QUANTUM D-MODULES OF TORIC VARIETIES AND
OSCILLATORY INTEGRALSHIROSHI IRITANI

Abstract. We review mirror symmetry for the quantum cohomology D-moduleof a compact weak-Fano toric manifold. We also discuss the relationship to theb
GKZ system, the Stanley-Reisner ring, the Mellin-Barnes integrals, and the Γintegral structure.

1. Introduction
The purpose of the present notes is to give a concise review on mirror symmetry
for the quantum D-modules of toric varieties, as proposed by Givental [26]. Our goal
will be very modest: we restrict to weak-Fano compact toric manifolds and describebtheir mirror symmetry concretely and explicitly. We will also discuss the Γ-integral
structure of the quantum D-modules and its role in mirror symmetry.
A mirror of a Fano manifold X is given by a Landau-Ginzburg model, that is, acomplex manifold Y equipped with a holomorphic function (potential) W : Y → C(see [3, 12, 26, 28, 39]). Under mirror symmetry, it is expected that the quantum
cohomology of X is isomorphic to the Jacobi ring of W , and that the quantum
cohomology D-module of X is isomorphic to the mirror D-module whose solutions
are oscillatory integrals associated with W (see Table 1). For a (weak-)Fano toric
manifold X, the mirror W is given by a family of Laurent polynomials whose Newton
polytopes are given by the fan diagram of X, and the mirror correspondence as in
Table 1 is by now well-established. We will explain the relationship between the
mirror D-module and the Gelfand-Kapranov-Zelevinsky (GKZ) system and how the(quantum) Stanley-Reisner ring arises as a limit of the mirror D-module. The GKZ
system or the Stanley-Reisner rings can be regarded as intermediate objects between
quantum cohomology of toric varieties and their mirrors.
Table 1. Mirror correspondence.
Fano manifold

Landau-Ginzburg

Xmodel W : Y → C

quantum cohomology

Jacobi ring

quantum Stanley-ring(or H(Y, (Ω•Y , dW )))Reisner ring

quantum D-module

D-module of oscillatory
R
integrals eW/z Ω

GKZ system

∂
zQ ∂Q
+ p?

toric cased I = 0

Even for Fano manifolds other than toric varieties, the mirror space Y is often
given by (a partial compactification of) the algebraic torus (C× )n and the potential
W is given by a Laurent polynomial on it; however, the coefficients of the mirror
1

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HIROSHI IRITANI

Laurent polynomial W are very special (see e.g. [13,48]). Many people have observed(see [6]) that, when a Fano manifold X admits a flat degeneration to a (possibly
singular) toric variety X 0 , the mirror of X is often given as a special subfamily of theLaurent polynomials that are mirror to a crepant resolution of X 0 . Such a subfamilycorresponds to a certain stratum in the discriminant loci of the GKZ system for X 0 .
Therefore, toric mirror symmetry can be viewed as a generic phenomenon whereas
mirror symmetry for more general Fano manifolds can be viewed as its specialization.
In this article, we do not delve into mirror symmetry for non-toric Fano manifolds; werefer the reader to the article of Golyshev [29] in the same volume for the differential
equations aspects of the story1.
b
In the last two sections, we will discuss the Γ-integral
structure [36, 38] in thebcontext of toric mirror symmetry. The Γ-integral
structure is a certain integral
blocal system underlying the quantum D-module, which is defined by the Γ-classand the topological K-group. Conjecturally this corresponds to the natural inteb
gral structure on the mirror. The Γ-classinvolves transcendental numbers such as
the Riemann ζ-values and its origin is quite mysterious from a viewpoint of curvecounting or symplectic topology. On the other hand, the existence of such a structure had been suggested since the beginning of mirror symmetry. Candelas-de laOssa-Green-Parkes [8] already observed that χ(X)ζ(3) appears in the asymptotics
of periods of mirror Calabi-Yau threefolds near the large complex structure limit(LCSL). Hosono-Klemm-Theisen-Yau [35] observed “remarkable identities” that relate certain characteristic numbers of complete intersection Calabi-Yau manifolds
with hypergeometric solutions of the Picard-Fuchs equation of the mirror family.
b
This observation led Libgober [42] to introduce the (inverse) Γ-classof a complex
manifold. Later, Hosono [34] made this connection more precise in terms of central
charges and homological mirror symmetry.
When a Fano manifold X is mirror to the Landau-Ginzburg model W : Y → C,b
the compatibility of the Γ-integral
structure and mirror symmetry would imply thefollowing “mirror symmetric Gamma conjecture” (cf. [23, 24], see §7)Z
Z−tWbX as t → 0.
eΩ∼t−c1 (X) · ΓΓ⊂YXThis says that the asymptotics of the mirror oscillatory integral is described in termsbof the Γ-classof X. The integration cycle Γ here should be a Lagrangian section of
the Strominger-Yau-Zaslow fibration [53] (i.e. mirror to the structure sheaf of X).
In the last section, we will introduce Mellin-Barnes integral representations for thebmirror oscillatory integrals (in the toric case) and explain these Γ-phenomena
fortoric varieties.
Acknowledgements
I thank Vasily Golyshev for many helpful discussions. This work is supported by
JSPS Kakenhi Grant Number 16K05127, 16H06335, 16H06337, and 17H06127. The
essential part of this work was done while I was in residence at the Hausdorff Research Institute for Mathematics at Bonn in January 2018, and at the Mathematical
Sciences Research Institute at Berkeley, California in April 2018. The latter stay
1This article was originally written as an appendix to his article, but became independent in

