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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 438648, 47 pages
http://dx.doi.org/10.1155/2008/438648
Research Article

Quantum Barnes Function as the Partition Function of the Resolved Conifold

Department of Mathematics, Northwestern University, Evanston, IL 60208, USA

Received 3 July 2008; Accepted 15 December 2008

Academic Editor: Alberto Cavicchioli

Copyright © 2008 Sergiy Koshkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a short new proof of large duality between the Chern-Simons invariants of the 3-sphere and the Gromov-Witten/Donaldson-Thomas invariants of the resolved conifold. Our strategy applies to more general situations, and it is to interpret the Gromov-Witten, the Donaldson-Thomas, and the Chern-Simons invariants as different characterizations of the same holomorphic function. For the resolved conifold, this function turns out to be the quantum Barnes function, a natural -deformation of the classical one that in its turn generalizes the Euler gamma function. Our reasoning is based on a new formula for this function that expresses it as a graded product of -shifted multifactorials.

1. Introduction

What is the topological string partition function of the resolved conifold? We should explain that heuristically one can assign string theories to each Calabi-Yau threefold and some of them such as topological A-models [1], only depend on its Kähler structure. Their topologically invariant amplitudes are then collected into a generating function called the partition function. Remarkably, this partition function may remain unchanged even if a threefold undergoes a topology changing transition [2].

A traditional approach is to interpret the string partition function as the Gromov-Witten partition function. For the resolved conifold it was originally computed by Faber-Pandharipande [3]; see also [4]Here, , , and , are known as the Kähler parameter and the string coupling constant, respectively. In mathematical terms, they are just formal variables andwhere is the Gromov-Witten invariant of genus degree holomorphic curves in the resolved conifold.

The incompleteness of this answer does not reveal itself until one considers dualities that relate Gromov-Witten invariants to other invariants of Calabi-Yau threefolds. One may notice that (1.2) is missing degree zero terms (hence the ). This is not a slip, they cannot be packaged into a form as nice as (1.1). This was not considered much of a problem until the Donaldson-Thomas theory [57] came about, since degree zero (constant) maps are trivial anyway. But apparently dualities have little tolerance for convenient omissions. For the Gromov-Witten/Donaldson-Thomas duality to hold, (1.1) has to be augmented aswhereis the MacMahon function, classically known as the generating function of plane partitions [8]. In all honesty, this is not quite true as has some spurious terms in its expansion at and only accounts for genus terms correctly (see Section 3). Also in the Donaldson-Thomas theory, one has , not . In a recent reformulation of the Donaldson-Thomas theory [9], the reduced partition function is even defined directly, and the MacMahon function is banished altogether. Let us disregard this minor discrepancy for now since even answer (1.3) is incomplete.

This becomes apparent in light of another duality of the Calabi-Yau threefolds, large duality. This one relates the Gromov-Witten invariants of the resolved conifold to the Chern-Simons invariants of the -sphere. The usual formulation defines the Chern-Simons theory as a gauge theory on a or bundle over a real -manifold . Less recognized despite the Witten famous paper [1] is the fact that it also gives invariants of the Calabi-Yau threefolds. As Witten pointed out, it can be viewed as a theory of open strings (holomorphic instantons at in his terminology) in the cotangent bundle ending on its zero section. is canonically a symplectic manifold (even Kähler if is real-analytic) with first Chern class , that is, the Calabi-Yau. In particular, is diffeomorphic to a quadric in . One of the reasons this interpretation did not get much currency is that the strings in question are very degenerate, they are represented by ribbon graphs, and are not honest holomorphic curves. In fact, there are no honest holomorphic curves in at all except for the constant ones [1, 10]. Another reason, perhaps, is that open the Gromov-Witten theory is still in its infancy and the powerful algebro-geometric techniques that dominate the field cannot be directly applied. There are successful approaches that replace open invariants with relative ones [11, 12] but only as a tool for computing closed invariants. In the other direction, there exists a detailed if only formal correspondence between geometry of real-oriented -manifold and the Calabi-Yau threefolds and the Donaldson-Thomas theory can be seen as a “holomorphization” of the Chern-Simons theory under this correspondence [13]. Thus, comparing the Chern-Simons partition function to promises some useful insight.

Once again, by ignoring some irrelevant prefactors, can be written as , where so that , andis the classical Euler generating function of ordinary partitions. At this point, it is appropriate to introduce notation that allows one to write and uniformly. Letbe the q-multifactorials then (see Section 6),Using and as variables, we see that After some thought one may sense a pattern here. We will see in Section 6 that it makes sense to join one more factor to the product and considerThis is the quantum Barnes function of Nishizawa [14], and our candidate for the partition function of the resolved conifold. All factors above are required to make it transforms aswhere is the Jackson quantum gamma function deforming the classical one. This in turn satisfies with the so-called quantum number. This makes a deformation of the classical Barnes function that satisfies (1.10) with -s removed.

The picture above is cute but not quite true, and clear-cut identities (1.8) are spoiled by pesky disturbances discussed in Sections 3 and 5. These disturbances are a large part of the reason why large duality is so hard to prove even in simple cases. Still, emerges as a common factor in the Gromov-Witten, the Donaldson-Thomas, and the Chern-Simons theories (Theorem 5.2). One may notice that we conspicuously omitted the most famous of the Calabi-Yau dualities, mirror symmetry. This is partly because local mirror symmetry is poorly developed, and partly because to the extent that its predictions can be divined [15] they match the Gromov-Witten ones completely. There is a structural prediction of mirror symmetry that seems relevant. For compact Calabi-Yau threefolds, is predicted to have modular properties [16], that is, obey transformation laws under and . For open threefolds like the resolved conifold, only the first one survives and is expressed by (1.10).

