IIntrinsic non-perturbative topological strings
Murad Alim ∗ Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146, Hamburg, Germany
Abstract
We study difference equations which are obtained from the asymptotic expansion of topolog-ical string theory on the deformed and the resolved conifold geometries as well as for topologicalstring theory on arbitrary families of Calabi-Yau manifolds near generic singularities at finitedistance in the moduli space. Analytic solutions in the topological string coupling to these equa-tions are found. The solutions are given by known special functions and can be used to extractthe strong coupling expansion as well as the non-perturbative content. The strong couplingexpansions show the characteristics of D-brane and NS5-brane contributions, this is illustratedfor the quintic Calabi-Yau threefold. For the resolved conifold, an expression involving boththe Gopakumar-Vafa resummation as well as the refined topological string in the Nekrasov-Shatashvili limit is obtained and compared to expected results in the literature. Furthermore, aprecise relation between the non-perturbative partition function of topological strings and thegenerating function of non-commutative Donaldson-Thomas invariants is given. Moreover, theexpansion of the topological string on the resolved conifold near its singular small volume locusis studied. Exact expressions for the leading singular term as well as the regular terms in thisexpansion are provided and proved. The constant term of this expansion turns out to be theknown Gromov-Witten constant map contribution. ∗ [email protected] a r X i v : . [ h e p - t h ] F e b ontents B.1
The A-model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26B.2
The B-model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Topological string theory bridges research in mathematics and physics and has been a richsource of insights for both areas. It provides a quantitative handle on physical dualities as wellas exact computations, see e. g. [1]. Within the context of mirror symmetry, topological stringtheory provides the tools to study higher genus mirror symmetry [2, 3, 4].Physically, the appeal of studying topological string theory originates from the fact thatit shares features of physical strings while at the same time possessing clearer mathematicalstructures which allow one to seek answers to difficult physical questions. One such aspect,which is the focus of this work, is the fact that the free energy of topological string theory is nly defined perturbatively as an asymptotic series in the topological string coupling. This isalso known to be true for physical string theories as well as for many quantum field theories,see [5, 6] and references therein.The topological string partition function of a given family of Calabi-Yau (CY) threefoldsis defined perturbatively in the topological string coupling λ . The free energies at genus g arerelated recursively to the free energies at lower genera by the holomorphic anomaly equations[7, 8]. The anomaly equations were identified in [9] as the projective flatness equations of a con-nection identifying Hilbert spaces obtained from polarization choices of a geometric quantizationproblem associated to the moduli space of the family. The topological string partition functionbecomes a flat section of this bundle of Hilbert spaces over the moduli space. Further detailsabout this quantum mechanical interpretation including a possible Hamiltonian as well as adescription of other states in this Hilbert space remain puzzling, see [10] for a the descriptionof other potential states in this Hilbert space.The OSV conjecture [11] relates the topological string wave-function to the partition func-tion of black holes. It provides an expectation that the non-perturbative content of topologicalstring theory should be captured by the partition function of objects which are defined non-perturbatively on the same Calabi-Yau family, namely the BPS states forming the black holes.A number of challenges, most sharply collected in [12], however limit the applicability of thisconnection for the study of the non-perturbative structure of topological string theory. Mathe-matically, the perturbative definition of topological string theory on one side of mirror symmetrycorresponds to the study Gromov-Witten (GW) theory, while the enumerative content of theBPS states forming the black holes is captured by Donaldson-Thomas (DT) invariants, theequivalence of the enumerative geometry content of the two is the MNOP conjecture [13, 14].The study of the non-perturbative structure of topological string theory has been much morepromising for non-compact CY manifolds which exhibit dualities with Chern-Simons theoryand matrix models, see e. g. [2]. This is also the context for which early all genus resultswere obtained including the all genus topological string theory on the deformed conifold, whosepartition function correspond to the c = 1 string as well as to the Gaussian matrix model[15]. The relevance of this result is that it also provides the expected universal behavior oftopological string theory near singularities in the moduli space where finitely many states of thecorresponding effective theory are becoming massless. Another all-genus result for topologicalstring theory comes from a large N duality with Chern Simons theory [16] and provides theperturbative expansion of topological string theory on the resolved conifold. The study of thenon-perturbative structure of topological string theory in these two cases as well as other non-compact CY manifolds has benefited a lot from their relation to matrix models, see e. g. [17, 6]and references therein.For non-compact CY geometries, another path gives further insights into the non-perturbativestructure of topological string theory. Considering mirror non-compact CY geometries whoserelevant data is captured by the mirror curve, a quantum mechanical problem was put forward n [18]. The curve equation in these cases is characterized by an algebraic equation in two com-plex variables which take values in C ∗ or C . These variables are identified with conjugate phasespace variables and the defining equation of the curve is interpreted as a Hamiltonian whoseeigenstates are wave-functions. This quantum curve approach is useful for the study of therelation of topological string theory to integrable systems. The approach of [18] was revisitedin the context of refined topological string theory in [19], shedding light on the relation of thequantization of integrable systems of supersymmetric theories and topological strings [20].Building on the quantum curve developments as well as on a series of insights from thestudy of ABJM theory [21] (see [22] and references therein), a proposal for the non-perturbativedefinition of topological string theory was put forward in [23] and further scrutinized in [24],using spectral properties of the quantum mechanical problem defined by the quantum curve.Interestingly, the wave functions obtained in this way were related to the Nekrasov-Shatashvili(NS) limit [20] of the refined topological string with the quantization parameter (cid:126) ∼ λ beingrelated to the inverse of the topological string coupling rather than (cid:126) ∼ λ as is expected fromthe wave-function interpretation of the anomaly equation [9], which suggests perhaps two dualquantum mechanical pictures for the topological string. The spectral properties of the quantummechanical system allowed the authors of [23, 24] however to extract both expansions in λ and λ . The quantum curve setting was also recently used in [25, 26] to propose non-perturbativepartition functions for topological strings on local CY manifolds related to the class S theoriesof [27].In the case of the all genus free energy of the deformed conifold, the Borel resummation givesthe Barnes