the end.

QUANTUM D-MODULES OF TORIC VARIETIES AND OSCILLATORY INTEGRALS

3

was supported by the National Science Foundation under Grant No. DMS-1440140.
I thank both institutes for providing excellent working conditions.
2. Preliminaries on toric varieties
A toric variety is a GIT quotient of a vector space Cm by a torus K ∼= (C× )k ,mwhere the torus K acts on C via an injective group homomorphism K → (C× )m .
Let D1 , . . . , Dm ∈ Hom(K, C× ) denote the characters defining the homomorphism
K → (C× )m . A GIT quotient is given by choosing a stability condition ω ∈ k∗R ,where kR denotes the Lie algebra of the maximally compact subgroup KR of K. We
assume thatP(a) ω lies in thePcone m
i=1 R≥0 Di ;(b) if ω lies in i∈I R>0 Di for a subset I ⊂ {1, . . . , m}, then {Di }i∈I spans k∗Rover R; P(c) the cone m
i=1 R≥0 Di is strictly convex.
Then the toric variety corresponding to ω is defined byXω := Uω /K
S I
m
where
U
:=
C
\
/
ω
I C with I ranging over subsets of {1, . . . , m} such that ω ∈Pi∈I R>0 Di . The conditions (a), (b), (c) respectively ensure that Xω is non-empty,that Xω has at worst orbifold singularities, and that Xω is compact. The space k∗Rof stability conditions has a fan structure, called the GKZ fan. A maximal cone of
the GKZ fan is given by the closure of a connected component of the set of ω ∈ k∗R
satisfying the conditions (a)–(c). The toric variety Xω depends only on the maximal
cone Aω to which ω belongs.
We assume that Xω is weak-Fano, or equivalently, that c1 (Xω ) is nef. For simplicity of notation and exposition, we will also assume that Xω is a smooth manifold (without orbifold singularities) and that {zi = 0} ∩ Uω is non-empty for alli = 1, . . . , m, where zi is the ith co-ordinate on Cm . With these assumptions, wehave k∗R ∼= H 2 (Xω , R) and the closure of the ample cone of Xω corresponds to the
maximal cone Aω of the GKZ fan. Under this isomorphism, the character Di corresponds to the class of the toric divisor {zi = 0} in Xω .
Remark 1. Although we restrict to smooth toric manifolds in this article, all the
results discussed in this paper can be extended to toric orbifolds [10, 11, 14, 36, 44].
In the orbifold case, it is important to allow {zi = 0} ∩ Uω = ∅ for some i as such
indices correspond to twisted sectors.
A toric variety can be also described in terms of a fan. Let kZ = Hom(C× , K)
denote the cocharacter lattice of K. Consider the natural map kZ → Zm induced by
the inclusion K → (C× )m and complete it to a short exact sequence
0 → kZ → Zm → N → 0
where N := Zm /kZ is a free abelian group of rank n = m − k. The fan Σω of Xω
is defined on the vector space NR = N ⊗ R: the imagePbi ∈ N of ei ∈ Zm gives agenerator of a 1-dimensional
cone of Σω , and the cone i∈I R≥0 bi belongs to Σω ifPand only if ω ∈ i∈I/ R>0 Di .