What are we to make of the above chain of augmentations? Perhaps, string theories on the Calabi-Yau threefolds are only partial reflections of some hidden master-theory. The Witten candidate for such a theory is the mysterious M-theory living on a seven-dimensional manifold with holonomy that projects to various string theories on the Calabi-Yau threefolds. Another unifying view of the Gromov-Witten and the Donaldson-Thomas theories, via noncommutative geometry, also emerged recently [17]. Different projections are equivalent even though they may live on topologically distinct threefolds and reflect the original each in its own way. So far, we ignored these ways relying instead on magical changes of variables. It is time to dwell upon them a bit. This will also serve as our justification for spending so much ink on the resolved conifold.

The relation between the Gromov-Witten and the Donaldson-Thomas invariants is very simple [6, 7]. For the resolved conifold, we havewith the same as in (1.2) and the Donaldson-Thomas invariants. In other words, in each degree, are simply the Taylor coefficients of at while are the Laurent coefficients in corresponding to with . The relation with the Chern-Simons invariants is more complicated. Traditionally, one has to take , where are the two parameters of the Chern-Simons theory, rank and level. They are positive integers making a root of unity. Not all roots of unity are covered in this way, but more sophisticated formulations allow one to include any root of unity. Naively, if the duality conjectures hold the Donaldson-Thomas invariants give us an expansion at , the Gromov-Witten invariants at and the Chern-Simons invariants give values at roots of unity of more or less the same function, but only naively. First of all, the Donaldson-Thomas generating functions are a priori only formal power series and may not have a positive radius of convergence. We need it to be at least to make a comparison. Things are nice in higher degree [9], but in degree zero it is exactly and every point of the unit circle is a singularity. This is remedied easily enough in the Gromov-Witten context since we can interpret as an asymptotic expansion at the natural boundary (Section 3). But the Chern-Simons invariants are not graded by degree, and the degree zero speck turns into a wooden beam spoiling the whole partition function that we wish to evaluate. With the resolved conifold being the simplest nontrivial case, we get a preview of the difficulties that will arise in general. This brings us to a paradox: for large duality to even make sense, the formal power series better converges to holomorphic functions extending to the unit circle or at least to roots of unity. This is not the case already for and an additional factor in appearing in (1.8) is needed to make it happen (see comments after Corollary 6.5). This is another reason to accept the quantum Barnes function as the completed partition function.

Since conjecturally for any Calabi-Yau threefold [6, 7] this phenomenon is likely to be general. The above discussion suggests that the master-invariant that manifests itself through dualities is a holomorphic function on the unit disk. The three theories we discussed showcase three different ways to package information about it. The dualities reduce to repackaging prescriptions. Physicists developed resummation techniques that transform generating functions one into another but they lead to unwieldy computations for the resolved conifold and do not produce conclusive results even for its cyclic quotients [18]. Since repackaging involves transcendental substitutions, analytic continuation and asymptotic expansions—things one does with functions and not with formal series—it makes sense to identify the underlying holomorphic functions to establish a duality. This is the strategy of this paper and it distinguishes it from previous approaches [2, 19, 20] that use double expansions in genus and degree. This makes for a cleaner comparison of partition functions with a clear view of what matches and what does not match in them (Theorem 5.2). It is also hoped that the idea generalizes to other threefolds.

The paper is organized as follows. Section 2 is a review of basic notions of the Gromov-Witten theory with emphasis on generating functions. In particular, we note that free energy is a shorthand for the Gromov-Witten potential restricted to divisor invariants. The well-known irregularities in degree zero are then naturally explained. In Section 3, the MacMahon function is examined in detail to determine to what extent it can be viewed as the degree zero partition function of the Gromov-Witten invariants. We describe resummation techniques used by physicists, and then recall an old but little-known asymptotic for it due to Ramanujan and Wright adapting it to our context. Sections 4 and 5 give a description of the topological vertex and the Reshetikhin-Turaev calculus, diagrammatic models that compute the Gromov-Witten and the Chern-Simons partition functions, respectively. Similarities between the two are specifically stressed. Section 5 ends by expressing both partition functions via the quantum Barnes function (Theorem 5.2). Since this function and its higher analogs are relatively recent (1995), we give a self-contained exposition of their theory in Section 6 different from the author's [14]. In particular, we prove the alternating formula (1.9) that connects to the Calabi-Yau partition functions and appears to be new (Theorem 6.3). In Conclusions, we point out the relations between the Calabi-Yau dualities and holography, and share some thoughts and conjectures inspired by the resolved conifold example. The appendix lists basic properties of the Stirling polynomials needed in Section 6.

2. Generating Functions of Gromov-Witten Invariants

There are a variety of generating functions appearing in the literature: the Gromov-Witten potential, prepotential, truncated potential, partition function, free energy, and so forth. In this section, we briefly review basic definitions from the Gromov-Witten theory and relationships among some of the above generating functions. Perhaps, the only unconventional notion is that of divisor potential which leads most naturally to the free energy and the partition function.

Stable Maps
Let be the Kähler manifold of complex dimension . We wish to consider holomorphic maps of the Riemann surfaces with marked points into that realize certain homology class . The space of such maps is denoted . There is a natural (Gromov) topology on this moduli space but it is not compact in it. To get the Gromov-Witten invariants, we need to integrate over the moduli so we have to compactify. The appropriate compactification was discovered by Kontsevich and its elements are called stable maps. They are holomorphic maps from prestable curves, that is, connected reduced projective curves with at worst ordinary double points (nodes) as singularities. A map is stable if its group of automorphisms is finite, that is, there are only finitely many biholomorphisms satisfying and , where are the marked points. Intuitively, we allow Riemann surfaces to degenerate by collapsing loops into points. Since only genus and curve have infinitely many automorphisms (Möbius transformations and translations, resp.), the stability condition is nonvacuous only for them and only if the map is trivial, that is, maps everything into a point. It requires then that each genus component has at least special points, nodes, or marked points. Under favorable circumstances, the space of stable maps up to reparametrization is itself a closed Kähler orbifold of dimensionFor instance, this is the case if and . Above is the first Chern class of the tangent bundle and the cohomology/homology pairing. The notation anticipates that in general the moduli are neither smooth nor have the expected dimension so (2.1) is called the virtual dimension. A deep result in the Gromov-Witten theory asserts that despite the complications, there is a cycle of expected dimension called the virtual fundamental class that one can integrate over.

Primary Invariants
Presence of marked points allows one to define evaluation maps:and pullback cohomology classes from to . These pullbacks are called the primary classes on [21, 22]. The primary Gromov-Witten invariants arewhere is the usual cup product and the integral denotes pairing with . Again under favorable circumstances, the primary invariants have an enumerative interpretation. Namely, is the number of genus holomorphic curves in a class passing through generic representatives of cycles Poincare dual to [19, 23]. In general, the enumerative interpretation fails and are only rational numbers, this is always the case for the Calabi-Yau manifolds. Most of the primary invariants are zero for dimensional reasons. Indeed, the complex degree of the integrand in (2.3) is and for the integral to be nonzero, it should be equal to the virtual dimension (2.1). There are other natural classes on that lead to more general Gromov-Witten invariants, gravitational descendants, and Hodge integrals [3, 19, 21], but we need not concern ourselves with them here.

It is convenient to arrange the primary invariants into a generating function [23]. To this end, we note that they are linear in insertions and we can recover all of them from and those with insertions chosen from an integral basis in . One may worry about torsion, but torsion classes are not represented by holomorphic curves and can be ignored. Thus, any is a linear combination of , where the “powers” stand for repeating that many times. Introduce formal variables for each element of the basis. Heuristically, they represent (minus) Kähler volumes of and are called Kähler parameters, especially in physics literature. Analogously, let be a linear basis in and let be the corresponding formal variables. We write with for short, when . The numbers are called degrees. Finally, we need one more variable , the string coupling constant, to incorporate genus. The primary Gromov-Witten potential (relative to the above bases choices) isThis particular choice of a generating function is by no means obvious and is inspired by two-dimensional topological quantum field theory. The power instead of just has in mind the Euler characteristic of a genus the Riemann surface. For Kähler is defined as at least a formal power series in [24]. Under a change of bases transforms as a tensor. One may entertain oneself by writing a tensor potential that is an invariant, see [23]. In [21, 22], a more general Gromov-Witten potential is considered that incorporates gravitational descendants and accordingly has more formal variables.

Let be the resolved conifold, the sum of two tautological line bundles over . Being a vector bundle over , it is homotopic to its base and has the same homology and cohomology. In particular, and , where is the Poincare dual to the class of a point in . Thus, is spanned by and is spanned by , the fundamental class of . Hence, we need only one and one variable. The primary potential simplifies to

Divisor Equation and Free Energy
We will be interested not even in all primary invariants but also in those corresponding to combinations of divisor classes, elements of . Divisor invariants turn out to be most relevant to large duality. In noncompact manifolds, the name is misleading since there is no Poincare duality. For example, the hyperplane class of is a divisor class in despite the fact that it is not dual to any divisor. But in closed manifolds, divisor classes are precisely Poincare duals to divisors, cycles of complex codimension one. Invariants containing only divisor classes can be reduced to using the so-called divisor equation. The latter is one of the universal relations among the Gromov-Witten invariants coming from universal relations among moduli spaces of stable maps with the same target . One of them is [21, 22]where is the map forgetting the last marked point. Its consequence is the divisor equationwhere and are arbitrary. There are two exceptions to the validity of (2.6) and hence (2.7), both in degree zero. If then consists of constant maps. The stability condition requires domains of stable maps in this case to be themselves stable, not just prestable. But when a stable curve must have at least marked points so the spaces of curves are empty. However, are not, and (2.6) fails for .

Since modulo torsion and form a basis in there are precisely basis elements in . We assume, without loss of generality, that are the ones and that they are dual to , that is . The divisor equation may now be used to flush all the insertions out of the divisor invariants. By induction from (2.7),assuming to avoid low-genus problems in degree zero. Define the truncated divisor potential as in (2.4) but restricting the sum to and . Using (2.8), we computeObviously, as far as divisor invariants go, are redundant and we can set them equal to . This naturally leads to another generating function [6, 7, 25].

Definition 2.1. The reduced Gromov-Witten-free energy isIts exponent is called the reduced the Gromov-Witten partition function. One writes when the target manifold needs to be indicated.The reduced-free energy is nonzero only if for some class , see (2.1). If is the Calabi-Yau, then and if, in addition, it is a threefold then also and the nontriviality condition holds for all classes and genera. For a toric Calabi-Yau the reduced partition function is the quantity directly computed by the topological vertex algorithm [12, 20, 26, 27].

Degree Zero
The moduli spaces consist of stable maps mapping stable curves into points. Therefore, they split [3]This reduces degree zero invariants to integrals over the spaces of curves and over . The divisor equation (2.7) still holds for for genus and for all in higher genus. Moreover, since now it directly implies that all the divisor invariants vanish except possibly for those that can no longer be reduced. Therefore, in genus the only surviving invariants are and in genus we are left with , and , respectively. There is automatically no dependence on , so the degree zero divisor potential is the same as the degree zero-free energy (cf. [28]):Note that degree zero genus terms are the only parts of the free energy depending on powers of rather than just their exponents . When is compact, these terms reflect its classical cohomology, namely [28]:

In particular, they vanish unless is a threefold. Higher genus contributions were computed in the celebrated paper of Faber-Pandharipande [3]:Here, as before are the Chern classes and are the Bernoulli numbers defined via a generating function [29]:The only nonzero odd-indexed number is and , , , .

One sees from (2.14) that higher genus contributions all vanish for nonthreefolds even when nondivisor invariants are taken into account because the Chern classes integrate to zero. However, genus terms may still survive if has cohomology classes of appropriate degree to cup with and each other. But the divisor invariants still vanish for dimensional reasons. Also note that (2.14) simplifies for the Calabi-Yau threefolds since and are the Euler characteristics of . Thus, for compacting Calabi-Yau threefolds,

When is noncompact but the moduli may still be compact. This usually happens if geometry forces images of stable maps to stay within a fixed compact subset of , for example, this is the case for the resolved conifold [10, 25]. Then, the virtual class is still defined and no new problems arise. However, if factorization (2.11) forces to be noncompact always. To the best of our knowledge, no virtual class theory exists for noncompact moduli so technically for noncompact are not defined at all.

Leaving the land of rigor and arguing like string theorists, we notice that for the Calabi-Yau threefolds, (2.16) still makes sense and can be taken as the “right” answer even for noncompact . This is consistent with a formal localization computation [19]. Unfortunately, for the invariants contain insertions and we really need to know how to interpret the integrals over in (2.13). In physics literature, it is suggested that they correspond to integrals over “noncompact cycles” [15] that can perhaps be interpreted as duals to compact cohomology cocycles [30]. We conclude that for the resolved conifold (), the degree zero-free energy has the formwhere are degree homogeneous polynomials with rational coefficients. We should mention that there are reasonable ways [15] of assigning values to at least for local curves (see [11]) from equivariant and mirror symmetry viewpoints. For the resolved conifold, they yieldand this function can be recovered from the mirror geometry. However, it appears that the Donaldson-Thomas and the Chern-Simons theories store classical cohomology information more crudely. We will see that in genus this answer or even the general template (2.17) is inconsistent with exact duality (see discussion after Corollary 3.2).

Definition 2.2. The (full) Gromov-Witten-free energy is and the (full) Gromov-Witten partition function is , where are reduced versions from Definition 2.1. As before, one writes to indicate the target manifold if necessary.For the resolved conifold, we get from (2.10)The positive degree part converges to a holomorphic function in an appropriate domain of (recall that is a negative Kähler volume). The same holds for all toric the Calabi-Yau threefolds and for them the partition function is given directly by the topological vertex [12, 20, 26, 27]. We will discuss the case of the resolved conifold in more detail in Section 4. But the degree zero part is not so well behaved. The sum in (2.17) diverges and fast. By a classical estimate for Bernoulli numbers,and the general term in (2.17) grows factorially for any . Coming up with a space of formal power series, where the sum lives is neither difficult nor helpful. A helpful insight comes from the conjectural duality with the Donaldson-Thomas theory [6, 7] that suggests to view (2.17) as an asymptotic expansion of a holomorphic function at a natural boundary point. The function in question is the MacMahon function , the point is and the relation to (2.17) is . We inspect this idea in Section 3.

3. The Donaldson-Thomas Theory and the MacMahon Function

In this section, we clarify the relationship between degree zero the Gromov-Witten invariants and the MacMahon function:This is a classical generating function for the number of plane partitions [31] [8, I.5.13]. More to the point, it appears in [57] in the generating function of degree zero the Donaldson-Thomas invariants of the Calabi-Yau threefolds.

The Donaldson-Thomas Invariants
The Donaldson-Thomas theory provides an alternative to the Gromov-Witten description of holomorphic curves in the Kähler manifolds, utilizing ideal sheaves instead of stable maps. Intuitively, an ideal sheaf is a collection of local holomorphic functions vanishing on a curve. This avoids counting multiple covers of the same curve separately and the Donaldson-Thomas invariants are integers unlike their Gromov-Witten cousins. Counting ideal sheaves is at least formally analogous to counting flat connections (i.e., locally constant sheaves) on a real -manifold, and the Donaldson-Thomas invariants are holomorphic counterparts of the Casson invariant in the Chern-Simons theory [13].

The genus of a stable map is replaced in the Donaldson-Thomas invariant by the holomorphic Euler characteristic of an ideal sheaf. As conjectured in [6, 7] and proved in [32], the degree zero partition function of the Calabi-Yau threefold is given bywhere as before is the classical Euler characteristic.

Since both kinds of invariants are meant to describe the same geometric objects, one expects a close relationship between them. Indeed, it is proved in [6, 7] for toric threefolds and conjectured for general ones that reduced partition functions of the Donaldson-Thomas and the Gromov-Witten theories are the same under a simple change of variables. This equality does not extend directly to degree zero but it is mentioned in [6, 7] that the Gromov-Witten is the asymptotic expansion of at (note the extra in the exponent).

A quick look at (2.17) tells one that even for the resolved conifold, this can be true at best for since no extra variables are involved in the Donaldson-Thomas function. We will see that this is the case but the complete asymptotic expansion involves some interesting extra terms that are perplexing from the Gromov-Witten point of view. However, the MacMahon factor is exactly reproduced in the Chern-Simons theory (Lemma 5.1). Moreover, with asymptotic expansions one has to specify not just a point but also a direction in the complex plane in which the expansion is taken, and the correct direction here is not the obvious (real positive) one.

-Resummation
To avoid imaginary numbers, we first consider instead of . For motivation, we start with a provocative “computation” that converts an expansion in powers of into one in powers of for a simpler function:The last two equalities are nonsense. Of course, the interchange of sums is illegitimate and is (very) divergent. It certainly does not converge to for positive , although by definition is the Riemann zeta function [29]. Nonetheless, the end result is almost correct. Indeed, by definition of Bernoulli numbers (2.15), [29], so In other words our “computation” (3.3) only missed the first term .

A similar feat can be performed with . First, we computeSo far, all the manipulations are legitimate assuming , although they would not be if we used instead of . Next, recall the Laurent expansion at zero of :One can now pull the same trick as in (3.3) of interchanging sums and replacing divergent power sums of integers with zeta values. Namely,where we used (3.5) in the last equality. This series is even more problematic than the one in (3.3) which at least made sense and converged for . Now, not only does it diverge factorially (see (2.20)) but also makes no sense at all, since has a pole at . Nonetheless, dropping the singular term , the “infinite constant” and formally replacing by in the sum, we get exactly the higher genus Gromov-Witten-free energy in degree zero (2.17).

The procedure used in (3.3), (3.9) can be traced back to Euler and in a more sophisticated guise is used in quantum field theory under the name of -resummation or -regularization [33]. The amazing fact is not that this is reasonable to do in physics (one can argue that has the same operational properties as the nonexistent ), but that it actually produces nearly mathematically correct answers. Unlike a physical situation, where a sensible answer is taken as a definition for an otherwise meaningless quantity, here we have an identity where both sides make perfect sense (as a holomorphic function and its asymptotic expansion, resp.) and only the passage from left to right is odious.

Mellin Asymptotics
A fix is a well-known Mellin transform technique that not only takes care of singular terms, divergent expansions, and infinite constants but even explains why the double blunder in (3.3) and (3.9) computes most of the asymptotic correctly [34]. We use it here to make the relationship between the degree zero invariants and the MacMahon function precise. Recall that given an integrable function on with a possible pole at and polynomial decay at its Mellin transform isThe transform is defined and holomorphic in the convergence strip, when at and at , assuming . It is most useful when admits a meromorphic continuation to the entire complex plane since location of the poles determines asymptotic behavior of the function at and (see [34] and below). For example, in extends meromorphically with the poles of the gamma function located at Analogously,extends with one additional zeta pole at .

The inverse Mellin transform recovers asassuming absolute integrability along . In the cases of interest to us, all the poles are located on the real axis to the left of . If the transform satisfies appropriate growth estimates, one can shift the integration contour in (3.12) to run counterclockwise along the real axis from to and back, Figure 1. This reduces (3.12) to a sum over residues at the poles by the Cauchy residue theorem:If are integers and the series converges, must be real-analytic on with at worst a pole at , and the residues give its Laurent coefficients at . For example, one can compute the Taylor expansion of at using that and the poles of are simple with the residues [29]. However, in most cases the series (3.13) diverges for all and (3.11) is such a case. Under analytic assumptions that we do not reproduce here, the following weakening of (3.13) is still true [34]:

438648.fig.001
Figure 1: Barnes contour for Mellin transforms.

Ifare orderpoles of (meromorphic continuation of)and its Laurent expansions athave the formthen an asymptotic expansion ofatis Now, it becomes clear where the extra in (3.6) came from. In addition to gamma poles in (3.11) that produce terms , there is also a simple pole of with residue that gives . Thus, (3.6) is at least an asymptotic expansion of at . The fact that it actually converges to the function is a rare bonus. In general, even if (3.15) does converge, it is not necessarily to the original function, see [34].

This technique extends to general Fourier sums (or harmonic sums) of the formbecause their Mellin transforms can be easily expressed in terms of those of the base function [34]. One can think of them as sums of generalized harmonics with amplitudes and frequencies, the usual ones corresponding to . Indeed,where is the Dirichlet series of the sum. If is entire and only has simple poles at thenIf, moreover, itself is entire and decays fast enough on , then , and . The same answer can be obtained by an (legitimate under the circumstances) interchange of sums in (3.16):In particular, this expansion is not just asymptotic but convergent. If is not entire but only meromorphic, the last two equalities fail. However, (3.15) still ensures that formal interchange of sums gives the regular part of the asymptotic expansion correctly as long as-poles are real-part positive. This is precisely what happened in (3.3).

Ramanujan-Wright Expansion
The situation in (3.9) is more complicated. We compute from (3.7):Now, assume that is large enough for the double series to converge absolutely, for example, , and proceedThe extra zeta poles occur at , that is, and becomes a double pole. Formula (3.15) now yields an asymptotic expansion for that we state as a theorem. This is a particular case of asymptotic expansions for analytic series obtained by Ramanujan who used a rough equivalent of the Mellin asymptotics, the Euler-Maclaurin summation (see [35, Theorem 6.12]). The Ramanujan considerations were heuristic and in any case remained unpublished until much later. The first rigorous asymptotic for is due to Wright [36]. We sketch a proof for the convenience of the reader.

Theorem 3.1 (Ramanujan-Wright). Let be the MacMahon function. Then has the Mellin transform , and its asymptotic expansion at along isProof. Recall that has “trivial zeros” at negative even integers [29]. Poles of at negative odd integers are therefore canceled by zeros of . Analytical assumptions needed for (3.15) to hold are satisfied here by the classical estimates for and [29]. The contributing poles are as follows.
(i) Gamma poles at with residues .(ii) Simple pole of at with residue .(iii) Double pole of at . We have from the first two items and (3.5)To take care of the double pole, we need more than just the residue. By the well-known properties of and where is the Euler constant. Thus,By (3.15), the corresponding terms in the asymptotic expansion are and it remains to combine the expressions.
In hindsight, it is amusing how much of (3.22) is visible in the naive expression (3.9), not just the regular part but also and even in front of the logarithm. The only hidden term is , sometimes called the Kinkelin constant [31], and for this reason perhaps it is usually missing in physical papers.

Stokes Phenomenon and the Natural Boundary
As already mentioned, the relationship between and is not . Replacing formally by in (3.22), we recover the infinite sum of (2.17) along with three extra terms:How legitimate is this substitution? Had (3.22) been a convergent Laurent expansion, there would be no such question. But it is asymptotic and represents only up to exponentially small terms (more precisely, “faster than polynomially small” but we follow the standard abuse of terminology). It is well-known that such expansions depend on a direction in the complex plane in which they are taken. As one crosses certain Stokes lines originating from the center of expansion, exponentially small terms may become dominant and change the expansion drastically. This change is commonly known as the Stokes phenomenon. Moreover, for an asymptotic expansion in some direction to exist, the function must be holomorphic in a punctured local sector containing this direction in its interior. Switching from to while keeping real positive forces us to approach along the upper arc of the unit circle, that is, along a purely imaginary direction. For an asymptotic expansion in this direction, we need to have analytically continued beyond the unit disk . But can it be continued?

Equation (3.1) does not look very promising. In fact, it strongly suggests that has a singularity at each root of unity. But roots of unity are dense on the circle making it a natural boundary for and no analytic continuation exists. It turns out to be quite hard to turn this observation into a proof, but Almkvist shows [31] that if is a proper irreducible fraction, thenfor real positive . Thus, every root of unity is indeed singular, and is the natural boundary.

This forces us to reconsider keeping real in . Should approach from the positive imaginary direction, we can set with and Theorem 3.1 gives us an asymptotic expansion in . We can rewrite it as an expansion in of course as long as it is understood that in it is positive imaginary. This may seem like an underhanded trick but it is not. The natural domain of is the upper half-plane, and the only distinguished direction in its interior is the positive imaginary one.

Corollary 3.2. Asymptotic expansion of at along is (taking the principal branch of the logarithm)Comparing this to (2.17), one ought to be somewhat perplexed. If we are to take (3.28) at face value then , (?!), and there is no space for at all. Aside from the fact that -s are supposed to be homogeneous polynomials of the corresponding degree, the numbers involved are not even rational, by the Apéry famous result. Nevertheless, the MacMahon factor appears as is in the Chern-Simons partition function, see Lemma 5.1.

The disappearance of extra variables and appearance of irrationals suggest that some kind of averaging is involved. It would not explain but we may guess, that averaging of is divergent and has to be regularized giving rise to an anomalous term. Why the Donaldson-Thomas theory does not reproduce the degree zero contributions in low genus is beyond our expertise. However, from the Chern-Simons vantage point this ought to be expected. The idea of large duality is that the same string theory is realized on manifolds with different topology [19, 20]. However, the degree zero terms in genus are exactly the ones that record the classical cohomology of the target manifold, see (2.13). Although some relation between topologies of manifolds supporting equivalent string theories may be expected, the entire cohomology ring is certainly too much to survive a geometric transition. Therefore, these classical terms cannot enter an invariant partition function except via averages that remain unchanged by such transitions.

4. Topological Vertex and Partition Function of the Resolved Conifold

This section and Section 5 are to be read in conjunction. We review the salient points of two combinatorial models, the topological vertex [12, 20, 26, 27], and the Reshetikhin-Turaev calculus [19, 37], highlighting the differences but more importantly the parallels between them. The former computes the Gromov-Witten invariants of toric Calabi-Yau threefolds, and the latter computes the Chern-Simons invariants of all closed manifolds. The reason to compare them is the conjectural large duality between the two. Both models encode their spaces into labeled diagrams and then assign values to them according to the Feynman-like rules. However, the encoding and the rules are quite different despite intriguing correspondences. The reason why we use the topological vertex instead of just summing up (2.19) as in [3] is that it directly gives the partition function in correct variables and in an appealing form. Comparing the answer to the Chern-Simons one, it becomes reasonable to express it in a closed form via the quantum Barnes function (Theorem 5.2).

Toric Webs
Just as the Reshetikhin-Turaev calculus [19, 37], the topological vertex is a diagrammatic state-sum model. This means that geometry of a space is encoded into a diagram, a graph enhanced by additional data, and the value of an invariant is computed by summing over all prescribed labelings of the diagram. In the Reshetikhin-Turaev calculus, the diagrams are link diagrams representing manifolds via surgery [37, 38]. In the topological vertex, they are toric webs representing toric Calabi-Yau threefolds.

A toric web is an embedding of a trivalent planar graph with compact and noncompact edges into that satisfies some integrality conditions [12, 20, 27]. Namely, vertices have integer coordinates, and direction vectors of edges can be chosen to have integer coordinates. Moreover, if the direction vectors are chosen primitive (without a common factor in coordinates), any pair of them meeting at a vertex forms a basis of , and every triple at a vertex if directed away from it adds up to zero. Examples for the resolved conifold and the local (i.e., the total space of ) are shown in Figure 2, where the primitive directions of noncompact edges are also indicated. Toric webs related by a transformation and an integral shift represent isomorphic threefolds. For this reason, we did not label the vertices in Figure 2, one may assume that one of them is and all compact edges have the unit length. The toric web is a complete invariant of a toric Calabi-Yau. Indeed, the moment polytope of the torus action can be recovered from it [12, 4.1] and therefore the threefold itself up to isomorphism by the Delzant classification theorem [39]. Analogously, a -manifold is recovered from its link diagram up to diffeomorphism by surgery on the link [37, 38]. Having toric webs rigidly embedded in is inconvenient, one would prefer to treat them as abstract graphs, perhaps with additional data. This is possible at least as far as the topological vertex is concerned although the resulting graphs may no longer be complete invariants.

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Figure 2: Toric webs of and local .

Tracing back the construction of a threefold from its web, one concludes that the vertices correspond to fixed points of the torus action and compact edges correspond to fixed rational curves (copies of ). Being rational curves sitting inside the Calabi-Yau threefold, their normal bundles are isomorphic to The framing number for each edge is assigned the value from the normal bundle type of the corresponding curve. This only determines up to sign, and the edge must be oriented to specify it. Although on their own these orientations are chosen arbitrarily, they must be aligned with the framing numbers, the exact rule is given in [12, 4.2].

If is an integral basis in as in Section 2, then each edge curve represents a homology class expressible as a linear combination , . One requires these homology relations to be attached to the edges as well. The result is a graph called the toric graph. Toric graphs for and the local are shown in Figure 3. Framing numbers and homology relations are the only data aside from the topology of the web used in the topological vertex. We emphasize that both can be recovered algorithmically from the web itself without any recourse to the original threefold [40], [12, 4.1].

438648.fig.003
Figure 3: Toric graphs of and local .

Partitions and the Schur Functions
We wish to briefly describe the topological vertex algorithm to see how -bifactorials naturally emerge from it. This requires some basic information about partitions [8] that appear in the Reshetikhin-Turaev calculus as well. Partitions serve as labels in state sums defining the invariants. A partition is an element of with only finitely many nonzero entries that are nonincreasing, that is,Let denote the set of all partitions. The number of nonzero entries is called the length of a partition, and the sum of all entries is called its size (or weight). Partitions are visualized by Young diagrams, rows of boxes stacked top down with boxes in th row, Figure 4. The conjugate partition is obtained visually by transposing the Young diagram along the main diagonal and analytically as . Note that and , . Another relevant characteristic of a partition, sometimes called its quadratic Casimir, isPartitions represent possible states of compact edges in a toric graph and a combination of partition labels for each edge represents a state of the graph [27]. The partition function is then obtained by summing over all possible states.

fig4
Figure 4: Young diagram and its conjugate.

Amplitudes (see Definition 4.2) of a labeled graph are defined via a specialization of the Schur functions indexed by partitions. They are symmetric “functions” in the sense of Macdonald [8], that is, formal infinite sums of monomials in countably many variables that become symmetric polynomials if all but finitely many variables are set equal to zero (more technically, if monomials containing any variable outside of a finite set are discarded from the sum). For instance, if then is the th elementary symmetric function:In general, are polynomials in the elementary symmetric functions given by the Jacobi-Trudy formula. For example,Since are homogeneous of degree the Jacobi-Trudy formula implies that are also homogeneous of degree , that is, . Moreover, form a linear basis in the space of symmetric functions, in particular . It turns out that are nonnegative integers that vanish unless . They are the famous Littlewood-Richardson coefficients [8].

Specializations of the Schur functions appearing in the topological vertex are obtained by specializing the formal variables to elements of a geometric series possibly modified at finitely many entries. Such specializations were extensively studied by Zhou [4]. Define the Weyl vector byNote that is not a partition. Introduce a new formal variable and for any vector set so, in particular, is a geometric series.

Definition 4.1. One-, two-, and three-point functions of the topological vertex are, respectively [4, 12],There is a shorter expression for the three-point function via the skew Schur functions [4, 27] but we do not need it here and (4.7) is somewhat reminiscent of the Verlinde formula in the Chern-Simons theory [41]. We assume and is then defined by the principal branch of the square root. One can see by inspection from (4.4) that converges for . Since the Schur “functions” are polynomials in , they are also well defined as honest functions of upon specializing to .

To be consistent with the usual basic hypergeometric notation [42], we wish to switch from to . This can be done using a symmetry of the two-point functions [4]This identity is a curious one since the two sides never converge simultaneously (both diverge for ). It has the same meaning as a more familiar identity:where the two sides never converge simultaneously either. In fact, are rational functions of and can be analytically continued to , (4.8) expresses this analytic continuation.

The appearance of -bifactorials in partition functions is due to the Cauchy identity for the Schur functions [4, 8]If , the right-hand side of (4.10) becomesNote that although (4.10) is a formal identity if both sides converge as in (4.11), it holds as a function identity.

Partition Functions as State Sums
Let us now inspect the state sums appearing in the topological vertex. Let and denote the sets of vertices and compact edges of a toric graph, respectively. Choose an arbitrary orientation for each element of , this determines the sign of the framing numbers. Assign a formal variable to each element of a basis and set for the corresponding edge curve . Finally, label all compact edges by arbitrarily chosen partitions and noncompact ones by the trivial partition . Triples of partitions are then assigned to each vertex according to the following rule.

Starting with any of the three edges and going counterclockwise around the vertex pick, the edge label if the arrow on the edge is outgoing and its conjugate if the arrow is incoming, Figure 5. Noncompact edges present no problem since . This determines the triple up to cyclic permutation which is enough since has cyclic symmetry.

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Figure 5: Partition triple at a vertex

Definition 4.2. Amplitude of a labeled toric graph relative to a basis is given by [12, 27]The main result of [12] can be stated as follows.

Theorem 4.3. The reduced Gromov-Witten partition function of a toric Calabi-Yau threefold relative to a basis is given by a state sum:assuming in the second sum that the edges are numbered and .

Partition Function of the Resolved Conifold
Here, is one-dimensional and . There is only one variable and only one edge. The amplitude for is (see Figure 3 and (4.8))Recalling that , and is homogeneous of degree , we compute furtherSuppose that is small enough for to converge then, we get by Theorem 4.3 and the Cauchy identity (4.10)If we accept the MacMahon function as the degree zero partition function of the resolved conifold (despite the issues discussed after Corollary 3.2), thenWe conclude that the (full) Gromov-Witten partition function of the resolved conifold isas used in Section 1.

5. Reshetikhin-Turaev Calculus and Partition Function of the -Sphere

As explained in the beginning of Section 4, this one is complementary to it. We briefly review the Reshetikhin-Turaev calculus [19, 37] in a form that invites analogies with the topological vertex. In particular, we forgo the usual terminology of dominant weights and irreducible representations of and rephrase everything directly in terms of partitions. The immediate goal is to compute the partition function of in a suitable form and compare it to the one for the resolved conifold (Theorem 5.2).

Whereas computation of the Gromov-Witten invariants in all degrees and genera is an open problem (beyond the cases of toric Calabi-Yau threefolds [12] and local curves [11]), the Reshetikhin-Turaev calculus provides an algorithm for computing the Chern-Simons invariants for arbitrary closed manifolds, especially effective for the Seifert-fibered ones [43]. This circumstance combined with explicit large dualities for the toric Calabi-Yau threefolds is the secret behind physical derivation of the topological vertex. To be sure, there is a catch. The Reshetikhin-Turaev model (or equivalently Atiyah-Turaev-Witten TQFT [37, 44]) is not a single model but a countable collection of them, one for each pair of positive integers known as level and rank. This would not be much of a hindrance if not for the tenuous connection between invariants for different and . As a rule, geometers are interested in asymptotic behavior for large [45] and physicists in both large and large behavior. The Reshetikhin-Turaev sums with ranges depending explicitly on and are not exactly custom-made for those types of questions. In fact, they require significant work even in simplest cases to be converted into asymptotic-friendly form. No general method exists; most common ad hoc procedures use the Poisson resummation [43] or finite group characters [19, 46].

The idea of the Reshetikhin-Turaev construction (related but different from the Witten original one [41] as formalized by Atiyah [44]) is to combine some deep topological results of Likorish-Wallace and Kirby with the representation theory of quantum groups [19, 37]. A theorem of Likorish and Wallace asserts that any closed -manifold can be obtained by surgery on a framed link in [38]. This is complemented by the Kirby characterization [47] of links that produce diffeomorphic manifolds as those related by a sequence of Kirby moves: blow up/down and handle-slide. Blow up/down adds/removes an unknotted unlinked component with a single twist and handle-slide pulls any component over any other one, Figure 6. Thus, if one can find an invariant of framed links that remains unchanged under Kirby moves, it automatically becomes an invariant of closed -manifolds via surgery.

438648.fig.006
Figure 6: Blow up/down and handle slide over a trefoil knot.

Hopf and Twist Matrices
Framed links can be represented up to isotopy by plane diagrams with under- and overcrossings and twists as in Figure 6 providing a combinatorial model of -manifolds. Slicing a link diagram bottom to top and avoiding slicing through cups, caps, twists, or crossings, one gets arrays of basic elements Figure 7 stacked on top of each other.

438648.fig.007
Figure 7: Basic elements and slicing of an Hopf link.

This decomposition fits nicely with structure of a linear representation category. Placing elements next to each other corresponds to tensoring and stacking corresponds to composition. It remains to find an object with representation category meeting all the invariance requirements. It turns out that it is extremely hard to find one producing nontrivial invariant. Classical Lie groups and algebras do not work unfortunately. One has to deform the universal enveloping algebras of, say, into quantum groups and then specialize the deformation parameter to a root of unity . As if that were not enough, the tensor product of representations has to be modified as well. The end result [19, 37, 46] is a representation-like category with only a finite number of irreducible representations. For at level they are indexed by partitions with the Young diagrams in the rectangle, that is,In the equivalent language of dominant weights, this corresponds to the weights in the Weyl alcove of the Cartan-Stiefel diagram of scaled by , see [19]. The Reshetikhin-Turaev invariants are computed as state sums over labelings of a link diagram with each link component labeled by a partition from , a finite set.

Thus, unlike in the topological vertex, where sums are taken over all partitions and are infinite, in the Reshetikhin-Turaev calculus sums are finite with explicit dependence on . Once a diagram is labeled, morphisms between irreducible representations and their tensor products are assigned to the elements from Figure 7, and then assembled by tensoring, composing, and eventually taking traces (corresponding to caps) to obtain numerical invariants. The hardest ones to compute are the crossing morphisms for they depend on the so-called -matrix of a quantum group [19, 37]. Good news is that for a large class of -manifolds, the Seifert-fibered ones and others, the use of crossing morphisms can be avoided entirely in computing the invariants [41, 43] (but not in proving their invariance). In terms of conformal field theory, they are completely determined by fusion rules without involving the braiding matrices [46]. This means that the only algebraic inputs are the Hopf and twist matrices and :The notation is as follows.

is the normalized quantum invariant of the Hopf link Figure 7 with components labeled by partitions (see more below); is the partition -dual to , for and for (not to be confused with the conjugate partition); denotes the coordinates of partition , , in particular ;