G-function and can be used to access the non-perturbative content of the partitionfunction as well as the corresponding matrix model, see for instance [17, 6]. A proposal forthe non-perturbative structure of topological string theory on the resolved conifold was givenin [28], making use of correspondence with supersymmetric indices. In [29] a modified Borelresummation was applied to the resolved conifold and the expected non-perturbative structureof [23, 30] was obtained. The expected non-perturbative structure of [29] was further obtainedin [31] from the exact duality with Chern-Simons theory.Remarkable recent progress in defining non-perturbative topological string theory was achievedrecently in mathematics [33, 34], inspired by [32]. In [33] a Riemann-Hilbert (RH) problem wasput forward which describes the wall-crossing phenomena in Donaldson-Thomas theory, Thiscorresponds physically to the wall-crossing phenomena of BPS states whose recent study hasbeen advanced by [27] and many others. In [33], the solution of the RH problem for the Argyres-Douglas A theory was given, this corresponds on the topological string side to the deformedconifold free energy The subsequent [34] solves the RH problem for the resolved conifold andsuggests the resulting Tau function as a non-perturbative definition of the topological stringtheory on the resolved conifold given its analytic properties and the fact that it contains as The observation of the link to the deformed conifold has not been made in [33], but is perhaps obvious to theexperts. n asymptotic expansion the Gromov-Witten theory of the resolved conifold. In a sense, thework of Bridgeland provides several missing links in the expectation that the BPS content ofa given geometry provides the non-perturbative definition of topological string theory on thatgeometry. The details of this program however become quickly very challenging since it requiresas an input the complete relevant BPS spectra and their wall-crossing behavior. This seemscurrently, especially for compact geometries, very challenging if not intractable.It is natural to wonder whether there is a more intrinsic path towards non-perturbativetopological string theory which does not rely on dualities to other physical or mathematicalproblems and is as such not limited in its scope of applicability. Given the asymptotic natureof the expansion of the free energies, the Borel resummation of the free energy as well as theapplication of resurgence techniques are such paths, see e. .g. [35, 36, 37] as well as [38] for amatching of the resurgence results with [24]. In the case of asymptotic series stemming fromdifferential equations with irregular singular points, the knowledge of the differential equationitself is often more powerful than the knowledge of the asymptotic expansion around singularpoints. Especially in problems of mathematical physics, the ODEs in question are often the oneswhich have been well-studied for a long time. Such a differential equation in the topologicalstring coupling is however not part of the defining data of topological strings. The quest forsuch differential equation in the string coupling was the motivation for [39]. In that paper,the holomorphic anomaly equations [8] as well as the polynomial structure of the higher genustopological string amplitudes [40, 41] were used to obtain a differential equation in the stringcoupling in a certain scaling limit. The relevant differential equation turned out to be theAiry equation. Apart from the expected asymptotic expansion in this limit the equation has asolution which is non-perturbative in the string coupling. The latter was subsequently relatedto non-perturbative resurgence effects of NS-branes [42].The aim of this work is to extend the study of the intrinsic characterization of the non-perturbative structure of topological string theory. We are in a fortunate situation where manypieces of the puzzle are already available in the recent physics and especially mathematicsliterature and can be readily used and put together. The starting point is a difference equationwhich was first proved in [43] for the free energies of the WKB analysis of the Weber curve.This curve is related to the deformed conifold. A similar difference equation was proved in[44] for the free energies of the resolved conifold. Both derivations only have the asymptoticexpansion as their input. A first aim of this work is to use the expected universal behavior oftopological string theory on arbitrary families of CY threefolds near singular loci in the modulispace where finitely many states of the effective theory become massless and derive a differenceequation for the topological string free energies in a limit around these loci. We next identifythe Barnes G-function as a solution for the difference equations of the deformed conifold as wellas for the universal behavior near the singularities. A solution of the difference equation for theresolved conifold [44] was identified in [45] using building blocks of Bridgeland’s Tau functionfor the resolved conifold [34]. The explicit analytic solutions can be used to obtain the strong oupling expansions of topological string theory as well as to express their non-perturbativecontent. The characteristic traits of non-perturbative effects due to D-branes and NS-branesare obtained. Moreover, for the resolved conifold an expression involving both the Gopakumar-Vafa resummation as well as the refined topological string in the Nekrasov-Shatashvili limit isobtained. The latter was put forward in [23, 29], we obtain a matching with their results up tosome factors which are discussed.The organization of this work is as follows. In sec. 2, the topological string free energies arerecalled as well as their Gromov-Witten and Gopakumar-Vafa expansions. We proceed witha discussion of the expected universal behavior of topological string theory near singularitieswhere finitely many states of the underlying effective theory in 4 d become massless. The explicitexpressions of the topological string free energies for the deformed and resolved conifold aregiven. In sec. 3, the difference equations for the deformed and resolved conifold geometriesare introduced and a similar equation for the universal behavior of the free energies in a limitaround singular points is derived. We proceed with discussing the analytic solutions in thestring coupling of the difference equations and extract their strong coupling expansion as wellas the non-perturbative content in sec. 4. We furthermore give an exact non-perturbativerelation between the topological string partition function and the generating function of non-commutative DT invariants. Finally we study in detail the expansion of the free energies ofthe resolved conifold near the locus where the P of the resolution shrinks to zero and thecorresponding coordinate t →
0. We prove an exact expression of the leading singular behavioras well as the sub-leading terms. In particular the constant terms in this expansion turn outto be the contributions of constant maps in Gromov-Witten theory, the higher order terms arepolynomials in the coordinate. Moreover, this provides a mathematical proof in this case of the gap condition which is expected on physical grounds and was used in [46, 47] in the study ofhigher genus mirror symmetry. We finish in sec. 5 with the conclusions.