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HIROSHI IRITANI

3. Quantum D-modules
The cocharacter lattice kZ = Hom(C× , K) is identified with H2 (Xω , Z) and thedual cone A∨ω ⊂ kR of Aω is identified with the cone of curves. The (small) quantum
product ? is a commutative and associative product on the space H ∗ (Xω )⊗C[[A∨ω ∩kZ ]]defined byX(α ? β, γ) =hα, β, γi0,3,d Qd
for all α, β, γ ∈ H ∗ (Xω ),
d∈A∨
ω ∩kZ

where hα, β, γi0,3,d is the genus-zero, three-points, degree-d Gromov-Witten invariant and Qd denotes the element of C[[A∨ω ∩ kZ ]] corresponding to d. The Dubrovinconnection is a flat connection on the trivial H ∗ (Xω )-bundle over Spec C[[A∨ω ∩ kZ ]]given by1∂
(1)
∇ξQ ∂ = ξQ
+ (ξ?),
ξ ∈ H 2 (Xω )
∂Q
∂Q z
∂
where z is a formal parameter and ξQ ∂Q
is a derivation of C[[A∨ω ∩ kZ ]] defined by∂(ξQ ∂Q)Qd = hξ, diQd . This is a flat connection with logarithmic singularities. The
Dubrovin connection z∇ (multiplied by z) acts on the space

QDM(Xω ) = H ∗ (Xω ) ⊗ C[z][[A∨ω ∩ kZ ]]which we call the quantum D-module. It is not known in general whether the quantum product ? converges or not. For toric varieties, it is known that the quantum
product converges and hence the quantum D-module extends to an actual analytic
neighbourhood of the origin “Q = 0” in Spec C[A∨ω ∩ kZ ]. The point Q = 0 is called
the large radius limit point.
Remark 2. The Dubrovin connection can be also extended in the z-direction. The
connection in the z-direction is given by1∂− c1 (Xω ) ? +µ∇z∂z = z∂z z∗where µ ∈ End(H (Xω )) is defined by µ(α) = (p − n2 )α for α ∈ H 2p (Xω ) withn = dimC Xω .
4. Mirror D-modules
We have the exact sequence of tori 1 → K → (C× )m → T → 1 where T :=(C× )m /K ∼= N ⊗ C× is a torus acting on Xω with an open dense orbit. Consider
the exact sequence 1 → Ť → (C× )m → Ǩ → 1 of dual tori. The mirror LandauGinzburg model of a toric variety Xω is given by the family of tori pr : (C× )m → Ǩ
together with a potential function W : (C× )m → C defined by W = u1 + · · · + um ,where ui denotes the ith co-ordinate on (C× )m .
(C× )m

W =u1 +···+um

/C

pr

Ǩ
Choosing a splitting of the sequence, we can also write Wq = W |pr−1 (q) as
Wq = q l1 xb1 + · · · + q lm xbm

QUANTUM D-MODULES OF TORIC VARIETIES AND OSCILLATORY INTEGRALS

5

where q ∈ Ǩ, x ∈ Ť and recall that bi are generators of 1-dimensional cones of the
fan Σω . By varying q ∈ Ǩ, Wq can represent any Laurent polynomial in x ∈ Ť
having b1 , . . . , bm as exponents. Hence the mirror of a toric variety can be thoughtof as generic Laurent polynomials.
Remark 3. We denote the B-model co-ordinates by q and the A-model co-ordinates
by Q. These co-ordinates are related by the mirror map ψ below.
Using the GKZ fan on k∗R and its ‘preimage fan’ on Rm (whose maximal conesare (R≥0 )m ∩ π −1 (A), where A is a maximal cone the GKZ fan and π : Rm → k∗R
is the natural map given by D1 , . . . , Dm ), we can partially compactify the family
pr : (C× )m → Ǩ to a map between toric varieties pr : Y → M and W extends to aregular function Y → C.
(C× )m 


