Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
UUSTC-ICTS/PCFT-20-17
Quantum Periods and Spectra inDimer Models and Calabi-Yau Geometries
Min-xin Huang ∗ , Yuji Sugimoto † , Xin Wang ‡ ∗† Interdisciplinary Center for Theoretical Study,University of Science and Technology of China, Hefei, Anhui 230026, China ∗† Peng Huanwu Center for Fundamental Theory,Hefei, Anhui 230026, China ‡ Bethe Center for Theoretical Physics, Universit¨at Bonn, D-53115, Bonn, Germany ‡ Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany
Abstract
We study a class of quantum integrable systems derived from dimer graphsand also described by local toric Calabi-Yau geometries with higher genus mir-ror curves, generalizing some previous works on genus one mirror curves. Wecompute the spectra of the quantum systems both by standard perturbationmethod and by Bohr-Sommerfeld method with quantum periods as the phasevolumes. In this way, we obtain some exact analytic results for the classical andquantum periods of the Calabi-Yau geometries. We also determine the differen-tial operators of the quantum periods and compute the topological string freeenergy in Nekrasov-Shatashvili (NS) limit. The results agree with calculationsfrom other methods such as the topological vertex. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] J un ontents C / Z case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.2 Y , case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.3 Y , case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.4 Y , case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.5 Y , case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The developments of various prosperous topics in mathematics and physics oftenintersect each others. Topological string theory on Calabi-Yau manifolds has beena fruitful branch of superstring theories that encompass many recurring themes inmathematical physics, see e.g. [1]. In the seminar work [2], Nekrasov and Shatashvili(NS) proposed a connection between the partition function of Seiberg-Witten gaugetheory on Ω background and certain quantum integrable systems. In the NS limit, weset one of the two Ω deformation parameters to vanish and identify the other as thePlanck constant of the quantum system. This relation can be uplifted to five dimen-sions, where the partition functions are computed by refined topological string theoryon corresponding Calabi-Yau spaces. The topological free energy in the NS limit canbe viewed as a quantum deformation of the prepotential, and is computed similarly bypromoting 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 quan-tization conditions of the quantum system are formulated as the Bohr-Sommerfeldquantization condition where the phase volumes are computed by quantum periods.In the five dimensional case, the quantum systems are often known as relativisticmodels due to the exponential kinetic and potential terms in the Hamiltonians from1uantizing the mirror curves of the local Calabi-Yau spaces. Inspired by earlier works[12, 13, 14], some novel non-perturbative contributions to the quantization conditionsare conjectured in [15, 16]. Various aspects of the quantization conditions, includingcomplex value Planck constant, resurgence, wave functions, etc are further exploredin e.g. [17, 18, 19, 20, 21]. The non-perturbative parts of the two types of exact quan-tization conditions in [15, 16] are related by certain constrains on the BPS invariantsknown as the blowup equations [22][23]. The blowup equations originally come fromstudies of Seiberg-Witten gauge theories [24] (see also [25, 26]), but have now becomea very effective tool for computing topological string amplitudes on various Calabi-Yau manifolds [27, 28, 29, 30, 31]. The exact quantization conditions have also beenapplied to related condensed matter systems, e.g. in [32, 33, 34, 35].Most examples of the early studies focus on geometries with mirror curves ofgenus one. The quantum periods and quantization conditions for quantum systemscorresponding to mirror curves of higher genus were subsequently considered, in e.g.[36, 37, 22, 38, 39]. A particularly interesting class of quantum integrable systemscan be constructed by dimer models on torus [40], and the quantization conditionsare studied in [41, 42]. The dimer models in this paper also correspond to local toricCalabi-Yau geometries and the mirror curves are encoded in the data of the bipartitedimer graphs. Some of Calabi-Yau spaces geometrically engineer 5d supersymmetricgauge theories, which are uplifts of the 4d SU ( N ) Seiberg-Witten theories consideredin [7]. There are a number of commuting Hamiltonians, and the multiple quantiza-tion conditions can be similarly derived from topological string free energy in the NSlimit on the corresponding Calabi-Yau spaces. The studies in [41, 42] mostly focuson numerical tests of the non-perturbative quantization conditions. However, in or-der to have a more insightful understanding of the interconnections between varioussubjects here, it is better to have some analytical results. In this paper we developsome analytic approaches to the problem, though mostly focusing on the perturbativeaspects.The paper is organized as followings. In Section 2 we review the constructions ofdimer models, and derive Hamiltonians of the quantum integrable systems based onprevious literatures. We shall study some examples with genus two mirror curves andcorrespondingly two commuting dynamical Hamiltonians. In Section 3 we study theperturbative quantum spectra of the Hamiltonians around minimal points of the phasespace. A useful technical ingredient is the symplectic transformations of the quantumcanonical coordinates, which are necessary to determine the energy eigenvalues of thequadratic terms. We find the symplectic transformations for the examples with simpleclassical minima, and further calculate the higher order spectra with standard pertur-bation methods in quantum mechanics. In Section 4 we systematically compute theperiods and topological string free energies for the Calabi-Yau geometries, summa-rizing the results in previous literature. We then compute the differential operators2hich exactly determine quantum corrections to classical periods, generalizing earlierworks [5, 6] to the situation of higher genus mirror curves. Similarly, the topologicalfree energy in the NS limit is determined by the quantum periods, and we show thatthis agrees with results from e.g. method of topological vertex. An interesting fea-ture is that the differential operators are the same for differential cycles of the highergenus mirror curves. Following earlier works [14], we perform some satisfying testsof our calculations by comparing the quantum spectra from direct perturbation andfrom Bohr-Sommerfeld quantization using quantum periods as phase volumes. Theseexercise provide some exact analytic results for the classical and quantum periods ofthe Calabi-Yau spaces, which are difficult to directly obtain. In [40], the authors proposed an infinite class of cluster integrable systems. Themost interesting ones among them are the cluster integrable systems for the dimermodels on a torus. The dimer model is the study of the set of perfect matching ofa graph, where the perfect matching is a subset of edges which covers each vertexexactly once. For a bipartite graph, the vertices are divided into two sets, the blackset and the white set. Every edge connects a white vertex to a black vertex. For amore 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 determinantof the Kasteleyn matrix, happens to be the mirror curve of the corresponding ToricCalabi-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 issgn( (cid:89) i e i ) = (cid:40) +1 , if ( − , if ( • Construct two loops γ X , γ Y along the two cycles of the torus, we draw them asred 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 theedge with positive orientation. Times each edge with a factor 1 /X or 1 /Y , ifthe