# Quantum Barnes Function as the Partition  Function

Authors: Sergiy Koshkin
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We give a short new proof of largeN 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.


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## Body

## 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 astopological 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
conifoldX:=𝒪(−1)⊕𝒪(−1), it was
originally computed by Faber-Pandharipande [3]; see also [4](1.1)ZX′(a;q)=∏n=1∞(1−aqn)n.Here, a=e−t, q=eix, and t, x are known as
the Kähler parameter and the string coupling constant, respectively. In
mathematical terms, they are just formal variables and(1.2)lnZX′=∑g=0∞∑d=1∞〈1〉g,dtdx2g−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 as(1.3)ZX=ZX0ZX′≈MZX′,where(1.4)M(q):=∏n=1∞(1−qn)−nis the MacMahon function,
classically known as the generating function of plane partitions [8]. In all honesty, this is
not quite true as lnM(eix) 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 ZXDT=M2ZX′DT, not ZX=MZX′. 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, largeN 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 UN or SUN 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 ZS3 to ZX promises some
useful insight.Once again, by ignoring some irrelevant prefactors,ZS3 can be written
as ZS3≈ℰ−zZX, where z=itx−1 so that a=qz, and(1.5)ℰ(q):=∏n=1∞(1−qn)−1is the classical Euler
generating function of ordinary partitions. At this point, it is appropriate to
introduce notation that allows one to write ZX′,M, and ℰ uniformly. Let(1.6)(a;q)∞(0):=1−a,(a;q)∞(d):=∏i1,…,id=0∞(1−aqi1+⋯+id)be the q-multifactorials then (see
Section 6),(1.7)ZX′(a;q)=(aq;q)∞(2),M(q)=1(q;q)∞(2),ℰ(q)=1(q;q)∞(1).Using q and z as variables,
we see that(1.8)ZX′=(qz+1;q)∞(2),ZX≈1(q;q)∞(2)(qz+1;q)∞(2),ZS3≈(q;q)∞(1)z1(q;q)∞(2)(qz+1;q)∞(2). 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 consider(1.9)Gq(z+1):=1(q;q)∞(0)(z(z−1)/2)(q;q)∞(1)z1(q;q)∞(2)(qz+1;q)∞(2).This Gq 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 as(1.10)Gq(z+1)=Γq(z)Gq(z),where Γq is the Jackson quantum gamma function deforming the
classical one. This in turn satisfies
Γq(z+1)=(z)qΓq(z) with the
so-called quantum number(z)q:=(1−qz)/(1−q). This makes Gq 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, Gq 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 hiddenmaster-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 have(1.11)lnZX′=∑n=0∞∑d=1∞(−1)nDn,dtdqnwith ZX′ the same as in
(1.2) and Dn,d the
Donaldson-Thomas invariants. In other words, in each degree, (−1)nDn,d are simply the
Taylor coefficients of ZX′ at q=0 while 〈1〉g,d are the Laurent
coefficients in x corresponding
to q=1 with q=eix. The relation with the Chern-Simons invariants is
more complicated. Traditionally, one has to take q=e2πi/(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 ZX0(eix)=∑g=0∞〈1〉g,0x2g−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 ZX and an
additional factor in ZS3 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 conjecturallyZY0=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. Section2 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 Gq 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
LetX 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 Mg,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(pi)=pi,  where p1,…,pn 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 dimension(2.1)dimℂvir⁢M¯g,n(X,α)=〈c1(X),α〉−(N−3)(g−1)+n.For instance, this is the case
if X=ℂℙN 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:(2.2)evi:M¯g,n(X,α)⟶Xf↦f(pi)and pullback cohomology classes γi 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 are(2.3)〈γ1⋯γn〉g,α:=∫[M¯g,n(X,α)]virev1*(γ1)∪⋯∪evn*(γn),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⋯γn〉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,…,γn [19, 23]. In general, the
enumerative interpretation fails and 〈γ1⋯γn〉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+⋯+degγn), 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 γi and we can
recover all of them from 〈1〉g,α and those with
insertions chosen from an integral basis h1,…,hm in H+(X,ℤ):=⊕n>0Hn(X,ℤ). One may worry about torsion, but torsion classes are
not represented by holomorphic curves and can be ignored. Thus, any 〈γ1⋯γn〉g,α is a linear
combination of 〈h1p1⋯hmpm〉g,α, where the
“powers” pi stand for
repeating hi that many
times. Introduce formal variables t1,…,tm for each
element of the basis. Heuristically, they represent (minus) Kähler volumes of h1,…,hm and are called Kähler parameters, especially in physics
literature. Analogously, let ξ1,…,ξk be a linear
basis in H2(X,ℤ), and let Q1,…,Qk
be the
corresponding formal variables. We write 〈h1p1⋯hmpm〉g,d→ with d→:=(d1,…,dk) for short, when α=d1ξ1+⋯+dkξk. The numbers d1,…dk 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) is(2.4)ℱ(t1,…,tm;Q1,…,Qk;x):=∑g=0∞∑p1,…,pm=0d1,…,dk=0∞〈h1p1⋯hmpm〉g,d→t1p1⋯tmpmp1!⋯pm!Q1d1⋯Qkdkx2g−2.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[tj,Qi,x] [24]. Under a change of bases 〈h1p1⋯hmpm〉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.LetX:=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 to(2.5)ℱ(t;Q;x):=∑p,d,g=0∞〈hp〉g,dtpp!Qdx2g−2.Divisor Equation and Free Energy
We will
be interested not even in all primary invariants but
also in those corresponding to combinations ofdivisor 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 〈h1p1⋯hmpm〉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](2.6)π*[M¯g,n−1(X,α)]vir=[M¯g,n(X,α)]vir,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 equation(2.7)〈hγ1⋯γn〉g,α=h(α)〈γ1⋯γn〉g,α,where h∈H2(X,ℤ) and γi 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).SinceH2(X,ℤ)≃H2(X,ℤ) modulo torsion
and ξ1,…,ξk 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,…,hk are the ones
and that they are dual to ξ1,…,ξk, that is hi(ξj)=δij. The divisor equation may now be used to flush all
the insertions out of the divisor invariants. By induction from (2.7),(2.8)〈h1p1⋯hkpk〉g,d→=d1p1⋯dkpk〈1〉g,d→,assuming d→≠0 to avoid
low-genus problems in degree zero. Define the truncated divisor potentialℱdiv′(t1,…,tk;Q1,…,Qk;x) as in (2.4) but
restricting the sum to p1,…,pk and d→≠0. Using (2.8), we compute(2.9)ℱdiv′(t1,…,tk;Q1,…,Qk;x)=∑g=0∞∑d→≠0〈1〉g,d→Q1d1⋯Qkdkx2g−2∑p1,…,pk=0∞(d1t1)p1⋯(dmtm)pmp1!⋯pm!=∑g=0∞∑d→≠0〈1〉g,d→Q1d1⋯Qkdkx2g−2ed1t1⋯edktk=∑g=0∞∑d→≠0〈1〉g,d→(Q1et1)d1⋯(Qketk)dkx2g−2.Obviously, as far as divisor
invariants go, Q1,…,Qk 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 is(2.10)F′(t1,…,tk;x):=∑g=0∞∑d→≠0〈1〉g,d→ed1t1⋯edktkx2g−2.Its exponent Z′(t1,…,tk;x):=exp(F′(t1,…,tk;x)) is called the
reduced the Gromov-Witten partition function. One
writes
FX′,ZX′ 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 ZX′ is the quantity
directly computed by the topological vertex algorithm [12, 20, 26, 27].Degree Zero
The moduli spacesM¯g,n(X,0) consist of
stable maps mapping stable curves into points. Therefore, they split [3](2.11)M¯g,n(X,0)=M¯g,n×X.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 〈hi3〉0,0,〈hi2hj〉0,0,〈hihjhl〉0,0, and 〈hi〉1,0, respectively. There is automatically no dependence
on Qi, so the degree zero divisor potential is the same as
the degree zero-free energy (cf. [28]):(2.12)F0(t1,…,tk;x):=ℱ0(t1,…,tk;x)=(∑i=1k〈hi3〉0,0ti36+∑i≠j〈hi2hj〉0,0ti2tj2+∑i≠j,j≠l,l≠i〈hihjhl〉0,0titjtl)1x2+∑i=1k〈hi〉1,0ti+∑g=2∞〈1〉g,0x2g−2.Note that degree zero genus 0(1) terms are the
only parts of the free energy depending on powers of ti rather than
just their exponents eti. When X is compact, these terms reflect its classical
cohomology, namely [28]:(2.13)〈hihjhl〉0,0=∫Xhi∪hj∪hl〈hi〉1,0=−124∫Xhi∪c2(X).In particular, they vanish
unlessX is a threefold.
Higher genus contributions were computed in the celebrated paper of
Faber-Pandharipande [3]:(2.14)〈1〉g,0=(−1)g|B2g||B2g−2|(2g−2)!2g(2g−2)⋅12∫X(c3(X)−c1(X)∪c2(X))=(−1)g−1(2g−1)B2gB2g−2(2g−2)(2g)!⋅12∫X(c3(X)−c1(X)∪c2(X)),g≥2.Here, ci(X) as before are
the Chern classes and Bn are the
Bernoulli numbers defined via a generating function [29]:(2.15)zez−1:=∑n=0∞Bnznn!.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,(2.16)〈1〉g,0=(−1)g−1(2g−1)B2gB2g−2(2g−2)(2g)!⋅χ(X)2,g≥2.WhenX 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⋯γn〉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 form(2.17)FO(−1)⊕O(−1)0(t;x)=p3(t)x2+p1(t)+∑g=2∞(−1)g−1(2g−1)B2gB2g−2(2g−2)(2g)!x2g−2,where pi 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 yield(2.18)FO(−1)⊕O(−1)0(t;x)=t361x2+t12+∑g=2∞(−1)g−1(2g−1)B2gB2g−2(2g−2)(2g−2)!x2g−2,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 isF:=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
FX,ZX to indicate the
target manifold if necessary.For the resolved
conifold, we get from (2.10)(2.19)FO(−1)⊕O(−1)(t;x)=FO(−1)⊕O(−1)0(t;x)+∑g=0d=1∞〈1〉g,dedtx2g−2.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,(2.20)(2g)!π2g22g−1<|B2g|<(2g)!π2g(22g−1−1),g≥1,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=eix. 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:(3.1)M(q):=∏n=1∞(1−qn)−n,|q|<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, anideal 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 genusg of a stable map
is replaced in the Donaldson-Thomas invariant Dκ,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 by(3.2)ZX0(q):=∑κ=0∞Dκ,0qκ=M(−q)χ(X),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(eix)(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 considerlnM(e−x) instead of lnM(eix). 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:(3.3)e−x1−e−x=∑n=1∞e−nx=∑n=1∞∑k=0∞(−nx)kk!⁢′=′∑k=0∞(−x)kk!∑n=1∞1n−k′=′∑k=0∞(−1)kζ(−k)k!xk.The last two equalities are
nonsense. Of course,  the interchange of sums is illegitimate and ∑n=1∞(1/n−k)=∑n=1∞nk is (very)
divergent. It certainly does not converge to ζ(−k) for positive k, although by definition ζ(s):=∑n=1∞(1/ns),Res>1 is the Riemann
zeta function [29].
Nonetheless, the end result is almost correct. Indeed, by definition of
Bernoulli numbers  (2.15),(3.4)e−x1−e−x=1xxex−1=1x∑j=0∞Bjxjj!,(3.5)ζ(−k)=−Bk+1k+1,k≥1;ζ(0)=−12 [29], so (3.6)e−x1−e−x=1x−12+∑j=2∞Bjj!xj−1=1x−12−∑k=1∞Bk+1(k+1)!xk=1x+∑k=0∞(−1)kζ(−k)k!xk.In other words our
“computation” (3.3) only missed the first term 1/x.A similar feat can be performed withlnM(e−x). First, we compute(3.7)lnM(e−x)=−∑n=1∞nln(1−e−nx)=∑n=1∞n⁢∑k=1∞(e−nx)kk=∑k=1∞1k∑n=1∞n(e−kx)n=∑k=1∞1ke−kx(1−e−kx)2=∑k=1∞1k1(ekx/2−e−(kx/2))2=∑k=1∞csch2(kx/2)4k.So far, all the manipulations
are legitimate assuming x>0, although they would not be if we used eix instead of e−x. Next, recall the Laurent expansion at zero of csch2:(3.8)csch2(z)=−∑g=0∞22g(2g−1)B2g(2g)!z2g−2.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,(3.9)lnM(e−x)=−∑k=1∞14k∑g=0∞4(2g−1)B2g(2g)!22g−2(kx2)2g−2′=′−∑g=0∞(2g−1)B2g(2g)!x2g−2∑k=1∞1k3−2g′=′−∑g=0∞(2g−1)B2gζ(3−2g)(2g)!x2g−2=ζ(3)x2−ζ(1)12+∑g=2∞(2g−1)B2gB2g−2(2g−2)(2g)!x2g−2,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=1∞nk), 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 is(3.10)Mf(s):=∫0∞xs−1f(x)dx.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)/ns in Res∈(0,∞) extends
meromorphically with the poles of the gamma function located at s=0,−1,−2,…. Analogously,(3.11)M[e−x1−e−x](s)=∑n=1∞M[e−nx](s)=∑n=1∞Γ(s)ns=Γ(s)ζ(s)inRes∈(1,∞)extends with one additional zeta
pole at s=1.Theinverse
Mellin transform recovers f as(3.12)f(x)=12πi∫c−i∞c+i∞Mf(s)x−sdsforc∈(α,β),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:(3.13)f(x)=∑n=0∞Ress=−γn[Mf(s)]xγn.If γn 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 −γn=−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−γnare ordermnpoles of
(meromorphic continuation of)𝔐f(s)and its Laurent
expansions at−γnhave the form(3.14)Mf(s)=An0s+γn+An1(s+γn)2+∑k=2mn−1Ank(s+γn)k+1,then an asymptotic expansion offatx=0is(3.15)f(x)∼∑n=0∞(An0−An1lnx+∑k=2mn−1(−1)kAnkk!lnkx)xγn.
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)xn, 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 generalFourier sums (or harmonic sums) of the
form(3.16)f(x)=∑k=1∞akg(ωkx)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 amplitudesak and frequenciesωk, the usual ones corresponding to g(x)=eix,ωk=±k. Indeed,(3.17)Mf(s)=∑k=1∞ak∫0∞xs−1g(ωkx)dx=∑k=1∞akωks∫0∞xs−1g(x)dx=D(s)Mg(s),where D(s):=∑k=1∞ak/ωks is the Dirichlet series of the sum. If D(s) is entire and 𝔐g(s) only has simple
poles at s=0,−1,−2,…, then(3.18)f(x)∼∑n=0∞Ress=−n[Mg(s)]D(−n)xn.If, moreover, g itself is
entire and decays fast enough on ℝ+, then g(x)=∑n=0∞gnxn, Ress=−n[𝔐g(s)]=gn and f(x)~∑n=0∞gnD(−n)xn. The same answer can be obtained by an (legitimate
under the circumstances) interchange of sums in (3.16):(3.19)f(x)=∑k=1∞ak⁢∑n=0∞gn(ωkx)n=∑n=0∞gn(∑k=1∞akωkn)xn=∑n=0∞gnD(−n)xn.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):(3.20)M[lnM(e−x)](s)=∑k=1∞1k∑n=1∞nM[e−nkx](s)=∑k=1∞1k∑n=1∞nΓ(s)(nk)s.Now, assume that Res is large enough
for the double series to converge absolutely, for example, Res>2, and proceed(3.21)=∑k,n=1∞Γ(s)ns−1ks+1=Γ(s)∑n=1∞1ns−1⋅∑k=1∞1ks+1=Γ(s)ζ(s−1)ζ(s+1).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).
LetM(q):=∏n=1∞(1−qn)−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+ is(3.22)lnM(e−x)∼ζ(3)x2+lnx12+ζ′(−1)+∑g=2∞(2g−1)B2gB2g−2(2g−2)(2g)!x2g−2.Proof.
Recall thatζ(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 ats=−2,−4,…,−2g,… with residues ((−1)2g/(2g)!)ζ(−2g−1)ζ(1−2g).(ii)
Simple pole ofζ(s−1) at s=2 with residue 1⋅Γ(2)ζ(3)=ζ(3).(iii)
Double pole ofΓ(s),ζ(s+1) at s=0.
We have from
the first two items and (3.5)(3.23)ζ(3)x2+∑g=1∞1(2g)!ζ(−2g−1)ζ(1−2g)x2g=ζ(3)x2+∑g=2∞(2g−1)B2gB2g−2(2g−2)(2g)!x2g−2.To take care of the double pole,
we need more than just the residue. By the well-known properties of Γ and ζ,(3.24)Γ(1+s)=1−γs+O(s2),ζ(s)=1s−1+γ+O(s−1), where γ is the Euler
constant. Thus,(3.25)Γ(s)ζ(s+1)=Γ(s+1)ζ(s+1)s=(1s−γ+O(s))(1s+γ+O(s))=1s2+O(1),Γ(s)ζ(s−1)ζ(s+1)=(1s2+O(1))(ζ(−1)+ζ′(−1)s+O(s2))=ζ(−1)s2+ζ′(−1)s+O(1)=−1/12s2+ζ′(−1)s+O(1).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 betweenq and x is q=eix 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:(3.26)−ζ(3)x2+ln(−ix)12+ζ′(−1).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=ei0=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, then(3.27)lnM(e2πi(a/b)e−x)∼ζ(3)b3x2+b12lnx+O(1)for real positive x. Thus, every root of unity is indeed singular, and |q|=1 is the natural
boundary.This forces us to reconsider keepingx real in lnM(eix). 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(eix) is the upper
half-plane, and the only distinguished direction in its interior is the
positive imaginary one.Corollary 3.2.
Asymptotic expansion oflnM(eix) at x=0 along iℝ+ is (taking the
principal branch of the logarithm)(3.28)lnM(eix)∼−ζ(3)x2+ln(−ix)12+ζ′(−1)+∑g=2∞(−1)g−1(2g−1)B2gB2g−2(2g−2)(2g)!x2g−2∼−ζ(3)x2+lnx12+ζ′(−1)−πi24+∑g=2∞(−1)g−1(2g−1)B2gB2g−2(2g−2)(2g)!x2g−2.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 pi-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 explainlnx/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 Section5
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.Atoric 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 T1,0(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 of𝒪(−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 ofℂℙ1). 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 numberne for each edge e is assigned the
value from the normal bundle type of the corresponding curve. This only
determines ne 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].Ifξ1,…,ξk is an integral
basis in H2(X,ℤ) as in Section 2, then each edge curve Ce represents a
homology class expressible as a linear combination [Ce]=m1ξ1+⋯+mkξk, mi∈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 of𝒪(−1)⊕𝒪(−1) and local ℂℙ1×ℂℙ1.Partitions and the Schur Functions
We
wish to briefly describe the topological vertex algorithm to see howq-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,(4.1)λ=(λ1,λ2,…,λN,0,…),λi∈ℤ+,⁢λ1≥λ2≥⋯≥λN≥0.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+⋯+λN is called its size (or weight). Partitions are
visualized by Young diagrams,
rows of boxes stacked top down with λi boxes in ith row,
Figure 4. The conjugate partitionλ′ is obtained
visually by transposing the Young diagram along the main diagonal and
analytically as λi′:=max{j∣λj≥i}. Note that λ′′=λ and λ′1=l(λ), |λ′|=|λ|. Another relevant characteristic of a partition,
sometimes called its quadratic Casimir, is(4.2)ϰ(λ):=∑i=1∞λi(λi−2i+1),ϰ(λ′)=−ϰ(λ),ϰ(λ)∈2ℤ.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.Young diagram and its conjugate.
(a)(b)Amplitudes (see Definition4.2) of a labeled
graph are defined via a specialization of the Schur functionssλ 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 (4.3)λ=(1n):=(1,1,…,1︸n⁢times,0,…) then s(1n) is the nth elementary symmetric function:(4.4)s(1n)(x)=en(x):=∑1≤i1<⋯<in<∞xi1⋯xin.In general, sλ are polynomials
in the elementary symmetric functions given by the Jacobi-Trudy formulasλ=det(eλi′−i+j),1≤i,j≤l(λ′)=λ1. For example,(4.5)s(2,1,0,…)(x)=|e2e0e3e1|=e1e2−e0e3=∑i=1∞xi⋅∑i<j=1∞xixj−∑i<j<k=1∞xixjxk.Since en are homogeneous
of degree n, the
Jacobi-Trudy formula implies that sλ are also
homogeneous of degree |λ|, that is, sλ(ax)=a|λ|sλ(x). Moreover, sλ,λ∈𝒫 form a linear basis in the
space of symmetric functions, in particular sλsμ=∑ν∈𝒫cμνλsν. It turns out that cμνλ 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 variablesxi 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(4.6)ρ:=(−12,−32,…)=(−i+12)i=1∞.Note that ρ is not a
partition. Introduce a new formal variable q and for any
vector ξ set qξ:=(qξ1,qξ2,…), so, in
particular, qρ=(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],(4.7)Wλ(q):=sλ(qρ)=Wλ0(q),Wλμ(q):=sλ(qρ)sμ(qλ+ρ),Wλμν(q):=q(ϰ(μ)+ϰ(ν))/2∑α,β,γ∈Pcαγλcγβν′Wμ′α(q)Wμβ′(q)Wμ0(q).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 q1/2 is then defined
by the principal branch of the square root. One can see by inspection from (4.4)
that en(qλ+ρ) converges for |q|>1. Since the Schur “functions” sμ are polynomials
in en, they are also well defined as honest functions of q upon
specializing to qλ+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](4.8)Wλμ(q)=(−1)|λ|+|μ|Wλ′μ′(q−1)=(−1)|λ|+|μ|sλ′(q−ρ)sμ′(q−λ′−ρ).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:(4.9)∑i=1∞qi=q1−q=−11−q−1=−∑i=0∞q−i=−q∑i=1∞q−i,where the two sides never
converge simultaneously either. In fact, 𝒲λμ(q) are rational
functions of q1/2 and can be analytically
continued to ℂ∖ℝ−, (4.8) expresses this analytic continuation.The appearance ofq-bifactorials
in partition functions is due to the Cauchy identity for the Schur functions
[4, 8](4.10)∑λ∈Psλ(x)sλ′(y)u|λ|=∏i,j=1∞(1+uxiyj).If xi=qi−1, yj=qj−1, the right-hand
side of (4.10) becomes(4.11)∏i,j=1∞(1+uqi−1qj−1)=∏i,j=0∞(1+uqi+j)=(u;q)∞(2).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. LetV and Ec denote the sets
of vertices and compact edges of a toric graph, respectively. Choose an
arbitrary orientation for each element of Ec, this determines the sign of the framing numbers.
Assign a formal variable ai to each element
of a basis ξi∈H2(X,ℤ), and set ae:=a1m1⋯akmk for the
corresponding edge curve [Ce]=m1ξ1+⋯+mkξk. Finally, label all compact edges by arbitrarily
chosen partitions λe∈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, Figure5.
Noncompact
edges present no problem since 0′=0. This determines the triple up to cyclic permutation
which is enough since 𝒲λ→v:=𝒲λ1λ2λ3 has cyclic
symmetry.Figure 5
Partition triple at a vertexλ→v:=(λ′,μ′,ν).Definition 4.2.
Amplitude of a labeled toric graph
relative to a basisξ1,…,ξk∈H2(X,ℤ) is given by
[12, 27](4.12)A{λe}(a1,…,ak;q):=∏e∈Ec(−1)|λe|(ne+1)qneϰ(λe)/2ae|λe|⋅∏v∈VWλ→v(q).The main result of [12] can be stated as follows.Theorem 4.3.
The reduced Gromov-Witten partition function of a toric Calabi-Yau threefoldX relative to a
basis ξ1,…,ξk∈H2(X,ℤ) is given by a
state sum:(4.13)ZX′(a1,…,ak;q)=∑{λe}λe∈PA{λe}(a1,…,ak;q)=∑λ1,…,λ|Ec|∈PAλ1,…,λ|Ec|(a1,…,ak;q),assuming in the second sum that
the edges are numbered and λi:=λei.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 Aλ for λ∈𝒫 is (see Figure 3 and
(4.8))(4.14)Aλ(a;q):=(−1)|λ|a|λ|⋅Wλ00(q)Wλ′00(q)=(−a)|λ|Wλ(q)Wλ′(q)=(−a)|λ|(−1)|λ|+|λ′|sλ′(q−ρ)sλ(q−ρ).Recalling that |λ′|=|λ|, −ρ=(i−(1/2))i=1∞ and sλ is homogeneous
of degree |λ|, we compute further(4.15)(−a)|λ|sλ′(qi−(1/2))sλ(qj−(1/2))=(−a)|λ|q(|λ|+|λ′|)/2sλ′((qi))sλ((qj))=(−aq−1)|λ|sλ′((qi))sλ((qj)).Suppose that a is small enough
for ∑λ∈𝒫Aλ(a;q) to converge then,  we get
by Theorem 4.3 and the Cauchy identity (4.10)(4.16)ZX′(a;q)=∑λ∈PWλ(q)Wλ′(q)(−a)|λ|=∑λ∈Psλ′((qi))sλ((qj))(−aq−1)|λ|,(4.17)=∏i,j=1∞(1+(−aq−1)qiqj)=∏i,j=0∞(1−aqqi+j)=(aq;q)∞(2).If we accept the MacMahon function as the degree
zero partition function of the resolved conifold (despite the issues discussed
after Corollary 3.2), then(4.18)ZX0(q)=M(q)χ(X)/2=1∏n=1∞(1−qn)n=1(q;q)∞(2).We conclude that the (full)
Gromov-Witten partition function of the resolved   conifold is(4.19)ZX(a;q)=ZX0(q)ZX′(a;q)=(aq;q)∞(2)(q;q)∞(2),as used in Section 1.


## 5. Reshetikhin-Turaev Calculus and Partition Function of the3-Sphere

As explained in the beginning of Section4, this one
is complementary to it. We briefly review the 𝔰𝔩Nℂ 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 𝔰𝔩Nℂ, 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 Figure6 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 producingnontrivial
invariant. Classical Lie groups and algebras do not
work unfortunately. One has to deform the universal enveloping algebras of,
say, 𝔰𝔩Nℂ into quantum
groups and then specialize the deformation parameter q to a root of
unity q=e2πi/(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 𝔰𝔩Nℂ at level k, they are
indexed by partitions with the Young diagrams in the (N−1)×krectangle, that is,(5.1)PN−1k:={λ∈P∣l(λ)≤N−1,⁢l(λ′)=λ1≤k}.In the equivalent language of
dominant weights, this corresponds to the weights in the Weyl alcove of the
Cartan-Stiefel diagram of 𝔰𝔩Nℂ 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 𝒫N−1k, 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 onk,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:(5.2)Sλμ=S00Wλμ;Tλμ=T00q(1/2)C2(λN)δλμ*.The notation is as follows.Wλμ 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 𝔰𝔩N-dual to μ, μi*:=μ1−μN−i+1 for 1≤i≤N−1 and μi*:=0 fori≥N (not to be
confused with the conjugate partitionμ′);λN denotes the 𝔤𝔩N coordinates of (N−1)×k partition λ, λiN:=λiN−|λ|/N, in particular ρiN:=(1/2)(N−2i+1);C2(λN):=λN⋅(λN+2ρN) is the
quadratic Casimir of λ as a dominant
weight closely related to the quadratic Casimir ϰ(λ) of a partition λ: (5.3)C2(λN)=ϰ(λ)+N|λ|−|λ|2N.
We will say
more about the normalization constants S00,T00 below.Formulas forWλμ 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,(5.4)Wλμ(N;q):=sλ(qρN)sμ(qλN+ρN)withq=e2πi/(k+N).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 𝒲λμ(q) are obtained as
some loosely interpreted limits of Wλμ(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 sλ,sμ before
specialization is infinite whereas here it is N<∞. This last circumstance dramatically simplifies many
computations with 𝒲λμ compared to
their analogs with Wλμ because
infinite specializations often have nice analytic expressions [8].The twist matrixT also has a
vertex counterpart in the form of the framing numbers ne that contribute
factors of qneϰ(λe)/2 to the
amplitude (4.12). Incidentally, this explains their name. In the
Reshetikhin-Turaev sums, Tλμ 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 symmetricWλμ=Wμλ, and the one-point functions Wλ:=Wλ0=W0λ are called the quantum dimensions of representations
indexed by λ. The quantum diameter is(5.5)D2:=∑λ∈PN−1kWλ(N;q)2,and S00 is simply its
inverse S00:=𝒟−1. Analogously, T00 is the inverse
of the charge factorζ:=e2πic/24, where c:=kdimℂ(𝔰𝔩Nℂ)/(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:(5.6)(ST)3=S2,S2T=TS2,S4=I.Note that unlike Wλμ and q(1/2)C2(λN) that depend on k only via q=e2πi/(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
LetJλ1,…,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,…,λn, see [19, 37] for specifics. Then the Reshetikhin-Turaev invariant of the 3-manifold M surgered on L from S3 is(5.7)τ(M):=ζ−3σD−n−1∑λ1,…,λn∈PN−1kJλ1,…,λn(L)Wλ1⋯Wλn,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 have(5.8)ZS3:=τ(S3)=D−1=S00=(∑λ∈PN−1kWλ(N;q)2)−(1/2).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 𝒫N−1k to all partitions. However,
unlike (4.16) that gives ZX′ directly
formula (5.5) gives ZS3−2, 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 ofS3
Although
we usedq above as much
as possible, one should not forget that in the context of the Chern-Simons
theory this is simply a shorthand for e2πi/(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 𝒫N−1k. 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 is(5.9)D2=(−1)N(N−1)/2N(k+N)N−1∏1≤i<j≤N(q(1/2)(j−i)−q−(1/2)(j−i))−2,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 Λr of 𝔰𝔩Nℂ, namely, of (k+N)1/2Λr. Ooguri and Vafa [49, 2.1] replace it with the volume of UN 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≤N1=N(N−1)/2 and ∑1≤i<j≤N(j−i)=N(N2−1)/12, we have(5.10)∏1≤i<j≤N(q(1/2)(j−i)−q−(1/2)(j−i))=q−(2N(N2−1)/24)∏1≤i<j≤N(1−qj−i).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 function(5.11)ZS3:=τ(S3)=iN(N−1)/2q−(N(N2−1)/12)(N(k+N)N−1)−(1/2)∏1≤i<j≤N(1−qj−i).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,(5.12)∏1≤i<j≤N(1−qj−i)=∏n=1N−1(1−qn)N−n=(q;q)∞N(qN+1;q)∞(2)(q;q)∞(2),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 ofq:  (5.13)∏1≤i<j≤N(1−qj−i)=∏n=1N−1(1−qn)∑i=1N−11{i∣n+i≤N}=∏n=1N−1(1−qn)N−n,where 1S 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(5.14)∏n=1N−1(1−qn)N−n=∏n=1∞(1−qn)N−n∏n=N∞(1−qn)N−n=∏n=1∞(1−qn)N−n∏n=0∞(1−qN+n)−n=∏n=1∞(1−qn)N∏n=1∞(1−qN+n)n∏n=N∞(1−qn)n=(q;q)∞N(qN+1;q)∞(2)(q;q)∞(2).Setting a=qN 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 functionGq. Its most characteristic property is the functional
equation Gq(z+1)=Γq(z)Gq(z), where Γq is the quantum
gamma function of Jackson [29]. Γq 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, Gq must be the
product(5.15)Gq(z+1)=(1−q)−(z(z−1)/2)(q;q)∞z(qz+1;q)∞(2)(q;q)∞(2),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 functionGq is the common
factor of the partition functions of the resolved conifold X and the 3-sphere,
namely,(5.16)(1−q)z(z−1)/2ZX(qz;q)=(q;q)∞−zGq(z+1),(5.17)(1−q)z(z−1)/2ZS3(z;q)=iz(z−1)/2q−(z(z2−1)/12)z−(1/2)(lnq2πi)−((z−1)/2)Gq(z+1).In (5.16), q is the usual
variable of the topological vertex, and a:=qz is the degree
variable [12]. In (5.17)
, q=e2πi/(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=eix,qz=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 Gq(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 equationΓ(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 normalizationG(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
Γq(z+1)=(z)qΓq(z) with (z)q:=(1−qz)/(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, (dd+1/dxd+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(6.1)(i)Gq(d)(1)=1,Gq(0)(z)=1−qz1−q,(ii)Gq(d)(z+1)=Gq(d−1)(z)Gq(d)(z),(iii)dd+1dxd+1lnGq(x)≥0forx≥0,⁢0<q<1. The last condition is only
required to hold for real positive q but it is
understood that Gq(d) are continued
analytically to other values of q.Definition 6.1 (Quantum Multigammas).
The
functionsGq(d) defined by (6.1)
are called q-multigamma
functions, in particular Γq:=Gq(1) is the Jackson'
quantum gamma function and Gq:=Gq(2) is the quantum Barnes (or  q-Barnes)
function.For |q|<1, we will show
the existence of Gq(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 Gq(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 classicalq-shifted
factorial as defined by Euler is (a;q)∞:=∏i=0∞(1−aqi). 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).
Aq-shifted d-factorial
is(6.2)(a;q)∞(0):=1−a(noq⁢dependence),(a;q)∞(d):=∏i1,…,id=0∞(1−aqi1+⋯+id),|q|<1,d=1,2,….For finite N, one sets(6.3)(a;q)N(d):=(a;q)∞(d)(aqN;q)∞(d).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 qk for each index.
Our definition is recovered if all qk are set equal
to q. We already used (a;q)∞(2) to write
partition functions in a closed form.
Unlike the cased=1, the finite
version (a;q)N(d) is no longer a
finite product of (1−aqi) unless d divides N. Nevertheless, we have(6.4)∏i1,…,id=0∞(1−aqi1+⋯+id)∏i1,…,id=0∞(1−aqN+i1+⋯+id)=∏id=0∞∏i1,…,id−1=0∞(1−aqi1+⋯+id)∏id=N∞∏i1,…,id−1=0∞(1−aqN+i1+⋯+id)=∏i=0∞(aqi;q)∞(d−1)∏i=N∞(aqi;q)∞(d−1)=∏i=0N−1(aqi;q)∞(d−1),and by definition(6.5)(a;q)N(d)=∏i=0N−1(aqi;q)∞(d−1),d≥1.A similar calculation gives
another useful identity(6.6)(aq;q)∞(d)=(a;q)∞(d)(a;q)∞(d−1).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 polynomials(6.7)(z0):=1(zn):=z(z−1)⋯(z−n+1)n!,n≥1,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),(6.8)∑i1,…,id=0∞ti1+⋯+id=∑n=0∞(∑i1+⋯+id=n1)tn=(∑i=0∞ti)d=(1−t)−d=∑n=0∞(d+n−1n)tn.In other words, (d+n−1n) is the number
of arrangements of d nonnegative
integers adding up to n. As a consequence, we have single-product representations:(6.9)(a;q)∞(d):=∏i1,…,id=0∞(1−aqi1+⋯+id)=∏n=0∞(1−aqn)(d+n−1n),(aq;q)∞(d)=∏n=1∞(1−aqn)(d+n−2n−1). In particular, 1/(q;q)∞(2)=∏n=1∞(1−qn)−(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 as(6.10)Lis(a;q):=∑k=1∞akk(1−qk)s−1.Indeed, by (A.4)(6.11)−ln(a;q)∞(d)=−∑n=0∞(d+n−1n)ln(1−aqn)=∑n=0∞(d+n−1n)∑k=1∞(aqn)kk=∑k=1∞akk∑n=0∞(d+n−1n)(qk)n=∑k=1∞akk(1−qk)−d=Lid+1(a;q).The identity holds for any a∈ℂ by analytic
continuation.Closed Formulas forGq(d)
Typically,
when a classical object isq-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 formula(6.12)Γ(z+1)=limn→∞(n+1)!(z+1)⋯(z+n+1)nz,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 forGq(d) and derive by
induction from (6.5) that for nonnegative integers N,(6.13)Gq(d)(N+1)=(q;q)∞(0)−(Nd)(q;q)∞(1)(Nd−1)⋯(q;q)∞(d)(−1)d+1(N0)(qN+1;q)∞(d)(−1)d=∏i=0d(q;q)∞(i)(−1)i+1(Nd−i)(qN+1;q)∞(d)(−1)d.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 qz:=ezlnq 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 Gq(d) of Definition 2.1. We now prove that this is the case.Theorem 6.3 (Alternating Formula).
Letq∈𝔻∖ℝ− with 𝔻 the open unit
disk in ℂ. Then Gq(d)(z) is an entire function of z given by(6.14)Gq(d)(z+1)=∏i=0d(q;q)∞(i)(−1)i+1(zd−i)(qz+1;q)∞(d)(−1)d,where (q;q)∞(i) are the q-multifactorials
(6.2), and (zn) are the
Stirling polynomials (6.7). In particular, for quantum gamma and the Barnes
functions (d=1,2),(6.15)Γq(z+1)=1(q;q)∞(0)z(q;q)∞(1)1(qz+1;q)∞(1),(6.16)Gq(z+1)=1(q;q)∞(0)(z(z−1)/2)(q;q)∞(1)z1(q;q)∞(2)(qz+1;q)∞(2).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 Gq(d) only as an
alias for this right-hand side. Normalizations (i) follow by direct
substitution of values. (ii) For the functional equation, we
compute(6.17)Gq(d−1)(z)Gq(d)(z)=∏i=0d−1(q;q)∞(i)(−1)i+1(z−1d−i−1)∏i=0d(q;q)∞(i)(−1)i+1(z−1d−i)⋅(qz;q)∞(d−1)(−1)d−1(qz;q)∞(d)(−1)d=∏i=0d−1(q;q)∞(i)(−1)i+1[(z−1d−i−1)+(z−1d−i)](q;q)∞(d)(−1)d+1((qz;q)∞(d)(qz;q)∞(d−1))(−1)d.By (A.6) with n=d−i, the sum in
brackets is just (zd−i) so the first
two factors combine into ∏i=0d(q;q)∞(i)(−1)i+1(zd−i). For the last factor, (6.6) with a=qz yields(6.18)(qz;q)∞(d)(qz;q)∞(d−1)=(qz+1;q)∞(d),so (6.17) becomes(6.19)Gq(d−1)(z)Gq(d)(z)=∏i=0d(q;q)∞(i)(−1)i+1(zd−i)(qz+1;q)∞(d)=Gq(d)(z+1).(iii) It remains to verify the
log-positivity condition. Taking the logarithm, we have(6.20)lnGq(d)(z+1)=∑i=0d(−1)i+1ln(q;q)∞(i)(zd−i)+(−1)dln(qz+1;q)∞(d).Since (zd−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 obtains(6.21)dd+1dzd+1lnGq(d)(z)=(−1)d+1dd+1dzd+1Lid+1(qz;q).It is helpful to notice that (d/dz)f(qz)=lnqx(df/dx)|x=qz and iterating(6.22)(−1)d+1dd+1dzd+1f(qz)=(−lnq)d+1(xddx)d+1f(x)|x=qz.Now, for |x|<1 by (6.10)(6.23)(xddx)d+1Lid+1(x;q)=∑n=1∞1n(1−qn)d(xddx)d+1xn=∑n=1∞nd+1n(1−qn)dxn,and therefore(6.24)dd+1dzd+1lnGq(d)(z)=(−lnq)d+1∑n=1∞(n1−qn)dqnzfor|qz|<1.If 0<q<1 and z is real
positive, then −lnq>0 and 0<qz<1 so the last
expression is positive by inspection.The
structure of Gq(d) is perhaps most
transparent in (6.20). We have the main
term(−1)dln(qz+1;q)∞(d) that depends on a=qz only and the anomaly term polynomial in z of degree d. For d=2, we recognize ln(qz+1;q)∞(2) as exactly the
reduced-free energy of the resolved conifold, see (4.17). The main/anomaly
structure of Gq(d) becomes even
more transparent if we introduce a generating function for the entire
hierarchy:(6.25)∑d=0∞lnGq(d)(z+1)td=(1+t)z∑d=0∞(−1)dLid+1(q;q)td−∑d=0∞(−1)dLid+1(qz+1;q)td.One can check that the sum
converges for |q|,|qz|<1, |t|<1−|q|. This formula displays very clearly the alternating
nature of Gq(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 a3-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:=logqa. Disregarding anomalies comes at a price. Fixing the
canonical 2-framing
would complicate the gluing rules, and Gq(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 Gq(d) originally
given by Nishizawa [14].Theorem 6.4 (Nishizawa Product).
Letq∈𝔻∖ℝ− then quantum
multigammas are given by(6.26)Gq(d)(z+1)=(1−q)−(zd)∏n=1∞(1−qn)(z−nd−1)(1−qz+n)(−nd−1)ford≥1.In particular, for quantum gamma
and Barnes functions (d=1,2):
(6.27)Γq(z+1)=(1−q)−z∏n=1∞(1−qz+n1−qn)−1,(6.28)Gq(z+1)=(1−q)−(z(z−1)/2)∏n=1∞(1−qz+n)n(1−qn)−z+n.Proof.
Applying (6.9) to the first product
in (6.14), we get(6.29)∏i=0d(q;q)∞(i)(−1)i+1(zd−i)=∏i=0d∏n=0∞(1−qn+1)(−1)i+1(n+i−1n)(zd−i)=∏n=0∞(1−qn+1)∑i=0d(−1)i+1(n+i−1n)(zd−i).As shown in the appendix (A.7),
when d≥1(6.30)∑i=0d(−1)i+1(n+i−1n)(zd−i)={(z−n−1d−1),n≥1(z−1d−1)−(zd),n=0.Splitting the n=0 and n≥1 factors in
(6.29), we see that it equals(6.31)(1−q)(z−1d−1)−(zd)∏n=1∞(1−qn+1)(z−n−1d−1)=(1−q)−(zd)∏n=0∞(1−qn+1)(z−n−1d−1)=(1−q)−(zd)∏n=1∞(1−qn)(z−nd−1).The second factor in (6.14) can be
transformed into a single product using (6.9) again(6.32)∏n=1∞(1−qz+n)(−1)d(d+n−2d−1)=∏n=1∞(1−qz+n)(−nd−1),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 Theorem6.3, 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 unityGq(1)=Γq 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=e2πiτ. Clearly, it is desirable to have a similar
construction for Gq(2)=Gq, especially when qis a root of unity. Indeed, this case is
most relevant to the Chern-Simons theory, where Gq 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 anyq. 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 𝔰𝔩Nℂ 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
integerN and any q∈ℂ, one has(6.33)Gq(d)(N+1)=(1−q)−(Nd)∏n=1N(1−qn)(N−nd−1).Proof.
By the
Nishizawa product forGq(d),(6.34)(1−q)(Nd)Gq(d)(N+1)=∏n=1∞(1−qn)(N−nd−1)∏n=1∞(1−qn+N)(−nd−1)=∏n=1∞(1−qn)(N−nd−1)∏n=N+1∞(1−qn)(−(n−N)d−1)=∏n=1N(1−qn)(N−nd−1).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) Gq(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 Gq 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 qz=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=e2πi/(k+N), which is where the Chern-Simons invariants are
defined.


## 7. Conclusions

Recently, the phenomenon ofholography 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=eix 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-formf 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 largeN 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 𝒫N−1k, and ad hoc tricks are
required to turn them into functions of q=e2πi/(k+N). Note that the Reshetikhin-Turaev building blocks Wλμ(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 𝒲λμ(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 auniversal 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=e2πi/(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(7.1)∑λ1,…,λn∈PN−1kWλλ1Wλ1λ2⋯Wλnμcan be represented as boundary
values of sums(7.2)∑λ1,…,λn∈PWλλ1Wλ1λ2⋯Wλnμr|λ1|+⋯+|λn|,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.



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*Source: 438648-2009-03-19.xml*