Ǩ 



/Y

W

/C



pr



/M

The maximal cone Aω defines a torus-fixed point 0ω ∈ M which is mirror to thelarge-radius limit point Q = 0 of the quantum cohomology of Xω . Let Yq denotethe fibre of q ∈ M and write Wq = W |Yq . Givental introduced oscillatory integralsZ(2)eWq /z ΩΓ⊂Yq

as mirrors of the quantum D-module, where Γ is a non-compact Morse cycle for<(Wq /z) and Ω is a holomorphic volume form on the fibre Yq . Introduce the log
structures on Y and M given by their toric boundaries and let Ω•Y/M denote therelative logarithmic de Rham complex. The integrands Ω of (2) can be naturally
viewed as elements of the twisted de Rham cohomology:
GM(W ) = pr∗ H top (Ω•Y/M [z], zd + dW ∧).
This is equipped with the (logarithmic) Gauss-Manin connection and the higher
residue pairing; such structures were introduced by K. Saito [50] in singularity theory. The Gauss-Manin connection is a map∇ : GM(W ) →1dz
GM(W ) ⊗OM Ω1M ⊕ GM(W ) 2zzwhich has the same pole structure along z = 0 as the Dubrovin connection (hereΩ1M denotes the sheaf of logarithmic 1-forms). When we choose a splitting of thesequence 1 → Ť → (C× )m → Ǩ → 1 and choose co-ordinates x = (x1 , . . . , xn ) onŤ ∼= (C× )n and q = (q1 , . . . , qk ) on Ǩ ∼= (C× )k , the connection ∇ is given by2
 

 
k 
X
∂a W
dqa
W
n
dz
f Ω0
+ z∂z f − f − f Ω0∇[f Ω0 ] =∂a f +
z
qa
z
2
z
a=1
2Here we twist the Gauss-Manin connection by − n dz so that it is compatible with the Dubrovin
2 z

connection.

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HIROSHI IRITANI

where ∂a = qa ∂q∂a , f = f (x, q, z) ∈ OY [z] and Ω0 is the standard relative volume
form of the family pr : Y → M:
(3)

Ω0 =

dxn
dx1
∧ ··· ∧.
x1
xn

The higher residue pairing is a map(−)∗ GM(W ) ⊗OM [z] GM(W ) → OM [[z]]which coincides with the residue pairing when restricted to z = 0, where (−) denotes
the automorphism of OM [z] sending f (q, z) to f (q, −z). For a concrete descriptionof the higher residue pairing, we refer the reader to [11, §6]. There exists a Zariski
open subset U of M containing 0ω such that GM(W ) is locally free and coherent(as an O-module) over U × Cz [36, 44, 47].
Theorem 4 ([11, 16, 26, 27, 36, 37, 44, 46, 47]). Over an analytic open neighbourhoodof 0ω , the mirror D-module GM(W ) is isomorphic to the pull-back of the quantum
D-module QDM(Xω ) by a map ψ ∈ Aut(C[[A∨ω ∩ kZ ]]), called the mirror map. Under
this isomorphism, the higher residue pairing corresponds to the Poincaré pairing.
The mirror map ψ above extends to a local isomorphism between analytic neighbourhoods of 0ω , and therefore the pull-back by ψ makes sense over an analytic
neighbourhood. If M contains two different large radius limit points 0ω1 , 0ω2 suchthat Xω1 , Xω2 are weak Fano, the theorem implies that the quantum D-modules ofXω1 and Xω2 are isomorphic under analytic continuation; this is an example where
the crepant transformation conjecture holds [9, 15].
Remark 5. The above theorem is essentially due to Givental [26, 27]. Givental
introduced the mirror potential function W (see also Hori-Vafa [33]) and expressed
solutions of the quantum D-module in terms of oscillatory integrals. Givental’s mirror theorem [27] (see also Lian-Liu-Yau [40,41]) is stated as the equality between the
J-function and the I-function, where the J-function is a solution to the quantum
D-module and the I-function is a solution to the mirror D-module (see the next section). Mirror symmetry as an isomorphism of D-modules has been studied in details
by [36, Proposition 4.8], [47, Theorem 4.11], [16, Theorem 5.1.1], [46, Theorem 7.43],[37, Theorem 4.2], [44, Theorem 6.4], [11, Theorem 1.1]. The logarithmic extension
across the boundary divisors of M has been studied by [16, 44, 46, 47] in terms of(GKZ) D-modules; the logarithmic extension of the mirror Landau-Ginzburg model
itself was discussed in [11, 37, 43]. In this paper, we restrict to the small quantum
cohomology, but we also have mirror symmetry for the big quantum cohomology,
see [4, 11, 17, 18, 37].
Example 6. The mirror family of Pn is given by the diagram
Cn+1