To a mirror family of CY threefolds, topological string theory associates the topological stringpartition function which is defined as an asymptotic series in the topological string coupling λ ,summing over the free energies associated to the world-sheets of genus g : Z top ( λ, t ) = exp ∞ (cid:88) g =0 λ g − F g − ( t ) . (2.1)Where t = ( t , . . . , t n ) is a set of distinguished local coordinates on the underlying moduli space M , which is of dim n = h , ( X t ) = h , ( ˇ X t ( z )). X t and ˇ X t ( z ) are a mirror pair of CY threefoldswhich correspond to the A-model and B-model sides of mirror symmetry. The map t ( z ) on theB-model side expressing the distinguished coordinates in terms of the more natural complex tructure coordinates z is the mirror map. It is useful to consider the total space of a linebundle L → M whose sections correspond to a distinguished vacuum state in the underlyingSCFT and which has a different geometric interpretation on both sides of mirror symmetry. M is a projective special K¨ahler manifold. The special geometry as well as the holomorphicanomaly equations of BCOV, together with the boundary conditions of sec. 2.2 can be usedto geometrically characterize the topological string free energies at each genus. The latter arein particular non-holomorphic sections of L − g [8]. A holomorphic limit can be considered bytaking the base point on M to i ∞ and expanding in canonical coordinates. In the holomorphic limit together with an expansion around a distinguished large volume pointin the moduli space, the topological string free energies become the generating functions ofhigher genus Gromov-Witten invariants on the A-model side of mirror symmetry. The GWpotential of X is the following formal power series: F ( λ, t ) = (cid:88) g ≥ λ g − F g ( t ) = (cid:88) g ≥ λ g − (cid:88) β ∈ H ( X, Z ) N gβ q β , (2.2)where q β := exp(2 πi (cid:104) t, β (cid:105) ) is a formal variable living in a suitable completion of the effectivecone in the group ring of H ( X, Z ).The GW potential can be furthermore written as: F = F β =0 + ˜ F , (2.3)where F β =0 denotes the contribution from constant maps and ˜ F the contribution from non-constant maps. The constant map contribution at genus 0 and 1 are t dependent and the highergenus constant map contributions take the universal form [48]: F gβ =0 = χ ( X )( − g − B g B g − g (2 g −
2) (2 g − , g ≥ , (2.4)where χ ( X ) is the Euler characteristic of X and the Bernoulli numbers B n are generated by: we w − ∞ (cid:88) n =0 B n w n n ! . (2.5)The Gopakumar-Vafa (GV) resummation of the GW potential [49, 50] reformulates the non-constant part of the GW potential in terms of the Gopakumar-Vafa invariants n gβ ∈ Z whichare given by a count of electrically charged M branes in an M-theory setup. The GW potentialcan thus be written as:˜ F ( λ, t ) = (cid:88) β> (cid:88) g ≥ n gβ (cid:88) k ≥ k (cid:18) (cid:18) kλ (cid:19)(cid:19) g − q kβ . (2.6) n particular ˜ F ( t ) = (cid:88) β> n β Li ( q β ) , q β = exp(2 πit β ) . Fixing a frame for L we will denote the functions, which are obtained from F g ∈ L g − by F g .The leading singular behavior of the free energy F g at a conifold locus has been determined in[7, 8, 15, 51, 49, 50] F g ( t c ) = b B g g (2 g − t g − c + O ( t c ) , g > . (2.7)Here t c ∼ ∆ m is the special coordinate at the discriminant locus ∆ = 0. For a conifoldsingularity b = 1 and m = 1. In particular the leading singularity in (2.7) as well as the absenceof subleading singular terms follows from the Schwinger loop computation of [49, 50], whichcomputes the effect of the extra massless hypermultiplet in the space-time theory [52]. Thesingular structure and the “gap” of subleading singular terms have been also observed in thedual matrix model [53] and were first used in [46, 47] to fix the holomorphic ambiguity at highergenus. The space-time derivation of [49, 50] is not restricted to the conifold case and appliesalso to the case m > b = n H − n V forthe coefficient of the leading singular term. A higher genus result with a singularity leading to b = 1 − The conifold singularity refers to a singular point in a threefold that locally looks like( x x − x x = 0) ⊂ C , (2.8)the singularity can be deformed by introducing a parameter a ∈ C ∗ , after changes of the localcoordinates in C this can be brought to the form: X a = (cid:8) x x + y = x − a (cid:9) (2.9)which defines the family of non-compact CY threefolds known as the deformed conifold. Notethat the addition of a complex parameter in the defining equation amounts to a change of thecomplex structure so it is natural to study this geometry using the B-model topological string ∆ is usually defined using the algebraic moduli of the problem, see e. g. appendix B. heory. The result of [15] is that the topological string free energy on this geometry obtainedfrom the relation to the c = 1 string has the form : F ( λ, a ) = λ − (cid:18) a a − a (cid:19) −
112 log a + ∞ (cid:88) g =2 B g g (2 g − a g − λ g − . (2.10)We note that in the quantum curve setting, the curve which is quantized is:Σ a := (cid:8) y = x − a ⊂ C (cid:9) , (2.11)and corresponds to the harmonic oscillator. In the exact WKB setting, this curve is known asthe Weber curve and is discussed, e. g. in [43]. Furthermore the WKB analysis encodes the BPScontent of the corresponding effective 4 d, N = 2 theory obtained from compactifying type IIBstring theory on this non-compact CY. In this case this gives the Argyres-Douglas A theory.The CY threefold given by the total space of the rank two bundle over the projective line: X t := O ( − ⊕ O ( − → P , (2.12)corresponds to the resolution of the conifold singularity in C and is known as the resolvedconifold. This geometry is defined on the A-model side of mirror symmetry and t correspondsto: t = (cid:90) C B + iω , (2.13)where B ∈ H ( X, R ) /H ( X, Z ) is the B-field, ω is the K¨ahler form and C corresponds to the P class in this geometry. The GW potential for this geometry was determined in physics [49, 16],and in mathematics [48] with the following outcome for the non-constant maps: ˜ F ( λ, t ) = ∞ (cid:88) g =0 λ g − ˜ F g ( t ) = 1 λ Li ( q ) + ∞ (cid:88) g =1 λ g − ( − g − B g g (2 g − − g ( q ) , (2.14)where q := exp(2 πi t ) and the polylogarithm ist defined by:Li s ( z ) = ∞ (cid:88) n =0 z n n s , s ∈ C . (2.15) In the following we will review the derivation of a difference equation which was obtained in [43]for the free energies of the Weber curve which correspond to the free energies of the deformedconifold geometry and which was adapted in [44] for the free energies of the resolved conifold. Compared to [15] we have added the λ dependence as well as the − a in order to match more recent results,such as [43]. See also [56] for the determination of F g from a string theory duality and the explicit appearance of the polylog-arithm expressions. heorem 3.1. [43, 44] The free energy of the deformed conifold [43] satisfies the followingdifference equation: F ( λ, a + λ ) − F ( λ, a ) + F ( λ, a − λ ) = ∂ ∂a F ( a ) , (3.1) with: F ( a ) = 12 a log a − a , (3.2) and the free energy of the resolved conifold satisfies [44]: ˜ F ( λ, t + ˇ λ ) − F ( λ, t ) + ˜ F ( λ, t − ˇ λ ) = (cid:18) π ∂∂t (cid:19) ˜ F ( t ) , ˇ λ = λ π . (3.3) with ˜ F ( t ) = Li ( q ) . (3.4)The two versions of the theorem were proved in [43, 57, 44]. The proof is included in theappendix A, a notable feature is that it only requires the asymptotic expansion.For the discussion of the universal structure of topological strings near finite distance sin-gularities, the following corollary of the above theorem is obtained: Corollary 3.2.
For X t and ˇ X t ( z ) , a mirror pair of CY threefolds which corresponding to the A − and B − sides of mirror symmetry and which can be thought of as the fibers of correspondingfamilies of over a base manifold M with dim M = n , where n = h , ( X t ) = h , ( ˇ X t ( z )) . Weconsider local coordinates t = (cid:8) t , . . . , t n (cid:9) . Let t c be a coordinate near a singularity of finitedistance in the special K¨ahler metric, We assume the following behavior of the topological stringfree energies, motivated by physical expectations: F ( t c ) = b (cid:18) t c log t c − t c (cid:19) + O ( t c ) , F ( t c ) = − b