loop γ X or γ Y get through the edge with positive orientation. For A type Toda systems, [43][44] have an equivalent but different description. ω ω (a) δ v ( ω , ω ) = − v ω ω (b) δ v ( ω , ω ) = v ω ω (c) δ v ( ω , ω ) = Figure 1: An illustration of δ v ( ω , ω ). If ω and ω are in the counterclockwiseorder, and with the same direction, δ v ( ω , ω ) = as in (c). Any change in theclockwise order or direction gives an extra sign, e.g. (a)(b). The arrows represent theorientations of the loops ω i .Then the Kasteleyn matrix is a matrix with row labeled by black vertices and columnlabeled by white vertices, with the entry as the weight between the connected blackand white vertices. The entry is 0, if two points are not connected. In this paper, weare interested in Y p,q system, the determinant of the Kasteleyn matrix has the form Y + X q Y + X p +2 + · · · + X + 1 = 0 (2.2)Following [40][48], the commutation relations and the Hamiltonians of the clusterintegrable systems can be read from the loops of the graph. Let ω i be the orientedloops on the graph, the Poisson bracket between cycles are defined as { ω i , ω j } = (cid:15) ω i ,ω j ω i ω j , (2.3)where (cid:15) ω i ,ω j := (cid:88) v sgn( v ) δ v ( ω i , ω j ) . (2.4)Here sgn( v ) = 1 for the white vertex v , and − δ v is a skewsymmetric bilinear form with δ v ( ω i , ω j ) = − δ v ( ω j , ω i ) = − δ v ( − ω i , ω j ) ∈ Z , as illus-trated in Figure 1. Though more general vertex is possible, for our examples of dimermodels we will only encounter cubic vertices.To construct the basis t i of all the loops, we can first fix an arbitrary perfectmatching as the reference perfect matching, then all the bases can be constructed fromthe 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 thesecycles t i with the condition that they do not overlap or touch at any vertex of thetiling. In this subsection, we give some examples for the dimer models of 5 d, N = 1 SU (3)gauge theories, with various Chern-Simons level m = 0 , , ,
3. The graphs of these4heories were appeared during the study of 4d N = 1 quiver gauge theories, wherethe 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 mergethe 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 , system is depicted in Figure 2a. Y , model We choose the loops to be t = 7 → → → → , t = 7 → → → → ,t = 8 → → → → , t = 8 → → → → ,t = 9 → → → → , t = 9 → → → → . (2.5)Only loops that are overlapped have non-vanishing Poisson brackets, they are { t , t } = − t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t . (2.6)The Hamiltonians can be read from the graph directly from the rules in precioussection: H = t + t + t + t + t + t ,H = t t + t t + t t + t t + t t + t t + t t + t t + t t ,H = t t t + t t t . (2.7)The Poisson brackets (2.6) can be enhanced to the quantum level as the commuta-tion relations, in terms of canonical variables { q i , p i = − i (cid:126) ∂∂q i } , we find a possiblecoordinates relation t = R e q , t = e p + q , t = R e q − q ,t = e p + q , t = R e − q , t = e − p − p − q − q . (2.8)The R is the radius of the compactification circle from 5d to 4d, which gives a non-trivial deformation to the integrable systems. It is related to the instanton countingparameter or mass parameters in the 5d gauge theory point of view. One can getthe brane tiling of Y , systems 2b (b) by merging the point 8 ,
11 and 2 , ,
12 and 3 ,
6, we get Y , Y , t i are the loops inherited from t i in Y , after merging the points. There is an independent but irrelevant zig-zag path 1 → → → → → →
47 10123 456789 101112 69 77899 (a) Y , (b) Y ,
44 1 45 8123 4567 837 15 1235677 (c) Y ,
14 123456 36 14 123456 (d) Y , Figure 2: Brane tiling for Y ,q , q = 3 , , ,
0, the unit cells are divided by the reddashed lines, which are the loops γ X,Y on the torus6 , model We choose the loops in Figure 2b t = 6 → → → → , t = 6 → → → → ,t = 7 → → , t = 7 → → → → ,t = 8 → → → → , t = 8 → → → → . (2.9)The non-vanishing Poisson brackets are { t , t } = − t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t . (2.10)In terms of canonical variables, t = R e q , t = e p + q , t = R e q − q ,t = e p + q − q , t = R e − q , t = e − p − p − q . (2.11)With the Hamiltonians H = t + t + t + t + t + t ,H = t t + t t + t t + t t + t t + t t + t t + t t ,H = t t t . (2.12) Y , model We choose the loops in Figure 2c t = 5 → → → → , t = 5 → → → → ,t = 6 → → , t = 6 → → → → ,t = 7 → → , t = 7 → → → → . (2.13)The non-vanishing Poisson brackets are { t , t } = − t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t . (2.14)In terms of canonical variables, t = R e q , t = e p + q , t = R e q − q ,t = e p + q − q , t = R e − q , t = e − p − p − q − q . (2.15)With the Hamiltonians H = t + t + t + t + t + t ,H = t t + t t + t t + t t + t t + t t + t t ,H = t t t . (2.16)7 , model We choose the loops in Figure 2d t = 4 → → , t = 4 → → → → ,t = 5 → → , t = 5 → → → → ,t = 6 → → , t = 6 → → → → . (2.17)The non-vanishing Poisson brackets are { t , t } = − t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = t t , { t , t } = − t t . (2.18)In terms of canonical variables, t = R e q , t = e p + q , t = R e q − q ,t = e p − q , t = R e − q , t = e − p − p . (2.19)With the Hamiltonians H = t + t + t + t + t + t ,H = t t + t t + t t + t t + t t + t t H = t t t . (2.20) In this section, we consider the perturbative energy spectra of the quantum integrablesystems described by genus two mirror curves, including the Y ,m models with m =0 , , ,
3, and C / Z model. Each model have two dynamical Hamiltonians, which arederived from dimer models. In the previous Section 2, we derived the Hamiltoniansfor the Y ,m models, where the case of m = 0 was also considered in [41]. TheHamiltonians of some orbifold models including C / Z are available in [42]. We alsonote that the Y , model is equivalent to the orbifold C / Z model in [42]. Wequantize the Hamiltonians by promoting the dynamical variables to operators withcanonical commutation relations [ q i , q j ] = [ p i , p j ] = 0 , [ q i , p j ] = i (cid:126) δ i,j with i, j = 1 , q , p , q , p ). Firstwe consider the Y , , Y , , C / Z models, for which the classical minima are simplylocated at the origin q = q = p = p = 0. We expand the Hamiltonians around theminimal point.First we study in details the C / Z model, whose Hamiltonians are H = e q + e p + e − q + q + e p + e − q − p − p , (3.1) H = e q + e q + p + e p + p + e − p − q + e − q − p − p . (3.2)8e expand the Hamiltonians up to quadratic order H i = 5 + 12 (cid:0) q q p p (cid:1) S i q q p p + O ( (cid:126) ) , i = 1 , , (3.3)where the S , S are real symmetric matrices S = − − , S = . (3.4)We would like to write the quadratic Hamiltonians as linear combinations of twoharmonic oscillators. We consider a linear transformation q q p p = M x x y y , (3.5)where M is a 4 × M must be a symplectic matrix M Σ M T = Σ, where Σ is the antisymmetricmatrix Σ = − − . (3.6)It turns out due to the special property that the Hamiltonians commute with eachother, we can find symplectic transformation M so that the quadratic terms can bewritten as linear combinations of the two harmonic oscillators H = 5 + 12 [ c ( x + y ) + c ( x + y )] + O ( (cid:126) ) ,H = 5 + 12 [ c ( x + y ) + c ( x + y )] + O ( (cid:126) ) . (3.7)There is a continuous 2-parameter family of solutions for the matrix M . Without lossof generality, we can use a particular solution M C / Z = − (5 − √ √ − (10 − √ √ ( + √ ) − (cid:113) (5 − √ − ( + √ ) / (1 + √ ) − (10+2 √ √ ( + √ ) − √ (1 − √ ) ( + √ ) − (25 − √ / √ − / (1 − √ ) − / (1 + √ ) , (3.8)9ith the linear coefficients c = ( 5 + √
52 ) , c = ( 5 − √