W =u0 +···+un

/C

pr

C
with q = pr(u0 , . . . , un ) = u0 u1 · · · un , where Y = Cn+1 and M = C. We can writeqWq = W |pr−1 (q) as Wq = x1 + · · · + xn + x1 ···xby setting xi := ui for 1 ≤ i ≤ n.
n

QUANTUM D-MODULES OF TORIC VARIETIES AND OSCILLATORY INTEGRALS

7

5. The GKZ system and hypergeometric solutions
The mirror D-module GM(W ) can be described in terms of the Gelfand-KapranovZelevinsky (GKZ) system [25]. To describe the GKZ system explicitly, we introducePka basis p1 , . . . , pk of k∗Z ∼= H 2 (Xω , Z) and write Di = a=1 mia pa . The elements
Qk
p1 , . . . , pk define co-ordinates q1 , . . . , qk on Ǩ ∼= (C× )k . We write q d = a=1 qapa ·d
for d ∈ kZ and set ∂a = qa ∂q∂a . Over the open torus Ǩ ⊂ M, the mirror D-module
GM(W ) is generated by the standard relative volume form Ω0 (see (3)) as a Dmodule, or more precisely, as a module over the ring OǨ [z]hz∂1 , . . . , z∂k i, where z∂aacts by the Gauss-Manin connection z∇∂a . All the D-module relations of Ω0 are
generated by d [Ω0 ] = 0 with d ∈ kZ , where!
!
kki ·d−1i ·d−1
Y DY
Y −DY
X
X
(4) d :=mia z∂a − jz − q dmia z∂a − jz .
i:Di ·d>0

j=0

i:Di ·d<0

a=1

j=0

a=1

This system of differential equations is called the GKZ system. Givental’s I-function[27] is a cohomology-valued function annihilated by d :m Q0XYj=−∞ (Di + jz)
d+p/z
I(q, z) =q
QDi ·dj=−∞ (Di + jz)
i=1d∈A∨ ∩kZ
ωQ
where q p/z = ka=1 epa log qa /z . The components of the I-function form a basis ofsolutions to the GKZ system near 0ω [36, 44, 47]. The mirror map ψ is determinedby the z −1 -expansion of the I-functionk

1X
I(q, z) = 1 +pa log ψa (q) + O(z −2 )z a=1
where we write the mirror map in the form Qa = ψa (q), a = 1, . . . , k using the A-model co-ordinates Q1 , . . . , Qk dual to p1 , . . . , pk . We have ψa (q) =qa + higher order terms; in the Fano case the mirror map is trivial ψa (q) = qa .
Remark 7 ([2, 25, 36]). The rank of the GKZ system (at a generic point in Ǩ)is equal to the normalized volume of the fan polytope, that is, the convex hull of
b1 , . . . , bm in NR ; this is also equal to dim H ∗ (Xω ).
Example 8 (continuation of Example 6). In the case of Pn , the mirror oscillatory
integral satisfies the differential equation:
Z 

q

x1 +···+xn + x ···x
/z dx1 · · · dxn
n+1
n1(zq∂q )−qe
= 0.
x1 · · · xnΓThe I-function is given byI(q, z) =∞Xq p/z+dQd

d=0

j=1 (p + jz)

n+1(with p = c1 (O(1))) and the mirror map is trivial: Q = q. The relationship between
these two solutions will be discussed in §7-8, see Remark 11 and Example 14.