12 log t c + O ( t c ) , (3.5) and F g ( t c ) = b B g g (2 g − t g − c + O ( t ) , g ≥ , (3.6) consider now Λ ∈ C ∗ and the rescaling: λ (cid:48) = λ · Λ , t (cid:48) c = t c · Λ , and define: F (cid:48) ( λ (cid:48) , t (cid:48) c ) := lim Λ →∞ (cid:18) F ( λ (cid:48) , t (cid:48) c ) + b ( t (cid:48) c ) − b
12 log Λ (cid:19) , then the following difference equation is satisfied by F (cid:48) : F (cid:48) ( λ (cid:48) , t ◦ , t (cid:48) c + λ (cid:48) ) − F (cid:48) ( λ (cid:48) , t ◦ , t (cid:48) c ) + F (cid:48) ( λ (cid:48) , t ◦ , t (cid:48) c − λ (cid:48) ) = b log t (cid:48) c . (3.7) In the following, w. l. o. g. we only highlight the dependence on the coordinate which corresponds to the singularity,for a study of the behavior of toplogical strings on higher dimensional moduli spaces near singularities in a compactsetting see, e. g. [58, 59]. roof. First consider the all-genus topological string free energy near t c → F ( λ, t c ) = ∞ (cid:88) g =0 λ g − F g ( t c ) = 1 λ (cid:18) b (cid:18) t c log t c − t c (cid:19) + O ( t c ) (cid:19) − b
12 log t c + O ( t c ) + b ∞ (cid:88) g =2 λ g − (cid:32) B g g (2 g − t g − c + O ( t c ) (cid:33) . (3.8)Inserting the rescaled t (cid:48) c and λ (cid:48) we obtain: F ( λ (cid:48) , t (cid:48) c ) = ∞ (cid:88) g =0 ( λ (cid:48) ) g − F g ( t (cid:48) c ) = 1( λ (cid:48) ) b (cid:18)
12 ( t (cid:48) c ) log t (cid:48) c −
34 ( t (cid:48) c ) (cid:19) − b ( t (cid:48) c ) O (1 / Λ) − b
12 log t (cid:48) c + b
12 log Λ + O (1 / Λ) + b ∞ (cid:88) g =2 ( λ (cid:48) ) g − (cid:18) B g g (2 g − t (cid:48) c ) g − + O (1 / Λ g − ) (cid:19) . (3.9)Hence we obtain for: F (cid:48) ( λ (cid:48) , t (cid:48) c ) := lim Λ →∞ (cid:18) F ( λ (cid:48) , t (cid:48) c ) + b ( t (cid:48) c ) − b
12 log Λ (cid:19) = b · F def ( λ (cid:48) , t (cid:48) c ) , (3.10)where F def is the free energy of the deformed conifold 2.10. The proof of the corollary thereforeproceeds as in the latter case which is given in the appendix. We proceed with the discussion of the solutions of the difference equations. We begin byintroducing the Barnes G-function, which is defined by: G ( z + 1) = Γ( z ) G ( z ) , z ∈ C ,G (1) = G (2) = G (3) = 1 , d dz log G ( z ) ≥ , z > . (3.11)One of the equivalent forms to express the G-function is the Weierstrass canonical product,see [60]: G ( z + 1) = (2 π ) z exp (cid:18) − z + z (1 + γ )2 (cid:19) ∞ (cid:89) k =1 (cid:16) zk (cid:17) k exp (cid:18) z k − z (cid:19) , (3.12)where γ is the Euler-Mascheroni constant. The logarithm Barnes G-function has moreover thefollowing Taylor expansion around z = 0:log G ( z + 1) = 12 (log 2 π − z − (1 + γ ) z ∞ (cid:88) n =3 ( − n − ζ ( n − z n n , (3.13) s well as asymptotic expansion for z → ∞ .log G ( z + 1) = z (cid:18) log z − (cid:19) −
112 log z − zζ (cid:48) (0) + ζ (cid:48) ( − − ∞ (cid:88) g =2 B g g (2 g − z g − , (3.14)where ζ is the ζ − function. We define: F np ( λ, a ) := log G (cid:16) aλ (cid:17) + a λ log λ + aλ ζ (cid:48) (0) + 112 log( λ ) + ζ (cid:48) ( − , (3.15)and obtain the following: Proposition 3.3. F np ( λ, a ) is the unique solution to the difference equation for the free energiesof the deformed conifold with asymptotic behavior fixed by (2.10) .Proof. Using the functional equation of the Barnes G -function we obtain:log G (cid:18) a + λλ (cid:19) + log G (cid:18) a − λλ (cid:19) − G (cid:16) aλ (cid:17) = log Γ (cid:16) aλ + 1 (cid:17) − log Γ (cid:16) aλ (cid:17) = log aλ . (3.16)From the additional terms in (3.15), only the quadratic term in a contributes to the r.h.s. ofthe difference equation. We obtain: F np ( λ, a + λ ) − F np ( λ, a ) + F np ( λ, a − λ ) = log a = ∂ ∂a F ( a ) . (3.17)For a proof of the uniqueness of the results one may follow exactly the same reasoning as in[45]. The solution of the difference equation for the resolved conifold was identified in [45], by adaptingbuilding blocks of Bridgeland’s Tau function for the resolved conifold [34]. The special functionsin [34] involve the multiple sine functions which are defined using the Barnes multiple Gammafunctions [61]. For a variable z ∈ C and parameters ω , . . . , ω r ∈ C ∗ these are defined by:sin r ( z | ω , . . . , ω r ) := Γ r ( z | ω , . . . , ω r ) · Γ r (cid:32) r (cid:88) i =1 ω i − z | ω , . . . , ω r (cid:33) ( − r , (3.18)for further definitions, see e. g. [34, 62] and references therein. We introduce furthermore thegeneralized Bernoulli polynomials, defined by the generating function: x r e zx (cid:81) ri =1 ( e ω i x −
1) = ∞ (cid:88) n =0 x n n ! B r,n ( z | ω , . . . , ω r ) . (3.19) onsider now the function G ( z | ω , ω ) of [34, Sec. 4.2], defined by: G ( z | ω , ω ) := exp (cid:18) πi · B , ( z + ω | ω , ω , ω ) (cid:19) · sin ( z + ω | ω , ω , ω ) , (3.20)and define a function F np ( λ, t ) := log G ( t | ˇ λ, . (3.21)It was shown in [45] that F np is the unique solution of the difference equation with the boundarycondition: lim λ → λ F np ( λ, t ) = ˜ F ( t ) = Li ( e πit ) , and that moreover it has the following asymptotic expansion [34, 45]: F np ( λ, t ) ∼ ∞ (cid:88) g =0 λ g − ˜ F g ( t ) , (3.22)where ˜ F g ( t ) are the non-constant parts of the conifold free energies defined in (2.14). We start by discussing the non-perturbative content of topological string theory on the deformedconifold, whose structure is universal for topological strings near finite distance singularities asdiscussed in sec. 2.2. We therefore consider the Taylor series expansion of the solution of thedifference equation as λ → ∞ using the Taylor series expansion of the Barnes G-function, weobtain: Z np ( λ, t ) = exp( F np ( λ, t ))= exp (cid:18) ζ (cid:48) ( −
1) + 112 log( λ ) − aλ − (1 + γ + log λ ) a λ (cid:19) × exp (cid:32) ∞ (cid:88) n =3 ( − n − ζ ( n − a n λ n n (cid:33) . (4.1)It is expected that non-perturbative effects due to D-branes are signaled by a factor of e − /g s and effects due to NS5-branes come with factors of e − /g s [63, 64, 65], see also [66] and referencestherein as well as [42] for a discussion in the resurgence and topological string context, where g s refers to the physical string theory coupling under consideration. It is perhaps reasonableto expect a similar structure in topological string theory using the topological string coupling A subscript 3 is added here to G compared to [34] to avoid confusion with the Barnes G-function. We have used ζ (cid:48) (0) = − log 2 π . . The discussion of which solitonic branes correspond to the factors requires a distinctionbetween the A- and B-model which would see non-perturbative objects of type IIA or IIBstring theory respectively. Recall that the result of the free energies of the deformed conifold athand is a B-model result. Although its all-genus structure is very clear, the corresponding A-model geometry is perhaps less clear. It would be interesting however to understand the preciseconnection between the non-perturbative topological string theory content in this case and themetric obtained in [67] by smoothening the conifold singularity through instanton effects. Suchas smoothening was expected in [68].We proceed here with a discussion of the non-perturbative content of the strong couplingexpansion in the case of the explicit example of the quintic and its mirror in the next subsection. By the corollary 3.2, the same difference equation as for the deformed conifold also holds fortopological string theory on an arbitrary family of Calabi-Yau manifolds near a locus wherefinitely many states of the effective field theory become massless. To exemplify this we considerthe quintic Calabi-Yau threefold, whose definition and mirror construction are reviewed in theappendix B. The singular conifold locus of the quintic and its mirror has been studied in manyworks, starting with [69]. The expectation of the physical behavior of topological string theorynear this singularity was in particular used in [47] to supplement the polynomial solution [40]of the holomorphic anomaly equations [8] with boundary conditions.In this case the good special coordinate in the B-model side of mirror symmetry is given bya ratio of two solutions of the Picard-Fuchs equation (B.16), when the latter is considered inthe coordinate: δ = 1 − z z , the outcome of [47] is that: t c ( δ ) = δ − δ + O ( δ ) . (4.2)This special coordinate is interpreted as the mirror map and thus gives the mass of the objecton the A-model side, which becomes massless in this limit. The object becoming massless inthis case has D6 brane charge which was obtained by the analytic continuation studied in [69]and revisited in [47] and more recently in [70]. By the physical reasoning reviewed in sec. 2.2,it was thus expected in [47] that: F g ( t c ) = B g g (2 g − t g − c + O ( t c ) , g > , F ( t c ) = 12 t c log t c − t c + O ( t c ) , F ( t c ) = −