52 ) ,c = (5 − √ , c = (5 + 2 √ . (3.9)Denoting the quantum levels of the harmonic oscillators ( x , y ) and ( x , y ) bytwo non-negative integers n , n , the quantum spectrum up to order (cid:126) is (cid:18) E E (cid:19) = 5 (cid:18) (cid:19) + (cid:18) c c c c (cid:19) (cid:18) n + n + (cid:19) (cid:126) + O ( (cid:126) ) (3.10)We can further compute the higher order corrections to energy spectra. We usethe time-independent perturbation theory well-known in quantum mechanics, whichseparates a Hamiltonian into a zero order part and a perturbation part H = H + H (cid:48) , (3.11)where the zero order part H corresponds to the Hamiltonians up to quadratic orderin (3.3), while the perturbation part H (cid:48) corresponds to the higher order terms.We denote the harmonic quantum states of the zero order Hamiltonians as | n , n (cid:105) .Then the first few order corrections to energy spectra are E ( n ,n ) = E (0)( n ,n ) + (cid:104) n , n |H (cid:48) | n , n (cid:105) + (cid:88) ( m ,m ) (cid:54) =( n ,n ) |(cid:104) m , m |H (cid:48) | n , n (cid:105)| E (0)( n ,n ) − E (0)( m ,m ) + · · · . (3.12)To compute the next (cid:126) order corrections, we need to expand the exponentials inthe Hamiltonians (3.1) to cubic and quartic orders, and rewrite the canonical coor-dinates in terms of the standard creation and annihilation operators. For the thefirst correction (cid:104) n , n |H (cid:48) | n , n (cid:105) , the cubic terms have no contribution since there isan odd number of creation and annihilation operators, while the quartic terms makean order (cid:126) contribution. The cubic terms have a (cid:126) order contribution in the morecomplicated second correction term in the above equation (3.12). After some com-plicated calculations, we find the (cid:126) order contributions to the quantum spectra. Forthe C / Z model, the results are E = 5 + [( 5 + √
52 ) n + ( 5 − √
52 ) n + 12 (5 + 2 √ ] (cid:126) +[7 + 2(3 + √ n (1 + n ) + 2(3 − √ n (1 + n )] (cid:126)
40 + O ( (cid:126) ) , (3.13) E = 5 + [(5 − √ n + (5 + 2 √ n + ( 5 + √
52 ) ] (cid:126) + [3 + √ n +2(2 − √ n + 4(1 + √ n + 2(2 + √ n + 4 √ n n ] (cid:126)
20 + O ( (cid:126) )10t is well known that the eigenvalues of a matrix do not change under a similaritytransformation 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 freedomof exchanging the two quantum numbers n ↔ n . This is easy to understand fromphysics point of view since the Hamiltonians are the same regardless of the choicesof the canonical coordinates. Furthermore, the linear coefficients (3.9) are indeedrelated to the eigenvalues of certain matrices. We note that for a general even-dimensional real symmetric matrix S , since det( S Σ − λI ) = det(Σ S − λI ) = det( S Σ + λI ) = det(Σ S + λI ), the eigenvalues of S Σ and Σ S are the same and always come inpairs with opposite signs. In our context, we find that for the matrices (3.4) in thequadratic Hamiltonians, the eigenvalues of S Σ and S Σ are always purely imaginaryand the positive imaginary parts are exactly the linear coefficients (3.9). Namely, theeigenvalues of S Σ are ± ic , ± ic and the eigenvalues of S Σ are ± ic , ± ic . This isalso true for the Y , and Y , models discussed below. In the Appendix A we give asimple general mathematical proof of this property.Similarly we find the symplectic transformations and the perturbative energy spec-tra for the Y , and Y , models in (2.20, 2.7). Again there is a continuous 2-parameterfamily of solutions for symplectic transformations. For the Y , model, we can usefor example a solution M Y , = 1 √ R (4 + R ) − R − R −√ R − R − R −√ R √ R + 4 − R − √ R +4+ R − √ ( √ R + 4 + R ) R − √ R +42 √ R + 4 − √ ( √ R + 4 + R ) √ R , and the perturbative energy spectrum is (cid:18) E E (cid:19) = 3(1 + R ) (cid:18) (cid:19) + √ R √ R ( n + n + 1) (cid:18) (cid:19) + R ( n − n ) (cid:18) − (cid:19) ] (cid:126) + (cid:8)(cid:2) n + 6 n + 15 n + 6 n + 18 n n ) + [5 + 6( n + n + n + n )] R −
724 + R (2 + 3 n + n + 3 n + n + 4 n n ) (cid:3) (cid:18) (cid:19) +6 R √ R ( n + n − n − n ) (cid:18) − (cid:19) (cid:9) (cid:126)
72 + O ( (cid:126) ) . (3.14)We see there is an apparent symmetry of the spectra. The spectra of the two Hamil-tonians E ↔ E are exchanged if the quantum levels are exchanged n ↔ n .For the Y , model, the results are M Y , = 2 · √ R − R − R −√ R − R − R −√ RR + 2 2 R − √ R −√ R − R + 2 −√ √ R , = 3(1 + R ) + √ R ( n + n + 1) (cid:126) + [ 1 + R n
12 (1 + n )(1 + R + R )+ n
12 (1 + n )(1 − R + R )] (cid:126) + O ( (cid:126) ) , (3.15) E = 3(1 + R + R ) + √ R [( n + n + 1)(1 + R ) − ( n − n ) R ] (cid:126) + (cid:8) [4 + 3( n + n + n + n )](1 + R ) − n − n )(1 + n + n ) R (1 + R )+2(8 + 15 n + 6 n + 15 n + 6 n + 18 n n ) R (cid:9) (cid:126)
36 + O ( (cid:126) ) . There is also an apparent symmetry that under a T-duality like transformation R → R , the energy spectra transforms as E → E R , E → E R .We need to be careful with a potential subtlety of perturbation theory here. Forthe first Hamiltonian of the Y , model, we see that the energy E are degenerate up to (cid:126) order for quantum states with the same n + n . It turns out that this does not affectthe calculations in formula (3.12), as we check that the off-diagonal elements of theperturbation in the degenerate space actually vanish, i.e. (cid:104) n + k, n − k |H (cid:48) | n , n (cid:105) = 0for k = ± , ±
2. The vanishing is trivial for cubic terms in the perturbation H (cid:48) , whilewe check by explicit computation that it is also true for quartic terms.For the remaining Y , and Y , models (2.16, 2.12), we need to determine theclassical minima by solving for the critical points of Hamiltonians ∂ q i H = ∂ p i H = 0for i = 1 ,
2. We find that the minima are located at the same points for the twoHamiltonians of the quantum system due to the special property that the Hamil-tonians commute withe each other. In these models it is much more complicatedto find the symplectic transformations that diagonalize the quadratic terms of theHamiltonians expanded around the minima. However, we can still use the formulain Appendix A to compute the (cid:126) order contributions to quantum spectra in terms ofthe eigenvalues of certain matrices from the quadratic terms.For the Y , model, the minima are at q = − r ) , q = − r ) , p = − log( r ) , p = − log( r + R ) , (3.16)where r is the only positive root of the polynomial equation, r + R r = 1 , (3.17)with numerical value e.g. r = 0 . R = 1. The quantum spectra are E = 3 r + 4 R r ( r + R ) + r R [(3 r + 2 R + 2 R √ r + R ) ( n + 12 )+(3 r + 2 R − R √ r + R ) ( n + 12 )] (cid:126) + O ( (cid:126) ) , (3.18) E = 3 r + 2 R r + r R ( r + R ) [(3 r + 4 R − R √ r + R ) ( n + 12 )+(3 r + 4 R + 4 R √ r + R ) ( n + 12 )] (cid:126) + O ( (cid:126) ) . Y , model, the minima are at q = − r ( r + R )] , q = − log[ r ( r + R )] ,p = − log( r ) , p = − log( r + R ) , (3.19)where r is now the only positive root of the equation r + 4 R r + 6 R r + 4 R r + R r = 1 , (3.20)with numerical value e.g. r = 0 . R = 1. The quantum spectra are E = 3 r + 5 R r ( r + R ) + r R ( r + R ) √ r + 5 R + R √ r + 5 R ) ( n + 12 )+(6 r + 5 R − R √ r + 5 R ) ( n + 12 )] (cid:126) + O ( (cid:126) ) , (3.21) E = 3 r + 7 rR + 5 R r ( r + R ) + r R ( r + R ) √ { [ r ( r + R ) (6 r + 25 r R + 30 rR + 10 R ) − R (5 r + 10 rR + 4 R ) (cid:112) r (4 r + 5 R )] ( n + 12 ) + [ r ( r + R ) (6 r + 25 r R +30 rR + 10 R ) + R (5 r + 10 rR + 4 R ) (cid:112) r (4 r + 5 R )] ( n + 12 ) } (cid:126) + O ( (cid:126) ) . Without solving the symplectic transformations for these two Y , and Y , models,there is an ambiguity of exchanging the quantum numbers n ↔ n in the spectra.This can be fixed by comparing with the derivatives of periods of the correspondingCalabi-Yau geometries. In this section we will show that the spectrum problem can be solved by utilizing thewell-known method in topological string theory. More precisely, we use the methoddeveloped in [4], and calculate the energy spectra by imposing the Bohr–Sommerfeldquantization condition on the quantum B-periods of the mirror curves. First we sum-marize some basic facts about the classical/quantum mirror curves, and the generalrelations between topological strings and the energy spectra. After that, we willdemonstrate how we calculate the energy spectra from the quantum periods in someconcrete models.
We consider topological string theory on the toric Calabi-Yau three-fold, where thetopological 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)13here z = ( z , z , ..., z s ) are the complex structure moduli parameters with s := b , ≥ g . Generally, there are g dynamical moduli corresponding to g compact A-and B-cycles of the Riemann surface, and the s − g remaining ones are known asnon-dynamical mass parameters. We can define two kinds of classical periods calledas A- and B-periods by integrating y = y ( x ; z ) around compact A-cycles and theirdual B-cycles,Π i ( z ) = (cid:73) A i y ( x ; z )d x, Π i,d ( z ) = (cid:73) B i y ( x ; z )d x, ( i = 1 , ..., g ) (4.2)where y ( x ; z ) is the solution of (4.1).The mirror maps connecting the K¨ahler moduli with the complex moduli can bewritten as linear combinations of A-periods and mass parameters t j ( z ) = g (cid:88) i =1 C ij Π i ( z ) + mass terms , j = 1 , , · · · s, (4.3)where the mass terms depend only on logarithm of mass parameters and will notappear in quantum corrections. Here C ij is the intersection matrix of compact divisorsand the base curves we have chosen. With a suitable choice of base curves, parts ofthe g × s matrix C ij happens to be the Cartan matrix of the gauge group in thecontext of geometric realizations of gauge theories.In the similar way, the dual B-periods give the derivatives of the genus-zero topo-logical string amplitude, so-called prepotential F ( t ),Π d,i ( z ) = ∂F ( t ( z )) ∂ Π i ( z ) = s (cid:88) j =1 C ij ∂F ( t ( z )) ∂t j ( z ) , i = 1 , , · · · g. (4.4)From the prepotential, we define the Bohr-Sommerfeld volumes as the derivatives ofprepotential with appropriate shift 4 π b NSi vol i ( z ) = s (cid:88) j =1 C ij (cid:18) ∂F ( t ) ∂t j + 4 π b NSj (cid:19) . (4.5)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]. Ingauge theory point of view, b NSi comes from the one loop contribution. For 5d N = 1pure SU ( N ) gauge theories with Chern-Simons level, denoting the t i , i ≤ N − A N group, and t N the instanton counting parameter. By setting b NSN = 0, wehave b NSi = b NSN − i = − ( N − i ) i , for i = 1 , , · · · , N . For the SU (3) models we considered,we always have b NS = b NS = − , b NS = 0.14or a toric Calabi-Yau three-fold, an efficient way to calculate A- and B-period isto solve the Picard–Fuchs equations defined by L α Π i = 0 , L α Π d,i = 0 , L α = (cid:89) Q αi > (cid:18) ∂∂x i (cid:19) Q αi − (cid:89) Q αi > (cid:18) ∂∂x i (cid:19) Q αi , (4.6)where Q αi is the charge vector and x i is the homogeneous coordinate of the toricvariety. The differential operator L α ’s are known as the Picard–Fuchs operators. Thevariables x i relate to z through the Batyrev coordinates z α = k +3 (cid:89) i =1 x Q αi i . (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 withthe canonical commutation relation, [ x , y ] = i (cid:126) . (4.8)Accordingly, the mirror curve is replaced by the difference equation, W (e x , e y )Ψ( x ) = 0 , (4.9)where Ψ( x ) is the wave function. We can solve the difference equation by utilizingthe WKB analysis, Ψ( x ) = exp (cid:18) i (cid:126) (cid:90) x w ( x (cid:48) ; (cid:126) )d x (cid:48) (cid:19) . (4.10)Then, we can define quantum version of two periods, called as quantum A- and B-period, Π i ( z ) → Π i ( z ; (cid:126) ) = ∞ (cid:88) n =0 Π ( n ) i (cid:126) n , Π ( n ) i = (cid:73) A w ( n ) ( x )d x, (4.11a)Π d,i ( z ) → Π i,d ( z ; (cid:126) ) = ∞ (cid:88) n =0 Π ( n ) i,d (cid:126) n , Π ( n ) i,d = (cid:73) B w ( n ) ( x )d x, (4.11b)where we expand w ( x ; (cid:126) ) in (cid:126) , w ( x ; (cid:126) ) = ∞ (cid:88) n =0 w ( n ) (cid:126) n . (4.12)In our example, w (2 n − , n ∈ Z > can be expressed as the total derivative of simplefunctions with no monodromy. Thus, its contour integral vanishes, and only (cid:126) n -corrections survive. 15he quantum corrected prepotential F ( t ; (cid:126) ), so-called NS free energy, is definedby refined topological string free energy in the NS limit F ( t , (cid:126) ) = ∞ (cid:88) n =0 F n ( t ) (cid:126) n . (4.13)It satisfies a similar equation as the prepotentialΠ d,i ( z ; (cid:126) ) = s (cid:88) j =1 C ij ∂F ( t ( z ; (cid:126) ) , (cid:126) ) ∂t j ( z , (cid:126) ) , (4.14)where t i ( z ; (cid:126) ) are the quantum mirror maps. Comparing both sides of (4.14), we canobtain the recursion relations which enable us to fix F i ( t ) completely, up to irrelevantconstants and mass parameters.The Bohr–Sommerfeld volumes (4.5) also have quantum corrections,vol i ( z ) → vol i ( z ; (cid:126) ) = (cid:88) n ≥ vol (2 n ) i ( z ) (cid:126) n . (4.15)In quantum mechanics, the phase volume should be quantized. In our case, theB-periods are quantized, (cid:73) B i w ( x, (cid:126) )d x = 2 π (cid:126) (cid:18) n i + 12 (cid:19) . n i ∈ Z ≥ . (4.16)By using (4.5), we can rewrite the quantization condition as follows,vol i ( z ; (cid:126) ) = 2 π (cid:126) (cid:18) n i + 12 (cid:19) , i = 1 , , ..., g. (4.17)The dynamical complex structure moduli will correspond to Hamiltonians of thequantum systems as we will see in concrete examples. As in the case of NS freeenergy, by expanding the quantum B-periods in (cid:126) , we can determine the quantumcorrections to the energy eigenvalues recursively. The B-periods have to vanish in theclassical limit at the minimal energy points, which correspond to the conifold pointsin the topological string moduli space. Thus, to solve the spectral problem from thetopological strings, we have to calculate the phase volumes at the conifold point. Itturns out that there is no logarithmic cut for the classical volumes (B-periods) at theconifold points, so they are the same as the mirror maps up to numerical factors,vol i (Coni; (cid:126) ) ∼ t i,c (Coni; (cid:126) ) , (4.18)where Coni denotes the conifold point, and t i,c ( z c ; (cid:126) ) is quantum mirror map expandedaround the conifold point. The numerical factor in the coefficients of t i,c ( z c ; (cid:126) ) canbe determined by comparing with the derivatives of the classical volumes at coni-fold point or the perturbative computation as we have done in the previous section.16herefore, we can calculate the eigenvalues only by using the quantum mirror mapsnear the conifold points.Now we move to the computation of the quantum periods. It is straightforwardto calculate the quantum A-periods from the definition by taking residues, whereasthe direct computations of B-periods are usually not so easy. Here we utilize thedifferential operator method proposed in [3], and developed in [5].The important fact is that the quantum A-periods can be given by the classicalperiods with differential operators as follows,Π k ( z ; (cid:126) ) = (cid:32) ∞ (cid:88) n =0 (cid:126) n D n (cid:33) Π k ( z ) , k = 1 , , · · · , g, (4.19)where D n = D n ( θ z , θ z , ...θ z s ) , θ z i = z i ∂∂z i . (4.20)and coefficients of θ z i are given by rational functions of z i . This means that we canobtain the differential operators in the conifold frame by transforming from largeradius frame to the conifold frame z i → z c,i . Then, by acting the operators on theclassical A-periods expanded near the conifold point, we can obtain the quantumcorrections in the conifold frame. Since the mass parameters are annihilated by thedifferential operators, they do not receive quantum corrections.According to (4.3), the quantum mirror maps are determined by the same differ-ential operators as t i ( z ; (cid:126) ) = (cid:32) ∞ (cid:88) n =0 (cid:126) n D n (cid:33) t i ( z ) , i = 1 , , · · · , s. (4.21)Interestingly, the differential operators that we will treat in our study do not dependon the choice of the cycles . Also, the classical mirror maps can be calculated from thePicard–Fuchs operators. Therefore, it is enough to calculate one of the quantum A-periods to derive the differential operators and determine the quantum mirror maps.By combining (4.18) with (4.21), the quantum corrections to the volumes andtheir derivatives with respect to the eigenvalues are given by ∂ p E ∂ p E · · · ∂ p s E s vol (2 n ) j ∼ ∂ p E ∂ p E · · · ∂ p s E s ( D n t c,j ) , (4.22)where p i ∈ Z ≥ , n ∈ Z > , and j = 1 , , ..., s . To calculate the right hand side, we use ∂ E i = (cid:80) gj =1 ( ∂ E i z c,j ) ∂ z c,j .Remarkably, this structure is the same as the quantum B-period; the quantumcorrections can be calculated by acting above operators on the classical B-periods,Π d,i ( z ; (cid:126) ) = (cid:32) ∞ (cid:88) n =0 (cid:126) n D n (cid:33) Π d,i ( z ) . (4.23) It would be interesting to confirm this property in general setup D n from the quantum A-period that we know how to calculate systematically, we can obtain the quantumB-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 freeenergy by expanding the equations (4.21, 4.14, 4.13). We can explicitly do this forthe first and second correction terms F , ( t ), which are determined by the differentialoperators D , D . In our examples, the differential operators will be a linear combi-nations of first and second derivatives of the complex structure moduli. Suppose D = (cid:88) i s (2) i θ i + (cid:88) i,j s (2) i,j θ i θ j , (4.24)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 θ k ( ∂ t i F ) = (cid:88) j θ k ( t j )( ∂ t i ∂ t j F ) , (4.25) θ k θ l ( ∂ t i F ) = (cid:88) j θ k θ l ( t j )( ∂ t i ∂ t j F ) + (cid:88) j,m θ k ( t j ) θ l ( t m )( ∂ t i ∂ t j ∂ t m F ) . So we have D ( ∂ t i F ) = (cid:88) j D ( t j )( ∂ t i ∂ t j F ) + (cid:88) j,k,l,m s (2) l,m θ l ( t j ) θ m ( t k )( ∂ t i ∂ t j ∂ t k F ) . (4.26)Combining the (cid:126) 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 s (cid:88) i =1 C n,i [ ∂ t i F − (cid:88) j,k,l,m s (2) l,m θ l ( t j ) θ m ( t k )( ∂ t i ∂ t j ∂ t k F )] = 0 , n = 1 , , · · · , 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 s (cid:88) i =1 C n,i [ ∂ t i F − (cid:88) j,k,l,m s (4) l,m θ l ( t j ) θ m ( t k )( ∂ t i ∂ t j ∂ t k F ) − (cid:88) j D ( t j )( ∂ t i ∂ t j F ) − (cid:88) j,k D ( t j ) D ( t k )( ∂ t i ∂ t j ∂ t k F )] = 0 , n = 1 , , · · · , g. (4.28)Again, the linear coefficients s (4) i cancel out. By using (4.27), if the matrix C isinvertible, we can eliminate F , and obtain the relation between F and F .18 .2 Examples We shall demonstrate the previous computations in some concrete models. In ourexamples, we focus on the genus-2 mirror curves: C / Z and Y ,m with m = 0 , , , C / Z case The mirror curve of C / Z is defined ase x + e − x − p + e − p + z / z e x + z − / = 0 . (4.29)The Picard–Fuchs operators are L = − θ θ + θ + z ( − θ + 3 θ − θ + 6 θ − θ θ + 9 θ θ + 27 θ − θ θ + 27 θ ) , L = θ − θ θ + z ( − θ − θ + θ + 4 θ θ − θ ) , L = θ θ + z z ( − θ + 2 θ + 7 θ θ − θ θ − θ + 24 θ θ − θ ) . (4.30)To provide the solutions of the Picard–Fuchs equations, we first define followingfunction, ω ( ρ i ) = (cid:88) l,m ≥ c ( l, m ; ρ ) z l + ρ z m + ρ (4.31)with c ( l, m ; ρ ) = Γ( ρ + 1) Γ( ρ + 1)Γ( ρ − ρ + 1)Γ( − ρ + ρ + 1)Γ( l + ρ + 1) Γ( m + ρ + 1)Γ( l − m + ρ − ρ + 1)Γ( − l + m − ρ + ρ + 1) . (4.32)We further define the derivatives of ω ( ρ i ), ω i = ∂ω ∂ρ i (cid:12)(cid:12)(cid:12)(cid:12) ρ , =0 , ω ij = ∂ ω ∂ρ i ∂ρ j (cid:12)(cid:12)(cid:12)(cid:12) ρ , =0 . (4.33)Then, the mirror maps are given by, t ( z ) = ω = log z − z − z + 45 z − z + O ( z i ) ,t ( z ) = ω = log z + 2 z + 2 z − z + 3 z + O ( z i ) . (4.34)The derivatives of the prepotential are ∂F ∂t = 2 ω , + 2 ω , + 3 ω , ,∂F ∂t = ω , + 6 ω , + 9 ω , . (4.35)19he 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, C = (cid:20) − − (cid:21) . (4.36)From the prepotential, the Bohr-Sommerfeld volumes arevol (0)1 ( z ) = 3 ∂F ∂t − ∂F ∂t − π , vol (0)2 ( z ) = − ∂F ∂t + 2 ∂F ∂t − π , (4.37)where the complex structure moduli z , z are related to the quantum systems of thedimer model by z = − E E , z = E E . (4.38)The classical volumes vanish at conifold point, z = − / , z = 1 /
5, or E = E = 5,which is checked numerically.Now let us consider the quantum periods. Correspondingly, the classical mirrorcurve is replaced by the difference equation,Ψ( x + i (cid:126) ) + e − x e i (cid:126) Ψ( x − i (cid:126) ) + (cid:16) z z e x + e x + z − (cid:17) Ψ( x ) = 0 . (4.39)According to [4], the quantum A-periods are given by taking the residue,Π( z ; (cid:126) ) = 15 log( z z ) + (cid:73) x = −∞ d xw ( x ; (cid:126) )= 15 log( z z ) − z z + 15 z − z − (cid:18) z z − z z (cid:19) (cid:126) + O ( (cid:126) , z i ) . (4.40)We note that as familiar from literature, the logarithmic term is not captured by theresidue calculations and is added by hand. We express the coefficients Π ( n ) by thedifferential operator method. Since the differential operators giving Π ( n ≥ is tediouslong expression, here we provide the differential operator giving the leading correctionto the classical periods as an example , D = 18 θ + 16 θ θ . (4.41)By using the operator, we can obtain the leading correction to the quantum mirrormaps t (2) i ( z ; (cid:126) ) and the quantum B-periods Π (2) d,i ( z ; (cid:126) ), t (2) i ( z ; (cid:126) ) = D t i ( z ) , Π (2) d,i ( z ; (cid:126) ) = D Π (0) d,i ( z ) , (4.42) We provide the results of differential operators giving higher order quantum corrections in the mathematica file. The results contain the differential operators of C / Z and Y ,m with m = 0 , , , i = 1 , F n ] inst. are given by [ F ] inst. = − Q Q −
16 65 Q Q − Q
16 + 7 Q Q Q Q Q − Q − Q O ( Q i ) , [ F ] inst. = − Q Q − Q Q − Q
64 + 29 Q Q
640 + 67 Q Q Q
640 + Q
180 + Q
360 + O ( Q i ) . (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 byvol i ( E , E ; (cid:126) ) = 2 π (cid:126) (cid:18) n i + 12 (cid:19) , i = 1 , , (4.44)where vol i ( E , E ; (cid:126) ) are the quantum corrected phase volumes. To obtain the quan-tum corrected spectrum, we define E i and vol i ( E , E ; (cid:126) ) as series of (cid:126) , E i = ∞ (cid:88) n =0 E ( n ) i (cid:126) n , vol i ( E , E ; (cid:126) ) = ∞ (cid:88) n =0 vol (2 n ) i ( E , E ) (cid:126) n . (4.45)The classical Bohr–Sommerfeld volumes have to vanish in the classical limit (cid:126) = 0 of(4.68) at the minimum E (0)1 = 5 =: E m , E (0)2 = 5 =: E m , (4.46)which corresponds to the conifold point. By expanding (4.68) in (cid:126) , we can obtain E ( n ) i as a function of vol ( n ) i ( E m , E m ), e.g., E (1)1 = 2 π (cid:110) ( n + ) ∂ E vol (0)2 − ( n + ) ∂ E vol (0)1 (cid:111)(cid:110) ∂ E vol (0)1 ∂ E vol (0)2 − ∂ E vol (0)1 ∂ E vol (0)2 (cid:111) ,E (1)2 = 2 π (cid:110) ( n + ) ∂ E vol (0)2 − ( n + ) ∂ E vol (0)1 (cid:111)(cid:110) ∂ E vol (0)1 ∂ E vol (0)2 − ∂ E vol (0)1 ∂ E vol (0)2 (cid:111) , (4.47)where we omit the arguments ( E m , E m ) of vol ( n ) i . By comparing (4.47) with per-turbative calculations (3.10), we obtain exact values of the E , -derivatives of phase We will use this expression for other models, where the arguments of vol ( n ) i in these models are( E m , E m , R ). (cid:32) ∂ E vol (0)1 ∂ E vol (0)1 ∂ E vol (0)2 ∂ E vol (0)2 (cid:33) = π (cid:32) − (cid:0) − √ (cid:1) / (cid:0) (cid:0) √ (cid:1)(cid:1) / (cid:0) √ (cid:1) / − (cid:0) − √ (cid:1) / (cid:33) . (4.48)With the change of variables (4.38), we find (cid:32) ∂ z vol (0)1 ( z ) ∂ z vol (0)1 ( z ) ∂ z vol (0)2 ( z ) ∂ z vol (0)2 ( z ) (cid:33) = π (cid:32) − (cid:112) √ (cid:112) − √ − (cid:112) − √ − (cid:112)
130 + 58 √ (cid:33) . (4.49)We check this is indeed true numerically.The classical mirror maps near the conifold point are t c, = − π (cid:16) (cid:16) √ (cid:17)(cid:17) / z c, − π (cid:18) − √ (cid:19) / z c, + 24 π (cid:18) − √ (cid:19) / z c, z c, + O ( z c,i ) ,t c, = − π (cid:16) − √ (cid:17) / z c, − π (cid:18)
13 + 22 √ (cid:19) / z c, + 24 π (cid:18) √ (cid:19) / z c, z c, + O ( z c,i ) , (4.50)where z = 125 + z c, , z = −
15 + z c, . (4.51)The coefficients of z c, , z c, in the classical mirror map are fixed by the relation (4.22).We can calculate the next leading order of the energy spectrum E (2)1 , by looking at (cid:126) -order term of (4.44). To obtain them, we need to calculate the second derivativesof the volumes and first quantum correction which can be calculated from the formula(4.22) with (4.41). After some computations, we find E (2)1 = 140 (cid:16) − (cid:16) √ − (cid:17) n ( n + 1) + 2 (cid:16) √ (cid:17) n ( n + 1) + 7 (cid:17) ,E (2)2 = 120 (cid:16) (cid:16) √ (cid:17) n − (cid:16) √ − (cid:17) n + 4 (cid:16) √ (cid:17) n + 4 n + 4 √ n n + √ (cid:17) . (4.52)These results agree with the perturbative computation (3.13). Y , case The mirror curve of Y , ise p + z z z e − p +3 x + z z e x + z e x + e x + 1 = 0 . (4.53)The Picard–Fuchs operators are L = ( θ − θ )( θ − θ ) − z ( − θ + θ − − θ + θ ) , L = ( θ − θ )( θ − θ ) − z ( − θ + θ − − θ + θ ) , L = θ − z ( θ − θ )( θ − θ ) , L = θ − z z z ( θ − θ )( θ − θ ) . (4.54)22ote that these operators are symmetric under exchange of z and z . To give thesolutions of Picard–Fuchs equation, we define following function, ω ( ρ i ) = (cid:88) l,m,n ≥ c ( l, m, n ; ρ i ) z l + ρ z m + ρ z n + ρ (4.55)with c ( l, m, n ; ρ i ) = 1Γ( n + ρ + 1) Γ( − n + l + ρ − ρ + 1)Γ( − n + m + ρ − ρ + 1) × l − m + ρ − ρ + 1)Γ( − l + m − ρ + ρ + 1) . (4.56)Then, the classical mirror maps and the derivatives of the prepotential are given by t ( z ) = ω = log z + 2 z + 3 z − z − z + z z − z z − z z z + 2 z z z + O ( z i ) ,t ( z ) = ω = t | z ↔ z ,t ( z ) = ω = log z , (4.57)and ∂F ∂t = ω + ω + 12 ω + 23 ω + 13 ω + 2 π ,∂F ∂t = ω + ω + 12 ω + 23 ω + 13 ω + 2 π , (4.58)where ω i = ∂ω ∂ρ i (cid:12)(cid:12)(cid:12)(cid:12) ρ , , =0 , ω ij = ∂ ω ∂ρ i ∂ρ j (cid:12)(cid:12)(cid:12)(cid:12) ρ , , =0 . (4.59)The classical B-periods Π d,i ( i = 1 ,
2) are given by formula (4.4) with C = (cid:20) − − (cid:21) , (4.60)where the first 2 × (0) i ( z ) = (cid:88) j =1 C ij ∂F ∂t j − π , i = 1 , , (4.61)where the complex structure moduli z , z , z are related to the quantum systems by z = E E , z = E E , z = − R . (4.62)The classical phase volumes should vanish at the classical minimum, z = z = R ) .We check numerically this is indeed true for e.g. R = 1.23ow let us consider the quantum mirror curve,Ψ( x − i (cid:126) ) + z z z e x e i (cid:126) Ψ( x + i (cid:126) ) + (cid:0) z z e x + z x + e x + 1 (cid:1) Ψ( x ) = 0 (4.63)By taking the residue of w ( x ; (cid:126) ), we find a quantum A-period,Π( z ; (cid:126) ) = −
13 log( z z ) + (cid:73) x = ∞ d xw ( x ; (cid:126) )= −
13 log( z z ) + (cid:18) − z − z − z z z + 4 z z + 2 z z z + 12 z z z (cid:19) − (cid:18) z z z + 72 z z z (cid:19) (cid:126) + O ( (cid:126) , z i ) . (4.64)The differential operator giving the first quantum correction is D = 112 z z (5 z + 4) θ + 112 z z (5 z + 4) θ + 124 ( − z z − z z z + 4) θ θ . (4.65)Then, we can obtain the (cid:126) correction to the quantum mirror map and quantumB-period by acting above differential operator on the classical periods, t (2) i = D t i , Π (2) d,i = D Π (0) d,i , i = 1 , . (4.66)We note that in this model, the t depends only on mass parameter R and receivesno quantum correction.To check the consistency, let us calculate the NS free energy near the large radiuspoint. In the similar way as in the C / Z case, we can obtain the NS free energyby solving the recursion relations (4.27) and (4.28) whose instanton parts [ F n ] inst. aregiven by [ F ] inst. = − Q − Q − Q − Q − Q Q − Q Q Q − Q Q + 78 Q Q Q + 78 Q Q Q + 56 Q Q Q − Q Q Q + O ( Q i ) , [ F ] inst. = 1360 Q + 1360 Q + 1180 Q + 1180 Q + 1360 Q Q + 75760 Q Q Q + 1180 Q Q + 29640 Q Q Q + 29640 Q Q Q + 371440 Q Q Q + 72880 Q Q Q + O ( Q i ) . (4.67)They agree with the topological vertex computations. Accidentally, it turns out thatthe derivatives with mass parameter ∂ t F also satisfies a similar equation althoughit 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 byvol i ( E , E , R ; (cid:126) ) = 2 π (cid:126) (cid:18) n i + 12 (cid:19) , i = 1 , , (4.68)24here vol i ( E , E , R ; (cid:126) ) are the quantum corrected phase volume. To obtain thequantum corrected spectrum, we define E i and vol i ( E , E , R ; (cid:126) ) as series of (cid:126) , E i = ∞ (cid:88) n =0 E ( n ) i (cid:126) n , vol i ( E , E , R ; (cid:126) ) = ∞ (cid:88) n =0 vol ( n ) i ( E , E , R ) (cid:126) n . (4.69)The classical Bohr–Sommerfeld volumes have to vanish in the classical limit (cid:126) = 0 of(4.68) at the minimum E (0) i = 3(1 + R ) =: E m , (4.70)which corresponds to the conifold point. In the following, we demonstrate the com-putation for R = 1. The leading corrections to the spectra are given by (4.47). Bycomparing them with direct perturbative calculations (3.14), we obtain the exactvalue of E , -derivatives of the volumes, (cid:32) ∂ E vol (0)1 ∂ E vol (0)1 ∂ E vol (0)2 ∂ E vol (0)2 (cid:33) = π (cid:32)(cid:0) (cid:0) √ (cid:1)(cid:1) / − (cid:0) (cid:0) − √ (cid:1)(cid:1) / (cid:0) (cid:0) − √ (cid:1)(cid:1) / (cid:0) (cid:0) √ (cid:1)(cid:1) / (cid:33) . (4.71)With the changes of variables, we find (cid:32) ∂ z vol (0)1 ( z ) ∂ z vol (0)1 ( z ) ∂ z vol (0)2 ( z ) ∂ z vol (0)2 ( z ) (cid:33) = − √ π − √ (cid:18) − √ − √ − √ − √ (cid:19) , (4.72)They agree with the direct computation numerically.To obtain the derivatives of the volumes, we use the classical periods near theconifold point, t c, = − π (5 + 9 √ √ z c, + 4 π (5 − √ √ z c, + z c, + 1312 π √ z c, z c, + 8 π (125 + 117 √ √ z c, z c, − π (125 − √ √ z c, z c, − π √ z c, z c, z c, + O ( z c,i ) ,t c, = t c, | z c, ↔ z c, ,t c, = log( − z c, ) , (4.73)where z = 16 + z c, , z = 16 + z c, , z = − z c, . (4.74)The coefficients of z c, and z c, are fixed by the relation (4.22).From them, we can obtain the next leading order of the quantum corrections tothe energy spectra by looking at (cid:126) -order of (4.68). After some computations, wehave E (2)1 = 1360 (cid:18) √ n + 6(19 − √ n + 6(13 + 5 √ n + 6(13 − √ n + 72 n n + 101 (cid:19) ,E (2)2 = E (2)1 | n ↔ n . (4.75)25hey agree with the perturbative computation given in (3.14). Y , case In this example, we sometimes use some of the notations and definitions given in4.2.2. The mirror curve of Y , is,e p + z e x − p + z z e x + z e x + e x + 1 = 0 . (4.76)The Picard–Fuchs operators are L = θ ( θ − θ − θ ) − z ( − θ + θ − − θ + θ ) , L = θ ( − θ + θ ) − z ( θ − θ − θ − θ − θ − θ ) , L = θ − z ( θ − θ − θ − θ − θ − θ ) , L = θ θ θ − z z z ( − θ + θ )( θ − θ − θ − θ − θ − θ − θ − θ − θ ) . (4.77)Then, the classical mirror maps and the derivatives of the prepotential are given by t ( z ) = ω = log z + 2 z − z − z + 3 z − z − z − z z − z z + 6 z z + z z − z z − z z − z z + 12 z z z + 90 z z z + O ( z i ) ,t ( z ) = ω = log z − z + 2 z + 2 z − z z + 3 z + 12 z z + z z − z z − z z + 60 z z + 60 z z + 630 z z − z z z − z z z + O ( z i ) ,t ( z ) = ω = log z + 13 { t ( z ) − log z ) + 4 ( t ( z ) − log z ) } , (4.78)and ∂F ∂t = 19 (4 ω − ω − ω + 6 ω + 3 ω ) ,∂F ∂t = 19 ( − ω − ω − ω + 3 ω + 6 ω ) ,∂F ∂t = 13 ( ω + ω + ω ) , (4.79)where ω i and ω ij are defined in (4.59), and ω are defined in (4.55) with the coefficient c ( l, m, n ; ρ i ) c ( l, m, n ; ρ i ) = Γ ( ρ + 1) Γ ( ρ + 1) Γ ( ρ + 1) Γ ( l + ρ + 1) Γ ( m + ρ + 1) Γ ( n + ρ + 1) × Γ ( − ρ + ρ + 1) Γ ( ρ − ρ − ρ + 1)Γ ( l + ρ − m + ρ ) − n + ρ ) + 1) Γ ( m − l + ρ ) + ρ + 1) . (4.80)The classical B-periods Π d,i ( i = 1 ,
2) are given by formula (4.4) with the non-invertible matrix C = (cid:20) − − (cid:21) . (4.81)26rom the prepotential, the Bohr-Sommerfeld volumes arevol (0) i ( z ) = (cid:88) j =1 C ij ∂F ∂t j − π , i = 1 , . (4.82)The complex structure moduli z , z , z are related to the dimer system as following, z = E E , z = E E , z = R E . (4.83)From (3.18), the classical phase volumes vanish at z = r + 2( r − , z = − r ( r − r + 2) , z = − ( r − ( r + 2) . (4.84)where we use the polynomial relation (3.17) to eliminate R and write the expressionspurely in terms of r . As an example, for r = 2 − / , we check numerically this isindeed true for (4.82).Now let us consider the quantum mirror curve,Ψ( x − i (cid:126) ) + e i (cid:126) z e x Ψ( x + i (cid:126) ) + (cid:0) z z e x + z e x + e x + 1 (cid:1) Ψ( x ) = 0 (4.85)By taking the residue of w ( x ; (cid:126) ), we find a quantum A-period,Π( z ; (cid:126) ) = −
13 log( z z ) + (cid:73) x = ∞ d xw ( x ; (cid:126) )= −
13 log( z z ) + (cid:18) − z − z − z z − z − z z z − z z − z z − z z + 90 z z z + 12 z z z (cid:19) − (cid:0) − z z − z z − z z − z z + 6 z z z + 90 z z z (cid:1) (cid:126) + O ( (cid:126) , z i ) . (4.86)The quantum corrections can be expressed as the classical part acted by the differ-ential operator, D = − z θ + (cid:18) − z (cid:19) θ + z θ + (cid:18) − z (cid:19) θ + (cid:18) −
112 + z − z z (cid:19) θ θ . (4.87)Then, we can obtain the quantum mirror map and quantum B-period by acting thedifferential operator on the classical periods, t (2) i = D t i , ( i = 1 , , , Π (2) d,k = D Π (0) d,k , ( k = 1 , . (4.88)The NS free energy near the large radius point can be calculated from the generalformulae (4.27) and (4.28) whose instanton parts [ F n ] inst. are given by[ F ] inst. = Q Q Q Q Q Q Q Q Q Q Q Q O ( Q i ) , [ F ] inst. = Q
360 + Q
360 + Q
360 + Q Q
360 + Q Q − Q Q − Q Q Q
40 + O ( Q i ) . (4.89)27hey 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 quantumcorrected spectra and volumes defined in (4.69). The classical Bohr–Sommerfeldvolumes have to vanish in the classical limit (cid:126) = 0 of (4.68) at the minimum E (0)1 = 4 − r r =: E m , E (0)2 = 2 + r r =: E m (4.90)which corresponds to the conifold point. In the followings, we do the computationsfor a particular case r = 2 − / . The leading corrections to the spectra are given by(4.47). By comparing with the perturbative computation (3.18), we find the exactvalues of E , -derivatives of the volumes at the conifold point, (cid:32) ∂ E vol (0)1 ∂ E vol (0)1 ∂ E vol (0)2 ∂ E vol (0)2 (cid:33) = π − − √ − / ( √ ) / ( √ ) / − / ( √ ) / − / ( √ ) (cid:16) / ( √ ) / − ( √ − ) / (cid:17) / ( √ ) / ( √ ) / − ( − √ ) / , (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 thecase of Y , , the classical mirror map is needed to obtain the higher order quantumcorrections to the (derivatives) of the volumes via the formulae (4.18), (4.22). Y , case In this example, we sometimes also use some of the notations and definitions givenin section 4.2.2. The mirror curve of Y , ise p + z e x − p + z z e x + z e x + e x + 1 = 0 . (4.92)The Picard–Fuchs operators are L = θ ( θ − θ − θ ) − z ( − θ + θ − − θ + θ ) , L = ( θ − θ )( − θ + θ ) − z ( θ − θ − θ − θ − θ − θ ) , L = θ − z ( θ − θ − θ )( θ − θ ) , L = θ θ − z z z ( θ − θ − θ − θ − θ − θ )( − θ + θ ) . (4.93)The classical mirror maps and the derivatives of the prepotential are given by t ( z ) = ω = log z + 2 z − z + 3 z − z z z − z z + z z + 12 z z − z z − z z z + 6 z z z + O ( z i ) ,t ( z ) = ω = log z − z + 2 z − z z − z z + z z − z z − z z + 30 z z + 12 z z z − z z z + O ( z i ) ,t ( z ) = ω = log z + 13 { ( t ( z ) − log z ) + 2 ( t ( z ) − log z ) } , (4.94)28nd ∂F ∂t = 118 (16 ω + 10 ω + ω + 12 ω + 6 ω ) ,∂F ∂t = 118 (5 ω + 2 ω + 2 ω + 6 ω + 12 ω ) ,∂F ∂t = 13 ( ω + ω + ω ) (4.95)where ω i and ω ij are defined in (4.59) with the coefficient c ( l, m, n ; ρ i ) given by c ( l, m, n ; ρ i ) = Γ ( ρ + 1) Γ ( ρ + 1) Γ ( ρ − ρ + 1)Γ ( l + ρ + 1) Γ ( n + ρ + 1) Γ ( m − n + ρ − ρ + 1) × Γ ( − ρ + ρ + 1) Γ ( ρ − ρ − ρ + 1)Γ ( m − l + ρ ) + ρ + 1) Γ ( l − n + ρ − m + ρ ) − ρ + 1) . (4.96)The classical B-periods Π d,i ( i = 1 ,
2) are given by (4.4) with the non-invertible matrix C ij , C = (cid:20) − − (cid:21) , (4.97)From the prepotential, the Bohr-Sommerfeld volumes arevol (0) i ( z ) = (cid:88) i =1 C ij ∂F ∂t j − π , i = 1 , . (4.98)The complex structure moduli z , z , z are related to the dimer model by z = E E , z = E E , z = − R E . (4.99)From (3.21), the classical phase volumes vanish at z = r / − r / + 5(2 r / − , z = 5 r / − r / ( r / − r / + 5) , z = ( r / − r / ( r / − r / + 5) , (4.100)where we use the polynomial relation (3.20) to eliminate R . We check numericallythat the B-periods vanish at this point for e.g. r = 2 − / .Now let us consider the quantum mirror curve,Ψ( x − i (cid:126) ) + e i (cid:126) z e x Ψ( x + i (cid:126) ) + (cid:0) z z e x + z e x + e x + 1 (cid:1) Ψ( x ) = 0 (4.101)By taking the residue of w ( x ; (cid:126) ), we find a quantum A-period, Π( z ; (cid:126) ) = −
13 log( z z ) + (cid:73) x = ∞ d xw ( x ; (cid:126) )= −
13 log( z z ) + (cid:18) − z − z z z + z z + 12 z z − z z − z z z + 6 z z z (cid:19) − (cid:18) z z z z − z z − z z z + 5 z z z (cid:19) (cid:126) + O ( (cid:126) , z i ) . (4.102)29he differential operator in this case is relatively long expression, D = − z (1 − z ) (cid:26) − z + 12 z z − z z z ) θ + 1 z (cid:0) − z + 5 z − z z − z z + 48 z z + 3 z z z − z z z (cid:1) θ + ( − z − z z + 13 z z z ) θ + (4 + 8 z − z + 12 z z − z z − z z z ) θ + 1 z (cid:0) − z − z + 4 z z + 20 z z − z z + 52 z z z (cid:1) θ θ (cid:27) . (4.103)Then, we can obtain the quantum mirror maps and quantum B-periods by actingabove operator on the classical periods, as in (4.88).We do not write down the NS free energy in this case since the computationprocess is completely the same as the case of Y , , but one can show that the NS freeenergy calculated from the differential operators agree with the topological vertexcomputations.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 quantumcorrected spectra and volumes defined in (4.69). The classical Bohr–Sommerfeldvolumes have to vanish in the classical limit (cid:126) = 0 of (4.68) at the classical minimum E (0)1 = 5 − r / r / =: E m , E (0)2 = 5 r / − r / + r =: E m , (4.104)which corresponds to the conifold point. For simplicity, we do the computation for r = 2 − / . The leading corrections to the spectra are given by (4.47). By comparingwith the perturbative computation, we find the exact values of E , -derivatives of thevolumes at the conifold point, (cid:32) ∂ E vol (0)1 ∂ E vol (0)1 ∂ E vol (0)2 ∂ E vol (0)2 (cid:33) = π (cid:32) / √ / √ − / − / (cid:33) . (4.105)which are consistent with the numerical computation.Similar to the Y , case, we do not calculate the classical mirror map aroundthe conifold point, but when one wants to calculate higher corrections to the energyspectra as in the case of Y , , the classical mirror map is needed to obtain the higherorder quantum corrections to the (derivatives) of the volumes via the formulae (4.18)(4.22). 