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HIROSHI IRITANI

6. The Jacobi ring and the Stanley-Reisner ring
A ring structure arises from these D-modules in the z → 0 limit. The z → 0 limitof the Dubrovin connection z∇ξQ ∂ is the quantum product by ξ ∈ H 2 (Xω ) (see∂Q(1)), and the quantum multiplication by divisors generate the quantum cohomology
ring. On the other hand, the z → 0 limit of the mirror D-module is given by the(logarithmic) Jacobi ring of W :
E
 .D
∂W
∂W
Jac(W ) = pr∗ OY
x1 ∂x1 , . . . , xn ∂xnwhere we choose C× -co-ordinates (x1 , . . . , xn ) of Ť ∼= (C× )n and xi ∂x∂ i defines a relative tangent vector field of the family Y → M. Theorem 4 induces an isomorphismψ ∗ QH ∗ (Xω ) ∼= Jac(W )between the quantum cohomology ring and the Jacobi ring. This isomorphism has
been studied by many people, see [5,20–22,31,45,49,52] together with the referencesin Remark 5.
Over the affine open chart Mω := Spec(C[A∨ω ∩ kZ ]) of M, the Jacobi ring is aquotient of the Pquantum Stanley-Reisner ring associated with the fan ΣPω . For v ∈ N ,v and write v = i∈I vi bi . We
choose a cone i∈I R≥0 bi of the
Q fan vΣi ω containing
0
∨
.
Then
H
(M
u
set vi = 0 for i ∈/ I. Set wv := m
ω , pr∗ OY ) is a free C[Aω ∩ kZ ]i=1 i
module generated by wv with v ∈ N . The product structure is given by(5)0

wv wv0 = q `(v,v ) wv+v0

where `(v, v 0 ) ∈ A∨ω ∩ kZ is the element given by the linear relation (vi + vi0 − (v +m
→ N → 0. This ringv 0 )i )m
i=1 among bi ’s via the exact sequence 0 → kZ → Z
0
H (Mω , pr∗ OY ) is called the quantum Stanley-Reisner ring. The Jacobian ideal
gives additional linear relations among u1 , . . . , um given by:
(6)

m
X

ui bi = 0.

i=1

The relations (5), (6) define the Jacobi ring Jac(W ); they are also known as therelations of Batyrev’s quantum ring [5]. At the large-radius limit q → 0ω , thequantum Stanley-Reisner relations (5) reduce to(
wv+v0 if v and v 0 lie in a common cone of Σω ;0wv wv =
0otherwise.
These relations define the Stanley-Reisner ring. As is well-known in toric geometry,
the Stanley-Reisner ring modulo the linear relations (6) is isomorphic to H ∗ (Xω ).
Remark 9. Over an analytic neighbourhood of 0ω , pr∗ OY is isomorphic to theT -equivariant quantum cohomology of Xω and the left-hand side of the linear relations (6) correspond to the T -equivariant parameters. Moreover, the multiplicationby the co-ordinates x1 , . . . , xn of Ť corresponds to the Seidel representation [51] on
the equivariant quantum cohomology. McDuff-Tolman [45] used the Seidel representation to determine a presentation of the quantum cohomology ring of a toric
manifold. In [37], we showed how equivariant mirror symmetry for toric manifolds
follows almost tautologically from the Seidel representation and shift operators [7].

QUANTUM D-MODULES OF TORIC VARIETIES AND OSCILLATORY INTEGRALS

9b
7. Γ-integral
structureb
The Γ-class[42] of an almost complex manifold X is defined to be the characteristic class
n
Y
b
ΓX =
Γ(1 + δi )
i=1