112 log t c + O ( t c ) . onsider now Λ ∈ C ∗ and the rescaling: λ (cid:48) = λ · Λ , t (cid:48) c = t c · Λ , and define: F (cid:48) ( λ (cid:48) , t (cid:48) c ) := lim Λ →∞ (cid:18) F ( λ (cid:48) , t (cid:48) c ) + ( t (cid:48) c ) −
112 log Λ (cid:19) , then all the ingredients of corollary 3.2 are met, and by the proof of that corollary F (cid:48) satisfiesthe difference equation 3.7. We can thus define the non-perturbative completion for topologicalstring theory on the quintic in the limit Λ → ∞ to be given by F np ( λ (cid:48) , t (cid:48) c ), where F np is definedin (3.15).For the strong coupling expansion we thus obtain: Z np ( λ (cid:48) , t (cid:48) ) = exp (cid:18) ζ (cid:48) ( −
1) + 112 log( λ (cid:48) ) − t (cid:48) c λ (cid:48) − (1 + γ + log λ ) ( t (cid:48) c ) λ (cid:48) ) (cid:19) × exp (cid:32) ∞ (cid:88) n =3 ( − n − ζ ( n −
1) ( t (cid:48) c ) n ( λ (cid:48) ) n n (cid:33) , (4.3)one may interpret the − /λ (cid:48) coefficient t (cid:48) c / t (cid:48) c ) is stillplausible as the volume in this limit we recall that the volume of the CY as determined bythe special geometry at large radius is typically given by a period of the mirror, which has theform : V ∼ F − tF t , in the limit t c → F ( t c ) − t c F t c = − t c , although geometric interpretations such as the volume become less clear once the large volumeregime of the moduli space is left. It would be interesting to interpret the higher order termsin this expansion. The strong coupling expansion for the non-perturbative free energy of the resolved conifold isobtained from the asymptotic expansion of log G , which is given in [34, Prop. 4.8], it is givenby: F np ( λ, t ) = − ζ (3)2 π ˇ λ − πi (cid:18) t − (cid:19) + πi ˇ λ (cid:18) t − t + t (cid:19) + ∞ (cid:88) k =2 B k ( t ) · B k − k ! · (2 πi ) 1ˇ λ k − , (4.4) See e. g. [4] and references therein. alid for for λ → ∞ and Im t > B n ( t ) are defined by thegenerating function: xe tx e x − ∞ (cid:88) n =0 B n ( t ) x n n ! , (4.5)this expansion leads to the following: Remark 4.1. • It is interesting to note that the asymptotic expansion for λ → q = exp(2 πit ) expansion,hence corresponds to a large volume expansion. The asymptotic expansion for λ → ∞ however is expressed in terms of t and is moreover manifestly polynomial! in t for everyorder in λ . This suggests that λ and t are not entirely independent expansion variablesbut should perhaps be thought to correspond to certain phases of the combined problemin λ and t , in analogy to the phases of [71]. • Although the volume of the resolved conifold is infinite since it is a non-compact CY, it isinteresting to observe that the λ term in the expansion comes with a factor t , which isthe classical CY volume expressed by the special geometry. It is then natural to speculatethat this term corresponds to the NS5 brane contribution, to obtain a negative sign, afactor of 1 / (2 πi ) should be included in the volume identification. For the resolved conifold we analyze the non-perturbative content of the solution of the differenceequation. We obtain the following:
Proposition 4.2. F np ( λ, t ) can be expressed as: F np ( λ, t ) = ∞ (cid:88) k =1 q k k (2 sin kλ/ − ˇ λ ∂∂ ˇ λ (cid:32) ˇ λ π ∞ (cid:88) k =1 e πik ( t − / / ˇ λ k sin( πk/ ˇ λ ) (cid:33) , ˇ λ = λ π . (4.6) Moreover, this expression can be written as: F np ( λ, t ) = F GV ( λ, t ) − ˇ λ ∂∂ ˇ λ (cid:0) ˇ λ F NS (1 / ˇ λ, ( t − / / ˇ λ ) (cid:1) , (4.7) where F GV ( λ, t ) = ∞ (cid:88) k =1 q k k (2 sin kλ/ , (4.8) is the Gopakumar-Vafa resummation of (2.6) for the resolved conifold and F NS ( (cid:126) , t ) = 14 π ∞ (cid:88) k =1 e πikt k sin( πk (cid:126) ) , (4.9) is the refined topological string free energy for the resolved conifold in the Nekrasov-Shatashvililimit as determined in [30, 29] following [23]. ×××× · πi · πi · πi · πi × × × × × · πi/ ˇ λ · πi/ ˇ λ · πi/ ˇ λ · πi/ ˇ λ Figure 1: Illustration of the simple and double poles as well as the contour C . Proof.
To prove this, we use the integral representation of the function G , discussed in [34,Prop. 4.2], based on [72, Prop. 2] obtaining: F np ( λ, t ) = − (cid:90) C e ( t +ˇ λ ) s ( e s − e ˇ λs − dss , (4.10)which is valid for Reˇ λ > − Reˇ λ < Re t < Re(ˇ λ + 1) and the contour C is following the realaxis from −∞ to ∞ avoiding 0 by a small detour in the upper half plane.To find the series expression in λ including the perturbative and non-perturbative piecesfrom the integral representation we close the contour in the upper half plane and analyze theresidues. In the upper half plane without zero, the integrand has two infinite sets of poles givenby: s = 2 πik , and s = 2 πik ˆ λ k ∈ N \ { } , (4.11)the first set corresponds to simple poles and the second set corresponds to double poles. Toavoid higher order poles we may assume that either Im λ (cid:54) = 0 or that ˇ λ / ∈ Q . The contribution of the simple poles gives the first factor of the r.h.s. of the proposition.We denote the contribution of the double poles by I db and obtain from Cauchy’s generalizedintegral formula: I db = 2 πi · ∞ (cid:88) k =1 dds f ( s ) | s =2 πik/ ˇ λ , (4.12) We note that there is nothing wrong with ˆ λ ∈ Q , it just requires a separate analysis of the poles of order three. here f ( s ) = − e ts s ( e s − , which gives: I db = − ˇ λ π ∞ (cid:88) k =1 (1 − πikt/ ˇ λ ) e πik ( t − / / ˇ λ k sin( πk/ ˇ λ ) − ˇ λ ∞ (cid:88) k =1 e πikt/ ˇ λ k (2 sin( πk/ ˇ λ )) , (4.13)the reformulation of the result follows after some substitution algebra. Remark 4.3. • We note that the comparison of (4.7) to the result of eq. (5.6) of [29] whichwas obtained by a generalized Borel resummation there is a difference of a relative factorof ˇ λ between the two contributions, it would be interesting to understand the source ofthis discrepancy. • There are differences of various relative factors of 2 πi compared to [29]. These are lessworrisome since they originate mostly from our definition of q = e πit (opposed to q = e − t ,which is often used). Including the 2 πi factors corresponds to the natural choice of variablegiving convergence of q series at large volume Im t → ∞ as well as having the periodicityin shifts of the B − field t → t + 1. Indeed, this makes the 1 / B -field shift in theargument of F NS , which was discussed in [23, 30, 29] very clear. • Since this computation reproduces results which mostly agree with [23], one may consult[23, Sec. 4.2] for comments and comparison to the proposal of [28]. • A similar computation using integral representations occurring in Chern-Simons theorywas done in [31].We conclude this subsection by noting that the special function determined by the differenceequation does indeed contain the non-perturbative structure of the resolved conifold which wasobtained by various different methods in [30, 29, 31]. This in particular confirms the expectationof [34], that the Tau function of the Riemann-Hilbert problem associated to wall-crossing ofDT invariants of the resolved conifold defines a non-perturbative topological string partitionfunction.
With the non-perturbative topological string partition function at hand one can revisit therelation between topological strings and the generating function of non-commutative Donaldson-Thomas invariants studied in [73]. We introduce therefore: Z NCDT ( λ, t ) := M ( q λ ) ∞ (cid:89) k =1 (cid:16) − e − πit q kλ (cid:17) k ∞ (cid:89) k =1 (cid:16) − e πit q kλ (cid:17) k , (4.14) Compared to [73], q λ has a different sign and 2 πi factors are added to t . here the MacMahon function is given by: M ( q ) = ∞ (cid:89) k =1 (1 − q k ) − k , (4.15)and q λ = e iλ . Defining the non-perturbative topological string free partition function: Z np ( λ, t ) := exp( F np ( λ, t )) , (4.16)as the exponential of (3.21), we obtain the following: Proposition 4.4.
The relation between the generating function of non-commutative DT invari-ants (4.14) and the non-perturbative topological string partition function (4.16) is given by: Z NCDT ( λ, t ) = M ( q λ ) · Z np ( λ, t + 1) · Z np ( − λ, − t ) , (4.17) Proof.
We consider the following property, proved in [34, Prop. 4.3] for the function G definedin (3.20): G ( z + ω | ω , ω ) · G ( z | ω , − ω ) = ∞ (cid:89) k =1 (1 − x q k ) k · ∞ (cid:89) k =1 (1 − x − q k ) k , (4.18)valid for Im( ω /ω ) > z ∈ C , where: x = exp(2 πiz/ω ) , q = exp(2 πiω /ω ) . The claim of the proposition follows by considering the property for G ( t | ˇ λ,
1) which defined F np and further noting that the function G is invariant under simultaneous rescaling of allthree arguments, see e. g. [34, Prop. 4.2]. Remark 4.5.
We remark that relations between Z NCDT and the partition function for theresolved conifold obtained by exponentiating the GV resummation were already observed in[73] and interpreted physically in e. g. [74, 75]. The result obtained here is however exact andnon-perturbative and therefore perhaps a little more surprising since proposition 4.3.2 showsthat the non-perturbative free energy contains more information than the GV resummation.
A natural question to ask is about the precise relation between the free energies of the resolvedand deformed conifold geometries. Relatedly, the free energy of the resolved conifold should bethe easiest example to explicitly test the universal behavior of topological string theory nearconifold type singularities. Since the resolved conifold corresponds to the small resolution of theconifold singularity, the singular locus corresponds to the locus where t , the K¨ahler parameterof the P shrinks to zero t →
0. Since the free energy of the resolved conifold (2.14) is given asa series expansion in q = e πit , which does not converge as t → heorem 4.6. For g > the analytic continuation of the genus g free energy of the resolvedconifold to the regime t → is given by: ˜ F g ( t ) = B g g (2 g −
2) 1(2 πt ) g − + ( − g B g B g − g (2 g − g − B g g (2 g − ∞ (cid:88) d = g ( − d +1 B d d (2 d − g + 2)! (2 πt ) d − g +2 . (4.19) Proof.
We begin by representing ˜ F g ( t ) for the resolved conifold as in (A.7):˜ F g ( t ) = ( − g − B g g (2 g − θ g − q Li ( q ) , g ≥ , (4.20)starting from Li ( q ) = q (1 − q ) and using the property: θ q Li s ( q ) = Li s − ( q ) , θ q := q ddq . (4.21)We introduce a = 2 πit , the expression becomes:˜ F g ( a ) = ( − g B g g (2 g − ∂ g − a (cid:18) a (cid:18) ae a − (cid:19)(cid:19) = ( − g B g g (2 g − ∂ g − a (cid:32) ∞ (cid:88) n =0 B n a n − n ! (cid:33) = ( − g B g g (2 g − a g − + ( − g B g B g − g (2 g − g − − g − B g g (2 g − ∞ (cid:88) d = g B d d (2 d − g + 2)! a d − g +2 , (4.22)where in the first and second line the expression Li ( q ) was brought into the form of thegenerating function of Bernoulli numbers. The result on the third line follows from consideringseparately the differential operator acting on the singular piece 1 /a of the r.h.s., the constantpiece comes from the differential operator acting on the term with power a g − , the rest iscollecting all the higher powers. We have furthermore used the vanishing of B n +1 , n >
0. Thestatement of the theorem follows by substituting back a = 2 πit . Remark 4.7. • This result is a mathematical proof of the gap condition used in [46, 47],based on the physical expectation outlined in sec. 2.2, as a condition imposed on the lo-cal behavior of topological string theory on arbitrary families of CY manifolds. In thecase at hand the result does not serve any computational purpose since the full exact and on-perturbative expression for topological string theory was already given, it is neverthe-less gratifying to see the conjectured behavior to hold mathematically rigorously in thisexample. • We think the result should have an interesting interpretation in enumerative geometry. Theleading singular term gives the Euler characteristic of genus g Riemann surfaces provedin [76], in the constant term, the contributions from constant maps determined in [48]make a somewhat surprising reappearance, although we had explicitly not included thesefrom the start. It is therefore natural to expect also the higher degree terms to have aninterpretation. This may open up new paths to study Gromov-Witten theory around theconifold point of arbitrary families of threefolds. See also [73] for speculation about theenumerative geometry content of the locus t → λ → i ∞ . • From a physical perspective, the singular behavior signals a hypermultiplet becomingmassless in the effective four dimensional theory in the limit t →
0. It is known that theresolved conifold supports only one BPS state from the M-theory perspective, this corre-sponds to the only non-vanishing GV invariant n = 1. However from the 4 d perspectivethere is an infinite set of BPS states supported on this geometry corresponding to theKaluza-Klein modes of the M D brane charge bound tothe D brane. It seems that the singular behavior of the topological string does not showthis tower BPS states in this limit. This could partly be due to D brane contributionsappearing separately on top of the topological string partition function as is clear fromthe relation between Z DT and Z top which differs by powers of the MacMahon function,see e. g. [13, 75]. It could also signal some wall-crossing phenomena which happen beforereaching the t → D state becomes massless in the limit.To fully test the universal behavior of topological string theory near a conifold singularityin this example, we need to analyze furthermore the behavior of the genus 0 and genus 1 freeenergies.We obtain the following: Proposition 4.8.
The analytic continuation of the genus 0 free energy (the prepotential) ˜ F ( t ) = Li ( q ) to the region t → is given by: ˜ F ( t ) = (2 π ) (cid:18) t πit ) − t (cid:19) − (2 πi ) t − ∞ (cid:88) n =1 B n n (2 n + 2)! ( − n +1 (2 πt ) n +2 . (4.23) Proof.
Since the prepotential ˜ F ( t ) is a locally defined function, we need to analytically continueit starting from functions which are geometrically globally defined. At genus zero such an object s the Yukawa coupling C ttt := 1(2 πi ) ∂ ∂t ˜ F ( t ) = θ q Li ( q ) = q − q , introducing again a = 2 πit , we thus have: C ttt = ∂ a ˜ F ( a ) = − − a (cid:18) aq − (cid:19) = − − ∞ (cid:88) n =0 B n a n − n ! , (4.24)which can be integrated to give:˜ F ( a ) = − (cid:18) a a − a (cid:19) − a − ∞ (cid:88) n =1 B n n (2 n + 2)! a n +2 . (4.25)The statement of the proposition follows by substituting a = 2 πit .For the genus 1 free energy we have:˜ F ( t ) = −
112 log(1 − q ) = −
112 log(2 πit ) −
112 log (cid:32) ∞ (cid:88) n =2 (2 πit ) n − n ! (cid:33) , (4.26)which shows that the assumptions of corollary 3.2 are met. We can now consider Λ ∈ C ∗ andthe rescaling: λ (cid:48) = λ · Λ , t (cid:48) = t · Λ , and define: F (cid:48) ( λ (cid:48) , t (cid:48) ) := lim Λ →∞ (cid:18) F ( λ (cid:48) , t ) + t (cid:18) Λ2 πi (cid:19) −
112 log (cid:18) Λ2 πi (cid:19)(cid:19) , one fine F (cid:48) ( λ (cid:48) , t (cid:48) ) = F def ( ˇ λ (cid:48) , t (cid:48) ) , ˇ λ (cid:48) = λ π , where F def is the free energy of the deformed conifold 2.10. The different normalization of λ isalso the reason for the 2 πi factors on the cutoff Λ. In this work, we addressed an intrinsic characterization of non-perturbative topological stringtheory. Hereby the topological string free energy is a solution to a difference equation mixingthe moduli and the string coupling. The difference equations are derived using the knowledgeof the asymptotic expansion and bypass any further physical or mathematical dualities. The This definition using only the q expansion part of F ( t ) misses some classical pieces which would correspond tothe classical triple intersection numbers. ifference equations as well as their solutions were obtained for the deformed and resolvedconifold geometries. The strong coupling expansion as well as the non-perturbative contentwere obtained in these cases and matched to known and expected results in the literature.Fortunately, the deformed conifold also gives the expected universal behavior of topologicalstring theory on any family of Calabi-Yau threefolds near a singularity where finitely manystates of the effective field theory in 4 d become massless. Both the difference equation as wellas its analytic solution in the string coupling can thus be obtained universally in a limit wherethe coordinate giving the mass of the particles becoming massless at the singularity as well asthe topological string coupling are rescaled and the rescaling is sent to ∞ .Although the full exact solution is lost in the rescaled limit, the remaining qualitative resultsare universal and should shed new lights on contexts where topological string theory connectsto problems of quantum gravity and thus requires the study of topological string theory onfamilies of compact CY manifolds. One such problem is the connection to black hole partitionfunctions [11]. It would for example be interesting to match the universal non-perturbativestructure which also holds for compact CY to computations of the black hole partition function,perhaps along the lines of [77]. Another quantum gravity context, where the precise knowledgeof the geometry of CY moduli spaces and their non-perturbative content is important is theswampland program which has attracted a lot of attention recently. We expect the quantitativehandle on the universal non-perturbative structure of topological strings to be relevant forexample in the context of the swampland distance conjecture [78]. Relatedly, one may alsoexpect the non-perturbative structure of topological string theory to be relevant in the study ofnon-perturbative corrections to quaternionic K¨ahler geometries [67, 66, 79].We note that difference equations provide the natural arena to study the connections be-tween topological strings and integrable hierarchies. This was indeed the context for [18]. Thedifference equations there were obtained from the quantization of the mirror curves, which areexpressed in terms of C ∗ variables. The difference equations and their solutions in this contextconnect however naturally to open topological string theory [18, 19]. The difference equationsstudied in this work are equations in the closed string moduli, they are closer in spirit to thedifference equation conjectured for instance in [80] for the Gromov-Witten potential of P andwhich provides the link to the Toda integrable hierarchy. Indeed, using the difference equa-tion for the resolved conifold, a conjecture [81] relating Gromov-Witten theory of the resolvedconifold to the Ablowitz-Ladik integrable hierarchy was proved in [45]. It is tempting to spec-ulate that there should be an underlying quantum mechanical problem that gives rise to thedifference equations studied here. The latter should be a quantization problem related to themoduli space of closed strings of the geometry, which may close the circle of ideas by relating itto the quantum mechanical interpretation of the closed string as a wave-function [9]. RecentlyQuantum K-theory was studied in e. g. [82] as a quantum deformation problem which can be I would like to thank Gabriel Lopes Cardoso for pointing this out, commenting on the potential use of [39] in thestudy of black holes. pplied to the Picard-Fuchs operators of any Calabi-Yau geometry, it would be interesting tounderstand potential links to the approach in this paper. A further context where the differenceequations studied in this work appear is as the equations characterizing the perturbative partof the Nekrasov-Okounkov partition functions of N = 2 gauge theories, see [83, Appendix A].The current work suggests that in addition to the perturbative piece, the difference equationsalso fix the non-perturbative content. It would be interesting to study the implications of thisfurther both in the gauge theory as well as in more general contexts. Acknowledgements
I would like to thank Arpan Saha for collaboration on related projects as well as Florian Beck andPeter Mayr for comments on the draft. I have benefited from many discussions with membersof the Emmy Noether research group on String Mathematics as well as from discussions withVicente Cort´es, J¨org Teschner, Iv´an Tulli, Timo Weigand and Alexander Westphal on relatedprojects within the quantum universe cluster of excellence. This work is supported through theDFG Emmy Noether grant AL 1407/2-1.
A Proof of the difference equation
We provide here the proof of [43, 44] of the difference equations for the free energies of thedeformed and the resolved conifold.
Proof.
The proof of the theorems relies on the following. Consider the generating function ofBernoulli numbers: we w − ∞ (cid:88) n =0 B n w n n ! . (A.1)Applying w ddw to both sides and rearranging gives: w e w ( e w − = B − ∞ (cid:88) n =2 B n n ( n − w n = 1 − ∞ (cid:88) g =1 B g g (2 g − w g , (A.2)where the last equality is obtained by noting that all B n +1 , n ∈ N \ { } vanish. This yields thefollowing: ( e w − e − w ) w − ∞ (cid:88) g =1 B g g (2 g − w g − = 1 . (A.3)In the next step, we replace on both sides of this equation the variable w by an operatoracting on functions of t namely: w → λ ∂∂a or the deformed conifold, we act with both sides on log a , obtaining:( e λ∂ a − · id + e − λ∂ a ) λ − ∂ − a − ∞ (cid:88) g =1 B g g (2 g − λ g − ∂ g − a log a = id · log a , (A.4)we use ∂ g − a log a = − (2 g − a − g , and interpret ∂ − a as an anti-derivative to obtain:( e λ∂ a − · id + e − λ∂ a ) F ( a ) −
112 log a + ∞ (cid:88) g =2 B g g (2 g − λ g − a g − = log a = ∂ ∂a F ( a ) . (A.5)For the resolved conifold we start from Li ( q ) = − log(1 − q ) and use the property: θ q Li s ( q ) = Li s − ( q ) , θ q := q ddq , (A.6)we write ˜ F g = ( − g − B g g (2 g − θ g − q Li ( q ) , g ≥ . (A.7)We make the replacement: w → ˇ λ ∂∂t = iλθ q . Acting with both sides on Li ( q ) we obtain:( e ˇ λ∂ t − · id + e − ˇ λ∂ t ) − λ − θ − q − ∞ (cid:88) g =1 ( − g − B g g (2 g − λ g − θ g − q Li ( q ) = id · Li ( q ) , (A.8)by using θ q ˜ F = θ q Li ( q ) = Li ( q ) and interpreting θ − q as an anti-derivative, we obtain:( e ˇ λ∂ t − · id + e − ˇ λ∂ t ) ˜ F ( λ, t ) = − θ q ˜ F ( q ) , (A.9)which proves the theorem.The following corollary was proved in [44] but also holds for the deformed conifold case sowe inlude it here as well: Corollary A.1.
