Degeneration of Topological String partition functions and Mirror curves of the Calabi-Yau threefolds $X_{N,M}$

In this paper we study certain degenerations of the mirror curves, associated with Calabi-Yau threefolds $X_{N,M}$, and the effect of these degenerations on the topological string partition function of $X_{N,M}$. We show that when the mirror curve degenerates and become the union of the lower genus curves the corresponding partition function factorizes into pieces corresponding to the components of the degenerate mirror curve. Moreoever we show that using degeneration of a generalised mirror curve it is possible to obtain the partition function corresponding to $X_{N,M-1}$ from $X_{N,M}$.

11d M-theory space-time The non-compact Calabi-Yau threefold (CY threefold) X N,M with N, M ∈ N [1,2,3,4,5,6,7,8] has the structure of a double elliptic fibration with an underlying SL(2, Z) × SL(2, Z) symmetry. One elliptic fibration has the Kodaira singularity of type I N −1 and the other elliptic fibration has I M −1 singularity. The topological string partition function on X N,M was computed in [1] and shown to be related to the Little string theories (LSTs) with eight supercharges. In the decompactification limit the low energy description of circle compactified LSTs of types (M, N ) and (N, M ) are described by quiver gauge theories with gauge groups U (M ) N and U (N ) M respectively. In the geometric engineering argument the M-theory compactification on a non-compact Calabi-Yau threefold Y is described at low energies by the 5d N = 1 SCFTs. These SCFTs are UV completions of the gauge theories we are interested in. The low energy gauge theory is completely specified by the requirement of supersymmetry, once the gauge group G, hypermultiplet representation R and the 5d Chern-Simons level k is fixed. In taking the QFT limit the gravitational interactions are tuned off. This is achieved by sending the volume of Y to infinity while keeping the volumes of compact four-cycles and two-cycles finite. This is equivalent to the non-compactness condition of the CY threefold. The coulomb branch of the SCFT is identical to the extended Kähler cone of the threefold Y [2, ?].
The CY Y can be understood as the singular limit of a smooth threefold Y in which certain number of compact four-cycles have shrunk to a point. The BPS states of the 5d theory correspond to M2-branes wrapping holomorphic two-cycles and M5-branes wrapping holomorphic four-cycles. The volume of the two-cycles and four-cycles correspond to the masses of the BPS states. At a generic point of the Coulomb branch the two-cycles and four-cycles have non-zero volumes and the BPS spectra is massive. At the origin of the Coulomb branch some of the cycles may shrink to a point and indicate a local singularity on the threefold. The refined topological type IIA string partition function Z N,M of X N,M can efficiently be computed using the refined topological vertex formalism [9]. The partition function Z N,M takes the form of an infinite series expansion. The expansion parameters depend on the choice of a preferred direction common to all vertices of the toric web diagram. Different choices of the preferred direction give equivalent but seemingly different representations of Z N,M [2,5,6].
Lately another powerful method of computing the partition function was proposed in [10] in terms of M-strings, which are one dimensional intersections of M5 and M2 branes. The table given in figure 1 summarises the coordinate labels and specifies the world volume directions of BPS M5-M2-M-string configuration. The M5-branes are separated along the compactified x 6 ∼ x 6 +2πR 6 dimension with the positions parameterised by scalars VEVs {a 1 , ..., a M } where M denotes the total number of M5-branes and a i − a i+1 are the VEVs of the scalars of 6d tensor multiplets. The M2-branes are stretched between these M5-branes. For the transverse space R 4 we can have only one stack of M2-branes between M5-branes. However it is possible to perform an orbifolding [11] of the transverse R 4 such that the mass deformation and supersymmetry remain preserved.
The orbifolding allows the multiple stacks of M2-branes with each stack charged under the orbifold action. For the M-string dual to (N, M ) web diagram there will be N stacks of M2-branes, with i-th stack consisting of k i number of them. In gauge theory k i characterises the instanton number. It was shown subsequently in [5]  For example the specific values M = 1, N = k correspond to a single M5-brane wrapped on parallel S 1 and k stack of M 2-branes wrapped on the transverse S 1 and ending on the M5-branes. The stack of M2-branes appear as coloured points in the R 4 || that resides inside the M5-brane world volume and transverse to the M-string world sheet. Thus for the configuration that involves n l number of M2-branes in the l-th stack, where l = 1, ..., k, the moduli space is obviously the product of Hilbert scheme of points as follows t3 tN The vector bundle V over H that is required for (2, 0) world sheet theory has been determined in [10] and turns out to be the following where I = (I 1 , I 2 , ..., I N ) ∈ H. Roughly speaking Ext groups count the massless open string states for strings that are stretched between D-branes wrapped on complex submanifolds of CY spaces. Note that each factor Ext 1 (Ir, Is) ⊗ L − 1 2 in the fibre denotes the contribution of a pair of stack of M2-branes ending on a single M5-brane from opposite sides. In other words there is an isomorphism between the degrees of freedom on the (N, M ) 5-branes web and the moduli space of M-strings, M(N, k). Using equivariant fixed point theorems one only needs to know the fibres of the bundle V(N, M ) over the fixed points.
The weights of V(N, M ) at the fixed points I (1) , I (2) , ..., I (M ) are given by the following Chern character expansion [5] weights e w = M p=1 N r,s=1 Qme i(ar−as) N label the fixed points. The elliptic genus is then given as follows where x i and x i denote the Chern roots respectively of the tangent bundle and vector bundle V(N, M ) as can be read from (1.4) and the theta function of first kind θ 1 (τ, z) is defined by More succinctly, the Nekrasov partition function of the gauge theory on the D5-branes of the web is identical to the appropriately normalised topological string partition function of CY threefold X N,M and it is also the generating function of the (2, 0) elliptic genus of the product of instanton moduli spaces M(N, k) on which the bundle V(N, M ) coupled to the right moving fermions exists.

Presentation of the paper
We summarised the type IIA/type IIB mirror symmetry conjecture in the introduction (1). In section (3) we construct the quantum mirror curve of X N,M and study the limits in which it can be reduced to a lower genus curve. In section (6) we show that in the splitting degeneration limit the partition function Z X N,M is recursively related to the partition function Z X N,M −1 and we show this degeneration pictorially. In the appendix we reproduce the proof of an identity used in the main text.

(p,q) webs and the mirror curves
We can consider [12,13] the A-model topological strings on a toric CY threefold M = C l+3 //U (1) l . Algebraically M is defined by the following set of constraints modulo the action of U (1) l , where each X i parameterizes a complex plane C and can be visualised as S 1 -fibrations over R + .
In this way M , as defined by (2.1), is a T 3 -fibration over a non-compact convex and linearly bounded subspace in R 3 , with T 3 parametrised by {θ i } coordinates. k a ∈ R + are called the Kähler parameters. The CY condition Inspecting equation (2.1) makes it clear that since Q a i ∈ Z, all toric CY threefolds are constrained to be non-compact. The second constraint (2.3) furnishes a representation of M as R + × T 2 fibered over R 3 . In this way the toric threefold M allows its construction by gluing patches of C 3 . To construct the mirror N of the threefold M, consider variable v 1 , v 2 ∈ C, and the homogeneous coordinates x i =: e yi ∈ C * , i = 1, ..., l + 3 related to X i by |x i | = e −|Xi| 2 . The variables x i are constrained by x i ∼ λx i for λ ∈ C * . The mirror geometry N is then given by the algebraic equation where x, y ∈ C * . The function h(x, y; r a , θa) can be decomposed into pant diagrams described by e x + e y + 1 = 0. (2.7) The last equation describes a conic bundle over C * × C * in which the fibers degenerate over two lines over the family of Riemann surfaces Σ : g(x, y; r a , θa) = 0 ∈ C * × C * . If the toric diagram of M is thickened, what emerges is nothing else but Σ ; the genus of Σ equals the number of closed meshes and the number of punctures equals the number of semi infinite lines in the toric diagram 1 . In the topological A-model the topological vertex computation can be interpreted as the states of a chiral boson on a three-punctured sphere. This chiral boson on each patch of the sphere is identified with the Kodaira Spencer field on the Riemann surface embedded in the CY threefold of mirror topological B-model [14,15,16,17,18,19,20]. The A-model closed topological strings on toric CY threefold, with or without D-branes, is computable by gluing cubic topological vertex expressions. On the mirror B-model the gluing rules are equivalent to the operator formation of the Kodaira Spencer theory on the Riemann surface.The elliptic Calabi-Yau threefold X N,M is dual to the brane web of type IIB M NS5-branes and N D5-branes wrapped on two S 1 s. We denote by {y 0 , y 1 , y 2 , y 3 , ..., y 9 } the coordinates of type IIB string theory vacuum R 1,9 . The common worldvolume of the 5-branes along {y 0 , y 1 , y 2 , y 3 , y 4 } gives rise to the gauge theory under consideration and the (p, q) brane web is arranged in the {y 5 , y 6 } plane which is compactified to a torus T 2 . The (p, q)-charges and their conservation encode the details of the five-dimensional mass deformed supersymmetric gauge theory.
The curve associated to a grid diagram is written as the zero locus of a sum of monomials, with each monomial associated to a vertex of the grid diagram. For example A kl X k Y l is a monomial that corresponds to the vertex (k, l). The modulus of the curve A kl is determined by imposing a set of condition: each link on the grid joining e.g. (k, l) to (u, v) uniquely corresponds to a link on the web, which is orthogonal to the former. If the link on the web is given by the line py = qx + α, the orthogonality condition is expressed as (2.8) and the constraint is given by In other words the mirror curves of toric CY threefolds are determined by the corresponding Newton polygons. The line in the web [21,22,9,23,24,25,26] orthogonal to the line in the Newton polygon joining the coordinates,let's call them (k 1 , 1 ) and (k 2 , 2 ) and passing through the point (x 0 , y 0 ) is given by , where ∆ = 2 − 1 and ∆k = k 2 − k 1 . Since the choice of (x 0 , y 0 ) is arbitrary, we get The equation of the Riemann surface in this patch is given by exponentiating and complexifying (x, y) to (u, v), where X = e u and Y = e v with u, v ∈ C and Re( α) = α. Since the imaginary part α is not determined, we have introduced a factor of −1 for later convenience. With this choice, α will be identified with the complexified Kähler parameters. In the mirror curve, we will have (2.14)