where
δ1 , . . . , δn are the Chern roots of the tangent bundle (so that c(T X) =Qn
(1
+ δi )) and Γ(x) is Euler’s Γ-function. The Γ-function Γ(1 + δ) here should
i=1
be expanded in Taylor series at δ = 0. For a toric variety X = Xω , it is given by
!
m
m
∞
Y
Y
X
ζ(k)
bXω =
Djk
Γ
Γ(1 + Dj ) =exp −γDj +
(−1)k
k
j=1
j=1
k=2
where γ is the Euler constant, ζ(s) is the Riemann ζ-function and recall that Dj isthe class of a prime toric divisor.
b
The Γ-integral
structure [36,38] is an integral lattice in the space of (multi-valued)
flat sections of the Dubrovin connection. For an element E ∈ K 0 (Xω ) of the topological K-group, there exists a unique flat section sE (Q, z) of the Dubrovin connection(which is flat also in the z-direction, see Remark 2) such that√1−p/z −µ c1 (Xω ) b
deg /2sE (Q, z) ∼Qz zΓXω ∪ (2π −1)ch(E)(2π)n/2Q
as Q → 0, where Q−p/z = ka=1 e−pa log Qa /z (we choose co-ordinates Q1 , . . . , Qk dual
to p1 , . . . , pk as in §5) and µ is as in Remark 2. These flat sections sE (Q, z) spanbthe Γ-integral
structure. Its important properties are as follows:(i) this lattice is invariant under local monodromy around the large radius
limit point; therefore it defines a Z-local system underlying the quantum
D-module;(ii) the Poincaré pairing of these flat√sections coincide with the Euler pairing on
the derived category: (sE (Q, e−π −1 z), sF (Q, z)) = χ(E, F ).
The second property follows from the identity Γ(1+x)Γ(1−x) = πx/ sin(πx) (whichbX is the “half” of the Todd class) and the Hirzebruch-Riemannimplies that the ΓRoch formula.
The mirror D-module GM(W ) from §4 has a natural integral structure3 dual to
the relative homology Hn (Yq , {<(Wq /z)  0}; Z) via the oscillatory integral (2).
Here note that an element of the relative homology gives an integration cycle in (2).
These two integral structures coincide under mirror symmetry.
b
Theorem 10 ([36]). The Γ-integral
structure coincides with the natural integral
structure on the mirror D-module under the mirror isomorphism in Theorem 4.
This theorem follows from the following identity of periods of both sides:ZZ
1
e−Wq /z Ω0(7)
sE (ψ(q), z) =
n/2
(2πz)
Γ(E)
Xω
where E 7→ Γ(E) is an isomorphism between K 0 (Xω ) and Hn (Yq , {<(Wq /z) 0}; Z); this correspondence should be a shadow of homological mirror symmetry,
3To be more precise, we twist the local system of relative cohomology by (−2πz)−n/2 so that it

is compatible with the Dubrovin connection.

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HIROSHI IRITANI

i.e. an equivalence between the derived category of coherent sheaves on Xω and the
Fukaya-Seidel category of Wq (see [19]):b(Xω ) ∼Dcoh
= F S(Y, Wq ).

When E is the structure sheaf O on Xω , the corresponding cycle Γ(O) is given by
the positive real locus (R>0 )n in Yq ∼= Ť ∼= (C× )n (when q lies in the positive real×
locus Hom(kZ , R>0 ) of Ǩ = Hom(kZ , C ) and z > 0). In this case, (7) yields thefollowing asymptotics:
Z
Z−Wq /z
bXω
(8)
e
Ω0 ∼
q −p z c1 (Xω ) ∪ Γ(R>0 )nXωb
as q → 0ω in the positive real locus and for z > 0. Namely, the Γ-classappears in
the q → 0ω asymptotics of the exponential period of the mirror. It would be very
interesting to study if such asymptotics hold for more general Fano manifolds andtheir mirrors.
Remark 11. The equality (7) can be restated in terms of the I-function as follows:
Z 
Z

√
deg
c1 (Xω ) degb22zz I(q, −z) ∪ ΓXω (2π −1) ch(E) =e−Wq /z Ω0 .
Xω

Γ(E)

