by (4.47). By comparing with the perturbative computation, we find the exact values of E 1,2 -derivatives of the volumes at the conifold point, which are consistent with the numerical computation. Similar to the Y 3,1 case, we do not calculate the classical mirror map around the conifold point, but when one wants to calculate higher corrections to the energy spectra as in the case of Y 3,0 , the classical mirror map is needed to obtain the higher order quantum corrections to the (derivatives) of the volumes via the formulae (4.18) (4.22).
where the complex structure moduli z 1 , z 2 , z 3 are related to the dimer model by , z 3 = R 6 (1 + R 6 ) 2 . (4.113) The Bohr-Sommerfeld volumes should vanish at the conifold point, (4.114) We check that the volumes vanish numerically for e.g. R = 1. Now let us quantize the mirror curve. Correspondingly, the classical mirror curve is replaced by the difference equation, parameters turning on, the Calabi-Yau threefolds are non-toric, it is interesting to study the differential operator approach for these cases [53].
Also, in [54], the authors pointed out that the quantum A-periods of D 5 del Pezzo geometry can be expressed as D 5 Weyl characters. The quantum mirror map of this curve would be given in the same way. Therefore, it would be interesting to clarify the relation between the Weyl group expression and the differential operators.
Recently, the authors in [55] provides the analytic results on black hole perturbation theory from the quantization conditions. They consider the quantization conditions for A-periods, not B-periods. Therefore, it would be interesting to clarify the physical implications of this quantization conditions in the integrable systems or topological strings.

A An eigenvalue formula
Suppose S is a real symmetric 2n × 2n matrix, and M is a real symplectic 2n × 2n matrix that diagonalizes the symmetric matrix, i.e. we have where C = diag{c 1 , c 2 , · · · , c n }, D = diag{d 1 , d 2 , · · · , d n } are real n × n diagonal matrices. Then we can show that the characteristic polynomial of the matrix SΣ (or ΣS) is So the eigenvalues of SΣ are ±i √ c k d k , k = 1, 2, · · · n. In the context of our physics problem, the two diagonal matrices are identical C = D, therefore the diagonal