Mirror curves and their degenerations
We start the discussion by giving an example of Resolved Conifold. In this case, the Newton polygon is shown in figure (3) and the corresponding mirror curve is given by, Let us choose the horizontal line in the web corresponding to the points (0, 0) and (0, 1) in the Newton polygon that goes through the origin so that α = 0 for this line. This gives Similarly A 10 = A 00 and A 10 = A 01 . The line in the web corresponding to (0, 1), (1, 1) has the equation x = T where T is the horizontal distance between the two vertices in the web. Note that the vertical distance is also T . Thus we get The mirror curve is then given by Recall that in the mirror construction the Riemann surface Σ is a part of the mirror CY threefold. For 6D theories the corresponding toric webs have no semi-infinite lines and hence no punctures. The periodicity of the web is taken into account by including all of its images under the periodic shift. Note that after the vertical and horizontal periodic identifications the toric diagram becomes non-planar. In this case the mirror curve is given by, Let's take the origin of the web to be the vertex of the web corresponding to the triangle coordinatized by (0, 0), (1, 0), (0, 1). With this choice the equation of the horizontal line in the web corresponding to (k, ) and (k, + 1) is given by where τ is the periodicity of the web in the vertical direction and t 1 is the horizontal distance between two consecutive vertices on the diagonal in the web given in figure (3). This gives where Im(τ ) = t1+t3 2π and Im(z) = t1 2π . The equation of the line in the web corresponding to (k, ), (k + 1, ) is given by where ρ is the periodicity of the web in the horizontal direction. We thus get Using the coefficients the mirror curve becomes If we define the genus two theta function by where the period matrix Ω(ρ, z, τ ) and the quadratic form Q(k, .) are given by the mirror curve can be written as It is interesting to note [27,28] that under the following identifications the theta function transforms covariantly and the curve (3.12) remains invariant 2 . Note that in the limit z → 0 the left side is factorized into the product of genus one theta functions 3.2 Mirror curve dual to X 1,2 Consider the periodic Newton polygon with vertices (0, 0), (1, 0), (2, 0), (2, 1), (1, 1), (0, 1) as shown in figure (4). The mirror curve is given by the theta function with characteristics given by satisfies the following identities under the shifts of z by lattice L Ω and a, b where the coefficients B k, can be determined in the same way as for the genus two case and are functions of the four Kähler parameters (τ, ρ, z, w). They are related to each other as follows: (3.16) These recursive relations have the following solution: Then the mirror curve is given by To see the factorisation we can write the last expression explicitly as It is easy to see that In the limit z → 0 we get the factorized form Consider the (N, M ) web shown in figure (5). The Kähler class ω of X N,M is parameterized by (m α,β , τ, ρ, T, t) = (m α,β , τ, ρ, m, For arbitrary (N, M ) values the factorisation properties of the mirror curve will in general be affected by the quantum corrections. The quantum corrected Kähler parameters are the solutions of the Picard-Fuchs equations [30]. After getting quantum corrections various Kähler parameters are mixed non-trivially and that renders the factorisation non-trivial as compared to the classical case discussed here. The mirror curve is given by a sum over the monomials associated with the Newton polygon. In this case the Newton polygon tiles the plane The coefficients A i,j depend on the length of the various line segments in the web which are the Kähler parameters of the corresponding Calabi-Yau threefolds. As discussed before the neighbouring pair of points in the Newton polygon connected by a line give a relation between the associated coefficients A i,j , where in the web diagram of X N,M , t i ∈ {t 1 , ..., t N } denotes the distance between i-th and i + 1-th vertical lines and T i ∈ {T 1 , ..., T M } denotes the distance between i-th and i + 1-th horizotntal lines and m a,b parametrize the diagonal finite line segments representing P 1 s. Using A 0,1 = A 0,0 = 1 we get the following solution The notation H N,M should not be confused with H which denotes the instanton moduli space in the introduction.
Thus the curve is given by Using the identifications = (ab + a(k+1) We define the genus two theta function as: The genus of the mirror curve   where A ij denote the moduli of the curve. This zero locus defines the mirror curve of genus M N + 1 and is the Riemann surface Σ. For the special case of M = 1 the mirror curve can be expressed in the following form where θ 1 is the Jacobi theta function and h(x) = N j=1 θ 1 (x − ξ j |ρ) with ξ j is the moduli of Σ. This can be reorganised into the following form whereΩ is the period matrix of the genus MN+1 curveΣ which is an unbranched cover of a genus 2 curve and in general is given byΩ It is easy to see from the following representation of genus g = M N + 1 theta function where Z, α, β, m are g-vectors and Ω is a g × g matrix with ImΩ > 0.
To study the decomposition of generalised theta function [33] defined on the Jacobian of a genus g = M curve, we start from the following Fourier representation where Ω is the period matrix and satisfies the following constraints This constraint encodes various periodicity properties. In other words we can decompose Ω as where Ω is the traceless part. Now redefine z i as follows Putting back these redefined variables in (3.43) we get where Θ i is the second summation factor in the first line of (3.47).