This formula expresses the oscillatory integral as an explicit linear combination ofcomponents of the I-function.
Remark 12. There is another version of “Gamma conjecture” for quantum cohomology of Fano manifolds formulated by Galkin, Golyshev and the author [23],which does not involve mirror symmetry. The quantum D-module of a Fano manifold has irregular singularities at “Q = ∞”, and the conjecture is about the Stokes
structure at Q = ∞ and the connection of solutions between Q = 0 and Q = ∞.
bThis version of the conjecture, if true, implies that the Γ-classcan be recovered fromthe quantum cohomology of a Fano manifold. See also [24, 30].
Remark 13. Abouzaid, Ganatra, Sheridan and the author [1] recently proposed
an approach to proving the asymptotics (8) (for Calabi-Yau mirror pairs) using
Strominger-Yau-Zaslow picture [53] and tropical geometry.
8. Mellin-Barnes integral representation
The oscillatory integrals (2) give an integral representation of solutions to the GKZ
system. There is another integral representation, Mellin-Barnes integral representation, which is “Gale dual” to (2). We shall regard p1 , . . . , pk ∈ k∗Z as co-ordinates on
kC = Lie(K) which are Mellin-dual to q1 , . . . , qk . Under the Mellin transformationRb
I(q) 7→ I(p)= I(q)q p dqq , the differential operator ∂a = qa ∂q∂a corresponds to themultiplication by −pa and the multiplication by qa corresponds to the shift operator
Ta : pb 7→ pb + δa,b . Thus the GKZ equations d I(q, z) = 0 (see (4)) with d ∈ kZ aretransformed into the difference equations:!
i ·d−1i ·d−1Y DY
Y DYb z) = 0(−Di − j)z − T d(−Di − j)z I(p,i:Di ·d>0

j=0

i:Di ·d<0

j=0

QUANTUM D-MODULES OF TORIC VARIETIES AND OSCILLATORY INTEGRALS

11

Pk
∗
where Di =of the standard coa=1 mia pa ∈ kZ is regarded as the pull-back
Qk
m
m
d
pa ·d
ordinates on C via the inclusion kC ,→ C and T = a=1 Ta . It is easy to checkthat this system has the following simple solution:b z) =I(p,mY(−z)Di Γ(Di ).

i=1

Q
ui /z
In fact,R Ib is (the restriction of) the Mellin transform of eW/z = m; here we
i=1 e∞ u/z D duD
recall 0 e u u = (−z) Γ(D) with z < 0 and D > 0. By the inverse Mellintransformation, we get a solution to the GKZ system:
!
Z
m
Y
1
√
(−z)Di Γ(Di ) dp1 · · · dpk(9)q −p(2π −1)k C⊂kC
i=1
where C ⊂ kC is a suitable (non-compact) k-cycle so that the integral Rconverges. Fora suitable choice of C, (9) should coincide with an oscillatory integral Γ eWq /z Ω0 (seeFigure 1 below), but the author does not know a precise choice of cycles in general.
b
Via the residue calculation, such a formula would explain the Γ-classappearing in
the leading asymptotics (8), see Example 14 below.

eW/z =

Qm

pr∗

ui /z
on (C× )m
i=1 e

O


Qm

/

R

eW/z Ω on Ǩ
O

MellinrestrictionDim
i=1 (−z) Γ(Di ) on CMellinb z) on kC/ I(p,

Figure 1. Oscillatory integral and its Mellin transform

Example 14 (continuation of Example 8). We consider the mirror oscillatory integral of Pn again. The following method is borrowed from [38]. The Mellin transformqR)/z dq−(x +···+xn + x ···xn1of the oscillatory integral I(q) = (R>0 )n e 1(with q, z > 0) gives
q
Z ∞
b
I(p)
=

q p I(q)Z0dq
q(u0 · · · un )p e−(u0 +···+un )/z=(R>0 )n+1=z(n+1)pdu0
dun∧ ··· ∧u0un

Γ(p)n+1 .

b −z) as expected. Then the Mellin inversion formula gives
This coincides with I(p,I(q) =1√2π −1Z c+√−1∞√bq −p I(p)dpc− −1∞

with c > 0. By closing the integration contour to the left, we can express the righthand side as the sum over residues at p = 0, −1, −2, . . . , arriving at the asymptotics

12

HIROSHI IRITANIin (8).
I(q) =∞
X

Resp=−d q −p z (n+1)p Γ(p)n+1 dp



d=0

 Z

n
−p (n+1)p
n+1 dpbPn .
q −p z c1 (P ) ∪ Γ=∼ Resp=0 q z
Γ(1 + p)n+1
pPnRemark 15. The Mellin-Barnes integral representations (9) appear in physics literature as hemisphere partition functions (studied for more general gauged linear
sigma models), see, e.g. [32].
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E-mail address: iritani@math.kyoto-u.ac.jp
Department of Mathematics, Graduate School of Science, Kyoto University,
Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan