Topological Strings on X N,M : Conifold Singularities and Degeneration of Mirror Curves
PPrepared for submission to JHEP
Topological Strings on X N,M : Conifold Singularitiesand Degeneration of Mirror Curves
Ambreen Ahmed, M.Nouman Muteeb, Abdus Salam School of Mathematical SciencesGovernment College University, Lahore, PAKISTAN. Abdus Salam School of Mathematical SciencesGovernment College University, Lahore, PAKISTAN
E-mail: [email protected],[email protected]
Abstract:
In this paper we study certain degenerations of the mirror curves, associated withCalabi-Yau threefolds X N,M , and the effect of these degenerations on the topological stringpartition function of X N,M . We show that when the mirror curve degenerates and becomethe union of the lower genus curves the corresponding partition function factorizes into piecescorresponding to the components of the degenerate mirror curve. Moreoever we show thatusing degeneration of a generalised mirror curve it is possible to obtain the partition functioncorresponding to X N,M − from X N,M . a r X i v : . [ h e p - t h ] S e p ontents X , X , X ( N,M ) ( N, M ) = (1 , ( N, M ) = (1 , M ) ( N, M ) Z ( M,N ) → Z N (1 , ( N, M ) = (1 , ( N, M ) = (1 , ( N, M ) = (2 , ( N, M ) X N,M : a quick review 36B (cid:80) N − a =0 m a , b is independent of b :proof 36 – 1 – Introduction: Topological strings, M-strings and quiver gauge theories
The non-compact Calabi-Yau 3-fold (CY3-fold) X N,M with
N, M ∈ N [1–6] has the structureof a double elliptic fibration with an underlying SL (2 , Z ) × SL (2 , Z ) symmetry. One ellipticfibration has the Kodaira singularity of type I N − and the other elliptic fibration has I M − singularity. The topological string partition function on X N,M was computed in [1] andshown to be related to the Little string theories (LSTs) with eight supercharges. In thedecompactification 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 3-fold Y is described at low energies by the 5d N = 1 SCFTs. TheseSCFTs are the UV completions of the gauge theories we are interested in. The low energygauge theory is completely specified by the requirement of supersymmetry, once the gaugegroup G , hypermultiplet representation R and the 5d Chern-Simons level k is fixed. In takingthe QFT limit the gravitational interactions are tuned off. This is achieved by sending thevolume of Y to infinity while keeping the volumes of compact four-cycles and two-cycles finite.This is equivalent to the non-comactness condition of the CY 3-fold. The coulomb branch ofthe SCFT is identical to the extended Kähler cone of the 3fold Y [2, 7]. Y can be understoodas the singular limit of a smooth 3-fold ˜ Y in which certain number of compact four-cycleshave shrunk to a point. The existence of a gauge theory description of ˜ Y implies that theabelian gauge algebra is isomorphic to the quotient H ( ˜ Y , R ) /H ( ˜ Y , Z ) . The later enhancesto a non-abelian gauge algebra in the singular limit ˜ Y → Y . The BPS states of the 5dtheory correspond to M2-branes wrapping holomorphic two-cycles and M5-branes wrappingholomorphic four-cylces. The volume of the two-cycles and four-cycles correspond to themasses 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 Coulombbranch some of the cycles may shrink to a point and indicate a local singularity on the 3-fold.The prepotential of the 5d gauge theory is related to the triple intersection of the divisors (four-cycles) in ˜ Y . Specifically for a given basis D i the Kähler forms J admits a linear expansion J = φ i D i for i = 1 , , ..., h , ( ˜ Y ) . The Kähler moduli φ i for i = 1 , .., r are associated tocompact 4-cycles, denote them by D i = M i , and parametrize the Coulomb branch. The restof the Kähler moduli φ i for i = r + 1 , .., h , ( ˜ Y ) are associated to the non-compact four-cycles D i = ˜ M i and parametrize the mass parameters of the 5d gauge theory. The tensions ofthe elementary monopole strings are proportional to the volumes of compact four-cycles andrelated [7] to the 5d gauge theory prepotential F as ∂ i F = vol ( M i ) = 12 (cid:90) ˜ Y J ∧ M i (1.1)– 2 –imilarly the volume of compact two-cycles and the triple intersection numbers are encodedin the prepotential F as follows ∂ i ∂ j F = vol ( M i ∩ M j ) = (cid:90) ˜ Y J ∧ M i ∧ M j ∂ i ∂ j ∂ k F = (cid:90) ˜ Y D i ∧ D j ∧ D k (1.2)This shows that the dynamics of the Coulomb branch of d SCFTs can be studied in termsof the mathematics of smooth 3-folds ˜ Y . The Coulomb branch of the 5d gauge theory ispartitioned into chambers by real codimension-one walls, along which some matter fieldsbecome massless. These chambers in gauge theory description are analogues of the relativeKähler cones of CY 3-fold, where the cones are related to each other through f lop transitions.The quantized Chern-Simons levels ,denoted by c lmn , of the 5d theory are discontinuousacross the codimension one walls as do the classical intersection numbers D l .D m .D n underflop transition. The refined topological type IIA string partition function Z N,M on X N,M canefficiently be computed using the refined topological vertex formalism. The partition function Z N,M takes the form of an infinite series expansion. The expansion parameters depend onthe choice of a preferred direction common to all vertices of the toric web diagram. Differentchoices of the preferred direction give equivalent but seemingly different representations of Z N,M [2]. The web diagram of X N,M contain either horizontal, vertical or diagonal directionsas the preferred ones. In this way the 5-brane web is seen as composed of vertical strips,horizontal strips and diagonal strips suitably glued together. This allows the Z N,M to beexpressed as three series representations denoted by Z ( N,M ) hor , Z ( N,M ) ver and Z ( N,M ) diag . The Kählermodui space of X N,M contains three special regions each of which can be interpreted as theweak coupling region of a quiver gauge theory related to either Z ( N,M ) hor , Z ( N,M ) ver or Z ( N,M ) diag . Therespective weak coupling regions are in general disjoint. The basis of independent parametersin each region is a complicated linear combinations of the Kähler parameters of the 5-branesweb. The Kähler parameters of X N,M are interpreted either as the gauge coupling constants,Coulomb branch parameters or hypermultiplet masses. Since the three decompositions of the5-brane web are mutually related through SL (2 , Z ) transformations, the authors suggested in[2] a triality symmetry between the three quiver gauge theories and further conjectured theequivalence of X N,M and X N (cid:48) ,M (cid:48) for M N = N (cid:48) M (cid:48) and gcd ( M, N ) = gcd ( N (cid:48) , M (cid:48) ) This trialitysymmetry is a generalization of T-daulity of the underlying LSTs. The elliptic Calabi-Yau3-fold X N,M is dual to the brane web of type IIB M NS5-branes and N D5-branes wrappedon two S s. Denote by { y , y , y , y , ..., y } the coordinates of type IIB string theory vacuum R , . The common worldvolume of the 5-branes along { y , y , y , y , y } gives rise to the gaugetheory under consideration. The ( p, q ) brane web is arranged in the { y , y } plane which iscompactified to a torus T . The ( p, q ) -charges and their conservation encode the details of thefive-dimensional mass deformed supersymmetric gauge theory. SL (2 , Z ) duality symmetry oftype IIB is translated to a duality between one gauge theory corresponding to M NS5-branesand N D5-branes and the second gauge theory corresponding to N NS5-branes and N D5-– 3 –ranes. This SL (2 , Z ) is the manifestation of the T-duality of underlying circle compactifiedLSTs as alluded to before. On lifting the type IIB superstring theory to F-theory the SL (2 , Z ) duality is encoded in the topological equivalence of X M,N and X N,M .The compactified 5-brane web gives rise to a five dimensional N = 2 supersymmetric gaugetheory on the common worldvolume. This 5-branes web can be deformed to include also (1 , N = 1 in five dimensions. Because of the toric compactificationof the 5-branes web one gets affine ˆ A N − quiver gauge theory with an SU ( N ) gauge groupat 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 M (cid:88) i =1 τ i = 1 R (1.3)where R is the radius of the S on which M5-brane theory is compactified. In geometricalterms each gauge coupling constant is related to the area of a distinct curve in CY M NS5-branesand N D5-branes the gauge theory on the D5-branes is given by gauge group : U (1) × SU ( N ) × SU ( N ) × ... × SU ( N ) M hypermultiplet representation : ⊕ Mi =1 (cid:18) ( N a , ¯ N a +1 ) ⊕ ( ¯ N a , N a +1 ) (cid:19) (1.4)where N a is the SU ( N ) fundamental representation of the a-th node and ¯ N a the complexconjugate one. Under the SL (2 , Z ) duality the corresponding web consists of N NS5-branesand M D5-branes, and the gauge theory is given by gauge group : U (1) × SU ( M ) × SU ( M ) × ... × SU ( M ) N hypermultiplet representation : ⊕ Ni =1 (cid:18) ( M a , ¯ M a +1 ) ⊕ ( ¯ M a , M a +1 ) (cid:19) (1.5)where M a is the SU ( N ) fundamental representation of the a-th node and ¯ M a the complexconjugate one. In the case that each bifundmental mass is different there are M N numberof them. Web compactification on T results in M + N − constraints. As a result the fivedimensional affine quiver gauge theory depends on M N + 2 number of parameters. As alludedto before there is a third quiver gauge theory description related to the (1.4) and (1.5) througha triality symmetry gauge group : U (1) × SU ( M Nk ) × SU ( M Nk ) × ... × SU ( M Nk ) k hypermultiplet representation : ⊕ ki =1 (cid:18) ( P a , ¯ P a +1 ) ⊕ ( ¯ P a , P a +1 ) (cid:19) (1.6)– 4 –1d M-theory space-time x x x x x x x x x x x M5-branes × × × × × ×
M2-branes × × ×
M-string × ×
Figure 1 : coordinates of the 11d M-theory space-timewhere k = gcd ( M, N ) . The UV completions of these gauge theories are described by LSTs.The equivalence of X M,N and X M (cid:48) ,N (cid:48) for M N = N (cid:48) M (cid:48) and gcd ( M, N ) = gcd ( N (cid:48) , M (cid:48) ) = k gives rise to a web of dualities between the quiver gauge theories (cid:26) U ( N ) M ∼ U ( M ) N ∼ U ( M Nk ) k (cid:27) ∼ (cid:26) U ( N (cid:48) ) M (cid:48) ∼ U ( M (cid:48) ) N (cid:48) ∼ U ( M (cid:48) N (cid:48) k ) k (cid:27) (1.7)for M N = N (cid:48) M (cid:48) and gcd ( M, N ) = gcd ( N (cid:48) , M (cid:48) ) = k .The partition function of the quiver gauge theories given in (1.4) and (1.5) can be computeddirectly by using Nekrasov instanton calculus as described in [3]. In doing so one has to takeinto account the non-trivial winding of strings on the compact direction transverse to the5-branes. In [8] it was proposed to consider the intersections of M2-branes and M5-branesas independent degrees of freedom and were called M- strings . The table given in figure 1summarises 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 ∼ x + 2 πR dimension with thepositions parametrised by scalars { a , ..., a M } where M denotes the total number of M5-branesand a i − a i +1 are the vev of the scalars of 6d tensor multiplets. The M2-branes are stretchedbetween these M5-branes. For the transverse space R we can have only one stack of M2-branes between M5-branes. However it is possible to perform an orbifolding of the transverse R such that the mass deformation and supersymmetry remain preserved. The orbifoldingallows the multiple stacks of M2-branes with each stack charged under the orbifold action. Forthe M-string dual to ( N, M ) web diagram there will be N stacks of M2-branes, with i- th stackconsisting of k i number of them. In gauge theory k i characterises the instanton number. Itwas shown subsequently in [3] that the M-string partition function Z ( N, M ) is the generatingfunction of the equivariant (2 , elliptic genus of the M-string world sheet, Z ( N, M ) = (cid:88) (cid:126)k Q k Q k ...Q k M M χ ell ( M ( N, (cid:126)k ) , V (cid:126)k ) (1.8)Its target space is the product of moduli spaces of U ( N ) instantons of charge k i on C : M ( N, (cid:126)k ) := M ( N, k ) × M ( N, k ) × ... × M ( N, k N ) along with a vector bundle V ( N, M ) on it. The mass deformation is taken care of by an extra U (1) m action with equivariantparameter m .The vector bundle is special in the sense that only right moving fermions coupleto it. M ( N, (cid:126)k ) is nothing other than the moduli space of M-strings. For example the specific– 5 – -- --- --- ------ - - -- - - - - - --- - - - = = = = M = = = = M − − − − − − N − − − − − − N Figure 2 : Web diagram of X N,M . t i ∈ { t , ..., t N } denotes the distance between i -th and i + 1 -th red lines and T i ∈ { T , ..., T M } denotes the distance between i -th and i + 1 -th bluelines. m denotes the Kähler parameter of the diagonal P s. The double and single bars || , | , = and − indicate periodic identifications.values M = 1 , N = k correspond to a single M5-brane wrapped on parallel S and k stackof M -branes wrapped on the transverse S and ending on the M5-branes. The stack of M2-branes appear as coloured points in the R || that resides inside the M5-brane world volume andtransverse to the M-string world sheet. Thus for the configuration that involves n l number ofM2-branes in the l − th stack, where l = 1 , ..., k , the moduli space is obviously the product ofHilbert scheme of points as follows H := Hilb n [ C ] × Hilb n [ C ] × ... × Hilb n k [ C ] (1.9)The vector bundle V over H that is required for (2 , world sheet theory has been determinedin [8] and turns out to be the following V I = ⊕ Nt,s =1 Ext ( I r , I s ) ⊗ L − (1.10)where I = ( I , I , ..., I N ) ∈ H . Roughly speaking Ext groups count the massless open stringstates for strings that are stretched between D-branes wrapped on complex submanifolds ofCY spaces. Note that each factor Ext ( I r , I s ) ⊗ L − in the fibre denotes the contribution ofa pair of stack of M2-branes ending on a single M5-brane from opposite sides. In other wordsthere is an isomorphism between the degrees of freedom on the ( N, M ) M ( N, (cid:126)k ) . Using equivariant fixed point theorems one only needsto know the fibres of the bundle V ( N, M ) over the fixed points.– 6 –he weights of V ( N, M ) at the fixed points (cid:126)I (1) , (cid:126)I (2) , ..., (cid:126)I ( M ) are given by the following Cherncharacter expansion [3] (cid:88) weights e w = M (cid:88) p =1 N (cid:88) r,s =1 Q m e i ( a r − a s ) (cid:18) (cid:88) ( i,j ) ∈ ν ( p ) r t ν t, ( p +1) s,j − i + q ν ( p ) r,i − j + + (cid:88) ( i,j ) ∈ ν ( p +1) s t − ν t, ( p ) r,j + i − q − ν ( p +1) s,i + j − (cid:19) (1.11)where ν (1)1 , ν (1)2 , ..., ν (1) N ; ν (1)1 , ..., ν (1) N label the fixed points. The elliptic genus is then given asfollows Z = = (cid:90) M (cid:89) i x i θ ( τ, ˜ x i + z ) θ ( τ, x i ) (1.12)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 (1.11) and the theta function of first kind θ ( τ, z ) is defined by θ ( τ ; z ) = − ie iπ ( e iπz − e − iπz ) ∞ (cid:89) k =1 (1 − e πikτ )(1 − e πikτ e πikz )(1 − e πikτ e − πikz ) . (1.13)More succinctly, the Nekrasov partition function of the gauge theory on the D5-branes of theweb is identical to the appropriately normalised topological string partition function of CY3-fold X N,M and it is the generating function of the (2 , elliptic genus of the product ofinstanton moduli spaces on which the bundle V ( N, M ) coupled to the right moving fermionsexists. Presentation of the paper
We summarising the type IIA/type IIB mirror symmetry conjecture in section ( ). In sections(3) we construct the quantum mirror curve of X N,M and study the limits in which it can bereduced to a lower genus curve. In section (5) we show that in the splitting degenerationlimit the partition function Z X N,M can be constructed from the partition function Z X N,M − and also elaborated on pictorially. In the last section (6) we briefly mention some physicalconsequences of the degenerations discussed in the previous sections. In the appendix wereproduce the proof of an identity used in the main text. Consdier the A-model topological strings on a toric CY 3-fold M = C l +3 //U (1) l . Algebraically M is defined by the following set of constraints l +3 (cid:88) i =1 Q ai | X i | = k a , a = 1 , ..., l (2.1)– 7 –odulo the action of U (1) l , where each X i parametrizes a complex plane C and can bevisualised as S -fibrations over R + . In this way M , as defined by (2.1), is a T -fibration overa non-compact convex and linearly bounded subspace in R , with T parametrised by { θ i } coordinates. k a ∈ R + are called the Kähler parameters. The CY condition c ( T M ) = 0 (2.2)holds iff l +3 (cid:88) i =1 Q ai = 0 , a = 1 , ..., l (2.3)Inspecting equation (2.1) makes it clear that since Q ai ∈ Z , all toric CY 3-folds are constrainedto be non-compact. The second constraint (2.3) furnishes a representation of M as a R + × T fibered over R . In this way the the toric three fold M allows its construction by gluing patchesof C .The toric diaram Γ M corresponding to M specifies the loci along which the S fibers degen-erate. The boundary of the region B is defined by X i = 0 . For each value of i this zero locusdefines a 2-plane in R whose normal vector satisfies l (cid:88) i =1 Q a (cid:126)n i = 0 (2.4)Obviously, the S parametrised by θ i shrinks at | X i | = 0 and at the intersection of two suchloci S ij = {| X i | = 0 } ∩ {| X j | = 0 } , two circles S s shrink to zero size. For S ij a closed line in ∂B the open S bundle over it is a P . For S ij a half open line it represents a non-compactdirection C . It is clear now that the relative position of X i , X j is determined by the length ofthe line segment S ij which is nothing other than the Kähler parameter of the corresponding P . The CY condition (2.3) and the T fibration structure allows to project the S ij s onto R in such a way that all the information about the geometry of M is contained in it. Projectingall the S ij s onto R in this way constitute the toric diagram Γ M .To construct the mirror N of the three fold M, consider variable v , v ∈ C , and the homa-geneous coordinates x i =: e y i ∈ C ∗ , i = 1 , ..., l + 3 related to X i by | x i | = e −| X i | . x i areconstrained by x i ∼ λx i for λ ∈ C ∗ . The mirror geometry N is then given by the algebraicequation v v = l +3 (cid:88) i =1 x i , (2.5)constrained by l +3 (cid:89) i =1 x Q ai i = e − r a − iθ a , a = 1 , ..., l (2.6)– 8 –ll of these equations can be combined into one equation v v = H ( x, y ; r a , θ a ) (2.7)where x, y ∈ C ∗ . H ( x, y ; r a , θ a ) can be decomposed into pant diagrams described by e x + e y + 1 = 0 . (2.8)The last equation describes a conic bundle over C ∗ × C ∗ in which the fibers degenerate overtwo lines over the family of Riemann surfaces Σ : H ( x, y ; r a , θ a ) = 0 ∈ C ∗ × C ∗ If the toricdiagram of M is thickened, what emerges is nothing else but Σ ; the genus of Σ equals thenumber of closed meshes and the number of punctures equals the number of semi infinitelines in the toric diagram. In the topological A-model the topological vertex computationcan be intrerepreted as the states of a chiral boson on a three-punctured sphere. This chiralboson on each patch of the sphere is identified with the Kodaira Spencer field on the Riemannsurface embedded in the the CY 3-fold of mirror topological B-model. The A-model closedtopological strings on toric CY 3-fold, with or without D-branes, is computable by gluingcubic topological vertex expressions. On the mirror B-model the gluing rules are equivalentto the operator formation of the Kodaira Spencer theory on the Riemann surface. KodairaSpencer theory describes the dynamics of complex deformation of the CY 3-fold.The holomorphic 3-form on the mirror CY is given by
Ω = dv dxdyv (2.9)In studying the variation of complex structure, if one is only interested in the perturbationsof H ( x, y ; r a , θ a ) with no variations in ( v , v ) , the problem gets reduced to one complexdimension. In this special case the CY 3-fold has the structure of a fibration over the ( x, y ) -plane, with fiber given by (2.5). This fibration develops a node on the locus H ( x, y ; r a , θ a ) = 0 (2.10)the mirror curve. The period integral over Ω over 3-cycles reduce, using Cauchy’s theorem, to (cid:90) S dxdy (2.11)where ∂S ⊂ Σ , which in turn reduces to (cid:90) β ydx (2.12)using Stoke’s theorem, where β denotes a one-cycle on Σ . This shows that the complex struc-ture deformation of H ( x, y ; r a , θ a ) = 0 depends on the 1-form λ = ydx (2.13) It is a standard in literature to call Σ the mirror curve. – 9 –nd λ is defined patch by patch. Solving H = 0 yields y = f ( x ) . Under complex structuredeformation we have y = f ( x ) + δf (2.14)and it correspondingly changes λ by δλ = δf dx (2.15)To put it in the context of QFT we can identify δf = ∂φ . This variation δλ is identified withthe Kodaira Spencer field in [9]. It also satisfies a consistency relation ¯ ∂ x δf = 0 (2.16)The Kodaira Spencer action has the following kinetic term (cid:90) CY ω∂ − ¯ ∂ω (2.17)where ω is a (2 , -form which represents the change in the complex structure of CY. In eachpatch ω = ∂ξ with ξ is a (1 , -form. (cid:90) CY ∂ξ ¯ ∂ξ (2.18)and locally in each patch we have (cid:90) Σ ∂φ ¯ ∂φ (2.19)the action for a free scalar field. For a higher genus Riemann surface, it is first decomposedinto pants, each pant having three boundaries. Near each of the boundary, a local coordinate x is chosen such that x → ∞ at the boundary. The complex structure variation is studiedin the limit x → ∞ at each boundary. The action given by (2.19) is free, however the theinteraction part is encoded in the gluing data of various patches. The CY 3fold X N,M is a double elliptic fibration of type A N − × A M − over a non-compactbase C . It is toric with a web diagram (5) which is drawn on a torus with radii of the twocircles of the torus being dual to the Kähler class of the elliptic fibers of X N,M . These 3foldswere studied by [1, 3–5, 10] as examples of toric varieties of infinite type.The toric CY 3fold X N,M can be obtained by Z N × Z M orbifolds of X , . This set up isdualizable to ( p, q ) 5 -brane webs and realise various five- and six-dimensional gauge theories.The 5-brane web is identical to the toric web underlying X N,M . The local patch dependence of this formulation is related to the framing ambiguity in topological A model – 10 –he mirror curves of toric CY 3folds are determined by the corresponding Newton polygon.The line in the web orthogonal to the line in the Newton polygon joining ( k , (cid:96) ) and ( k , (cid:96) ) is given by (passing through ( x , y ) ), (∆ (cid:96) ) y + (∆ k ) x = (∆ (cid:96) ) y + (∆ k ) x (3.1)where ∆ (cid:96) = (cid:96) − (cid:96) and ∆ k = k − k . Since ( x , y ) is arbitrary therefore we get (∆ (cid:96) ) y + (∆ k ) x = α (3.2)The equation of the Riemann surface in this patch is given by exponentiating and complexi-fying ( x, y ) to ( u, v ) , X ∆ k Y ∆ (cid:96) = − e (cid:101) α , (3.3)where X = e u and Y = e v with u, v ∈ C and Re ( (cid:101) α ) = α . Since the imaginary part (cid:101) α is notdetermined, we have introduced a factor of − (shifting the imaginary part by iπ ) for laterconvenience. With this choice, (cid:101) α will be identified with the complexified Kähler parameters.In the mirror curve, we will have A k (cid:96) X k Y (cid:96) + A k (cid:96) X k Y (cid:96) = 0 (3.4)which implies X ∆ k Y ∆ (cid:96) = − A k (cid:96) A k (cid:96) = ⇒ A k (cid:96) = A k (cid:96) e − (cid:101) α (3.5) In this case, the Newton polygon is shown in figure (3) and the corresponding mirror curve isgiven by, A + A X + A Y + A XY = 0 (3.6)Let us choose so that the horizontal line in the web corresponding to (0 , and (0 , pointsin the Newton polygon goes through the origin so that α = 0 for this line, which gives, A = A (3.7)Similarly A = A and A = A . The line in the web corresponding to (0 , , (1 , hasthe equation x = T where T is the horizontal distance between the two vertices in the web(the vertical distance is also T ). Thus we get A = A e − t where Re ( t ) = T , thus the mirrorcurve is given by X + Y + e πit ∗ X Y = 0 (3.8)where t ∗ = i π t = i π T − Im ( t )2 π so that Im ( t ∗ ) > . This is the notation that we will use in therest of the section. – 11 – origin ρ + 2 t A , = A , e πi (2 τ +2 t ) t t τ = t + t ρ = t + t Figure 3 : tessellation of Newton polygons and web diagram of X , X , In this case the mirror curve is given by, (cid:88) ( k,(cid:96) ) ∈ Z A k,(cid:96) X k Y (cid:96) = 0 . (3.9)Lets take the origin of the web to be the vertex of the web corresponding to the triangle (0 , , (1 , , (0 , as shown in the figure below.With this choice the equation of the horizontal line in the web corresponding to ( k, (cid:96) ) and ( k, (cid:96) + 1) is given by y = (cid:96) ( t + t ) + k t (3.10)where τ is the periodicity of the web in the vertical direction and t is the horizontal distancebetween two consecutive vertices on the diagonal in the web (see figure above). This gives A k,(cid:96) +1 = A k,(cid:96) e πi ( (cid:96)τ + k z ) = ⇒ A k,(cid:96) +1 = A k, e πi ( τ (cid:96) ( (cid:96) +1)2 +( (cid:96) +1) k z ) (3.11)where Im ( τ ) = t + t π and Im ( z ) = t π . The equation of the line in the web corresponding to ( k, (cid:96) ) , ( k + 1 , (cid:96) ) is given by x = k ( t + t ) + (cid:96)t where ρ is the periodicity of the web in thehorizontal direction. This gives A k +1 ,(cid:96) = A k,(cid:96) e πi ( kρ + (cid:96)z ) = ⇒ A k +1 ,(cid:96) = A ,(cid:96) e πi ( ρ k ( k +1)2 +( k +1) (cid:96)z ) (3.12)From Eq(3.11) and Eq(3.12) it follows that: A k,(cid:96) = A , e πi ( (cid:96) ( (cid:96) − τ + k ( k − ρ + (cid:96)kz ) (3.13)– 12 –his gives the following mirror curve: (cid:88) k,(cid:96) ∈ Z e πi ( (cid:96) ( (cid:96) − τ + k ( k − ρ + (cid:96)kz ) X k Y (cid:96) = 0 (3.14)Note that in the limit z → we give the factorisation of this curve (cid:0) (cid:88) k, ∈ Z e πi ( k ( k − ρ ) X k (cid:1)(cid:0) (cid:88) (cid:96) ∈ Z e πi ( (cid:96) ( (cid:96) − τ ) Y (cid:96) (cid:1) = 0 (3.15) X , Consider the periodic Newton polygon with vertices (0 , , (1 , , (2 , , (2 , , (1 , , (0 , asshown in figure (4). The mirror curve is given by (cid:88) k,(cid:96) ∈ Z B k(cid:96) X k Y (cid:96) = 0 (3.16)where the coefficients B k,(cid:96) can be determined in the same way as for the genus two case andare functions of the four Kähler parameters ( τ, ρ, z, w ) (see figure below). They are related to origin A , = A , e πi (2 ρ +3 z + w ) τ ρ + 3 z + w ρw Figure 4 : tessellation of Newton polygons and web diagram of X , each other as follows: B k +2 ,(cid:96) = B k +1 ,(cid:96) e πi ( kρ +( (cid:96) +1) z + w ) , B k +1 = B k,(cid:96) e πi ( kρ + (cid:96)z ) (3.17) B k,(cid:96) +1 = B k,(cid:96) e πi ( (cid:96) τ + k z ) (3.18)– 13 –hese recursive relations have the following solution: B k,(cid:96) = exp (cid:104) πi (cid:16) k ( k − ρ + (cid:96) ( (cid:96) − τ + 2 k(cid:96)z + kz + kw (cid:17)(cid:105) B k +1 ,(cid:96) = exp (cid:104) πi (cid:16) k ρ + (cid:96) ( (cid:96) − τ + (2 k + 1) (cid:96)z + k ( z + w ) (cid:17)(cid:105) If we define Θ (cid:16) Ω( ρ, z, τ ) | ( u, v ) (cid:17) = (cid:88) k,(cid:96) exp (cid:16) πiQ ( k, (cid:96) ) / (cid:17) X k Y (cid:96) (3.19)where Ω( ρ, z, τ ) = (cid:32) ρ zz τ (cid:33) , Q ( k, (cid:96) ) = ( k (cid:96) )Ω (cid:32) k(cid:96) (cid:33) (3.20)then the mirror curve is given by Θ (cid:16) Ω(2 ρ, z, τ ) | (2 u − ρ + z + w, v − τ ) (cid:17) + e πiu Θ (cid:16) Ω(2 ρ, z, τ ) | (2 u + z + w, v − τ + z ) (cid:17) = 0 (3.21)To see the factorisation we rewrite the curve as (cid:80) k,(cid:96) ∈ Z (cid:18) exp (cid:104) πi (cid:16) k ( k − ρ + (cid:96) ( (cid:96) − τ + 2 k(cid:96)z + kz + kw (cid:17)(cid:105) X k Y (cid:96) + exp (cid:104) πi (cid:16) k ρ + (cid:96) ( (cid:96) − τ + (2 k + 1) (cid:96)z + k ( z + w ) (cid:17)(cid:105) X k +1 Y (cid:96) (cid:19) = 0 (3.22)In the limit z → we get (cid:80) k,(cid:96) ∈ Z (cid:18) exp (cid:104) πi (cid:16) k ( k − ρ + (cid:96) ( (cid:96) − τ + kw (cid:17)(cid:105) X k Y (cid:96) + exp (cid:104) πi (cid:16) k ρ + (cid:96) ( (cid:96) − τ + kw (cid:17)(cid:105) X k +1 Y (cid:96) (cid:19) = 0 (3.23)Taking out the common factor we get the factorised form (cid:88) (cid:96) ∈ Z (cid:18) exp (cid:104) πi (cid:16) (cid:96) ( (cid:96) − τ (cid:17)(cid:105) Y (cid:96) (cid:19)(cid:18) (cid:88) k ∈ Z X k (cid:18) exp (cid:104) πi (cid:16) k ( k − ρ + kw (cid:17)(cid:105) + exp (cid:104) πi (cid:16) k ρ + kw (cid:17)(cid:105) X (cid:19)(cid:19) = 0 (3.24). In the mirror construction this Riemann surface Σ is a part of the mirror CY 3-fold.Obviously for D theories the corresponding toric webs have no semi-infinite lines and henceno punctures. The periodicity of the web is taken into account by including all of its imagesunder the periodic shift. – 14 – -- --- --- ------ - - -- - - - - - --- - - - = = = = M = = = = M − − − − − − N − − − − − − N Figure 5 : Web diagram of X N,M . t i ∈ { t , ..., t N } denotes the distance between i -th and i + 1 -th red lines and T i ∈ { T , ..., T M } denotes the distance between i -th and i + 1 -th bluelines. m a , b parametrize the diagonal P s. X ( N,M ) Consider the ( N, M ) web shown in figure (5). The Kähler class ω of X N,M is parametrized by ( m α,β , τ, ρ, T , t ) = ( m α,β , τ, ρ, m, T , T , · · · , T M − , t , t , · · · , t N − ) with τ = (cid:80) Mi =1 T i and ρ = (cid:80) Nj =1 t j . In the partition function Z N,M the Kähler parameter are quantum corrected whereasin the mirror curve the Kähler parameter have to be quantum corrected. The factorisationproperties of the mirror curve will in general be affected by the quantum corrections. Themirror curve is given by a sum over the monomials associated with the Newton polygon. Inthis case the Newton polygon tiles the plane therefore, H N,M ( X, Y ) := (cid:88) ( i,j ) ∈ Z A i,j X i Y j . (3.25)The coefficients A i,j depend on the length of the various line segments in the web whichare the Kähler parameters of the corresponding Calabi-Yau threefolds. As discussed beforeneighboring pair of points in the Newton polygon connected by a line give a relation betweenthe associated coefficients A i,j , A i,k +1 A i,k = e (cid:80) k − j =1 T j + (cid:80) i − α =0 m α,k (3.26) A i +1 ,k A i,k = e (cid:80) i − j =1 t j + (cid:80) k − α =0 m i,α – 15 – i +1 ,k +1 = A i +1 , e T +( T + T )+( T + T + T )+ ··· +( T + ··· + T k − )+ (cid:80) kβ =1 (cid:80) iα =0 m α,β (3.27) = A i +1 , e (cid:80) k − γ =1 ( k − γ ) T γ + (cid:80) kβ =1 (cid:80) iα =0 m α,β = A , e t +( t + t )+ ··· +( t + t + ··· + t i − ) e (cid:80) k − γ =1 ( k − γ ) T γ + (cid:80) kβ =0 (cid:80) iα =0 m α,β Since A , = A , = 1 we get, A i +1 ,k +1 = e (cid:80) i − γ =1 ( i − γ ) t γ + (cid:80) k − γ =1 ( k − γ ) T γ + (cid:80) kβ =0 (cid:80) iα =0 m α,β (3.28)Thus the curve is given by H N,M ( X, Y ) = (cid:88) ( i,k ) ∈ Z A i +1 ,k +1 X i +1 Y k +1 (3.29) = N − ,M − (cid:88) i =0 ,k =0 W i,k ( X, Y ) W i,k ( X, Y ) = (cid:88) ( a,b ) ∈ Z A Na + i +1 ,Mb + k +1 X Na + i +1 Y Mb + k +1 A Na + i +1 ,Mb + k +1 = e (cid:80) Na + i − γ =1 ( Na + i − γ ) t γ + (cid:80) Mb + k − γ =1 ( Mb + k − γ ) T γ + (cid:80) Mb + kβ =0 (cid:80) Na + iα =0 m α,β (3.30)Using t γ = t γ (cid:48) if γ ≡ γ (cid:48) ( mod N ) (3.31) T γ = T γ (cid:48) if γ ≡ γ (cid:48) ( mod M ) m α ,β = m α ,β if α ≡ α ( mod N ) and β ≡ β ( mod M ) – 16 –e get Na + i − (cid:88) γ =1 ( N a + i − γ ) t γ = N (cid:88) γ =1 ( N a + i − γ ) t γ + N (cid:88) γ = N +1 ( N a + i − γ ) t γ + · · · (3.32) + Na (cid:88) γ = N ( a − ( N a + i − γ ) t γ + Na + i − (cid:88) γ = Na +1 ( N a + i − γ ) t γ = N (cid:88) γ =1 (cid:104) ( N a + i − γ ) + ( N ( a −
1) + i − γ ) + ( N ( a −
2) + i − γ ) + · · · +( N + i − γ ) (cid:105) t γ + i − (cid:88) γ =1 ( i − γ ) t γ = N (cid:88) γ =1 (cid:104) N a ( a +1)2 + a ( i − γ ) (cid:105) t γ + i − (cid:88) γ =1 ( i − γ ) t γ = (cid:104) N a ( a +1)2 + ai (cid:105) τ − N (cid:88) γ =1 γ t γ + i − (cid:88) γ =1 ( i − γ ) t γ Similarly Mb + k − (cid:88) γ =1 ( M b + k − γ ) T γ = (cid:104) M b ( b +1)2 + bk (cid:105) ρ − M (cid:88) γ =1 γ T γ + k − (cid:88) γ =1 ( k − γ ) T γ (3.33) Mb + k (cid:88) β =0 Na + i (cid:88) α =0 m α,β = Mb + k (cid:88) β =0 (cid:104) N − (cid:88) α =0 m α,β + N − (cid:88) α = N m α,β + · · · + Na − (cid:88) α = N ( a − m α,β + Na + i (cid:88) α = Na m α,β (cid:105) (3.34) = Mb + k (cid:88) β =0 (cid:104) a N − (cid:88) α =0 m α,β + i (cid:88) α =0 m α,β (cid:105) = a N − (cid:88) α =0 (cid:104) b M − (cid:88) β =0 m α,β + k (cid:88) β =0 m α,β (cid:105) + i (cid:88) α =0 (cid:104) b M − (cid:88) β =0 m α,β + k (cid:88) β =0 m α,β (cid:105) = ab N − (cid:88) α =0 M − (cid:88) β =0 m α,β + a N − (cid:88) α =0 k (cid:88) β =0 m α,β + b i (cid:88) α =0 M − (cid:88) β =0 m α,β + i (cid:88) α =0 k (cid:88) β =0 m α,β Since (cid:80) N − α =0 m α,β is independent of β by Lemma 5.4 of [10] therefore Mb + k (cid:88) β =0 Na + i (cid:88) α =0 m α,β = ( ab + a ( k +1) M + b ( i +1) N ) N − (cid:88) α =0 M − (cid:88) β =0 m α,β + i (cid:88) α =0 k (cid:88) β =0 m α,β (3.35) = ( ab + a ( k +1) M + b ( i +1) N ) m + m i,k We reproduce the proof in appendix (B) – 17 – a + i − (cid:88) γ =1 ( N a + i − γ ) t γ + Mb + k − (cid:88) γ =1 ( M b + k − γ ) T γ + Mb + k (cid:88) β =0 Na + i (cid:88) α =0 m α,β + (3.36) z ( N a + i + 1) + z ( M b + k + 1) = (cid:104) N a ( a +1)2 + ai (cid:105) τ − N (cid:88) γ =1 γ t γ + i − (cid:88) γ =1 ( i − γ ) t γ + (cid:104) M b ( b +1)2 + bk (cid:105) ρ − M (cid:88) γ =1 γ T γ + k − (cid:88) γ =1 ( k − γ ) T γ +( ab + a ( k +1) M + b ( i +1) N ) m + m i,k + z ( N a + i + 1) + z ( M b + k + 1)= G i,kN,M ( t , T , m ) + ( a + i +1 N , b + k +1 M ) (cid:32) N τ mm M ρ (cid:33) (cid:32) a + i +1 N b + k +1 M (cid:33) + aτ ( N −
1) + bρ ( M − − ( i +1)( k +1) MN m − ( i +1 N ) N τ − ( k +1 M ) M ρ + N z ( a + i +1 N ) + M z ( b + k +1 M )= G i,kN,M ( t , T , m ) + ( n + u ) t Ω( n + u ) + ( n + u ) · ( (cid:98) z + v ) where G i,kN,M ( t , T , m ) = − ( k +1)( M + k − M ρ − ( i +1)( N + i − N τ + m i,k − N (cid:88) γ =1 γ t γ + i − (cid:88) γ =1 ( i − γ ) t γ (3.37) − M (cid:88) γ =1 γ T γ + k − (cid:88) γ =1 ( k − γ ) T γ (cid:98) z = ( N z , M z ) u = ( i +1 N , k +1 M ) v = ( τ ( N − , ρ ( M − We define the genus two theta function as: Θ (cid:126)u,(cid:126)v ( (cid:126)z, Ω) = (cid:88) (cid:126)n ∈ Z e ( (cid:126)n + (cid:126)u ) t Ω( (cid:126)n + (cid:126)u )+( (cid:126)n + (cid:126)u ) · ( (cid:126)z + (cid:126)v ) (3.38)Then W i,k ( X, Y ) = e G i,kN,M Θ (cid:126)u,(cid:126)v ( (cid:126)z, Ω) (3.39)The genus of the mirror curve N − ,M − (cid:88) i =0 ,k =0 W i,k ( X, Y ) = 0 (3.40)is
M N + 1 . The underlying abelian surface has polarisation ( N, M ) with the period Ω = (cid:32)
N τ mm M ρ (cid:33) . The theta functions form a basis of this ( N, M ) -polarization of the abeliansurface. The mirror curve for general values of N and M cannot be factorized in the limit z → . – 18 – .5 Geometric interpretation of the mirror curve An interesting way to visualise the mirror curve Σ is to see it as N copies of the base torusglued together by N-1 branch cuts. The one cycles, A and B, of the base torus are lifted toa basis of 1-cycles A i , B i , i = 1 , ..., N on Σ . Riemann-Hurwitz theorem is used to computethe genus of Σ and is equal to N. Riemann-Roch theorem is handy in the computation of thenumber of moduli of Σ , which is equal to N in this case.In the case under consideration, the genus N Riemann surface is seen as defined by thetadivisor. A polarised abelian variety U admits a line bundle L with c ( L ) = ω where ≤ y i ≤ define the abelian variety where ω = [ N dy ∧ dy + M dy ∧ dy ] (3.41) ω is a (1 , -form if the period matrix Ω of Σ is symmetric and Im (Ω) > . The line bundle L admits M N holomorphic sections. In the case of an abelian surface these sections are givenby genus theta functions Θ (cid:34) iM jN (cid:35) ( z | Ω) 0 ≤ i < M, ≤ j < N. (3.42)A theta divisor is the zero locus of a linear combination of the above set of theta functions M (cid:88) i N (cid:88) j A ij Θ (cid:34) iM jN (cid:35) ( (cid:126)z | Ω) = 0 (3.43)where A ij are 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 expressedin the following form (cid:88) n =0 n ! ( m πi ) n ∂ nz θ ( z | τ ) ∂ nx h ( x ) = 0 (3.44)where θ is the jacobi theta function and h ( x ) = (cid:81) Nj =1 θ ( x − ξ j | ρ ) with ξ j is the moduli of Σ . This can be reorganised into the following form Θ [ ,..., ] , [ ,..., ] ( z, N β π ( x − ξ i ) | ˆΩ) = 0 (3.45)where ˆΩ is the period matrix of the genus MN+1 curve ˆΣ which is an unbranched cover of agenus 2 curve and in general is given by ˆΩ = τ βm πi βm πi βm πi ... βm MN πiβm πi ρ ... βm πi ρ ... . . . .. . . .. . . . βm MN πi ...ρ (3.46)– 19 –t is easy to see from the following representation of genus g = M N + 1 theta function Θ (cid:34) αβ (cid:35) ( Z | ˆΩ) = (cid:88) m ∈ Z g exp (cid:18) πi ( m + α ) . ˆΩ . ( m + α ) + 2 πi ( Z + β ) . ( m + α ) (cid:19) (3.47)where Z, α, β, m are g-vectors and Ω is a g × g matrix with Im Ω > . In the limit m N → the genus N + 1 theta function gets split into the product of genus N theta function and ajocobi theta function ( g = 1) with characteristics. For instance the mirror curve for X , isgiven by Θ (cid:16) Ω(2 ρ, z, τ ) | (2 u − ρ + z + w, v − τ ) (cid:17) + e πiu Θ (cid:16) Ω(2 ρ, z, τ ) | (2 u + z + w, v − τ + z ) (cid:17) = 0 (3.48)In the limit of Im ( u ) → ∞ this curve collapses to Θ (cid:16) Ω(2 ρ, z, τ ) | (2 u − ρ + z + w, v − τ ) (cid:17) = 0 (3.49)which, after performing Sp (2 , Z ) transformations, is the curve mirror to X , . To study the decomposition of generalised theta function [11] defined on the Jaobian of agenus g = M curve, we start from the following Fourier representation Θ(Ω | (cid:126)z ) = (cid:88) m ∈ Z M e πi (cid:80) Mi =1 m i z i + iπ (cid:80) Mi,j =1 m i Ω ij m j (3.50)where Ω is the period matrix and satisfies the following constraints M (cid:88) i =1 Ω ij = τ, M (cid:88) j =1 Ω ij = τ (3.51)This constraint encodes various periodicity properties. In other words we can decompose Ω as Ω = τM + Ω (cid:48) (3.52)where Ω (cid:48) is the traceless part. Now redefine z i as follows z i = zM + z (cid:48) i such that M (cid:88) i =1 z (cid:48) i = 0 (3.53)– 20 –utting back these redefine variable in (3.50) we get Θ(Ω | (cid:126)z ) = (cid:88) m ∈ Z M e πi zM (cid:80) Mi =1 m i + iπ τM ( (cid:80) Mi =1 m i ) +2 πi (cid:80) Mi =1 m i z (cid:48) i + πi (cid:80) Mi,j =1 m i Ω (cid:48) ij m i (3.54)Next we use a trick, essentially a redefinition of indices, to write the exponential in a suggestiveform. To this end we decompose the set of indices m into two parts. First we impose theconstraint that (cid:80) Mi =1 m i = l which effectively reduces the set { m , ..., m M } to { m , ..., m M − } .Secondly we perform a sum over l . Θ(Ω | (cid:126)z ) = (cid:88) l ∈ Z e πi kM z + πi l M τ (cid:88) m ∈ Z M ; (cid:80) Mi =1 m i = l e i πi (cid:80) Mi =1 m i z (cid:48) i + πi (cid:80) Mi,j =1 m i Ω (cid:48) ij m i (3.55)To be able to write the first summation as a Jocobi theta function with characteristics, wemake another redefinition l = M s + i where s ∈ Z and i ∈ Z N , resulting in Θ(Ω | (cid:126)z ) = (cid:88) i ∈ Z M ,s ∈ Z e πi ( s + iM ) z + πiτ ( s + iM ) (cid:88) m ∈ Z M , (cid:80) Mj =1 m j = i e i π (cid:80) Mp =1 m p z (cid:48) p + πi (cid:80) Mp,q =1 m p Ω (cid:48) pq m q = (cid:88) i ∈ Z M θ (cid:34) iM (cid:35) ( M τ | z )Θ i (Ω (cid:48) | (cid:126)z (cid:48) ) (3.56)where Θ i is the second summation factor in the first line of (3.56).The splitting of theta functions has important consequences for M5-branes partition function.On general grounds [12] the partition functions of M5-branes on a six-manifold X are actuallysections of a line bundle L over the intermediate Jacobian J X = H ( X, R ) /H ( W, Z ) . Theintermediate Jacobian for our CY 3-fold is an abelian surface. The line bundle L is uniquelyspecified by its first Chern class c ( L ) = ω , where ω ∈ H ( J, Z ) gives the principal polarisation.For the case at hand we have M M5 branes that are probing the transverse space S × C / Z N .The partition function of this theory correspond to sections of a line bundle L of polarisation ( M, N ) and readily given by eq.(3.38) Θ (cid:126)u,(cid:126)v ( (cid:126)z, Ω) = (cid:88) (cid:126)n ∈ Z e ( (cid:126)n + (cid:126)u ) t Ω( (cid:126)n + (cid:126)u )+( (cid:126)n + (cid:126)u ) · ( (cid:126)z + (cid:126)v ) (3.57)and there are M N of them. The eq.(3.56) then shows that a genus g theta function splitsinto a product of genus g − theta function and an ordinary theta function. Therefore samemust be true of the theta functions (3.57) that describe M5-branes partition functions.– 21 – Degenerations and their Effect on the Partition Function
The partition function for the ( N, M ) configuration of the CY 3fold which we denoted by X N,M is given by Z ( N,M ) ( τ, ρ, (cid:15) , , m, t ) = (cid:88) α ia N (cid:89) i =1 Q | α ( i ) | i N (cid:89) i =1 M (cid:89) a =1 ϑ α i +1 a α ia ( m ) ϑ α ia α ia ( (cid:15) + ) (cid:89) ≤ a
0) = 0 , therefore ϑ µν (0) = 0 if h µ ( i, j ) + ν tj − µ tj = 0 . Since, h µ ( i, j ) (cid:54) = 0 , therefore ν tj (cid:54) = µ tj If µ = ν then ϑ µµ (0) = (cid:89) ( i,j ) ∈ µ ϑ ( ρ, h µ ( i, j )) h µ ( i, j ) is non zero therefore ϑ µµ (0) (cid:54) = 0 . To prove that µ (cid:54) = ν implies ϑ µν (0) = 0 i.e., Either h µ ( i, j ) + ν tj − µ tj = 0 or h ν ( i, j ) + µ tj − ν tj = 0 . Because h µ ( i, j ) (cid:54) = 0 therefore ν tj (cid:54) = µ tj . Aninteresting property of ϑ µν ( x ) which we will use extensively in later sections is the following, ϑ µν (0) = δ µ ν (cid:89) ( i,j ) ∈ µ ϑ ( q h µ ( i,j ) ) ϑ ( q − h µ ( i,j ) ) (4.5)where δ µ ν is the kronecker delta function. ( N, M ) = (1 , We begin by looking at the case of X , . The unrefined partition function is given by, Z (1 , ( τ, ρ, m, t, (cid:15) ) = (cid:88) α , Q | α | + | α | ϑ α α ( m ) ϑ α α ( m ) ϑ α α (0) ϑ α α (0) ϑ α α ( t − m ) ϑ α α ( t + m ) ϑ α α ( t ) (4.6)– 23 –ere, t − m = t − m and t + m = t + m . The above defined partition function Z (1 , in (4.6) in thelimit t (cid:55)→ m changes to Z (1 , ( τ, ρ, m, t = m, (cid:15) ) = (cid:88) α , Q | α | + | α | ϑ α α ( m ) ϑ α α ( m ) ϑ α α (0) ϑ α α (0) ϑ α α (0) ϑ α α (2 m ) ϑ α α ( m ) Using the property of ϑ µν ( x ) defined in eq.(4.5) we get Z (1 , ( τ, ρ, m, t = m, (cid:15) ) = (cid:88) α Q | α | ϑ α α (2 m ) ϑ α α (0) (4.7) = Z (1 , (2 τ, ρ, m, (cid:15) ) (4.8) ( N, M ) = (1 , M ) The partition function defined in (4.1) for N = 1 has the following expression Z (1 ,M ) ( τ, ρ, (cid:15) , , m, t ) = (cid:88) α , , ··· ,M Q | α | + ··· + | α M | M (cid:89) a =1 ϑ α a α a ( m ) ϑ α a α a ( (cid:15) + ) (cid:89) ≤ a
1) + ( N −
1) +
M N − ( M − − ( N −
1) + 2=
M N + 2 (4.15)In general we can have three different series representations of Z ( M,N ) according to whetherthe toric web diagram of X M,N is sliced into horizontal strips, vertical strips and diagonalstrips Z ( M,N ) ( t , T , m , (cid:15) , (cid:15) ) = Z pert ( T , m ) (cid:88) (cid:126)k e − (cid:126)k. t Z (cid:126)k ( T , m ) Z ( M,N ) ( t , T , m , (cid:15) , (cid:15) ) = Z pert ( t , m ) (cid:88) (cid:126)k e − (cid:126)k. T Z (cid:126)k ( t , m ) Z ( M,N ) ( t , T , m , (cid:15) , (cid:15) ) = Z pert ( T , t ) (cid:88) (cid:126)k e − (cid:126)k. m Z (cid:126)k ( T , t ) (4.16)– 25 – -- --- --- ------ - - -- - - - - - --- - - - = = = = M = = = = M − − − − − − N − − − − − − N degenerates to M m ρM τ N
Figure 7 : Pictorial representation of (4.14): ( M, N ) web degenerating to N copies of the ( M, webThese expansion have been interpreted as instanton expansions of three gauge theories whichare dual to each other. For these to be consistent expansions it is assumed that there existsa region of the moduli space of X ( M,N ) in which either either T or t or m become infinite ,with all the rest of parameters kept finite. This region of the moduli space corresponds to theweak coupling limit of gauge theories.At the special point in the moduli space where t a,a +1 = m , we are left with three independentKähler parameters, τ, ρ, m . Moreover due to the weak coupling expansion { T → ∞} , N – 26 –orizontal strips gets decoupled and we get Z N , . Remark 1:
After normalisation by the gauge theory perturbative part , the partition function Z (1 , ( τ, ρ, m ) can be written as [13] Z (1 , ( τ, ρ, m ) = e − πi ( τ + ρ + m )12 (cid:89) ( k,l,m ) > (1 − e πi ( kτ + lρ + pm ) ) − c (4 kl − p ) = 1Φ ( τ, ρ, m ) (4.17)where c (4 kl − p ) is the Fourier coefficient of the elliptic genus of K χ ( K , τ, z ) = (cid:88) h ≥ ,m ∈ Z c (4 h − m ) e πi ( hτ + mz ) (4.18)and Φ ( τ, ρ, m ) is the unique weight automorphic form of Sp (2 , Z ) . We have implicit usedthe fact that the large radius limit (universal part) of the Taub-NUT elliptic genus matcheswith the elliptic genus of C [14]. This allows us to write Z ( N,M ) ( τ, ρ, t a,a +1 = m ) in thefollowing way Z ( N,M ) ( τ, ρ, t a,a +1 = m ) = e − Nπi ( τ + ρ + m )12 (cid:89) ( k,l,m ) > (1 − e πi ( Mkτ + lρ + pMm ) ) − Nc (4 kl − p ) = 1Φ ( M τ, ρ, M m ) N (4.19) Remark 2:
The CY 3-fold X , has a nice interpretation in terms of the so-called banana curves [15]. Abanana configuration of curves in the CY 3-fold is a union of three curves C i ≡ P with thenormal bundle given by O ( − ⊕ O ( − . Moreover C ∩ C = C ∩ C = C ∩ C = { x, y } fordistinct point x, y ∈ CY C i arealong the coordinate axis.In other words the refined topological string partition function Z X N,M ( ω, (cid:15) , (cid:15) ) is factored intoa product of N copies of Z X , ( τ, ρ, m ) , where the later is the topological partition functionon on a CY 3-fold with a single banana configuration of curves. In the case of splitting degeneration we consider the following partition function for ( N, M ) Z ( N,M ) ( τ, ρ, m a,b , (cid:15) , , t ab ) = (cid:88) α ia N (cid:89) i =1 Q | α ( i ) | i N (cid:89) i =1 M (cid:89) a =1 ϑ α i +1 a α ia ( m a ) ϑ α ia α ia ( (cid:15) + ) (cid:89) ≤ a
Recall the following degeneration (4.14) Z ( N,M ) ( τ, ρ, t a,a +1 = m, (cid:15) ) = Z (1 , ( M τ, ρ, M m, (cid:15) ) N (6.1)This degeneration corresponds to a U ( M ) N quiver gauge theory degenerating to a U (1) N gaugetheory. Moreover the gauge coupling constant τ and the hypermultiplet mass parameter m are scaled to M τ and
M m under the degeneration. This rescaling corresponds to multiplewrapping number of the D-branes along the τ and m directions.Similarly the second degeneration of the Z N,M (5.15) that we discussed and is given by Z ( N,M ) ( τ, ρ, m i , t ab , (cid:15) ) = (cid:88) α ia F α ia ( τ, ρ, m c,d , t cd ) Z α i ,α i ,...,α iM − ( N,M − ( τ, ρ, m p,q , t pq , (cid:15) ) (6.2)has an interesting physical interpretation. The limit m i → corresponds to supersymme-try enhancement to N = 4 and we get a decoupling factor of η ( τ ) . This is true only for N ∈ N , M = 1 . For N ∈ N , M ∈ N ≥ the factorisation is only partial. This paper explored some interesting consequences of the mirror symmetry of the local CY3-fold X N,M . We investigated some important properties of the type A topological stringpartition function on X N,M in special regions of the Kähler moduli space. We have calledthese degenerate limits, because in these limits the partition functions on X N,M collapse tothose on X N,M − in various ways. In accordance with mirror symmetry the degenerationbehaviour on the type A side is reproduced on the type B side in the degeneration of thequantum mirror curves into lower genus curves.For future directions it would be interesting to study the analogous properties of Z N,M andquantum mirror curves for the general Ω -background .i.e. (cid:15) (cid:54) = 0 and/or (cid:15) (cid:54) = 0 and (cid:15) (cid:54) = (cid:15) and at an arbitrary point of the Kähler moduli space of X N,M . It will also be interestingto study the modular properties of the free energy log( ˆ Z ( N,M ) ( τ, ρ, (cid:15), m, t )) and the singleparticle free energy [18] P Log ( ˆ Z ( N,M ) ( τ, ρ, (cid:15), m, t )) along the lines of [19]. We hope to reporton these matters in future. Also see recent interesting work [20, 21]. The authors are grateful to Amer Iqbal for crucial discussions and gratefully acknowledge thesupport of the Abdus Salam School of Mathematical Sciences, Lahore.– 35 –
Geometry of X N,M : a quick review
The non-compact CY 3-fold X , is defined as the partial compactification [3, 10] of theresolved conifold geometry. The later is given by C × × C × fibered over the z -plane. Thepartial compactification is achieved by compactifying each of the two C × fibers to a T fiber.Of the three Kähler parameters τ, ρ, m of the CY 3-fold X , , ρ and τ correspond to the ellipticfibers and m corresponds to the curve class of the exceptional P of the resolved conifold. Wewill define the non-compact CY 3-fold X N,M for
N, M ∈ N as the Z N × Z M orbifold of X , .In toric geometry the equation of the conifold z z − z z = 0 , z , z , z , z ∈ C (A.1)is translated to an equation on integer latices parametrised by 3-vectors v , v , v , v v + v − v − v = 0 . (A.2)The CY condition constrains the geometry to a plane. The irreducible toric rational curvesof the 2-dimensional cone are given by C a,b ) : = R ≥ Conv ( { ( a + 1 , b, , ( a, b + 1 , } ) , C a,b ) := R ≥ Conv ( { ( a, b, , ( a, b + 1 , } ) ,C a,b ) : = R ≥ Conv ( { ( a, b, , ( a + 1 , b, } ) . (A.3)for all a, b ∈ Z . These curve classes satisfy the following relations C a − ,b ) + C a − ,b ) = C a,b − + C a − ,b ) ,C a − ,b ) + C a,b ) = C a,b − + C a,b − . (A.4)For the local CY 3-fold X N,M a modular covariant basis of generators can be given by C m, ( a,b ) = C a,b ) , C τ, ( a,b ) = C a,b ) + C a,b ) ,C ρ, ( a,b ) = C a,b ) + C a,b ) (A.5)where a, b ∈ Z . In the fundamental domain of the ( N, M ) -web there are M N toric rationalcurves where a ∈ Z N , b ∈ Z M . Due to the N M constraints in (A.5) and torus periodicitythe effective rank is
M N + 2 . B (cid:80) N − a =0 m a , b is independent of b :proof Note that in our notation the curve classes C a,b ) are represented by the Kähler parameters m a , b . Using the first relation in eq.(A.5), we can write the following summation p − (cid:88) a =0 ( C a − ,b ) + C a − ,b ) ) = p − (cid:88) a =0 ( C a,b − + C a − ,b ) ) , (B.1)– 36 –ue to the compactification of web diagram on a torus there is periodicity relation C − ,b ) = C p − ,b ) . After simplification the second term cancels on both sides and we get p − (cid:88) a =0 ( C a − ,b ) ) = p − (cid:88) a =0 ( C a,b − ) , (B.2)Expanding the left side p − (cid:88) a =0 ( C − ,b ) + C ,b ) + C ,b ) + ... + C p − ,b ) + C p − ,b ) ) = p − (cid:88) a =0 ( C a,b − ) , (B.3)Rearranging the terms after using Using C − ,b ) = C p − ,b ) , we obtain the desired relation p − (cid:88) a =0 C a,b ) = p − (cid:88) a =0 C a,b − . (B.4) References [1] S. Hohenegger, A. Iqbal, and S.-J. Rey,
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