30 .2.5 Y , case As the final example, we consider the Y , model. We again sometimes use some ofthe notation and definition given in section 4.2.2. The mirror curve of Y , ise p + e − p + e x + e x z / z / z / + e x z / z / z / + 1 z / = 0 . (4.106)The Picard–Fuchs operators are L = θ ( θ − θ − θ ) + 4 θ θ − z (2 θ − θ + 1) (2 θ − θ ) , L = θ ( θ − θ ) + z (2 θ − θ + 1) (2 θ − θ ) , L = θ + z (2 θ − θ + 1) (2 θ − θ ) , L = θ θ + z z z ( θ − θ )( θ − θ )(2 θ − θ ) . (4.107)The solutions provide the mirror maps and the derivatives of prepotential, t ( z ) = ω = log z + 2 z − z − z + 3 z − z − z − z z + 6 z z + z z − z z z + O ( z i ) ,t ( z ) = ω = log z − z + 2 z − z z + z z − z z − z z + 2 z z z + O ( z i ) ,t ( z ) = ω = log( z ) + 2 z + 3 z + O ( z i ) , (4.108)and ∂F ∂t = 23 ω + 23 ω + 23 ω + 23 ω + 13 ω + 2 π ,∂F ∂t = 13 ω + 43 ω + 13 ω + 43 ω + 23 ω + 2 π , (4.109)where ω i and ω ij are given in (4.59) with the coefficient c ( l, m, n ; ρ i ), c ( l, m, n ; ρ i ) = 1Γ(1 + l − m + ρ − ρ ) Γ(1 − l + m − ρ + ρ )Γ(1 + l − n + ρ − ρ ) × m + ρ )Γ(1 + n + ρ ) . (4.110)For the third mirror map t , we can calculate the summation exacty, t ( z ) = log z − (cid:18) − √ − z (cid:19) (4.111)The classical B-periods are completely the same form as the one of Y , since the ma-trices C ij of Y , and Y , are the same. From the prepotential, the Bohr-Sommerfeldvolumes are vol (0) i ( z ) = (cid:88) j =1 C ij ∂F ∂t j − π , i = 1 , , (4.112)31here the complex structure moduli z , z , z are related to the dimer model by z = (1 + R ) E E , z = E E , z = R (1 + R ) . (4.113)The Bohr-Sommerfeld volumes should vanish at the conifold point, z = (1 + R )(1 + R )3(1 + R + R ) , z = 1 + R + R R ) , z = − R (1 + R ) . (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 curveis replaced by the difference equation,Ψ( x + i (cid:126) ) + Ψ( x − i (cid:126) ) + (cid:32) e x + e x z / z / z / + e x z / z / z / + 1 z / (cid:33) Ψ( x ) = 0 , (4.115)By taking the residue of w ( x ; (cid:126) ), we find a quantum A-period,Π( z ; (cid:126) ) = −
13 log( z z z ) + (cid:73) x = ∞ d xw ( x ; (cid:126) )= −
13 log( z z z ) + (cid:18) z z z + z z − z z − z − z (cid:19) − (cid:0) z z − z z z (cid:1) (cid:126) + O ( (cid:126) , z i ) . (4.116)The (cid:126) -corrections can be expressed as the classical part acted by the differentialoperator, D = 112( − z ) (cid:26) z (15 z − z + 4) − z θ + z (9 z − z + 8) + 5 z − θ + ( z (cid:0) − z + 6 z − (cid:1) + z ) θ − z (36 z − z + 2) − z + 12 θ (cid:27) . (4.117)Then, we can obtain the quantum B-period by acting above differential operator onthe classical B-period.We do not write down the NS free energy in this case since the computationprocess is completely the same as the case of Y , , but one can show that the NS freeenergy calculated from the differential operators agrees with the topological vertexcomputations.Now we are ready to calculate the quantum corrections to the energy spectra. Inthe following computation, we consider the particular case of R = 1 for simplicity.The Bohr–Sommerfeld quantization condition is given by (4.68) with the quantumcorrected spectra and volumes defined in (4.69).The classical Bohr–Sommerfeld volumes have to vanish in the classical limit (cid:126) = 0of (4.68) at the classical minimum E (0)1 = 6 =: E m , E (0)2 = 9 =: E m , (4.118)32hich corresponds to the conifold point. The leading corrections to the spectra aregiven by (4.47). By comparing them with direct perturbative calculations (3.15), wefind the exact values of E , -derivatives of the volumes at conifold point, (cid:32) ∂ E vol (0)1 ∂ E vol (0)1 ∂ E vol (0)2 ∂ E vol (0)2 (cid:33) = (cid:32) √ − √ − √ √ (cid:33) . (4.119)With the change of variables to complex structure moduli, we find (cid:32) ∂ z vol (0)1 ( z ) ∂ z vol (0)1 ( z ) ∂ z vol (0)2 ( z ) ∂ z vol (0)2 ( z ) (cid:33) = √ π (cid:18) − − (cid:19) . (4.120)The classical A-periods near the conifold point are t c, = − π √ z c, − πz c, √ − πz c, √ πz c, z c, √ πz c, z c, √ πz c, z c, √ − πz c, z c, z c, √ O ( z c,i ) ,t c, = − π √ z c, − πz c, √ − πz c, √ πz c, z c, √ πz c, z c, √ πz c, z c, √ πz c, z c, z c, √ O ( z c,i ) ,t c, = − − (cid:112) − z c, ) + log(1 + 4 z c, ) , (4.121)where z = 16 + z c, , z = 16 + z c, , z = − z c, . (4.122)The coefficients of z c, and z c, are fixed by the relation (4.22).Repeating the computation for (cid:126) -order, we find, E (2)1 = 136 (9 n ( n + 1) + 3 n ( n + 1) + 8) ,E (2)2 = 12 n + n + 3 n n n + 23 . (4.123)These results agree with (3.15) with R = 1. In this paper, we studied the analytic connections between genus-2 mirror curves and Y p,q cluster integrable systems, which are generalizations of affine A -type relativisticToda systems. It is interesting to consider the more higher genus mirror curves andthe application to other types affine Toda systems.In the topic of the differential operator method, there are still interesting issues tobe clarify. For example, it would be interesting to consider the genus-1 mirror curvesfor local E n del Pezzo surfaces, where the global symmetries are E n groups. Suchcurves are considered in [52, 6] with some mass parameters turning off. With all mass33arameters turning on, the Calabi-Yau threefolds are non-toric, it is interesting tostudy the differential operator approach for these cases [53].Also, in [54], the authors pointed out that the quantum A-periods of D del Pezzogeometry can be expressed as D Weyl characters. The quantum mirror map of thiscurve would be given in the same way. Therefore, it would be interesting to clarifythe relation between the Weyl group expression and the differential operators.Recently, the authors in [55] provides the analytic results on black hole pertur-bation theory from the quantization conditions. They consider the quantization con-ditions for A-periods, not B-periods. Therefore, it would be interesting to clarifythe physical implications of this quantization conditions in the integrable systems ortopological strings.
Acknowledgements
We would like to thank Sheldon Katz, Albrecht Klemm for helpful discussions and/orstimulating collaborations on related papers. Some of the computation based on math-ematica were carried out on the computer sushiki at Yukawa Institute for TheoreticalPhysics in Kyoto University. The work of MH and YS was supported by the nationalNatural Science Foundation of China (Grants No.11675167 and No.11947301).
A An eigenvalue formula
Suppose S is a real symmetric 2 n × n matrix, and M is a real symplectic 2 n × n matrix that diagonalizes the symmetric matrix, i.e. we haveΣ = (cid:18) I n − I n (cid:19) , M T Σ M = Σ , M T SM = (cid:18) C D (cid:19) , (A.1)where C = diag { c , c , · · · , c n } , D = diag { d , d , · · · , d n } are real n × n diagonalmatrices. Then we can show that the characteristic polynomial of the matrix S Σ (orΣ S ) is det( S Σ − λI ) = n (cid:89) k =1 ( λ + c k d k ) . (A.2)So the eigenvalues of S Σ are ± i √ c k d k , k = 1 , , · · · n . In the context of our physicsproblem, the two diagonal matrices are identical C = D , therefore the diagonalelements are completely determined by the symmetric matrix S , are thus independentof the choice of the symplectic matrix M .The calculations are straightforward. Noticing Σ = − I and ( − Σ M T )(Σ M ) = I ,so the characteristic polynomial isdet( S Σ − λI ) = det( − Σ M T S Σ M − λI )= det( (cid:18) D − C (cid:19) − λI ) . (A.3)34t is now simple to verify the determinant is indeed the polynomial in the right handside of (A.2). References [1] M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mari˜no, and C. Vafa, “Topologicalstrings and integrable hierarchies,”
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