Degenerations and their Effect on the Partition Function
The partition function of the CY threefold X N,M is given by [5] where the sum is over N partitions of α (a) = {α , Q i = e bi+1−bi , t ab = t a,a+1 + t a+1,a+2 + · · · + t a+b−(a+1),b , b i+1 − b i is the distance between vertical lines (or M5 branes) and moreover the factorisation degeneration takes place when all the mass parameters ma are taken equal to m. The expressions of partition functions after degeneration becomes particularly simple at the special point in the Kähler moduli space where Q i := Q := e 2πiτ and in the unrefined limit of the Ω-background parameters 1 = − 2 = .
We define where |α (a) | is the size of the partition α (a) which is the sum of the parts of partition. To study the degeneration of partition function we have to study the x → 0 limit of ϑµν (x). For two integer partitions µ and ν, theta function ϑµν in the above partition function (4.1) is defined as Here t = e −i 2 , q = e i 1 , ν t represents the transpose of the partition ν and product (i,j)∈ν means that the product is over all the boxes of the Young diagram corresponding to the partition ν having length (ν) (i, j) ∈ ν , implies that 1 ≤ i ≤ (ν), 1 ≤ j ≤ ν i . The Jacobi theta function ϑ(ρ, y) for y = e 2πiz is defined as For x = 0 and in unrefined case where hµ(i, j) = µ i + µ t j − i − j + 1 is the hook length of the partition µ. Since, the Jacobi theta function ϑ(ρ, z) is an odd function w.r.t. z i.e., ϑ(ρ, 0) = 0, therefore hµ(i, j) is non zero therefore ϑµµ(0) = 0. In other words µ = ν implies ϑµν (0) = 0 i.e. either hµ(i, j) + ν t j − µ t j = 0 or hν (i, j) + µ t j − ν t j = 0. Because hµ(i, j) = 0 therefore ν t j = µ t j . We thus arrive at the useful property of ϑµν (x) at x = 0 given by: where δµ ν is the kronecker delta function and hµ(i, j) = µ i + µ t j − i − j + 1 is the hook length of the partition µ. This identity is useful for studying different degenerations of the partition functions.

Degeneration 1:Factorization
This type of degeneration corresponds to taking both the vertical and horizontal 'distances' between the 5-branes equal to m, which is the Kähler parameter corresponding to the exceptional curve or (1, 1) brane in the web diagram fig.5.

(N, M ) = (1, 2)
We begin by looking at the case of X 1,2 . The unrefined partition function is given by, Here, t − m = t − m and t + m = t + m. The partition function Z (1,2) in the limit t → m reduces to Using the property of ϑµν (x) defined in Eq.(4.5) we get The partition function defined in (4.1) for N = 1 has the following expression For t a a+1 = m we get t ab = t a a+1 + t a+1 a+2 + · · · + t a+b−(a+1) b = (b − a)m. In this case the unrefined Z (1,M ) partition function ( + → 0) becomes Since ϑα aαb (0) = 0 for αa = α b as shown in the previous section, we get This shows self-similarity behaviour of the partition function Z (1,M ) (τ, ρ, t = m, ) upto the rescaling of τ and m. In other words as far as the partition function is concerned the the CY-3fold X N,M is equivalent to the CY-3fold X 1,1 upto the rescaling of some kähler parameters. This self-similarity structure is actually followed by the partition function Z (N,M ) (τ, ρ, t = m, ) for general values of N and M as shown below.

General (N, M )
By generalising to the CY-threefold X N,M , we get the following result So, In general This corresponds to degenerating the web diagram of X N,M to the disconnected union of N rescaled web diagrams of X 1,1 as shown in figure 6. The CY threefold X 1,1 has a nice interpretation in terms of the so-called banana curves [34]. A banana configuration of curves in the CY threefold is a union of three curves C i ≡ P 1 with the normal bundle given by O(−1) ⊕ O(−1). Moreover C 1 ∩ C 2 = C 2 ∩ C 3 = C 3 ∩ C 1 = {x, y} for distinct point x, y ∈ CY3-fold and there exists a preferred coordinate patch in which C i are along the coordinate axis.
In other words the topological string partition function Z X N,M (ω, ) is factored [4,35] into a product of N copies of Z X1,1 (τ, ρ, m), where the later is the topological partition function on a CY threefold with a single banana configuration of curves.
where the Kähler parameters T i from T= {T 1 , T 2 , ..., T M } represent the distance between vertical lines , t i from t= {t 1 , t 2 , ..., t N } represent the distance between horizontal lines and m denote the diagonal lines of the web diagram in figure 5. These expansion have been interpreted as instanton expansions of three gauge theories which are dual to each other. For these to be consistent expansions it is assumed that there exists a region of the moduli space of X (M,N ) in which either T or t or m become infinite, with all the rest of parameters kept finite. This region of the moduli space corresponds to the weak coupling limit of gauge theories.
At the special point in the moduli space where t a,a+1 = m, we are left with three independent Kähler parameters τ, ρ, m.
Moreover due to the weak coupling expansion {T → ∞}, N horizontal strips gets decoupled and we get Z N 1,1 .
• t ab → 0 : Previous subsections discuss the cases when N = 1 and now we generalize to the case of N = 2. Explicitly the partition function is of the form For the unrefined case 1 = − 2 = , we consider the degenerate limit m 3 = 0. Using the identity (4.5) we get Recognizing the Z (2,2) (τ, ρ, m 1,2 , t ab , ) part, the last expression can be written more succinctly as Similar degenerations follow by taking the limit m 2 = 0 or m 1 = 0.

General (N, M )
The previous sections discuss the cases when N was taken equal to one. In this section we generalize the argument to generic values of M and N. For the unrefined case Specializing to N = 1, Q i = Q and in the limit m 1 = 0 the last expression reduces to where t ab and m i do not include the moduli which are tuned to zero. More generally and at the same point Q i = Q in the moduli space we expect similar structure for Z (N,M ) In the limit m 1 → 0

Discussions
The compactified 5-brane web given in fig.5 gives rise to a five dimensional N = 2 supersymmetric gauge theory on the common worldvolume. This 5-branes web can be deformed to include also (1, 1) 5-branes. In string theory this is interpreted as the splitting of the D5-branes on the NS5-brane world volume. In other words the string tension is turned on for the strings that are stretched between D5-branes. It gives rise to the mass deformation of the bifundamental hypermultiplets in the five dimensional gauge theory. The mass deformation results in the breaking of supersymmetry to N = 1 in five dimensions. Because of the toric compactification of the 5-branes web one gets affineÂ N −1 quiver gauge theory with an SU (N ) gauge group at each node and one bifundamental matter stretched between adjacent nodes. There are M coupling constants τ i , i = 1, ..., M for each node such that where R 1 is the radius of the S 1 on which M5-brane theory is compactified. In geometrical terms each gauge coupling constant is related to the area of a distinct curve in CY threefold. If there are more than one, though equivalent, choices of these curves, this gives rise to dual gauge theory formulations of the same system. In other words for the web of M NS5-branes and N D5-branes the gauge theory on the D5-branes is given by where Na is the SU (N ) fundamental representation of the a-th node andNa the complex conjugate one. The partition function of the quiver gauge theories given in (7.2) can be computed directly by using Nekrasov instanton calculus as described in [5]. In doing so one has to take into account the non-trivial winding of strings on the compact direction transverse to the 5-branes. There is interesting physical interpretations of these degenerations. In the previous sections we have discussed how various degenerations of the mirror curve is related to certain degeneration of the corresponding partition functions Z (N,M ) . Recall the following degeneration (5.8) Similarly the second degeneration of the Z N,M (6.10) that we discussed and is given by has an interesting physical interpretation. The limit m i → 0 corresponds to supersymmetry enhancement to N = 4 and we get a decoupling factor of η(τ ).

Conclusions
This paper explored some interesting consequences of the mirror symmetry of the local CY threefold X N,M . We investigated some interesting properties of the type A topological string partition function of X N,M in special regions of the Kähler moduli space. We have called these degenerate limits, because in these limits the partition functions on X N,M collapse to those on X N,M −1 in various ways. In accordance with mirror symmetry the degeneration behaviour on the type A side is reproduced on the type B side in the degeneration of the mirror curves into lower genus curves. For future directions it would be interesting to study the analogous properties of Z N,M and quantum mirror curves for the general Ω-background .i.e. 1 = 0 and/or 1 = 0 and 1 = 2 and at an arbitrary point of the Kähler moduli space of X N,M . It will also be interesting to study the modular properties of the free energy log(Ẑ (N,M ) (τ, ρ, , m, t)) and the single particle free energy [43] P Log(Ẑ (N,M ) (τ, ρ, , m, t)) along the lines of [44]. It is also interesting to generalise the quantisation of classical DELL system as done in [45] to the case where the underlying abelian variety has (M,N) polarization.
The CY condition constrains the geometry to a plane. The irreducible toric rational curves of the 2-dimensional cone are given by defines a curve with genus N M + 1 with (N, M ) polarisation. For illustration, consider the CY3-fold X 1,1 , for which the cone of effective curves is given by R ≥0 {C 1 , C 2 , C 3 }. To make the modularity of the system manifest, we redefine the curve classes as for which the corresponding Kähler parameters are denoted as qτ = q 1 q 2 = e 2πiτ , qρ = q 1 q 3 = e 2πiρ , qσ = q 1 = e 2πiσ . Then following the SYZ program, the SYZ mirror of X 1,1 is given by Moreover it turns out that the right hand side can be re-written in terms of theta function as where Θ 2 is the genus 2 theta function and Ω = N τ σ σ M ρ is the period matrix of the following genus 2 curve Here we prove the identity used in subsection 3.3. Note that in our notation the curve classes C 1 (a,b) are represented by the Kähler parameters m a,b . Using the first relation in eq.(A.11), we can write the following summation