Quantum Barnes Function as the Partition Function of the Resolved Conifold

Abstract

We give a short new proof of large 𝑁 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 𝑞-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 𝑞-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 𝑋∶=𝒪(−1)⊕𝒪(−1), it was originally computed by Faber-Pandharipande [3]; see also [4]𝑍^′_𝑋(𝑎;𝑞)=^∞_𝑛=11−𝑎𝑞^𝑛𝑛. (1.1)Here, 𝑎=𝑒^−𝑡, 𝑞=𝑒^𝑖𝑥, and 𝑡, 𝑥 are known as the Kähler parameter and the string coupling constant, respectively. In mathematical terms, they are just formal variables andln𝑍^′_𝑋=^∞∞_𝑔=0𝑑=1⟨1⟩_𝑔,𝑑𝑡^𝑑𝑥^2𝑔−2, (1.2)where ⟨1⟩_𝑔,𝑑 is the Gromov-Witten invariant of genus 𝑔 degree 𝑑 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 [5–7] 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 as𝑍_𝑋=𝑍⁰_𝑋𝑍^′_𝑋≈𝑀𝑍^′_𝑋, (1.3)where𝑀(𝑞)∶=^∞_𝑛=11−𝑞^𝑛−𝑛 (1.4)is the MacMahon function, classically known as the generating function of plane partitions [8]. In all honesty, this is not quite true as ln𝑀(𝑒^𝑖𝑥) has some spurious terms in its expansion at 𝑥=0 and only accounts for genus 𝑔≥2 terms correctly (see Section 3). Also in the Donaldson-Thomas theory, one has 𝑍^DT_𝑋=𝑀²𝑍^′DT_𝑋, not 𝑍_𝑋=𝑀𝑍^′_𝑋. In a recent reformulation of the Donaldson-Thomas theory [9], the reduced partition function𝑍^′DT 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 𝑁 duality. This one relates the Gromov-Witten invariants of the resolved conifold to the Chern-Simons invariants of the 3-sphere. The usual formulation defines the Chern-Simons theory as a gauge theory on a 𝑈_𝑁 or 𝑆𝑈_𝑁 bundle over a real 3-manifold 𝑀. 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 𝑇^∗𝑀 ending on its zero section. 𝑇^∗𝑀 is canonically a symplectic manifold (even Kähler if 𝑀 is real-analytic) with first Chern class 𝑐₁(𝑇^∗𝑀)=0, that is, the Calabi-Yau. In particular, 𝑇^∗𝑆³ is diffeomorphic to a quadric 𝑥²+𝑦²+𝑧²+𝑤²=1 in ℂ⁴. 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 𝑇^∗𝑀 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 3-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 𝑍_𝑆³ to 𝑍_𝑋 promises some useful insight.

Once again, by ignoring some irrelevant prefactors, 𝑍_𝑆³ can be written as 𝑍_𝑆³≈ℰ^−𝑧𝑍_𝑋, where 𝑧=𝑖𝑡𝑥⁻¹ so that 𝑎=𝑞^𝑧, andℰ(𝑞)∶=^∞_𝑛=11−𝑞^𝑛−1 (1.5)is the classical Euler generating function of ordinary partitions. At this point, it is appropriate to introduce notation that allows one to write 𝑍^′_𝑋,𝑀, and ℰ uniformly. Let(𝑎;𝑞)_∞⁽⁰⁾∶=1−𝑎,(𝑎;𝑞)_∞^(𝑑)∶=^∞_{𝑖1,…,𝑖𝑑=0}1−𝑎𝑞^{𝑖1+⋯+𝑖𝑑} (1.6)be the q-multifactorials then (see Section 6),𝑍^′_𝑋(𝑎;𝑞)=(𝑎𝑞;𝑞)_∞⁽²⁾1,𝑀(𝑞)=(𝑞;𝑞)_∞⁽²⁾1,ℰ(𝑞)=(𝑞;𝑞)_∞⁽¹⁾. (1.7)Using 𝑞 and 𝑧 as variables, we see that𝑍^′_𝑋=𝑞^𝑧+1;𝑞_∞⁽²⁾,𝑍_𝑋≈1(𝑞;𝑞)_∞⁽²⁾𝑞^𝑧+1;𝑞_∞⁽²⁾,𝑍_𝑆³≈(𝑞;𝑞)_∞^(1)𝑧1(𝑞;𝑞)_∞⁽²⁾𝑞^𝑧+1;𝑞_∞⁽²⁾. (1.8) 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 consider𝐺_𝑞1(𝑧+1)∶=(𝑞;𝑞)_∞^{(0)(𝑧(𝑧−1)/2)}(𝑞;𝑞)_∞^(1)𝑧1(𝑞;𝑞)_∞⁽²⁾𝑞^𝑧+1;𝑞_∞⁽²⁾. (1.9)This 𝐺_𝑞 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 as𝐺_𝑞(𝑧+1)=Γ_𝑞(𝑧)𝐺_𝑞(𝑧), (1.10)where Γ_𝑞 is the Jackson quantum gamma function deforming the classical one. This in turn satisfies Γ_𝑞(𝑧+1)=(𝑧)_𝑞Γ_𝑞(𝑧) with the so-called quantum number(𝑧)_𝑞∶=(1−𝑞^𝑧)/(1−𝑞). This makes 𝐺_𝑞 a deformation of the classical Barnes function that satisfies (1.10) with 𝑞-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 𝑁 duality is so hard to prove even in simple cases. Still, 𝐺_𝑞 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, 𝑍 is predicted to have modular properties [16], that is, obey transformation laws under 𝑧↦𝑧+1 and 𝑧↦−1/𝑧. 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 𝐺₂ 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 haveln𝑍^′_𝑋=^∞∞_𝑛=0𝑑=1(−1)^𝑛𝐷_𝑛,𝑑𝑡^𝑑𝑞^𝑛 (1.11)with 𝑍^′_𝑋 the same as in (1.2) and 𝐷_𝑛,𝑑 the Donaldson-Thomas invariants. In other words, in each degree, (−1)^𝑛𝐷_𝑛,𝑑 are simply the Taylor coefficients of 𝑍^′_𝑋 at 𝑞=0 while ⟨1⟩_𝑔,𝑑 are the Laurent coefficients in 𝑥 corresponding to 𝑞=1 with 𝑞=𝑒^𝑖𝑥. The relation with the Chern-Simons invariants is more complicated. Traditionally, one has to take 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)}, where 𝑘,𝑁 are the two parameters of the Chern-Simons theory, rank and level. They are positive integers making 𝑞 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 𝑞=0, the Gromov-Witten invariants at 𝑞=1 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 1 to make a comparison. Things are nice in higher degree [9], but in degree zero it is exactly 1 and every point of the unit circle is a singularity. This is remedied easily enough in the Gromov-Witten context since we can interpret 𝑍⁰_𝑋(𝑒^𝑖𝑥)=∑^∞_𝑔=0⟨1⟩_𝑔,0𝑥^2𝑔−2 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 𝑁 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 𝑍_𝑋 and an additional factor in 𝑍_𝑆³ 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 𝑍⁰_𝑌=𝑀^{(1/2)𝜒(𝑌)} for any Calabi-Yau threefold 𝑌 [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 𝐺_𝑞 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 𝑋 be the Kähler manifold of complex dimension 𝑁. We wish to consider holomorphic maps 𝑓∶Σ→𝑋 of the Riemann surfaces with 𝑛 marked points into 𝑋 that realize certain homology class 𝛼∈𝐻₂(𝑋,ℤ). The space of such maps is denoted 𝑀_𝑔,𝑛(𝑋,𝛼). 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 𝜎∶Σ→Σ satisfying 𝑓∘𝜎=𝑓 and 𝜎(𝑝_𝑖)=𝑝_𝑖, where 𝑝₁,…,𝑝_𝑛 are the marked points. Intuitively, we allow Riemann surfaces to degenerate by collapsing loops into points. Since only genus 0 and 1 curve have infinitely many automorphisms (Möbius transformations and translations, resp.), the stability condition is nonvacuous only for them and only if the map 𝑓 is trivial, that is, maps everything into a point. It requires then that each genus 0(1) component has at least 3(1) special points, nodes, or marked points. Under favorable circumstances, the space of stable maps 𝑀_𝑔,𝑛(𝑋,𝛼) up to reparametrization is itself a closed Kähler orbifold of dimensiondim^vir_ℂ𝑀_𝑔,𝑛(𝑋,𝛼)=𝑐₁(𝑋),𝛼−(𝑁−3)(𝑔−1)+𝑛. (2.1)For instance, this is the case if 𝑋=ℂℙ^𝑁 and 𝑔=0. Above 𝑐₁(𝑋) 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 [𝑀_𝑔,𝑛(𝑋,𝛼)]^vir called the virtual fundamental class that one can integrate over.

Primary Invariants
Presence of marked points allows one to define evaluation maps:ev_𝑖∶𝑀_𝑔,𝑛𝑝(𝑋,𝛼)⟶𝑋𝑓↦𝑓_𝑖 (2.2)and pullback cohomology classes 𝛾_𝑖 from 𝑋 to 𝑀_𝑔,𝑛(𝑋,𝛼). These pullbacks are called the primary classes on 𝑀_𝑔,𝑛(𝑋,𝛼) [21, 22]. The primary Gromov-Witten invariants are𝛾₁⋯𝛾_{𝑛𝑔,𝛼}∶=_{[𝑀𝑔,𝑛(𝑋,𝛼)]vir}ev^∗₁𝛾₁∪⋯∪ev^∗_𝑛𝛾_𝑛, (2.3)where ∪ is the usual cup product and the integral denotes pairing with [𝑀_𝑔,𝑛(𝑋,𝛼)]^vir. Again under favorable circumstances, the primary invariants have an enumerative interpretation. Namely, ⟨𝛾₁⋯𝛾_𝑛⟩_𝑔,𝛼 is the number of genus 𝑔 holomorphic curves in a class 𝛼∈𝐻₂(𝑋,ℤ) passing through generic representatives of cycles Poincare dual to 𝛾₁,…,𝛾_𝑛 [19, 23]. In general, the enumerative interpretation fails and ⟨𝛾₁⋯𝛾_𝑛⟩_𝑔,𝛼 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 (1/2)(deg𝛾₁+⋯+deg𝛾_𝑛), and for the integral to be nonzero, it should be equal to the virtual dimension (2.1). There are other natural classes on 𝑀_𝑔,𝑛(𝑋,𝛼) 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 𝛾_𝑖 and we can recover all of them from ⟨1⟩_𝑔,𝛼 and those with insertions chosen from an integral basis ℎ₁,…,ℎ_𝑚 in 𝐻⁺(𝑋,ℤ)∶=⊕_𝑛>0𝐻^𝑛(𝑋,ℤ). One may worry about torsion, but torsion classes are not represented by holomorphic curves and can be ignored. Thus, any ⟨𝛾₁⋯𝛾_𝑛⟩_𝑔,𝛼 is a linear combination of ⟨ℎ^𝑝1₁⋯ℎ^𝑝𝑚_𝑚⟩_𝑔,𝛼, where the “powers” 𝑝_𝑖 stand for repeating ℎ_𝑖 that many times. Introduce formal variables 𝑡₁,…,𝑡_𝑚 for each element of the basis. Heuristically, they represent (minus) Kähler volumes of ℎ₁,…,ℎ_𝑚 and are called Kähler parameters, especially in physics literature. Analogously, let 𝜉₁,…,𝜉_𝑘 be a linear basis in 𝐻₂(𝑋,ℤ), and let 𝑄₁,…,𝑄_𝑘 be the corresponding formal variables. We write ⟨ℎ^𝑝1₁⋯ℎ^𝑝𝑚_𝑚⟩_𝑔,^→_𝑑 with ^→𝑑∶=(𝑑₁,…,𝑑_𝑘) for short, when 𝛼=𝑑₁𝜉₁+⋯+𝑑_𝑘𝜉_𝑘. The numbers 𝑑₁,…𝑑_𝑘 are called degrees. Finally, we need one more variable 𝑥, the string coupling constant, to incorporate genus. The primary Gromov-Witten potential (relative to the above bases choices) isℱ𝑡₁,…,𝑡_𝑚;𝑄₁,…,𝑄_𝑘;𝑥∶=^∞∞_{𝑔=0𝑝1,…,𝑝𝑚𝑑=01,…,𝑑𝑘=0}ℎ^𝑝1₁⋯ℎ^𝑝𝑚_𝑚𝑔,^→_𝑑𝑡^𝑝1₁⋯𝑡^𝑝𝑚_𝑚𝑝₁!⋯𝑝_𝑚!𝑄^𝑑1₁⋯𝑄^𝑑𝑘_𝑘𝑥^2𝑔−2. (2.4)This particular choice of a generating function is by no means obvious and is inspired by two-dimensional topological quantum field theory. The power 2𝑔−2 instead of just 𝑔 has in mind the Euler characteristic −(2𝑔−2) of a genus 𝑔 the Riemann surface. For 𝑋 Kähler ℱ is defined as at least a formal power series in ℚ[𝑡_𝑗,𝑄_𝑖,𝑥] [24]. Under a change of bases ⟨ℎ^𝑝1₁⋯ℎ^𝑝𝑚_𝑚⟩_𝑔,^→_𝑑 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 𝑋∶=𝒪(−1)⊕𝒪(−1) be the resolved conifold, the sum of two tautological line bundles over ℂℙ¹. Being a vector bundle over ℂℙ¹, it is homotopic to its base and has the same homology and cohomology. In particular, 𝐻₂(𝑋,ℤ)=ℤ[ℂℙ¹] and 𝐻^•(𝑋,ℤ)=ℤ[ℎ]/(ℎ²), where ℎ is the Poincare dual to the class of a point in ℂℙ¹. Thus, 𝐻⁺(𝑋,ℤ) is spanned by ℎ and 𝐻₂(𝑋,ℤ) is spanned by 𝜉∶=[ℂℙ¹], the fundamental class of ℂℙ¹. Hence, we need only one 𝑡 and one 𝑄 variable. The primary potential simplifies toℱ(𝑡;𝑄;𝑥)∶=^∞_{𝑝,𝑑,𝑔=0}ℎ^𝑝_𝑔,𝑑𝑡^𝑝𝑄𝑝!^𝑑𝑥^2𝑔−2. (2.5)

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 𝐻²(𝑋,ℤ). Divisor invariants turn out to be most relevant to large 𝑁 duality. In noncompact manifolds, the name is misleading since there is no Poincare duality. For example, the hyperplane class of ℂℙ¹ is a divisor class in 𝒪(−1)⊕𝒪(−1), 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 ⟨ℎ^𝑝1₁⋯ℎ^𝑝𝑚_𝑚⟩_𝑔,^→_𝑑 containing only divisor classes can be reduced to ⟨1⟩_𝑔,^→_𝑑 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 𝑋. One of them is [21, 22]𝜋^∗𝑀_{𝑔,𝑛−1}(𝑋,𝛼)^vir=𝑀_𝑔,𝑛(𝑋,𝛼)^vir, (2.6)where 𝑀_𝑔,𝑛(𝑋,𝛼)→_𝜋𝑀_{𝑔,𝑛−1}(𝑋,𝛼) is the map forgetting the last marked point. Its consequence is the divisor equationℎ𝛾₁⋯𝛾_{𝑛𝑔,𝛼}=ℎ(𝛼)𝛾₁⋯𝛾_{𝑛𝑔,𝛼}, (2.7)where ℎ∈𝐻²(𝑋,ℤ) and 𝛾_𝑖 are arbitrary. There are two exceptions to the validity of (2.6) and hence (2.7), both in degree zero. If 𝛼=0 then 𝑀_𝑔,𝑛(𝑋,0) consists of constant maps. The stability condition requires domains of stable maps in this case to be themselves stable, not just prestable. But when 𝑔=0(1), a stable curve must have at least 3(1) marked points so the spaces of curves 𝑀_0,0,𝑀_0,1,𝑀_0,2,𝑀_1,0 are empty. However, 𝑀_0,3,𝑀_1,1 are not, and (2.6) fails for (𝑔,𝑛)=(0,3),(1,1).

Since 𝐻²(𝑋,ℤ)≃𝐻₂(𝑋,ℤ) modulo torsion and 𝜉₁,…,𝜉_𝑘 form a basis in 𝐻₂(𝑋,ℤ), there are precisely 𝑘 basis elements in 𝐻²(𝑋,ℤ). We assume, without loss of generality, that ℎ₁,…,ℎ_𝑘 are the ones and that they are dual to 𝜉₁,…,𝜉_𝑘, that is ℎ_𝑖(𝜉_𝑗)=𝛿_𝑖𝑗. The divisor equation may now be used to flush all the insertions out of the divisor invariants. By induction from (2.7),ℎ^𝑝1₁⋯ℎ^𝑝𝑘_𝑘𝑔,^→_𝑑=𝑑^𝑝1₁⋯𝑑^𝑝𝑘_𝑘⟨1⟩_𝑔,^→_𝑑, (2.8)assuming ^→𝑑≠0 to avoid low-genus problems in degree zero. Define the truncated divisor potentialℱ^′_div(𝑡₁,…,𝑡_𝑘;𝑄₁,…,𝑄_𝑘;𝑥) as in (2.4) but restricting the sum to 𝑝₁,…,𝑝_𝑘 and ^→𝑑≠0. Using (2.8), we computeℱ^′_div𝑡₁,…,𝑡_𝑘;𝑄₁,…,𝑄_𝑘;𝑥=^∞_{𝑔=0→𝑑≠0}⟨1⟩_𝑔,^→_𝑑𝑄^𝑑1₁⋯𝑄^𝑑𝑘_𝑘𝑥^∞2𝑔−2_{𝑝1,…,𝑝𝑘=0}(𝑑₁𝑡₁)^𝑝1⋯(𝑑_𝑚𝑡_𝑚)^𝑝𝑚𝑝₁!⋯𝑝_𝑚!=^∞_{𝑔=0→𝑑≠0}⟨1⟩_𝑔,^→_𝑑𝑄^𝑑1₁⋯𝑄^𝑑𝑘_𝑘𝑥^2𝑔−2𝑒^𝑑1𝑡1⋯𝑒^{𝑑𝑘𝑡𝑘}=^∞_{𝑔=0→𝑑≠0}⟨1⟩_𝑔,^→_𝑑(𝑄₁𝑒^𝑡1)^𝑑1⋯(𝑄_𝑘𝑒^𝑡𝑘)^𝑑𝑘𝑥^2𝑔−2. (2.9)Obviously, as far as divisor invariants go, 𝑄₁,…,𝑄_𝑘 are redundant and we can set them equal to 1. This naturally leads to another generating function [6, 7, 25].

Definition 2.1. The reduced Gromov-Witten-free energy is𝐹^′𝑡₁,…,𝑡_𝑘;𝑥∶=^∞_{𝑔=0→𝑑≠0}⟨1⟩_𝑔,^→_𝑑𝑒^𝑑1𝑡1⋯𝑒^{𝑑𝑘𝑡𝑘}𝑥^2𝑔−2. (2.10)Its exponent 𝑍^′(𝑡₁,…,𝑡_𝑘;𝑥)∶=exp(𝐹^′(𝑡₁,…,𝑡_𝑘;𝑥)) is called the reduced the Gromov-Witten partition function. One writes 𝐹^′_𝑋,𝑍^′_𝑋 when the target manifold needs to be indicated.The reduced-free energy is nonzero only if ⟨𝑐₁(𝑋),𝛼⟩−(𝑁−3)(𝑔−1)=0 for some class 𝛼≠0, see (2.1). If 𝑋 is the Calabi-Yau, then 𝑐₁(𝑋)=0 and if, in addition, it is a threefold then also 𝑁=3 and the nontriviality condition holds for all classes and genera. For a toric Calabi-Yau 𝑋, the reduced partition function 𝑍^′_𝑋 is the quantity directly computed by the topological vertex algorithm [12, 20, 26, 27].

Degree Zero
The moduli spaces 𝑀_𝑔,𝑛(𝑋,0) consist of stable maps mapping stable curves into points. Therefore, they split [3]𝑀_𝑔,𝑛(𝑋,0)=𝑀_𝑔,𝑛×𝑋. (2.11)This reduces degree zero invariants to integrals over the spaces of curves and over 𝑋. The divisor equation (2.7) still holds for 𝑛≥4(2), for genus 𝑔=0(1), and for all 𝑛 in higher genus. Moreover, since 𝛼=0, now it directly implies that all the divisor invariants vanish except possibly for those that can no longer be reduced. Therefore, in genus 𝑔≥2, the only surviving invariants are ⟨1⟩_𝑔,0 and in genus 0,1, we are left with ⟨ℎ³_𝑖⟩_0,0,⟨ℎ²_𝑖ℎ_𝑗⟩_0,0,⟨ℎ_𝑖ℎ_𝑗ℎ_𝑙⟩_0,0, and ⟨ℎ_𝑖⟩_1,0, respectively. There is automatically no dependence on 𝑄_𝑖, so the degree zero divisor potential is the same as the degree zero-free energy (cf. [28]):𝐹⁰(𝑡₁,…,𝑡_𝑘;𝑥∶=ℱ⁰(𝑡₁,…,𝑡_𝑘=;𝑥_𝑘𝑖=1ℎ³_𝑖0,0𝑡³_𝑖6+_𝑖≠𝑗ℎ²_𝑖ℎ_𝑗0,0𝑡²_𝑖𝑡_𝑗2+_{𝑖≠𝑗,𝑗≠𝑙,𝑙≠𝑖}ℎ_𝑖ℎ_𝑗ℎ_𝑙0,0𝑡_𝑖𝑡_𝑗𝑡_𝑙1𝑥²+^𝑘_𝑖=1ℎ_𝑖1,0𝑡_𝑖+^∞_𝑔=2⟨1⟩_𝑔,0𝑥^2𝑔−2. (2.12)Note that degree zero genus 0(1) terms are the only parts of the free energy depending on powers of 𝑡_𝑖 rather than just their exponents 𝑒^𝑡𝑖. When 𝑋 is compact, these terms reflect its classical cohomology, namely [28]:ℎ_𝑖ℎ_𝑗ℎ_𝑙0,0=_𝑋ℎ_𝑖∪ℎ_𝑗∪ℎ_𝑙ℎ_𝑖1,01=−24_𝑋ℎ_𝑖∪𝑐₂(𝑋). (2.13)

In particular, they vanish unless 𝑋 is a threefold. Higher genus contributions were computed in the celebrated paper of Faber-Pandharipande [3]:⟨1⟩_𝑔,0=(−1)^𝑔|𝐵_2𝑔||𝐵_2𝑔−2|⋅1(2𝑔−2)!2𝑔(2𝑔−2)2_𝑋𝑐₃(𝑋)−𝑐₁(𝑋)∪𝑐₂=(𝑋)(−1)^𝑔−1(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2⋅1(2𝑔−2)(2𝑔)!2_𝑋𝑐₃(𝑋)−𝑐₁(𝑋)∪𝑐₂(𝑋),𝑔≥2. (2.14)Here, 𝑐_𝑖(𝑋) as before are the Chern classes and 𝐵_𝑛 are the Bernoulli numbers defined via a generating function [29]:𝑧𝑒^𝑧−1∶=^∞_𝑛=0𝐵_𝑛𝑧^𝑛.𝑛! (2.15)The only nonzero odd-indexed number is 𝐵₁=−1/2 and 𝐵₀=1, 𝐵₂=1/6, 𝐵₄=−1/30, 𝐵₆=1/42.

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 0(1) terms may still survive if 𝑋 has cohomology classes of appropriate degree to cup with 𝑐₂(𝑋) and each other. But the divisor invariants still vanish for dimensional reasons. Also note that (2.14) simplifies for the Calabi-Yau threefolds since 𝑐₁(𝑋)=0 and ∫_𝑋𝑐₃(𝑋)=𝜒(𝑋) are the Euler characteristics of 𝑋. Thus, for compacting Calabi-Yau threefolds,⟨1⟩_𝑔,0=(−1)^𝑔−1(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2⋅(2𝑔−2)(2𝑔)!𝜒(𝑋)2,𝑔≥2. (2.16)

When 𝑋 is noncompact but 𝛼≠0, the moduli 𝑀_𝑔,𝑛(𝑋,𝛼) may still be compact. This usually happens if geometry forces images of stable maps to stay within a fixed compact subset of 𝑋, 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 𝛼=0 factorization (2.11) forces 𝑀_𝑔,𝑛(𝑋,0) to be noncompact always. To the best of our knowledge, no virtual class theory exists for noncompact moduli so technically ⟨𝛾₁⋯𝛾_𝑛⟩_𝑔,0 for noncompact 𝑋 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 𝑋. This is consistent with a formal localization computation [19]. Unfortunately, for 𝑔=0,1, the invariants contain insertions and we really need to know how to interpret the integrals over 𝑋 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 (𝜒(𝑋)=2), the degree zero-free energy has the form𝐹⁰_{𝒪(−1)⊕𝒪(−1)}𝑝(𝑡;𝑥)=³(𝑡)𝑥²+𝑝₁(𝑡)+^∞_𝑔=2(−1)^𝑔−1(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2𝑥(2𝑔−2)(2𝑔)!^2𝑔−2, (2.17)where 𝑝_𝑖 are degree 𝑖 homogeneous polynomials with rational coefficients. We should mention that there are reasonable ways [15] of assigning values to 𝑝₃,𝑝₁ at least for local curves (see [11]) from equivariant and mirror symmetry viewpoints. For the resolved conifold, they yield𝐹⁰_{𝒪(−1)⊕𝒪(−1)}𝑡(𝑡;𝑥)=³61𝑥²+𝑡+12^∞_𝑔=2(−1)^𝑔−1(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2𝑥(2𝑔−2)(2𝑔−2)!^2𝑔−2, (2.18)and 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 0,1 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 𝐹∶=𝐹⁰+𝐹^′ and the (full) Gromov-Witten partition function is 𝑍∶=exp(𝐹)=𝑍⁰𝑍′, where 𝐹′,𝑍′ are reduced versions from Definition 2.1. As before, one writes 𝐹_𝑋,𝑍_𝑋 to indicate the target manifold if necessary.For the resolved conifold, we get from (2.10)𝐹_{𝒪(−1)⊕𝒪(−1)}(𝑡;𝑥)=𝐹⁰_{𝒪(−1)⊕𝒪(−1)}(𝑡;𝑥)+^∞_𝑔=0𝑑=1⟨1⟩_𝑔,𝑑𝑒^𝑑𝑡𝑥^2𝑔−2. (2.19)The positive degree part converges to a holomorphic function in an appropriate domain of 𝑡,𝑥 (recall that 𝑡 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,(2𝑔)!𝜋^2𝑔2^2𝑔−1<𝐵_2𝑔<(2𝑔)!𝜋^2𝑔(2^2𝑔−1−1),𝑔≥1, (2.20)and the general term in (2.17) grows factorially for any 𝑥≠0. 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 𝑀(𝑞), the point is 𝑞=1 and the relation to (2.17) is 𝑞=𝑒^𝑖𝑥. 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:𝑀(𝑞)∶=^∞_𝑛=11−𝑞^𝑛−𝑛,|𝑞|<1. (3.1)This is a classical generating function for the number of plane partitions [31] [8, I.5.13]. More to the point, it appears in [5–7] 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 3-manifold, and the Donaldson-Thomas invariants are holomorphic counterparts of the Casson invariant in the Chern-Simons theory [13].

The genus 𝑔 of a stable map is replaced in the Donaldson-Thomas invariant 𝐷_𝜅,𝛼 by the holomorphic Euler characteristic𝜅 of an ideal sheaf. As conjectured in [6, 7] and proved in [32], the degree zero partition function of the Calabi-Yau threefold 𝑋 is given by𝑍⁰_𝑋(𝑞)∶=^∞_𝜅=0𝐷_𝜅,0𝑞^𝜅=𝑀(−𝑞)^𝜒(𝑋), (3.2)where as before 𝜒(𝑋) 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 𝐹⁰ is the asymptotic expansion of ln𝑀(𝑒^𝑖𝑥)^{(1/2)𝜒(𝑋)} at 𝑥=0 (note the extra 1/2 in the exponent).

A quick look at (2.17) tells one that even for the resolved conifold, this can be true at best for 𝑔≥2 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.

𝜁-Resummation
To avoid imaginary numbers, we first consider ln𝑀(𝑒^−𝑥) instead of ln𝑀(𝑒^𝑖𝑥). For motivation, we start with a provocative “computation” that converts an expansion in powers of 𝑒^−𝑥 into one in powers of 𝑥 for a simpler function:𝑒^−𝑥1−𝑒^−𝑥=^∞_𝑛=1𝑒^−𝑛𝑥=^∞∞_𝑛=1𝑘=0(−𝑛𝑥)^𝑘𝑘!′=′^∞_𝑘=0(−𝑥)^𝑘𝑘!^∞_𝑛=11𝑛^−𝑘′=′^∞_𝑘=0(−1)^𝑘𝜁(−𝑘)𝑥𝑘!^𝑘. (3.3)The last two equalities are nonsense. Of course, the interchange of sums is illegitimate and ∑^∞_𝑛=1(1/𝑛^−𝑘)=∑^∞_𝑛=1𝑛^𝑘 is (very) divergent. It certainly does not converge to 𝜁(−𝑘) for positive 𝑘, although by definition 𝜁(𝑠)∶=∑^∞_𝑛=1(1/𝑛^𝑠),Re𝑠>1 is the Riemann zeta function [29]. Nonetheless, the end result is almost correct. Indeed, by definition of Bernoulli numbers (2.15),𝑒^−𝑥1−𝑒^−𝑥=1𝑥𝑥𝑒^𝑥=1−1𝑥^∞_𝑗=0𝐵_𝑗𝑥^𝑗,𝐵𝑗!𝜁(−𝑘)=−^𝑘+11𝑘+1,𝑘≥1;𝜁(0)=−2 (3.4)(3.5) [29], so 𝑒^−𝑥1−𝑒^−𝑥=1𝑥−12+^∞_𝑗=2𝐵_𝑗𝑥𝑗!^𝑗−1=1𝑥−12−^∞_𝑘=1𝐵_𝑘+1𝑥(𝑘+1)!^𝑘=1𝑥+^∞_𝑘=0(−1)^𝑘𝜁(−𝑘)𝑥𝑘!^𝑘. (3.6)In other words our “computation” (3.3) only missed the first term 1/𝑥.

A similar feat can be performed with ln𝑀(𝑒^−𝑥). First, we computeln𝑀(𝑒^−𝑥)=−^∞_𝑛=1𝑛ln(1−𝑒^−𝑛𝑥)=^∞_𝑛=1𝑛^∞_𝑘=1(𝑒^−𝑛𝑥)^𝑘𝑘=^∞_𝑘=11𝑘^∞_𝑛=1𝑛(𝑒^−𝑘𝑥)^𝑛=^∞_𝑘=11𝑘𝑒^−𝑘𝑥(1−𝑒^−𝑘𝑥)²=^∞_𝑘=11𝑘1(𝑒^𝑘𝑥/2−𝑒^{−(𝑘𝑥/2)})²=^∞_𝑘=1csch²(𝑘𝑥/2).4𝑘 (3.7)So far, all the manipulations are legitimate assuming 𝑥>0, although they would not be if we used 𝑒^𝑖𝑥 instead of 𝑒^−𝑥. Next, recall the Laurent expansion at zero of csch²:csch²(𝑧)=−^∞_𝑔=02^2𝑔(2𝑔−1)𝐵_2𝑔𝑧(2𝑔)!^2𝑔−2. (3.8)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,ln𝑀𝑒^−𝑥=−^∞_𝑘=114𝑘^∞_𝑔=04(2𝑔−1)𝐵_2𝑔2(2𝑔)!^2𝑔−2𝑘𝑥2^2𝑔−2′=′−^∞_𝑔=0(2𝑔−1)𝐵_2𝑔𝑥(2𝑔)!^∞2𝑔−2_𝑘=11𝑘^3−2𝑔′=′−^∞_𝑔=0(2𝑔−1)𝐵_2𝑔𝜁(3−2𝑔)𝑥(2𝑔)!^2𝑔−2=𝜁(3)𝑥²−𝜁(1)+12^∞_𝑔=2(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2𝑥(2𝑔−2)(2𝑔)!^2𝑔−2, (3.9)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 |𝑥|<2𝜋. Now, not only does it diverge factorially (see (2.20)) but also 𝜁(1) makes no sense at all, since 𝜁 has a pole at 1. Nonetheless, dropping the singular term 𝜁(3)/𝑥², the “infinite constant” −(𝜁(1)/12) and formally replacing 𝑥 by −𝑖𝑥 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 𝜁-resummation or 𝜁-regularization [33]. The amazing fact is not that this is reasonable to do in physics (one can argue that 𝜁(−𝑘) has the same operational properties as the nonexistent ∑^∞_𝑛=1𝑛^𝑘), 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 (0,∞) with a possible pole at 0 and polynomial decay at ∞, its Mellin transform is𝔐𝑓(𝑠)∶=^∞₀𝑥^𝑠−1𝑓(𝑥)𝑑𝑥. (3.10)The transform is defined and holomorphic in the convergence stripRe𝑠∈(𝛼,𝛽), when 𝑓(𝑥)∼𝑂(𝑥^−𝛼) at 0 and ∼𝑂(𝑥^−𝛽) at ∞, assuming 𝛼<𝛽. It is most useful when 𝔐𝑓 admits a meromorphic continuation to the entire complex plane since location of the poles determines asymptotic behavior of the function at 0 and ∞ (see [34] and below). For example, 𝔐[𝑒^−𝑛𝑥](𝑠)∶=Γ(𝑠)/𝑛^𝑠 in Re𝑠∈(0,∞) extends meromorphically with the poles of the gamma function located at 𝑠=0,−1,−2,…. Analogously,𝔐𝑒^−𝑥1−𝑒^−𝑥(𝑠)=^∞_𝑛=1𝔐[𝑒^−𝑛𝑥](𝑠)=^∞_𝑛=1Γ(𝑠)𝑛^𝑠=Γ(𝑠)𝜁(𝑠)inRe𝑠∈(1,∞) (3.11)extends with one additional zeta pole at 𝑠=1.

The inverse Mellin transform recovers 𝑓 as1𝑓(𝑥)=2𝜋𝑖^𝑐+𝑖∞_{𝑐−𝑖∞}𝔐𝑓(𝑠)𝑥^−𝑠𝑑𝑠for𝑐∈(𝛼,𝛽), (3.12)assuming absolute integrability along Re𝑠=𝑐. In the cases of interest to us, all the poles are located on the real axis to the left of 𝛼. 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 𝛼 and back, Figure 1. This reduces (3.12) to a sum over residues at the poles by the Cauchy residue theorem:𝑓(𝑥)=^∞_𝑛=0Res_{𝑠=−𝛾𝑛}𝔐𝑓(𝑠)𝑥^𝛾𝑛. (3.13)If 𝛾_𝑛 are integers and the series converges, 𝑓 must be real-analytic on (0,∞) with at worst a pole at 0, and the residues give its Laurent coefficients at 0. For example, one can compute the Taylor expansion of 𝑒^−𝑥 at 0 using that 𝔐[𝑒^−𝑥](𝑠)∶=Γ(𝑠), and the poles −𝛾_𝑛=−𝑛 of Γ are simple with the residues (−1)^𝑛 [29]. However, in most cases the series (3.13) diverges for all 𝑥≠0 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]:

Figure 1 

Barnes contour for Mellin transforms.

If−𝛾_𝑛are order𝑚_𝑛poles of (meromorphic continuation of)𝔐𝑓(𝑠)and its Laurent expansions at−𝛾_𝑛have the form𝐴𝔐𝑓(𝑠)=^𝑛0𝑠+𝛾_𝑛+𝐴_𝑛1(𝑠+𝛾_𝑛)²+^𝑚𝑛−1_𝑘=2𝐴_𝑛𝑘(𝑠+𝛾_𝑛)^𝑘+1, (3.14)then an asymptotic expansion of𝑓at𝑥=0is𝑓(𝑥)∼^∞_𝑛=0𝐴_𝑛0−𝐴_𝑛1ln𝑥+^𝑚𝑛−1_𝑘=2(−1)^𝑘𝐴_𝑛𝑘𝑘!ln^𝑘𝑥𝑥^𝛾𝑛. (3.15) Now, it becomes clear where the extra 1/𝑥 in (3.6) came from. In addition to gamma poles in (3.11) that produce terms ((−1)^𝑛/𝑛!)𝜁(−𝑛)𝑥^𝑛, there is also a simple pole of 𝜁(𝑠) with residue 1 that gives Γ(1)⋅1/𝑥=1/𝑥. Thus, (3.6) is at least an asymptotic expansion of 𝑒^−𝑥/(1−𝑒^−𝑥) at 0. 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 form𝑓(𝑥)=^∞_𝑘=1𝑎_𝑘𝑔𝜔_𝑘𝑥 (3.16)because their Mellin transforms can be easily expressed in terms of those of the base function 𝑔 [34]. One can think of them as sums of generalized harmonics with amplitudes𝑎_𝑘 and frequencies𝜔_𝑘, the usual ones corresponding to 𝑔(𝑥)=𝑒^𝑖𝑥,𝜔_𝑘=±𝑘. Indeed,𝔐𝑓(𝑠)=^∞_𝑘=1𝑎_𝑘^∞₀𝑥^𝑠−1𝑔(𝜔_𝑘𝑥)𝑑𝑥=^∞_𝑘=1𝑎_𝑘𝜔^𝑠_𝑘^∞₀𝑥^𝑠−1𝑔(𝑥)𝑑𝑥=𝐷(𝑠)𝔐𝑔(𝑠), (3.17)where 𝐷(𝑠)∶=∑^∞_𝑘=1𝑎_𝑘/𝜔^𝑠_𝑘 is the Dirichlet series of the sum. If 𝐷(𝑠) is entire and 𝔐𝑔(𝑠) only has simple poles at 𝑠=0,−1,−2,…, then𝑓(𝑥)∼^∞_𝑛=0Res_𝑠=−𝑛𝔐𝑔(𝑠)𝐷(−𝑛)𝑥^𝑛. (3.18)If, moreover, 𝑔 itself is entire and decays fast enough on ℝ₊, then 𝑔(𝑥)=∑^∞_𝑛=0𝑔_𝑛𝑥^𝑛, Res_𝑠=−𝑛[𝔐𝑔(𝑠)]=𝑔_𝑛 and 𝑓(𝑥)∼∑^∞_𝑛=0𝑔_𝑛𝐷(−𝑛)𝑥^𝑛. The same answer can be obtained by an (legitimate under the circumstances) interchange of sums in (3.16):𝑓(𝑥)=^∞_𝑘=1𝑎_𝑘^∞_𝑛=0𝑔_𝑛(𝜔_𝑘𝑥)^𝑛=^∞_𝑛=0𝑔_𝑛^∞_𝑘=1𝑎_𝑘𝜔^𝑛_𝑘𝑥^𝑛=^∞_𝑛=0𝑔_𝑛𝐷(−𝑛)𝑥^𝑛. (3.19)In particular, this expansion is not just asymptotic but convergent. If 𝐷(𝑠) 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 as𝐷-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):𝔐ln𝑀(𝑒^−𝑥)(𝑠)=^∞_𝑘=11𝑘^∞_𝑛=1𝑛𝔐𝑒^{−𝑛𝑘𝑥}(𝑠)=^∞_𝑘=11𝑘^∞_𝑛=1𝑛Γ(𝑠)(𝑛𝑘)^𝑠. (3.20)Now, assume that Re𝑠 is large enough for the double series to converge absolutely, for example, Re𝑠>2, and proceed=^∞_𝑘,𝑛=1Γ(𝑠)𝑛^𝑠−1𝑘^𝑠+1=Γ(𝑠)^∞_𝑛=11𝑛^𝑠−1⋅^∞_𝑘=11𝑘^𝑠+1=Γ(𝑠)𝜁(𝑠−1)𝜁(𝑠+1). (3.21)The extra zeta poles occur at 𝑠−1,𝑠+1=1, that is, 𝑠=0,2, and 𝑠=0 becomes a double pole. Formula (3.15) now yields an asymptotic expansion for ln𝑀(𝑒^−𝑥) 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 ln𝑀(𝑒^−𝑥) is due to Wright [36]. We sketch a proof for the convenience of the reader.

Theorem 3.1 (Ramanujan-Wright). Let 𝑀(𝑞)∶=∏^∞_𝑛=1(1−𝑞^𝑛)^−𝑛,|𝑞|<1 be the MacMahon function. Then ln𝑀(𝑒^−𝑥) has the Mellin transform 𝔐[ln𝑀(𝑒^−𝑥)](𝑠)=Γ(𝑠)𝜁(𝑠−1)𝜁(𝑠+1),Re𝑠>2, and its asymptotic expansion at 𝑥=0 along ℝ₊ isln𝑀(𝑒^−𝑥)∼𝜁(3)𝑥²+ln𝑥12+𝜁^′(−1)+^∞_𝑔=2(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2𝑥(2𝑔−2)(2𝑔)!^2𝑔−2. (3.22)Proof. Recall that 𝜁(𝑠) has “trivial zeros” at negative even integers [29]. Poles of Γ(𝑠) at negative odd integers are therefore canceled by zeros of 𝜁(𝑠−1). Analytical assumptions needed for (3.15) to hold are satisfied here by the classical estimates for Γ and 𝜁 [29]. The contributing poles are as follows.
(i) Gamma poles at 𝑠=−2,−4,…,−2𝑔,… with residues ((−1)^2𝑔/(2𝑔)!)𝜁(−2𝑔−1)𝜁(1−2𝑔).(ii) Simple pole of 𝜁(𝑠−1) at 𝑠=2 with residue 1⋅Γ(2)𝜁(3)=𝜁(3).(iii) Double pole of Γ(𝑠),𝜁(𝑠+1) at 𝑠=0. We have from the first two items and (3.5)𝜁(3)𝑥²+^∞_𝑔=11(2𝑔)!𝜁(−2𝑔−1)𝜁(1−2𝑔)𝑥^2𝑔=𝜁(3)𝑥²+^∞_𝑔=2(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2𝑥(2𝑔−2)(2𝑔)!^2𝑔−2. (3.23)To take care of the double pole, we need more than just the residue. By the well-known properties of Γ and 𝜁,Γ(1+𝑠)=1−𝛾𝑠+𝑂𝑠²,1𝜁(𝑠)=𝑠−1+𝛾+𝑂(𝑠−1), (3.24) where 𝛾 is the Euler constant. Thus,Γ(𝑠)𝜁(𝑠+1)=Γ(𝑠+1)𝜁(𝑠+1)𝑠=1𝑠1−𝛾+𝑂(𝑠)𝑠=1+𝛾+𝑂(𝑠)𝑠²1+𝑂(1),Γ(𝑠)𝜁(𝑠−1)𝜁(𝑠+1)=𝑠²+𝑂(1)𝜁(−1)+𝜁^′(−1)𝑠+𝑂(𝑠²)=𝜁(−1)𝑠²+𝜁^′(−1)𝑠+𝑂(1)=−1/12𝑠²+𝜁^′(−1)𝑠+𝑂(1). (3.25)By (3.15), the corresponding terms in the asymptotic expansion are ln𝑥/12+𝜁^′(−1), 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 𝜁(3)/𝑥² and even 1/12 in front of the logarithm. The only hidden term is 𝜁^′(−1), 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 𝑞 and 𝑥 is 𝑞=𝑒^𝑖𝑥 not 𝑞=𝑒^−𝑥. Replacing formally 𝑥 by −𝑖𝑥 in (3.22), we recover the infinite sum of (2.17) along with three extra terms:−𝜁(3)𝑥²+ln(−𝑖𝑥)12+𝜁^′(−1). (3.26)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 ln𝑀(𝑒^−𝑥) 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 𝑥 to −𝑖𝑥 while keeping 𝑥 real positive forces us to approach 𝑞=𝑒^𝑖0=1 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 𝑀(𝑞) analytically continued beyond the unit disk |𝑞|<1. But can it be continued?

Equation (3.1) does not look very promising. In fact, it strongly suggests that 𝑀(𝑞) has a singularity at each root of unity. But roots of unity are dense on the circle making it a natural boundary for 𝑀(𝑞) 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 𝑎/𝑏 is a proper irreducible fraction, thenln𝑀(𝑒^{2𝜋𝑖(𝑎/𝑏)}𝑒^−𝑥)∼𝜁(3)𝑏³𝑥²+𝑏12ln𝑥+𝑂(1) (3.27)for real positive 𝑥. Thus, every root of unity is indeed singular, and |𝑞|=1 is the natural boundary.

This forces us to reconsider keeping 𝑥 real in ln𝑀(𝑒^𝑖𝑥). Should 𝑥 approach 0 from the positive imaginary direction, we can set 𝑥=𝑖𝑦 with 𝑦>0, and Theorem 3.1 gives us an asymptotic expansion in 𝑦. We can rewrite it as an expansion in 𝑥 of course as long as it is understood that 𝑥 in it is positive imaginary. This may seem like an underhanded trick but it is not. The natural domain of ln𝑀(𝑒^𝑖𝑥) is the upper half-plane, and the only distinguished direction in its interior is the positive imaginary one.

Corollary 3.2. Asymptotic expansion of ln𝑀(𝑒^𝑖𝑥) at 𝑥=0 along 𝑖ℝ₊ is (taking the principal branch of the logarithm)ln𝑀(𝑒^𝑖𝑥)∼−𝜁(3)𝑥²+ln(−𝑖𝑥)12+𝜁^′(−1)+^∞_𝑔=2(−1)^𝑔−1(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2𝑥(2𝑔−2)(2𝑔)!^2𝑔−2∼−𝜁(3)𝑥²+ln𝑥12+𝜁^′(−1)−𝜋𝑖+24^∞_𝑔=2(−1)^𝑔−1(2𝑔−1)𝐵_2𝑔𝐵_2𝑔−2𝑥(2𝑔−2)(2𝑔)!^2𝑔−2. (3.28)Comparing this to (2.17), one ought to be somewhat perplexed. If we are to take (3.28) at face value then 𝑝₃(𝑡)=−𝜁(3), 𝑝₁(𝑡)=𝜁′(−1)−𝜋𝑖/24 (?!), and there is no space for ln𝑥/12 at all. Aside from the fact that 𝑝_𝑖-s are supposed to be homogeneous polynomials of the corresponding degree, the numbers involved are not even rational, 𝜁(3) 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 ln𝑥/12, but we may guess, that averaging of 𝑝₁(𝑡) 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 𝑁 duality is that the same string theory is realized on manifolds with different topology [19, 20]. However, the degree zero terms in genus 0,1 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 3 manifolds. The reason to compare them is the conjectural large 𝑁 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 3 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 ℝ² 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 ℤ², and every triple at a vertex if directed away from it adds up to zero. Examples for the resolved conifold 𝒪(−1)⊕𝒪(−1) and the local ℂℙ¹×ℂℙ¹ (i.e., the total space of 𝑇^1,0(ℂℙ¹×ℂℙ¹)) are shown in Figure 2, where the primitive directions of noncompact edges are also indicated. Toric webs related by a 𝐺𝐿₂ℤ 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 (0,0), 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 3-manifold is recovered from its link diagram up to diffeomorphism by surgery on the link [37, 38]. Having toric webs rigidly embedded in ℝ² 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.

Figure 2 

Toric webs of 𝒪(−1)⊕𝒪(−1) and local ℂℙ1×ℂℙ1.

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 ℂℙ¹). Being rational curves sitting inside the Calabi-Yau threefold, their normal bundles are isomorphic to 𝒪(𝑛−1)⊕𝒪(−𝑛−1),𝑛=±1,±2,…. The framing number𝑛_𝑒 for each edge 𝑒 is assigned the value from the normal bundle type of the corresponding curve. This only determines 𝑛_𝑒 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 𝜉₁,…,𝜉_𝑘 is an integral basis in 𝐻₂(𝑋,ℤ) as in Section 2, then each edge curve 𝐶_𝑒 represents a homology class expressible as a linear combination [𝐶_𝑒]=𝑚₁𝜉₁+⋯+𝑚_𝑘𝜉_𝑘, 𝑚_𝑖∈ℤ. 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 𝒪(−1)⊕𝒪(−1) and the local ℂℙ¹×ℂℙ¹ 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].

Figure 3 

Toric graphs of 𝒪(−1)⊕𝒪(−1) and local ℂℙ1×ℂℙ1.

Partitions and the Schur Functions
We wish to briefly describe the topological vertex algorithm to see how 𝑞-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 partition𝜆 is an element of ℤ^∞₊ with only finitely many nonzero entries that are nonincreasing, that is,𝜆=𝜆₁,𝜆₂,…,𝜆_𝑁,0,…,𝜆_𝑖∈ℤ₊,𝜆₁≥𝜆₂≥⋯≥𝜆_𝑁≥0. (4.1)Let denote the set of all partitions. The number of nonzero entries 𝑙(𝜆) is called the length of a partition, and the sum of all entries |𝜆|∶=𝜆₁+𝜆₂+⋯+𝜆_𝑁 is called its size (or weight). Partitions are visualized by Young diagrams, rows of boxes stacked top down with 𝜆_𝑖 boxes in 𝑖th row, Figure 4. The conjugate partition𝜆^′ is obtained visually by transposing the Young diagram along the main diagonal and analytically as 𝜆^′_𝑖∶=max{𝑗∣𝜆_𝑗≥𝑖}. Note that 𝜆^′′=𝜆 and 𝜆^′₁=𝑙(𝜆), |𝜆′|=|𝜆|. Another relevant characteristic of a partition, sometimes called its quadratic Casimir, is𝜘(𝜆)∶=^∞_𝑖=1𝜆_𝑖𝜆_𝑖−2𝑖+1,𝜘(𝜆^′)=−𝜘(𝜆),𝜘(𝜆)∈2ℤ. (4.2)Partitions 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.

(a)

(b)

Figure 4 

Young diagram and its conjugate.

Amplitudes (see Definition 4.2) of a labeled graph are defined via a specialization of the Schur functions𝑠_𝜆 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 𝜆=(1^𝑛)∶=1,1,…,1_𝑛times,0,… (4.3) then 𝑠₍₁^𝑛₎ is the 𝑛th elementary symmetric function:𝑠₍₁^𝑛₎(𝑥)=𝑒_𝑛(𝑥)∶=_{1≤𝑖1<⋯<𝑖𝑛<∞}𝑥_𝑖1⋯𝑥_𝑖𝑛. (4.4)In general, 𝑠_𝜆 are polynomials in the elementary symmetric functions given by the Jacobi-Trudy formula𝑠_𝜆=det(𝑒_{𝜆′𝑖−𝑖+𝑗}),1≤𝑖,𝑗≤𝑙(𝜆′)=𝜆₁. For example,𝑠_(2,1,0,…)𝑒(𝑥)=²𝑒₀𝑒₃𝑒₁=𝑒₁𝑒₂−𝑒₀𝑒₃=^∞_𝑖=1𝑥_𝑖⋅^∞_𝑖<𝑗=1𝑥_𝑖𝑥_𝑗−^∞_{𝑖<𝑗<𝑘=1}𝑥_𝑖𝑥_𝑗𝑥_𝑘. (4.5)Since 𝑒_𝑛 are homogeneous of degree 𝑛, the Jacobi-Trudy formula implies that 𝑠_𝜆 are also homogeneous of degree |𝜆|, that is, 𝑠_𝜆(𝑎𝑥)=𝑎^|𝜆|𝑠_𝜆(𝑥). Moreover, 𝑠_𝜆,𝜆∈𝒫 form a linear basis in the space of symmetric functions, in particular 𝑠_𝜆𝑠_𝜇=∑_𝜈∈𝒫𝑐^𝜆_𝜇𝜈𝑠_𝜈. It turns out that 𝑐^𝜆_𝜇𝜈 are nonnegative integers that vanish unless |𝜈|=|𝜆|+|𝜇|. 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 𝑥_𝑖 to elements of a geometric series possibly modified at finitely many entries. Such specializations were extensively studied by Zhou [4]. Define the Weyl vector𝜌 by−1𝜌∶=23,−2=1,…−𝑖+2^∞_𝑖=1. (4.6)Note that 𝜌 is not a partition. Introduce a new formal variable 𝑞 and for any vector 𝜉 set 𝑞^𝜉∶=(𝑞^𝜉1,𝑞^𝜉2,…), so, in particular, 𝑞^𝜌=(𝑞^−(1/2),𝑞^−(3/2),…) is a geometric series.

Definition 4.1. One-, two-, and three-point functions of the topological vertex are, respectively [4, 12],𝒲_𝜆(𝑞)∶=𝑠_𝜆(𝑞^𝜌)=𝒲_𝜆0𝒲(𝑞),_𝜆𝜇(𝑞)∶=𝑠_𝜆(𝑞^𝜌)𝑠_𝜇𝑞^𝜆+𝜌,𝒲_𝜆𝜇𝜈(𝑞)∶=𝑞^{(𝜘(𝜇)+𝜘(𝜈))/2}_{𝛼,𝛽,𝛾∈𝒫}𝑐^𝜆_𝛼𝛾𝑐^𝜈′_𝛾𝛽𝒲_𝜇′𝛼(𝑞)𝒲_𝜇𝛽′(𝑞)𝒲_𝜇0.(𝑞) (4.7)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 𝑞∈ℂ⧵ℝ₋ and 𝑞^1/2 is then defined by the principal branch of the square root. One can see by inspection from (4.4) that 𝑒_𝑛(𝑞^𝜆+𝜌) converges for |𝑞|>1. Since the Schur “functions” 𝑠_𝜇 are polynomials in 𝑒_𝑛, they are also well defined as honest functions of 𝑞 upon specializing to 𝑞^𝜆+𝜌.

To be consistent with the usual basic hypergeometric notation [42], we wish to switch from |𝑞|>1 to |𝑞|<1. This can be done using a symmetry of the two-point functions [4]𝒲_𝜆𝜇(𝑞)=(−1)^{|𝜆|+|𝜇|}𝒲_{𝜆′𝜇′}(𝑞⁻¹)=(−1)^{|𝜆|+|𝜇|}𝑠_𝜆′(𝑞^−𝜌)𝑠_𝜇′𝑞^{−𝜆′−𝜌}. (4.8)This identity is a curious one since the two sides never converge simultaneously (both diverge for |𝑞|=1). It has the same meaning as a more familiar identity:^∞_𝑖=1𝑞^𝑖=𝑞11−𝑞=−1−𝑞⁻¹=−^∞_𝑖=0𝑞^−𝑖=−𝑞^∞_𝑖=1𝑞^−𝑖, (4.9)where the two sides never converge simultaneously either. In fact, 𝒲_𝜆𝜇(𝑞) are rational functions of 𝑞^1/2 and can be analytically continued to ℂ⧵ℝ₋, (4.8) expresses this analytic continuation.

The appearance of 𝑞-bifactorials in partition functions is due to the Cauchy identity for the Schur functions [4, 8]_𝜆∈𝒫𝑠_𝜆(𝑥)𝑠_𝜆′(𝑦)𝑢^|𝜆|=^∞_𝑖,𝑗=11+𝑢𝑥_𝑖𝑦_𝑗. (4.10)If 𝑥_𝑖=𝑞^𝑖−1, 𝑦_𝑗=𝑞^𝑗−1, the right-hand side of (4.10) becomes^∞_𝑖,𝑗=11+𝑢𝑞^𝑖−1𝑞^𝑗−1=^∞_𝑖,𝑗=01+𝑢𝑞^𝑖+𝑗=(𝑢;𝑞)_∞⁽²⁾. (4.11)Note 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 𝑉 and 𝐸_𝑐 denote the sets of vertices and compact edges of a toric graph, respectively. Choose an arbitrary orientation for each element of 𝐸_𝑐, this determines the sign of the framing numbers. Assign a formal variable 𝑎_𝑖 to each element of a basis 𝜉_𝑖∈𝐻₂(𝑋,ℤ), and set 𝑎_𝑒∶=𝑎^𝑚1₁⋯𝑎^𝑚𝑘_𝑘 for the corresponding edge curve [𝐶_𝑒]=𝑚₁𝜉₁+⋯+𝑚_𝑘𝜉_𝑘. Finally, label all compact edges by arbitrarily chosen partitions 𝜆_𝑒∈𝒫 and noncompact ones by the trivial partition 0∈𝒫. Triples of partitions ^→𝜆_𝑣∶=(𝜆₁,𝜆₂,𝜆₃) 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 0^′=0. This determines the triple up to cyclic permutation which is enough since 𝒲^→_𝜆𝑣∶=𝒲_{𝜆1𝜆2𝜆3} has cyclic symmetry.

Figure 5 

Partition triple at a vertex →𝜆𝑣∶=(𝜆′,𝜇′,𝜈).

Definition 4.2. Amplitude of a labeled toric graph relative to a basis 𝜉₁,…,𝜉_𝑘∈𝐻₂(𝑋,ℤ) is given by [12, 27]𝐴_{𝜆𝑒}𝑎₁,…,𝑎_𝑘;𝑞∶=_{𝑒∈𝐸𝑐}(−1)^{|𝜆𝑒|(𝑛𝑒+1)}𝑞^{𝑛𝑒𝜘(𝜆𝑒)/2}𝑎^|𝜆𝑒|_𝑒⋅_𝑣∈𝑉𝒲^→_𝜆𝑣(𝑞). (4.12)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 𝑋 relative to a basis 𝜉₁,…,𝜉_𝑘∈𝐻₂(𝑋,ℤ) is given by a state sum:𝑍^′_𝑋𝑎₁,…,𝑎_𝑘;𝑞=_{{𝜆𝑒}𝜆𝑒∈𝒫}𝐴_{𝜆𝑒}𝑎₁,…,𝑎_𝑘;𝑞=_{𝜆1,…,𝜆𝑐||𝐸∈𝒫}𝐴_{𝜆1,…,𝜆𝑐||𝐸}𝑎₁,…,𝑎_𝑘;𝑞, (4.13)assuming in the second sum that the edges are numbered and 𝜆_𝑖∶=𝜆_𝑒𝑖.

Partition Function of the Resolved Conifold
Here, 𝐻₂(𝑋,ℤ) is one-dimensional and 𝜉=[ℂℙ¹]. There is only one 𝑎 variable and only one edge. The amplitude 𝐴_𝜆 for 𝜆∈𝒫 is (see Figure 3 and (4.8))𝐴_𝜆(𝑎;𝑞)∶=(−1)^|𝜆|𝑎^|𝜆|⋅𝒲_𝜆00(𝑞)𝒲_𝜆′00(𝑞)=(−𝑎)^|𝜆|𝒲_𝜆(𝑞)𝒲_𝜆′(𝑞)=(−𝑎)^|𝜆|(−1)^{|𝜆|+|𝜆′|}𝑠_𝜆′(𝑞^−𝜌)𝑠_𝜆(𝑞^−𝜌). (4.14)Recalling that |𝜆^′|=|𝜆|, −𝜌=(𝑖−(1/2))^∞_𝑖=1 and 𝑠_𝜆 is homogeneous of degree |𝜆|, we compute further(−𝑎)^|𝜆|𝑠_𝜆′𝑞^𝑖−(1/2)𝑠_𝜆𝑞^𝑗−(1/2)=(−𝑎)^|𝜆|𝑞^{(|𝜆|+|𝜆′|)/2}𝑠_𝜆′𝑞^𝑖𝑠_𝜆𝑞^𝑗=−𝑎𝑞^−1|𝜆|𝑠_𝜆′𝑞^𝑖𝑠_𝜆𝑞^𝑗. (4.15)Suppose that 𝑎 is small enough for ∑_𝜆∈𝒫𝐴_𝜆(𝑎;𝑞) to converge then, we get by Theorem 4.3 and the Cauchy identity (4.10)𝑍^′_𝑋(𝑎;𝑞)=_𝜆∈𝒫𝒲_𝜆(𝑞)𝒲_𝜆′(𝑞)(−𝑎)^|𝜆|=_𝜆∈𝒫𝑠_𝜆′𝑞^𝑖𝑠_𝜆𝑞^𝑗−𝑎𝑞^−1|𝜆|,=^∞_𝑖,𝑗=11+−𝑎𝑞⁻¹𝑞^𝑖𝑞^𝑗=^∞_𝑖,𝑗=01−𝑎𝑞𝑞^𝑖+𝑗=(𝑎𝑞;𝑞)_∞⁽²⁾. (4.16)(4.17)If we accept the MacMahon function as the degree zero partition function of the resolved conifold (despite the issues discussed after Corollary 3.2), then𝑍⁰_𝑋(𝑞)=𝑀(𝑞)^𝜒(𝑋)/2=1∏^∞_𝑛=1(1−𝑞^𝑛)^𝑛=1(𝑞;𝑞)_∞⁽²⁾. (4.18)We conclude that the (full) Gromov-Witten partition function of the resolved conifold is𝑍_𝑋(𝑎;𝑞)=𝑍⁰_𝑋(𝑞)𝑍^′_𝑋(𝑎;𝑞)=(𝑎𝑞;𝑞)_∞⁽²⁾(𝑞;𝑞)_∞⁽²⁾, (4.19)as used in Section 1.

5. Reshetikhin-Turaev Calculus and Partition Function of the 3-Sphere

As explained in the beginning of Section 4, this one is complementary to it. We briefly review the 𝔰𝔩_𝑁ℂ 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 𝔰𝔩_𝑁ℂ, and rephrase everything directly in terms of partitions. The immediate goal is to compute the partition function of 𝑆³ 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 3 manifolds, especially effective for the Seifert-fibered ones [43]. This circumstance combined with explicit large 𝑁 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 𝑘,𝑁 known as level and rank. This would not be much of a hindrance if not for the tenuous connection between invariants for different 𝑘 and 𝑁. As a rule, geometers are interested in asymptotic behavior for large 𝑘 [45] and physicists in both large 𝑘 and large 𝑁 behavior. The Reshetikhin-Turaev sums with ranges depending explicitly on 𝑘 and 𝑁 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 3-manifold can be obtained by surgery on a framed link in 𝑆³ [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 3-manifolds via surgery.

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 3-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.

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, 𝔰𝔩_𝑁ℂ into quantum groups and then specialize the deformation parameter 𝑞 to a root of unity 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)}. 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 𝔰𝔩_𝑁ℂ at level 𝑘, they are indexed by partitions with the Young diagrams in the (𝑁−1)×𝑘rectangle, that is,𝒫^𝑘_𝑁−1∶={𝜆∈𝒫∣𝑙(𝜆)≤𝑁−1,𝑙(𝜆^′)=𝜆₁≤𝑘}. (5.1)In the equivalent language of dominant weights, this corresponds to the weights in the Weyl alcove of the Cartan-Stiefel diagram of 𝔰𝔩_𝑁ℂ scaled by 𝑘, 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 𝒫^𝑘_𝑁−1, 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 𝑘,𝑁. 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 𝑅-matrix of a quantum group [19, 37]. Good news is that for a large class of 3-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 matrices𝑆 and 𝑇:𝑆_𝜆𝜇=𝑆₀₀𝑊_𝜆𝜇;𝑇_𝜆𝜇=𝑇₀₀𝑞^{(1/2)𝐶2(𝜆𝑁)}𝛿_𝜆𝜇∗. (5.2)The notation is as follows.

𝑊_𝜆𝜇 is the normalized quantum invariant of the Hopf link Figure 7 with components labeled by partitions 𝜆,𝜇 (see more below);𝜇^∗ is the partition 𝔰𝔩_𝑁-dual to 𝜇, 𝜇^∗_𝑖∶=𝜇₁−𝜇_{𝑁−𝑖+1} for 1≤𝑖≤𝑁−1 and 𝜇^∗_𝑖∶=0 for𝑖≥𝑁 (not to be confused with the conjugate partition𝜇^′); 𝜆^𝑁 denotes the 𝔤𝔩_𝑁 coordinates of (𝑁−1)×𝑘 partition 𝜆, 𝜆^𝑁_𝑖∶=𝜆^𝑁_𝑖−|𝜆|/𝑁, in particular 𝜌^𝑁_𝑖∶=(1/2)(𝑁−2𝑖+1);𝐶₂(𝜆^𝑁)∶=𝜆^𝑁⋅(𝜆^𝑁+2𝜌^𝑁) is the quadratic Casimir of 𝜆 as a dominant weight closely related to the quadratic Casimir 𝜘(𝜆) of a partition 𝜆: 𝐶₂𝜆^𝑁=𝜘(𝜆)+𝑁|𝜆|−|𝜆|²𝑁. (5.3) We will say more about the normalization constants 𝑆₀₀,𝑇₀₀ below.

Formulas for 𝑊_𝜆𝜇 were originally obtained by Kac and Peterson in 1984 in the context of affine Lie algebras. Their relevance to the Chern-Simons theory was discovered by Witten [41]. In 2001, Lukac [48] realized that they are specializations of the Schur functions of 𝑁 variables, namely,𝑊_𝜆𝜇(𝑁;𝑞)∶=𝑠_𝜆𝑞^𝜌𝑁𝑠_𝜇𝑞^{𝜆𝑁+𝜌𝑁}with𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)}. (5.4)This should be compared to the two-point functions (4.7) of the topological vertex. This realization led among other things to the physical derivation of the topological vertex, where 𝒲_𝜆𝜇(𝑞) are obtained as some loosely interpreted limits of 𝑊_𝜆𝜇(𝑁;𝑞) [20, 26, 40]. Let us emphasize the differences though. In (4.7), 𝑞 is a formal variable whereas in (5.4) it is a number. Moreover, the number of variables in 𝑠_𝜆,𝑠_𝜇 before specialization is infinite whereas here it is 𝑁<∞. This last circumstance dramatically simplifies many computations with 𝒲_𝜆𝜇 compared to their analogs with 𝑊_𝜆𝜇 because infinite specializations often have nice analytic expressions [8].

The twist matrix 𝑇 also has a vertex counterpart in the form of the framing numbers 𝑛_𝑒 that contribute factors of 𝑞^{𝑛𝑒𝜘(𝜆𝑒)/2} to the amplitude (4.12). Incidentally, this explains their name. In the Reshetikhin-Turaev sums, 𝑇_𝜆𝜇 factors account for twists in link diagrams that in their turn represent framing of a link, that is, trivialization of its normal bundle up to homotopy. If one thinks of strands as thin ribbons, the signed number of twists gives exactly the signed number of full twists in a ribbon.

The two-point functions are symmetric 𝑊_𝜆𝜇=𝑊_𝜇𝜆, and the one-point functions 𝑊_𝜆∶=𝑊_𝜆0=𝑊_0𝜆 are called the quantum dimensions of representations indexed by 𝜆. The quantum diameter is𝒟²∶=_{𝜆∈𝒫𝑘𝑁−1}𝑊_𝜆(𝑁;𝑞)², (5.5)and 𝑆₀₀ is simply its inverse 𝑆₀₀∶=𝒟⁻¹. Analogously, 𝑇₀₀ is the inverse of the charge factor𝜁∶=𝑒^{2𝜋𝑖𝑐/24}, where 𝑐∶=𝑘dim_ℂ(𝔰𝔩_𝑁ℂ)/(𝑘+𝑁) is the so-called central charge from conformal field theory [46]. Thus, explicitly, 𝑇₀₀∶=𝜁⁻¹=𝑞^{−𝑘(𝑁2−1)/24}. These normalizations are needed to make the 𝑆 and 𝑇 matrices satisfy the defining relations of 𝑆𝐿₂ℤ∶(𝑆𝑇)³=𝑆²,𝑆²𝑇=𝑇𝑆²,𝑆⁴=𝐼. (5.6)Note that unlike 𝑊_𝜆𝜇 and 𝑞^{(1/2)𝐶2(𝜆𝑁)} that depend on 𝑘 only via 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)} this is no longer the case for the normalizing constants 𝑆₀₀,𝑇₀₀, and this causes major problems in relating the Chern-Simons expressions to the Gromov-Witten ones [19].

Reshetikhin-Turaev Invariants
Let 𝐽_{𝜆1,…,𝜆𝑛}(𝐿) denote the colored HOMFLY polynomial of an 𝑛-component link 𝐿, that is, the amplitude of the link diagram computed as outlined above after labeling the link components by partitions (colors) 𝜆₁,…,𝜆_𝑛, see [19, 37] for specifics. Then the Reshetikhin-Turaev invariant of the 3-manifold 𝑀 surgered on 𝐿 from 𝑆³ is𝜏(𝑀)∶=𝜁^−3𝜎𝒟^−𝑛−1_{𝜆1,…,𝜆𝑛∈𝒫𝑘𝑁−1}𝐽_{𝜆1,…,𝜆𝑛}(𝐿)𝑊_𝜆1⋯𝑊_𝜆𝑛, (5.7)where 𝜎 is the signature of the linking matrix of 𝐿 [38]. Since 𝑆³ can be obtained from itself by surgery on the empty link ∅ with 𝐽(∅)=1, 𝑛=0 components and 0 linking matrix, we have𝑍_𝑆³∶=𝜏(𝑆³)=𝒟⁻¹=𝑆₀₀=_{𝜆∈𝒫𝑘𝑁−1}𝑊_𝜆(𝑁;𝑞)^2−(1/2). (5.8)This is the Chern-Simons partition function of the3-sphere to be identified after Witten [1] and Gopakumar-Vafa [2] with the string partition function of 𝑇^∗𝑆³. Note that (5.5) bears some resemblance to the expression (4.16) for the reduced partition function of the resolved conifold especially if we interpret (−𝑎)^|𝜆| as a convergence factor needed to extend the sum from 𝒫^𝑘_𝑁−1 to all partitions. However, unlike (4.16) that gives 𝑍^′_𝑋 directly formula (5.5) gives 𝑍_𝑆⁻²₃, so naively one does not expect these partition functions to be nearly equal.

The normalization adopted in (5.7) is the one that makes 𝜏(𝑀) into an honest invariant. It is due to Reshetikhin and Turaev and is known to differ from the physical normalization of Witten [41] coming from the path integral. Although the physical normalization gives better agreement with the Gromov-Witten theory [49], it has not been consistently defined in general. In the known examples, it is discerned by comparing 𝜏(𝑀) to heuristic path integral expansions [45].

Partition Function of 𝑆³
Although we used 𝑞 above as much as possible, one should not forget that in the context of the Chern-Simons theory this is simply a shorthand for 𝑒^{2𝜋𝑖/(𝑘+𝑁)} and all the relevant quantities are only defined for positive integers 𝑘,𝑁. Large 𝑁 duality predicts among other things that it should be possible to interpret 𝜏(𝑀) as restrictions of holomorphic in 𝑞 functions to these special values, perhaps up to some parasite factors stemming form misnormalizations. It is these holomorphic functions that should correspond to the Gromov-Witten generating functions in 𝑞. According to this philosophy, we should try to transform the right-hand side of (5.5) into an explicit function of 𝑞 and 𝑁 as far as possible. A clean way of doing this is not known to the author, the problem being the rogue range of summation 𝒫^𝑘_𝑁−1. The known ways via Poisson resummation [43, 4.2] or group characters [19, Theorem 10] are rather down and dirty and we do not reproduce them here. An intermediate answer is𝒟²=(−1)^{𝑁(𝑁−1)/2}𝑁(𝑘+𝑁)^𝑁−1_{1≤𝑖<𝑗≤𝑁}𝑞^{(1/2)(𝑗−𝑖)}−𝑞^{−(1/2)(𝑗−𝑖)−2}, (5.9)and it fulfils our wish only partially. One could replace 𝑘+𝑁 by ln𝑞/2𝜋𝑖 but this does not lead to anything useful.

In fact, 𝑁(𝑘+𝑁)^𝑁−1 is the volume of the fundamental parallelepiped of a rescaled root lattice Λ_𝑟 of 𝔰𝔩_𝑁ℂ, namely, of (𝑘+𝑁)^1/2Λ_𝑟. Ooguri and Vafa [49, 2.1] replace it with the volume of 𝑈_𝑁 in their normalization. Thus, we ignore 𝑁(𝑘+𝑁)^𝑁−1 as a parasite factor and focus on the product following it instead. Since ∑_{1≤𝑖<𝑗≤𝑁}1=𝑁(𝑁−1)/2 and ∑_{1≤𝑖<𝑗≤𝑁}(𝑗−𝑖)=𝑁(𝑁²−1)/12, we have_{1≤𝑖<𝑗≤𝑁}𝑞^{(1/2)(𝑗−𝑖)}−𝑞^{−(1/2)(𝑗−𝑖)}=𝑞^{−(2𝑁(𝑁2−1)/24)}_{1≤𝑖<𝑗≤𝑁}1−𝑞^𝑗−𝑖. (5.10)Note that 𝑞^{−(2𝑁(𝑁2−1)/24)}=𝜁^{−2𝑁/𝑘}, and a power of 𝜁 appears explicitly in the Reshetikhin-Turaev normalization (5.7). We get for the partition function𝑍_𝑆³∶=𝜏𝑆³=𝑖^{𝑁(𝑁−1)/2}𝑞^{−(𝑁(𝑁2−1)/12)}𝑁(𝑘+𝑁)^{𝑁−1−(1/2)}_{1≤𝑖<𝑗≤𝑁}1−𝑞^𝑗−𝑖. (5.11)The first two factors are 1 in absolute value and can be regarded as framing corrections and we already discussed the third (volume) factor. In any case, it turns out that only the last product is relevant in the context of large 𝑁 duality as we now demonstrate (cf. [50, appendix]).

Lemma 5.1. For any |𝑞|<1,_{1≤𝑖<𝑗≤𝑁}1−𝑞^𝑗−𝑖=^𝑁−1_𝑛=11−𝑞^{𝑛𝑁−𝑛}=(𝑞;𝑞)^𝑁_∞(𝑞^𝑁+1;𝑞)_∞⁽²⁾(𝑞;𝑞)_∞⁽²⁾, (5.12)where (𝑎;𝑞)_∞∶=(𝑎;𝑞)_∞⁽¹⁾ is the usual 𝑞-shifted factorial [42], and (𝑎;𝑞)_∞^(𝑑) was defined in Section 1 (see (1.6).Proof. First, we arrange the factors according to the powers of 𝑞: _{1≤𝑖<𝑗≤𝑁}1−𝑞^𝑗−𝑖=^𝑁−1_𝑛=11−𝑞^{𝑛∑𝑁−1𝑖=11{𝑖∣𝑛+𝑖≤𝑁}}=^𝑁−1_𝑛=11−𝑞^{𝑛𝑁−𝑛}, (5.13)where 1_𝑆 denotes the characteristic function of a set 𝑆. A finite product can be written as a ratio of two infinite ones as long as the latter converge, in particular^𝑁−1_𝑛=11−𝑞^{𝑛𝑁−𝑛}=∏^∞_𝑛=1(1−𝑞^𝑛)^𝑁−𝑛∏^∞_𝑛=𝑁(1−𝑞^𝑛)^𝑁−𝑛=∏^∞_𝑛=1(1−𝑞^𝑛)^𝑁−𝑛∏^∞_𝑛=0(1−𝑞^𝑁+𝑛)^−𝑛=^∞_𝑛=1(1−𝑞^𝑛)^𝑁∏^∞_𝑛=1(1−𝑞^𝑁+𝑛)^𝑛∏^∞_𝑛=𝑁(1−𝑞^𝑛)^𝑛=(𝑞;𝑞)^𝑁_∞(𝑞^𝑁+1;𝑞)_∞⁽²⁾(𝑞;𝑞)_∞⁽²⁾. (5.14)Setting 𝑎=𝑞^𝑁 in (5.12) and comparing to the partition function of the resolved conifold (4.19), we see that the Chern-Simons theory reproduces both the reduced partition function and the MacMahon factor. It also produces an additional one (𝑞;𝑞)^𝑁_∞ aside from the parasite factors discussed above. This extra factor appears naturally though in the quantum Barnes function𝐺_𝑞. Its most characteristic property is the functional equation 𝐺_𝑞(𝑧+1)=Γ_𝑞(𝑧)𝐺_𝑞(𝑧), where Γ_𝑞 is the quantum gamma function of Jackson [29]. Γ_𝑞 is a 𝑞-deformation of the classical Euler Γ, and the classical Barnes function satisfies the same equation with 𝑞-s removed [14]. As we derive in Section 6 (Theorem 6.3) to have this property, 𝐺_𝑞 must be the product𝐺_𝑞(𝑧+1)=(1−𝑞)^{−(𝑧(𝑧−1)/2)}(𝑞;𝑞)^𝑧_∞(𝑞^𝑧+1;𝑞)_∞⁽²⁾(𝑞;𝑞)_∞⁽²⁾, (5.15)that is, reproduce the right-hand side of (5.12) with 𝑧=𝑁 up to a power of 1−𝑞. Emergence of the same expressions from a simple functional equation is quite intriguing as well as the fact that they are the ones common in the Chern-Simons and the Gromov-Witten theories. We summarize the observations made in Sections 4 and 5 as a theorem.

Theorem 5.2. Quantum Barnes function 𝐺_𝑞 is the common factor of the partition functions of the resolved conifold 𝑋 and the 3-sphere, namely,(1−𝑞)^{𝑧(𝑧−1)/2}𝑍_𝑋𝑞^𝑧;𝑞=(𝑞;𝑞)_∞^−𝑧𝐺_𝑞(𝑧+1),(1−𝑞)^{𝑧(𝑧−1)/2}𝑍_𝑆³(𝑧;𝑞)=𝑖^{𝑧(𝑧−1)/2}𝑞^{−(𝑧(𝑧2−1)/12)}𝑧^−(1/2)ln𝑞2𝜋𝑖^{−((𝑧−1)/2)}𝐺_𝑞(𝑧+1). (5.16)(5.17)In (5.16), 𝑞 is the usual variable of the topological vertex, and 𝑎∶=𝑞^𝑧 is the degree variable [12]. In (5.17) , 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)} and 𝑧=𝑁, where 𝑘,𝑁 are level and rank. Equivalently, if 𝑡 is the Kähler parameter and 𝑥 the string coupling constant of [20], then 𝑞=𝑒^𝑖𝑥,𝑞^𝑧=𝑒^−𝑡.

6. Quantum Multigamma Hierarchy

In this section, we give a self-contained introduction into the theory of the quantum Barnes function and its higher order analogs, quantum multigammas. The main result is the alternating formula (6.14) for 𝐺_𝑞^(𝑑) that displays its graded product structure. Along the way, we describe connections to other special functions that came into the spotlight lately, 𝑞-shifted multifactorials and quantum polylogarithms.

Quantum Multigammas
Quantum multigammas emerge naturally if one iterates two classical constructions. One is the construction of factorials from natural numbers with subsequent analytic continuation to complex values via the functional equation Γ(𝑧+1)=𝑧Γ(𝑧). The other is a 𝑞-deformation of Γ first performed by Jackson, thoroughly forgotten and then revived by Askey in the 1970s, see [29]. The Euler construction was iterated by Kinkelin in the 1860s who turned Γ(𝑧) into a new 𝑧, that is, considered the equation 𝐺(𝑧+1)=Γ(𝑧)𝐺(𝑧) but only for positive integers 𝑧. Barnes in the 1900s introduced 𝐺(𝑧) from entirely different considerations that define it for complex values directly and lead to a hierarchy of functions with 𝐺⁽⁰⁾(𝑧)=𝑧,𝐺⁽¹⁾(𝑧)=Γ(𝑧),𝐺⁽²⁾(𝑧)=𝐺(𝑧),…,𝐺^(𝑑)(𝑧+1)=𝐺^(𝑑−1)(𝑧)𝐺^(𝑑)(𝑧). Nowadays, 𝐺(𝑧) is known as the (classical) Barnes function and 𝐺^(𝑑) as multigamma functions [51].

It is clear that the hierarchy of functional equations together with a normalization 𝐺^(𝑑)(1)=1 uniquely define 𝐺^(𝑑) on all natural numbers. Extension to complex values cannot be unique even for Γ because there are entire functions that vanish on all integers, sin(𝜋𝑧), for example. However, Bohr and Mollerup proved in the 1920s that if a log-concavity condition is added (𝑑²/𝑑𝑥²)Γ(𝑥)≥0 for 𝑥≥0, then the Euler Γ is the only possibility [29]. In 1963 Dufresnoy and Pisot generalized the Bohr-Mollerup existence and uniqueness result to a wide class of functional equations of the type 𝑓(𝑧+1)−𝑓(𝑧)=𝜙(𝑧). It constructs not only 𝐺^(𝑑) but also the Jackson deformation of the Euler Γ that satisfies Γ_𝑞(𝑧+1)=(𝑧)_𝑞Γ_𝑞(𝑧) with (𝑧)_𝑞∶=(1−𝑞^𝑧)/(1−𝑞) called the quantum number [8, 42]. For higher order multigammas, the log-concavity condition has to be replaced with log-positivity of order 𝑑+1, that is, (𝑑^𝑑+1/𝑑𝑥^𝑑+1)lnΓ(𝑥)≥0 for 𝑥≥0. After 𝑞-deformations of the Barnes multigammas appeared in the context of integrable hierarchies [52] Nishizawa realized that the same iteration works for them as well [14]. In other words, there exists a unique hierarchy of meromorphic functions satisfying(i)𝐺_𝑞^(𝑑)(1)=1,𝐺_𝑞⁽⁰⁾(𝑧)=1−𝑞^𝑧,1−𝑞(ii)𝐺_𝑞^(𝑑)(𝑧+1)=𝐺_𝑞^(𝑑−1)(𝑧)𝐺_𝑞^(𝑑)𝑑(𝑧),(iii)^𝑑+1𝑑𝑥^𝑑+1ln𝐺_𝑞(𝑥)≥0for𝑥≥0,0<𝑞<1. (6.1) The last condition is only required to hold for real positive 𝑞 but it is understood that 𝐺_𝑞^(𝑑) are continued analytically to other values of 𝑞.

Definition 6.1 (Quantum Multigammas). The functions 𝐺_𝑞^(𝑑) defined by (6.1) are called 𝑞-multigamma functions, in particular Γ_𝑞∶=𝐺_𝑞⁽¹⁾ is the Jackson' quantum gamma function and 𝐺_𝑞∶=𝐺_𝑞⁽²⁾ is the quantum Barnes (or q-Barnes) function.For |𝑞|<1, we will show the existence of 𝐺_𝑞^(𝑑) independently by deriving explicit expressions for it. In addition to integrable hierarchies, these functions also appear in analytic number theory, see [53, Remark 3].

𝑞-Multifactorials
In [14], Nishizawa derives an explicit combinatorial formula for 𝐺_𝑞^(𝑑) (see (6.26)). A more illuminating formula for our purposes can be expressed in terms of 𝑞-shifted multifactorials, 𝑞-multifactorials for short (Theorem 6.3). We streamline the Nishizawa approach by introducing them and systematically using generating functions.
The classical 𝑞-shifted factorial as defined by Euler is (𝑎;𝑞)_∞∶=∏^∞_𝑖=0(1−𝑎𝑞^𝑖). This notation is widely used in the theories of basic hypergeometric series [42], modular forms, and partitions [8]. A natural generalization essentially due to Appell is as follows.
Definition 6.2 (𝑞-Multifactorials). A 𝑞-shifted 𝑑-factorial is(𝑎;𝑞)_∞⁽⁰⁾∶=1−𝑎(no𝑞dependence),(𝑎;𝑞)_∞^(𝑑)∶=^∞_{𝑖1,…,𝑖𝑑=0}1−𝑎𝑞^{𝑖1+⋯+𝑖𝑑},|𝑞|<1,𝑑=1,2,…. (6.2)For finite 𝑁, one sets(𝑎;𝑞)_𝑁^(𝑑)∶=(𝑎;𝑞)_∞^(𝑑)(𝑎𝑞^𝑁;𝑞)_∞^(𝑑). (6.3)Our indexing convention is in line with [53] but differs from [54], our (𝑎;𝑞)_∞^(𝑑) is the Nishizawa (𝑎;𝑞)_∞^(𝑑−1). Also one should not confuse our 𝑞-multifactorials with multiple q-factorials that have a separate variable 𝑞_𝑘 for each index. Our definition is recovered if all 𝑞_𝑘 are set equal to 𝑞. We already used (𝑎;𝑞)_∞⁽²⁾ to write partition functions in a closed form.
Unlike the case 𝑑=1, the finite version (𝑎;𝑞)_𝑁^(𝑑) is no longer a finite product of (1−𝑎𝑞^𝑖) unless 𝑑 divides 𝑁. Nevertheless, we have∏^∞_{𝑖1,…,𝑖𝑑=0}(1−𝑎𝑞^{𝑖1+⋯+𝑖𝑑})∏^∞_{𝑖1,…,𝑖𝑑=0}(1−𝑎𝑞^{𝑁+𝑖1+⋯+𝑖𝑑})=∏^∞_𝑖𝑑=0∏^∞_{𝑖1,…,𝑖𝑑−1=0}(1−𝑎𝑞^{𝑖1+⋯+𝑖𝑑})∏^∞_{𝑖𝑑=𝑁}∏^∞_{𝑖1,…,𝑖𝑑−1=0}(1−𝑎𝑞^{𝑁+𝑖1+⋯+𝑖𝑑})=∏^∞_𝑖=0(𝑎𝑞^𝑖;𝑞)_∞^(𝑑−1)∏^∞_𝑖=𝑁(𝑎𝑞^𝑖;𝑞)_∞^(𝑑−1)=^𝑁−1_𝑖=0𝑎𝑞^𝑖;𝑞_∞^(𝑑−1), (6.4)and by definition(𝑎;𝑞)_𝑁^(𝑑)=^𝑁−1_𝑖=0𝑎𝑞^𝑖;𝑞_∞^(𝑑−1),𝑑≥1. (6.5)A similar calculation gives another useful identity(𝑎𝑞;𝑞)_∞^(𝑑)=(𝑎;𝑞)_∞^(𝑑)(𝑎;𝑞)_∞^(𝑑−1). (6.6)Also 𝑞-multifactorials can be expressed as a single product if we group the factors by powers of 𝑞. To this end, it is convenient to introduce the Stirling polynomials𝑧0𝑧𝑛∶=1∶=𝑧(𝑧−1)⋯(𝑧−𝑛+1)𝑛!,𝑛≥1, (6.7)that reduce to the binomial coefficients when 𝑧=𝑁≥𝑛 is a positive integer. Some of their properties are reviewed in the appendix. From a generating function for Stirling polynomials (A.4),^∞_{𝑖1,…,𝑖𝑑=0}𝑡^{𝑖1+⋯+𝑖𝑑}=^∞_{𝑛=0𝑖1+⋯+𝑖𝑑=𝑛}1𝑡^𝑛=^∞_𝑖=0𝑡^𝑖𝑑=(1−𝑡)^−𝑑=^∞_𝑛=0𝑛𝑡𝑑+𝑛−1_𝑛. (6.8)In other words, (_𝑛^{𝑑+𝑛−1}) is the number of arrangements of 𝑑 nonnegative integers adding up to 𝑛. As a consequence, we have single-product representations:(𝑎;𝑞)_∞^(𝑑)∶=^∞_{𝑖1,…,𝑖𝑑=0}1−𝑎𝑞^{𝑖1+⋯+𝑖𝑑}=^∞_𝑛=01−𝑎𝑞^{𝑛(𝑛)𝑑+𝑛−1},(𝑎𝑞;𝑞)_∞^(𝑑)=^∞_𝑛=11−𝑎𝑞^{𝑛()𝑑+𝑛−2𝑛−1}. (6.9) In particular, 1/(𝑞;𝑞)_∞⁽²⁾=∏^∞_𝑛=1(1−𝑞^𝑛)^𝑛1)−(=𝑀(𝑞) as we claimed before. They also imply that −ln(𝑎;𝑞)_∞^(𝑑) is nothing other than the quantum polylogarithm of Kirillov [55, Example 2.5.8], [54] defined for |𝑎|,|𝑞|<1 asLi_𝑠(𝑎;𝑞)∶=^∞_𝑘=1𝑎^𝑘𝑘(1−𝑞^𝑘)^𝑠−1. (6.10)Indeed, by (A.4)−ln(𝑎;𝑞)_∞^(𝑑)=−^∞_𝑛=0𝑛𝑑+𝑛−1ln1−𝑎𝑞^𝑛=^∞_𝑛=0𝑛𝑑+𝑛−1^∞_𝑘=1(𝑎𝑞^𝑛)^𝑘𝑘=^∞_𝑘=1𝑎^𝑘𝑘^∞_𝑛=0𝑛𝑞𝑑+𝑛−1_𝑘𝑛=^∞_𝑘=1𝑎^𝑘𝑘1−𝑞^𝑘−𝑑=Li_𝑑+1(𝑎;𝑞). (6.11)The identity holds for any 𝑎∈ℂ by analytic continuation.

Closed Formulas for 𝐺_𝑞^(𝑑)
Typically, when a classical object is 𝑞-deformed, its theory becomes more complicated. Refreshingly, 𝑞-multigammas for |𝑞|<1 are an exception. Their theory is much simpler than its classical counterpart. The underlying reason is that it is possible to write finite products as ratios of infinite ones (6.3), the latter having a straightforward extension from integers 𝑁 to complex values 𝑧. An analogous attempt to write 𝑁!=(1⋅2⋅3⋯)/((𝑁+1)⋅(𝑁+2)⋅(𝑁+3)⋯) leads to a nonsensical product of all natural numbers. The closest classical imitation is the Gauss product formulaΓ(𝑧+1)=lim_𝑛→∞(𝑛+1)!𝑛(𝑧+1)⋯(𝑧+𝑛+1)^𝑧, (6.12)that requires the rather involved theory of the Weierstrass products. One can fruitfully turn things around and derive product formulas for Γ and classical higher multigammas 𝐺^(𝑑) from the 𝑞-deformed ones via a limiting procedure [51].

We leave as an exercise to the reader to iterate the functional equation for 𝐺_𝑞^(𝑑) and derive by induction from (6.5) that for nonnegative integers 𝑁,𝐺_𝑞^(𝑑)(𝑁+1)=(𝑞;𝑞)^𝑁𝑑)_∞⁽⁰⁾⁻⁽(𝑞;𝑞)^𝑁)_∞^(1)(𝑑−1⋯(𝑞;𝑞)^{(𝑑)(−1)𝑑+1(𝑁0)}_∞(𝑞^𝑁+1;𝑞)^{(𝑑)(−1)𝑑}_∞=^𝑑_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1(𝑁)}_∞^𝑑−𝑖(𝑞^𝑁+1;𝑞)^{(𝑑)(−1)𝑑}_∞. (6.13)Now, 𝑁 can be painlessly replaced with any complex 𝑧 as long as we stipulate that 𝑞∈𝔻⧵ℝ₋. This way first the infinite products converge since 𝑞 is inside the unit disk 𝔻, and second 𝑞^𝑧∶=𝑒^𝑧ln𝑞 is defined by choosing the principal branch of the logarithm. Of course, a priori, there is no guarantee that the function so extended coincides with 𝐺_𝑞^(𝑑) of Definition 2.1. We now prove that this is the case.

Theorem 6.3 (Alternating Formula). Let 𝑞∈𝔻⧵ℝ₋ with 𝔻 the open unit disk in ℂ. Then 𝐺_𝑞^(𝑑)(𝑧) is an entire function of 𝑧 given by𝐺_𝑞^(𝑑)(𝑧+1)=^𝑑_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1(𝑧)}_∞^𝑑−𝑖(𝑞^𝑧+1;𝑞)^{(𝑑)(−1)𝑑}_∞, (6.14)where (𝑞;𝑞)_∞^(𝑖) are the 𝑞-multifactorials (6.2), and (^𝑧_𝑛) are the Stirling polynomials (6.7). In particular, for quantum gamma and the Barnes functions (𝑑=1,2),Γ_𝑞1(𝑧+1)=(𝑞;𝑞)_∞^(0)𝑧(𝑞;𝑞)_∞⁽¹⁾1(𝑞^𝑧+1;𝑞)_∞⁽¹⁾,𝐺_𝑞1(𝑧+1)=(𝑞;𝑞)_∞^{(0)(𝑧(𝑧−1)/2)}(𝑞;𝑞)_∞^(1)𝑧1(𝑞;𝑞)_∞⁽²⁾𝑞^𝑧+1;𝑞_∞⁽²⁾. (6.15)(6.16)Recall that (𝑞;𝑞)_∞⁽⁰⁾∶=1−𝑞,(𝑞;𝑞)_∞⁽¹⁾∶=(𝑞;𝑞)_∞.Proof. We have to verify that the right-hand side of (6.14) satisfies all three conditions of (6.1). For the duration of the proof, we use 𝐺_𝑞^(𝑑) only as an alias for this right-hand side. Normalizations (i) follow by direct substitution of values. (ii) For the functional equation, we compute𝐺_𝑞^(𝑑−1)(𝑧)𝐺_𝑞^(𝑑)(𝑧)=^𝑑−1_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1()}_∞^{𝑑𝑧−1𝑑−𝑖−1}_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1()}_∞^{𝑧−1𝑑−𝑖}⋅𝑞^𝑧;𝑞^{(𝑑−1)(−1)𝑑−1}_∞𝑞^𝑧;𝑞^{(𝑑)(−1)𝑑}_∞=^𝑑−1_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1}_∞^{[(𝑧−1𝑑−𝑖−1)+(𝑧−1𝑑−𝑖)]}(𝑞;𝑞)^{(𝑑)(−1)𝑑+1}_∞(𝑞^𝑧;𝑞)_∞^(𝑑)(𝑞^𝑧;𝑞)_∞^{(𝑑−1)(−1)𝑑}. (6.17)By (A.6) with 𝑛=𝑑−𝑖, the sum in brackets is just (^𝑧_𝑑−𝑖) so the first two factors combine into ∏^𝑑_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1(𝑧)}_∞^𝑑−𝑖. For the last factor, (6.6) with 𝑎=𝑞^𝑧 yields(𝑞^𝑧;𝑞)_∞^(𝑑)(𝑞^𝑧;𝑞)_∞^(𝑑−1)=𝑞^𝑧+1;𝑞_∞^(𝑑), (6.18)so (6.17) becomes𝐺_𝑞^(𝑑−1)(𝑧)𝐺_𝑞^(𝑑)(𝑧)=^𝑑_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1(𝑧)}_∞^𝑑−𝑖𝑞^𝑧+1;𝑞_∞^(𝑑)=𝐺_𝑞^(𝑑)(𝑧+1). (6.19)(iii) It remains to verify the log-positivity condition. Taking the logarithm, we haveln𝐺_𝑞^(𝑑)(𝑧+1)=^𝑑_𝑖=0(−1)^𝑖+1ln(𝑞;𝑞)_∞^(𝑖)𝑧𝑑−𝑖+(−1)^𝑑ln𝑞^𝑧+1;𝑞_∞^(𝑑). (6.20)Since (^𝑧_𝑑−𝑖) is a polynomial in 𝑧 of degree 𝑑−𝑖, the sum above is a polynomial of degree 𝑑 and its 𝑑+1 st derivative vanishes. Recalling (6.11), one obtains𝑑^𝑑+1𝑑𝑧^𝑑+1ln𝐺_𝑞^(𝑑)(𝑧)=(−1)^𝑑+1𝑑^𝑑+1𝑑𝑧^𝑑+1Li_𝑑+1𝑞^𝑧;𝑞. (6.21)It is helpful to notice that (𝑑/𝑑𝑧)𝑓(𝑞^𝑧)=ln𝑞𝑥(𝑑𝑓/𝑑𝑥)|_{𝑥=𝑞𝑧} and iterating(−1)^𝑑+1𝑑^𝑑+1𝑑𝑧^𝑑+1𝑓𝑞^𝑧=(−ln𝑞)^𝑑+1𝑥𝑑𝑑𝑥^𝑑+1𝑓(𝑥)_{𝑥=𝑞𝑧}. (6.22)Now, for |𝑥|<1 by (6.10)𝑥𝑑𝑑𝑥^𝑑+1Li_𝑑+1(𝑥;𝑞)=^∞_𝑛=11𝑛(1−𝑞^𝑛)^𝑑𝑥𝑑𝑑𝑥^𝑑+1𝑥^𝑛=^∞_𝑛=1𝑛^𝑑+1𝑛(1−𝑞^𝑛)^𝑑𝑥^𝑛, (6.23)and therefore𝑑^𝑑+1𝑑𝑧^𝑑+1ln𝐺_𝑞^(𝑑)(𝑧)=(−ln𝑞)^∞𝑑+1_𝑛=1𝑛1−𝑞^𝑛𝑑𝑞^𝑛𝑧for𝑞^𝑧<1. (6.24)If 0<𝑞<1 and 𝑧 is real positive, then −ln𝑞>0 and 0<𝑞^𝑧<1 so the last expression is positive by inspection.The structure of 𝐺_𝑞^(𝑑) is perhaps most transparent in (6.20). We have the main term(−1)^𝑑ln(𝑞^𝑧+1;𝑞)_∞^(𝑑) that depends on 𝑎=𝑞^𝑧 only and the anomaly term polynomial in 𝑧 of degree 𝑑. For 𝑑=2, we recognize ln(𝑞^𝑧+1;𝑞)_∞⁽²⁾ as exactly the reduced-free energy of the resolved conifold, see (4.17). The main/anomaly structure of 𝐺_𝑞^(𝑑) becomes even more transparent if we introduce a generating function for the entire hierarchy:^∞_𝑑=0ln𝐺_𝑞^(𝑑)(𝑧+1)𝑡^𝑑=(1+𝑡)^𝑧∞_𝑑=0(−1)^𝑑Li_𝑑+1(𝑞;𝑞)𝑡^𝑑−^∞_𝑑=0(−1)^𝑑Li_𝑑+1𝑞^𝑧+1;𝑞𝑡^𝑑. (6.25)One can check that the sum converges for |𝑞|,|𝑞^𝑧|<1, |𝑡|<1−|𝑞|. This formula displays very clearly the alternating nature of 𝐺_𝑞^(𝑑) and the distinction between the main and the anomaly terms. The dependence of the anomaly on 𝑧 is regulated by a universal term (1+𝑡)^𝑧 that explains the convolution structure of the anomaly sum in (6.20).

This structure is quite common for functions appearing in quantum field theory. The main terms usually reflect the expected symmetry of a system while anomalies require additional choices to be defined. For instance, the Witten Chern-Simons path integral is not a 3-manifold invariant because of the framing anomaly [41, 44] but can be turned into one by choosing the canonical 2-framing to cancel the anomaly [37]. In our case, the obvious extra choice is that of a logarithm branch to define 𝑧∶=log_𝑞𝑎. Disregarding anomalies comes at a price. Fixing the canonical 2-framing would complicate the gluing rules, and 𝐺_𝑞^(𝑑) would not obey a simple functional equation. The Reshetikhin-Turaev normalization may be to blame for nasty prefactors in (5.17) that spoil the duality. It might be of interest to find an analog of 2-framings on the Gromov-Witten side and compare answers without making choices, even canonical ones.

From the alternating formula (6.14), we now derive a single-product formula for 𝐺_𝑞^(𝑑) originally given by Nishizawa [14].

Theorem 6.4 (Nishizawa Product). Let 𝑞∈𝔻⧵ℝ₋ then quantum multigammas are given by𝐺_𝑞^(𝑑)(𝑧+1)=(1−𝑞)^{𝑧𝑑)∞−(}_𝑛=1(1−𝑞^𝑛)^{()𝑧−𝑛𝑑−1}(1−𝑞^𝑧+𝑛)^{()−𝑛𝑑−1}for𝑑≥1. (6.26)In particular, for quantum gamma and Barnes functions (𝑑=1,2): Γ_𝑞(𝑧+1)=(1−𝑞)^∞−𝑧_𝑛=11−𝑞^𝑧+𝑛1−𝑞^𝑛−1,𝐺_𝑞(𝑧+1)=(1−𝑞)^{∞−(𝑧(𝑧−1)/2)}_𝑛=1(1−𝑞^𝑧+𝑛)^𝑛(1−𝑞^𝑛)^−𝑧+𝑛. (6.27)(6.28)Proof. Applying (6.9) to the first product in (6.14), we get^𝑑_𝑖=0(𝑞;𝑞)^{(𝑖)(−1)𝑖+1(𝑧)}_∞^𝑑−𝑖=^𝑑∞_𝑖=0𝑛=01−𝑞^{𝑛+1(−1)𝑖+1(𝑛𝑧)𝑛+𝑖−1)(𝑑−𝑖}=^∞_𝑛=01−𝑞^{𝑛+1∑𝑑𝑖=0(−1)𝑖+1(𝑛𝑧)𝑛+𝑖−1)(𝑑−𝑖}. (6.29)As shown in the appendix (A.7), when 𝑑≥1^𝑑_𝑖=0(−1)^𝑖+1𝑛𝑧=−𝑧𝑑𝑛+𝑖−1𝑑−𝑖𝑧−𝑛−1𝑑−1,𝑛≥1𝑧−1𝑑−1,𝑛=0. (6.30)Splitting the 𝑛=0 and 𝑛≥1 factors in (6.29), we see that it equals(1−𝑞)^{(𝑧𝑑)∞𝑧−1𝑑−1)−(}_𝑛=11−𝑞^{𝑛+1()𝑧−𝑛−1𝑑−1}=(1−𝑞)^{𝑧𝑑)∞−(}_𝑛=01−𝑞^{𝑛+1()𝑧−𝑛−1𝑑−1}=(1−𝑞)^{𝑧𝑑)∞−(}_𝑛=11−𝑞^{𝑛()𝑧−𝑛𝑑−1}. (6.31)The second factor in (6.14) can be transformed into a single product using (6.9) again^∞_𝑛=11−𝑞^{𝑧+𝑛(−1)𝑑()𝑑+𝑛−2𝑑−1}=^∞_𝑛=11−𝑞^{𝑧+𝑛()−𝑛𝑑−1}, (6.32)where we also applied (A.3) and (A.6) to get the second expression. Multiplying the right-hand sides of the last two formulas gives the claim.Formula (6.27) is the standard expression given for the quantum gamma function, see, for example, [29], and (6.28) is its closest quantum Barnes analog. Although the Nishizawa product (6.26) may appear prettier than the alternating formula (6.14), it completely hides the graded structure of 𝑞-multigammas.

What About |𝑞|=1?
By Theorem 6.3, 𝑞-multigammas are entire functions of 𝑧 for |𝑞|<1. Things change if we venture onto the unit circle. As one can judge by the example of the classical Euler Γ, where 𝑞=1, (6.1) still defines a unique function of 𝑧 but this time it is only meromorphic. The appearance of poles prevents infinite products (6.26) from converging. When |𝑞|=1 but 𝑞is not a root of unity𝐺_𝑞⁽¹⁾=Γ_𝑞 is constructed explicitly in [56] via the Shintani double sine function (see [53]). This case is complementary to the classical one 𝑞=1, and the poles are located at the points −𝑛−𝑚/𝜏 with 𝑛,𝑚∈ℤ, and 𝑞=𝑒^{2𝜋𝑖𝜏}. Clearly, it is desirable to have a similar construction for 𝐺_𝑞⁽²⁾=𝐺_𝑞, especially when 𝑞is a root of unity. Indeed, this case is most relevant to the Chern-Simons theory, where 𝐺_𝑞 is essentially the partition function of 𝑆³ by Theorem 5.2, and probably appears as a factor in partition functions of other manifolds.

There is a small window through which we can peak at what happens on the unit circle for any 𝑞. When 𝑧=𝑁 is a nonnegative integer, the product (6.26) terminates. In the context of the Chern-Simons theory, 𝑁 is the rank of 𝔰𝔩_𝑁ℂ and we already saw this phenomenon for the quantum Barnes function in Lemma 5.1. This also generalizes the fact that although Γ(𝑧+1) cannot be simply expressed as an infinite product for complex 𝑧, we have nonetheless Γ(𝑁+1)=1⋯𝑁.

Corollary 6.5 (Nishizawa). For a nonnegative integer 𝑁 and any 𝑞∈ℂ, one has𝐺_𝑞^(𝑑)(𝑁+1)=(1−𝑞)^{𝑁𝑑)𝑁−(}_𝑛=11−𝑞^{𝑛()𝑁−𝑛𝑑−1}. (6.33)Proof. By the Nishizawa product for 𝐺_𝑞^(𝑑),(1−𝑞)^(𝑁𝑑)𝐺_𝑞^(𝑑)∏(𝑁+1)=^∞𝑛=1(1−𝑞^𝑛)^{()𝑁−𝑛𝑑−1}∏^∞_𝑛=1(1−𝑞^𝑛+𝑁)^{()−𝑛𝑑−1}=∏^∞_𝑛=1(1−𝑞^𝑛)^{()𝑁−𝑛𝑑−1}∏^∞_𝑛=𝑁+1(1−𝑞^𝑛)^{()−(𝑛−𝑁)𝑑−1}=^𝑁_𝑛=11−𝑞^{𝑛()𝑁−𝑛𝑑−1}. (6.34)This gives a proof for |𝑞|<1 but since the right-hand side is a polynomial in 𝑞, the general case follows by analytic continuation.Note that by (6.33) 𝐺_𝑞^(𝑑)(𝑁+1)=0 for 𝑁>𝑛 if 𝑞 is an 𝑛th root of unity but this never affects the Chern-Simons invariants since for them 𝑛=𝑘+𝑁 with positive integer 𝑘. This termination phenomenon sheds some light on why so far the Chern-Simons invariants have only been defined for integral 𝑧=𝑁, when it can be interpreted as the rank of a quantum group or an affine Lie algebra. To include complex 𝑧, one should probably study more involved algebraic/analytic structures. Reversing the perspective, we observe that termination is what makes 𝐺_𝑞 meaningful in the Chern-Simons theory that in its current form produces values only at roots of unity. Note that with the exception of (1−𝑞)^{𝑧𝑑)−(}, all anomaly terms in (6.16) are required for termination. In particular, the ratio (𝑎𝑞;𝑞)_∞⁽²⁾/(𝑞;𝑞)_∞⁽²⁾ with 𝑞^𝑧=𝑎, that we recover from the Gromov-Witten/Donaldson-Thomas theory, does not suffice for duality to even make sense. It does not terminate for 𝑧=𝑁 and becomes meaningless for 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)}, which is where the Chern-Simons invariants are defined.

7. Conclusions

Recently, the phenomenon of holography has become prominent in physics [57]. Roughly, the idea is that to every theory on a bulk space there corresponds an equivalent theory living on its boundary. The most famous example is the AdS/CFT correspondence of Maldacena but one can certainly trace the analogy back to the classical potential theory, where a harmonic function is recovered from its boundary values. We see a toy example of holography playing out on the ranges of the Calabi-Yau invariants. The master-invariant is a holomorphic function in the bulk (the unit disk) and the Donaldson-Thomas theory comes closest to being a bulk theory by giving its Taylor coefficients at 0. The Chern-Simons theory gives its values on the boundary (the unit circle) and the Gromov-Witten theory also lives on the boundary but via asymptotic coefficients at 1. Recall that 𝑞=𝑒^𝑖𝑥 and in terms of the string coupling constant 𝑥, we are dealing with periodic holomorphic functions on the upper half-plane. Their Fourier coefficients are given by the Donaldson-Thomas invariants and values at the cusps of𝑆𝐿₂ℤ are given by the Chern-Simons invariants. This brings to mind the classical modular forms [58] and indeed the relationship between them and the corresponding boundary objects was interpreted recently as an example of the holographic correspondence [59]. Although our functions are not modular in the classical sense, they do have some modular transformation properties [60]. This links large 𝑁 duality to a well-known problem in analytic number theory and perhaps this can aid in its proof.

There are some serious difficulties to be resolved. First, values of a holomorphic function at roots of unity are not enough to recover it uniquely. There are plenty of holomorphic functions on the unit disk that vanish at all roots of unity. Take a modular cusp-form 𝑓 on the upper half-plane [58], for example, and consider 𝑓(ln𝑞/2𝜋𝑖). The same is true of an asymptotic expansion at a point on the unit circle or even a collection of asymptotic expansions at every root of unity [60]. In other words, absent extra data the Chern-Simons and the Gromov-Witten theories do not recover the master-invariant. The Donaldson-Thomas theory has a converse problem. It sure gives a function on the unit disk but one that is singular at each point of the unit circle. For the resolved conifold, we reconciled the results by “completing a pattern” by hand but this will not work in general. A large part of the problem in proving equality of partition functions is that as mathematically defined they are not quite equal.

We contrived to make large 𝑁 duality work for the resolved conifold. What about other cases? Similarities between the topological vertex and the Reshetikhin-Turaev calculus outlined in Sections 4 and 5 are promising but there is a vast difference between toric webs and link diagrams representing the dual objects. This ought to be expected of course since not all large 𝑁 duals to 3-manifolds are toric and certainly most toric Calabi-Yaus are not dual to any 3-manifold. However, not all is lost. Link invariants are known to extend to invariants of knotted trivalent graphs [61], and these include toric graphs that are also trivalent but planar. Then there is the case of lens spaces (cyclic quotients of 𝑆³ [38]), where the duals are known and toric [18, 40]. Moreover, there exist physical generalizations of the topological vertex to nontoric threefolds [62] some of which are dual to noncyclic spherical quotients.

Even if diagrammatic presentations are reconciled, there is yet another hurdle to wrestle with. Topological vertex expressions are infinite sums over all partitions. The Reshetikhin-Turaev expressions are finite sums over partitions in the (𝑁−1)×𝑘 rectangle 𝒫^𝑘_𝑁−1, and ad hoc tricks are required to turn them into functions of 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)}. Note that the Reshetikhin-Turaev building blocks 𝑊_𝜆𝜇(𝑁;𝑞) are already expressed in a desirable form. It is when the gluing rules are applied that 𝑞 has to be specialized to a root of unity to render the sums finite. Analogous infinite sums naively diverge but so would sums of 𝒲_𝜆𝜇(𝑞) in the topological vertex without the convergence factors like (−𝑎)^|𝜆| in (4.16). After the sum is performed, it is a again possible to turn it into a function of 𝑁,𝑞 up to framing and volume factors. Recall that values at roots of unity do not in themselves determine a holomorphic function on the unit disk appearing in the dualities, and the Donaldson-Thomas theory as a bulk theory is incomplete. An enticing possibility is that the Chern-Simons theory itself can be turned into a bulk theory.

Conjecture 1. There exists a universal Chern-Simons TQFT that assigns holomorphic functions of 𝑞,𝑧 to links and 3-manifolds with 𝑞∈𝔻. These functions extend to roots of unity in 𝑞 and specialize to ordinary Chern-Simons invariants at rank 𝑁 and level 𝑘 upon specializing to 𝑧=𝑁, 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)} up to normalization factors. Partitions serve as colors of link components and the gluing rules are expressed via sums over all partitions. The Donaldson-Thomas invariants of the dual Calabi-Yau threefolds are the Taylor coefficients at 𝑞=0 of these functions, and the Gromov-Witten invariants are their asymptotic coefficients at 𝑞=1.

This conjecture formalizes the idea that extra extension data for invariants comes from algebraic restrictions required by the axioms of TQFT [44]. A corollary of this conjecture that seems to be within reach is that sums of the type_{𝜆1,…,𝜆𝑛∈𝒫𝑘𝑁−1}𝑊_𝜆𝜆1𝑊_𝜆1𝜆2⋯𝑊_𝜆𝑛𝜇 (7.1)can be represented as boundary values of sums_{𝜆1,…,𝜆𝑛∈𝒫}𝑊_𝜆𝜆1𝑊_𝜆1𝜆2⋯𝑊_𝜆𝑛𝜇𝑟^{|𝜆1|+⋯+|𝜆𝑛|}, (7.2)where 𝑟 is a convergence factor. These and similar sums appear in the Chern-Simons invariants of the Seifert-fibered 3-manifolds [43, 45], and such representation would facilitate a proof of large 𝑁 duality for them.

Appendix

Stirling Polynomials

The Stirling polynomials [8, I.2.11] are defined as𝑧0𝑧𝑛∶=1∶=𝑧(𝑧−1)⋯(𝑧−𝑛+1)𝑛!,𝑛≥1 (A.1)and reduce to the binomial coefficients when 𝑧=𝑁≥𝑛 is a positive integer. Their generating function is^∞_𝑛=0𝑧𝑛𝑡^𝑛=(1+𝑡)^𝑧. (A.2)In agreement with (A.2), we always assume (^𝑁_𝑛)=0 for 𝑛<0 or 𝑛>𝑁. Negative 𝑧 is also allowed:𝑛−𝑧=(−1)^𝑛𝑛𝑧+𝑛−1. (A.3)Combining with the generating function (A.2), we have^∞𝑛=0𝑛𝑡𝑧+𝑛−1_𝑛=(1−𝑡)^−𝑧. (A.4)The coefficients 𝑠(𝑛,𝑘) in𝑧𝑛=^𝑛_𝑘=0𝑠(𝑛,𝑘)𝑧𝑛!^𝑘,𝑛=𝑧+𝑛−1^𝑛_𝑘=0(−1)^𝑛−𝑘𝑠(𝑛,𝑘)𝑧𝑛!^𝑘 (A.5)are known as the Stirling numbers of the first kind [8, I.2.11]. The following identity is easily derived via generating functions:𝑧+𝑧𝑛=𝑛−1𝑛𝑧+1. (A.6)For the convenience of the reader, we prove a harder one needed in Theorem 6.4, namely,^𝑑_𝑖=0(−1)^𝑖+1𝑛𝑧=−𝑧𝑑𝑛+𝑖−1𝑑−𝑖𝑧−𝑛−1𝑑−1,𝑛≥1,𝑧−1𝑑−1,𝑛=0. (A.7)

Proof. Consider the generating function^∞_𝑑=0𝑡^𝑑𝑑_𝑖=0(−1)^𝑖+1𝑛𝑧=𝑛+𝑖−1𝑑−𝑖^{∞𝑖,𝑗=0}𝑡^𝑖+𝑗(−1)^𝑖+1𝑛𝑧𝑗=𝑛+𝑖−1_∞𝑖=0(−1)^𝑖+1𝑛𝑡𝑛+𝑖−1_𝑖⋅^∞_𝑗=0𝑧𝑗𝑡^𝑗. (A.8)The second factor is obviously (1+𝑡)^𝑧 by (A.2). Assuming 𝑛>0 so that (_𝑛^𝑛−1)=0, the first factor can be rewritten as𝑡^∞_𝑖=1𝑖−1+(𝑛+1)−1𝑖−1(−𝑡)^𝑖−1=𝑡(1+𝑡)^−𝑛−1, (A.9)by (A.4). Multiplying them and expanding in the powers of 𝑡,𝑡(1+𝑡)^{𝑧−𝑛−1}=^∞_𝑑=1𝑡𝑧−𝑛−1𝑑−1^𝑑. (A.10)We may conclude for 𝑛≥1,^𝑑_𝑖=0(−1)^𝑖+1𝑛𝑧=𝑛+𝑖−1𝑑−𝑖𝑧−𝑛−1𝑑−1,𝑑≥1,0,𝑑=0. (A.11)The case 𝑛=0 is special since (₀^𝑖−1)=1 by definition. Therefore,^∞_𝑖=0(−1)^𝑖+10𝑡𝑖−1_𝑖=−(1+𝑡)⁻¹, (A.12)and expanding as above, we obtain for 𝑛=0 and all 𝑑^𝑑_𝑖=0(−1)^𝑖+10𝑧𝑑0+𝑖−1𝑑−𝑖=−𝑧−1. (A.13)By assumption, we have 𝑑≥1 so applying (A.6),−𝑑=−𝑧𝑑𝑧−1𝑧−1𝑑−1. (A.14)

Acknowledgments

The author wishes to thank D. Auckly and D. Karp for sharing their thoughts and notes on the Gromov-Witten and the Chern-Simons theories and also for suggesting numerous corrections and improvements to the original draft. He is also indebted to M. Mariño for his extensive email comments on the physical background of the Calabi-Yau dualities. His influence can be felt in the overall mindset of this paper.