For every g ≥ the difference equation gives a recursive differential equationwhich determines ∂ ∂t F g ( t ) g ≥ by: g (cid:88) k =0 g − k + 2)! (cid:18) π ∂∂t (cid:19) g − k +2 F k ( t ) = 0 , g ≥ . (A.10) Proof.
This follows from expanding the L.H.S. of the theorem in λ and then matching thecoefficients of λ g on both sides. The quintic and its mirror
In the following the quintic and its mirror are reviewed. The exposition follows [4], more detailscan be found in [69, 84]. The quintic X denotes the CY manifold defined by X := { P ( x ) = 0 } ⊂ P , (B.1)where P is a homogeneous polynomial of degree 5 in 5 variables x , . . . , x . The mirror quin-tic ˇ X can be constructed using the Greene-Plesser construction [85]. Equivalently it may beconstructed using Batyrev’s dual polyhedra [86] in the toric geometry language . In the Greene-Plesser construction the family of mirror quintics is the one parameter family of quintics definedby { p ( ˇ X ) = (cid:88) i =1 x i − ψ (cid:89) i =1 x i = 0 } ∈ P , (B.2)after a ( Z ) quotient and resolving the singularities.In the following, the mirror construction following Batyrev will be outlined. The mirrorpair of CY 3-folds ( X, ˇ X ) is given as hypersurfaces in toric ambient spaces ( W, ˇ W ). The mirrorsymmetry construction of Ref. [86] associates the pair ( X, ˇ X ) to a pair of integral reflexivepolyhedra (∆ , ˇ Delta ). B.1
The A-model geometry
The polyhedron ∆ is characterized by k relevant integral points ν i lying in a hyperplane ofdistance one from the origin in Z , ν will denote the origin following the conventions of Refs. [86,89]. The k integral points ν i (∆) of the polyhedron ∆ correspond to homogeneous coordinates u i on the toric ambient space W and satisfy n = h , ( X ) linear relations: k − (cid:88) i =0 l ai ν i = 0 , a = 1 , . . . , n . (B.3)The integral entries of the vectors l a for fixed a define the weights l ai of the coordinates u i underthe C ∗ actions u i → ( λ a ) l ai u i , λ a ∈ C ∗ . The l ai can also be understood as the U (1) a charges of the fields of the gauged linear sigmamodel (GLSM) construction associated with the toric variety [71]. The toric variety W is definedas W (cid:39) ( C k − Ξ) / ( C ∗ ) n , where Ξ corresponds to an exceptional subset of degenerate orbits.To construct compact hypersurfaces, W is taken to be the total space of the anti-canonicalbundle over a compact toric variety. The compact manifold X ⊂ W is defined by introducinga superpotential W X = u p ( u i ) in the GLSM, where x is the coordinate on the fiber and For a review of toric geometry see Refs. [87, 88]. ( u i ) a polynomial in the u i> of degrees − l a . At large K¨ahler volumes, the critical locus is at u = p ( u i ) = 0 [71].The quintic is the compact geometry given by a section of the anti-canonical bundle over P . The charge vectors for this geometry are given by: u u u u u u l = ( − . (B.4)The vertices of the polyhedron ∆ are given by: ν = (0 , , , , , ν = (1 , , , , , ν = (0 , , , , ,ν = (0 , , , , , ν = (0 , , , , , ν = ( − , − , − , − , . (B.5) B.2
The B-model geometry
The B-model geometry ˇ X ⊂ ˇ W is determined by the mirror symmetry construction of Refs. [90,86] as the vanishing locus of the equation p ( ˇ X ) = k − (cid:88) i =0 a i y i = (cid:88) ν i ∈ ∆ a i X ν i , (B.6)where a i parameterize the complex structure of ˇ X , y i are homogeneous coordinates [90] onˇ W and X m , m = 1 , . . . , C ∗ ) ⊂ ˇ W and X ν i := (cid:81) m X ν i,m m [91]. The relations (B.3) impose the following relations on the homogeneouscoordinates k − (cid:89) i =0 y l ai i = 1 , a = 1 , . . . , n = h , ( ˇ X ) = h , ( X ) . (B.7)The important quantity in the B-model is the holomorphic (3 ,
0) form which is given by:Ω( a i ) = Res p =0 p ( ˇ X ) (cid:89) i =1 dX i X i . (B.8)Its periods π α ( a i ) = (cid:90) γ α Ω( a i ) , γ α ∈ H ( ˇ X ) , α = 0 , . . . , h , + 1 , (B.9)are annihilated by an extended system of GKZ [92] differential operators L ( l ) = (cid:89) l i > (cid:18) ∂∂a i (cid:19) l i − (cid:89) l i < (cid:18) ∂∂a i (cid:19) − l i , (B.10) Z k = k − (cid:88) i =0 ν i,j θ i , j = 1 , . . . , . Z = k − (cid:88) i =0 θ i + 1 , θ i = a i ∂∂a i , (B.11) here l can be a positive integral linear combination of the charge vectors l a . The equation L ( l ) π ( a i ) = 0 follows from the definition (B.8). The equations Z k π α ( a i ) = 0 express theinvariance of the period integral under the torus action and imply that the period integrals onlydepend on special combinations of the parameters a i π α ( a i ) ∼ π α ( z a ) , z a = ( − ) l a (cid:89) i a l ai i , (B.12)the z a , a = 1 , . . . , n define local coordinates on the moduli space M of complex structures ofˇ X . The charge vector defining the A-model geometry in Eq. (B.4) gives the mirror geometrydefined by: p ( ˇ X ) = (cid:88) i =0 a i y i = 0 , (B.13)where the coordinates y i are subject to the relation y y y y y = y . (B.14)Changing the coordinates y i = x i , i = 1 , . . . , ψ − = − a a a a a a =: z . (B.15)Furthermore, the following Picard-Fuchs (PF) operator annihilating ˜ π α ( z i ) = a π α ( a i ) isfound: L = θ − z (cid:89) i =1 (5 θ + i ) , θ = z ddz . (B.16)The discriminant of this operator is ∆ = 1 − z . (B.17)and the Yukawa coupling can be computed: C zzz = 5 z ∆ . (B.18)The PF operator gives a differential equation which has three regular singular points whichcorrespond to points in the moduli space of the family of quintics where the defining equationbecomes singular or acquires additional symmetries, these are the points: • z = 0 , the quintic at this value corresponds to the quotient of (cid:81) i =1 x i = 0 which isthe most degenerate Calabi-Yau and corresponds to large radius when translated to theA-model side. z = 5 − this corresponds to a discriminant locus of the differential equation (B.16) and alsoto the locus where the Jacobian of the defining equation vanishes. This type of singularityis called a conifold singularity. • z = ∞ , this is known as the Gepner point in the moduli space of the quintic and itcorresponds to a non-singular CY threefold with a large automorphism group. This isreflected by a monodromy of order 5. Bibliography [1] A. Neitzke and C. Vafa, “Topological strings and their physical applications,” arXiv:hep-th/0410178 .[2] M. Marino,
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