GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO MANIFOLDS: GAMMA CONJECTURES

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GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO

MANIFOLDS: GAMMA CONJECTURES

SERGEY GALKIN, VASILY GOLYSHEV, AND HIROSHI IRITANI

Abstract. We propose Gamma Conjectures for Fano manifolds which can be thought ofas a square root of the index theorem. Studying the exponential asymptotics of solutions tothe quantum differential equation, we associate a principal asymptotic class AF to a Fanomanifold F . We say that F satisfies Gamma Conjecture I if AF equals the Gamma class

ΓF . When the quantum cohomology of F is semisimple, we say that F satisfies GammaConjecture II if the columns of the central connection matrix of the quantum cohomology

are formed by ΓF Ch(Ei) for an exceptional collection Ei in the derived category of coher-

ent sheaves Dbcoh(F ). Gamma Conjecture II refines part (3) of Dubrovin’s conjecture [20].

We prove Gamma Conjectures for projective spaces, toric manifolds, certain toric completeintersections and Grassmannians.

Contents

1. Introduction 21.1. Gamma class 21.2. Gamma conjectures 31.3. When the Gamma Conjectures hold 41.4. Limit formula and Apery’s irrationality 51.5. Mirror symmetry 51.6. Plan of the paper 62. Quantum cohomology, quantum connection and solutions. 62.1. Quantum cohomology 62.2. Quantum connection 72.3. Fundamental solution around the regular singular point z = ∞ 82.4. Formal fundamental solution around the irregular singular point z = 0 92.5. Laplace transform of ∇ 112.6. Asymptotically exponential flat section 122.7. Changing integration paths: mutation and Stokes matrix 132.8. Isomonodromic deformation and marked reflection system 162.9. Deformation of repeated eigenvalues 183. Gamma Conjecture I 203.1. Conjecture O 203.2. Flat sections with the smallest asymptotics 213.3. Gamma Conjecture I: statement 223.4. The leading asymptotics of dual flat sections 243.5. A limit formula for the principal asymptotic class 273.6. Apery limit 293.7. Quantum cohomology central charges 31

Date: April 28, 2014.

1

http://arxiv.org/abs/1404.6407v1

2 GALKIN, GOLYSHEV, AND IRITANI

4. Gamma Conjecture II 324.1. Semiorthonormal basis and Gram matrix 324.2. Marked reflection system 324.3. Mutation of MRSs 334.4. Exceptional collections 344.5. Asymptotic basis and MRS of a Fano manifold 354.6. Dubrovin’s (original) conjecture and Gamma conjecture II 354.7. Quantum cohomology central charges 365. Remarks on toric case and quantum Lefschetz principle 375.1. Toric manifolds 375.2. Toric complete intersections 385.3. Compatibility with the quantum Lefschetz principle 406. Gamma conjectures for projective spaces 416.1. Quantum connection of projective spaces 416.2. Frobenius solutions and Mellin solutions 426.3. Monodromy transformation and mutation 447. Gamma conjectures for Grassmannians 467.1. Statement 467.2. Quantum Pieri and quantum Satake 467.3. The wedge product of the big quantum connection of P 497.4. The wedge product of MRS 527.5. MRS of Grassmannian 537.6. The wedge product of Gamma basis 547.7. The abelian/non-abelian correspondence and the Gamma basis 557.8. Gamma Conjecture I for Grassmannians 56References 57

1. Introduction

1.1. Gamma class. The Gamma class of a complex manifold X is the cohomology class

ΓX =

n∏

i=1

Γ(1 + δi) ∈ H (X,R)

where δ1, . . . , δn are the Chern roots of the tangent bundle TX and Γ(x) is Euler’s Gammafunction. A well-known Taylor expansion for the Gamma function implies that it is expandedin the Euler constant Ceu and the Riemann zeta values ζ(k), k = 2, 3, . . . :

ΓX = exp

−Ceuc1(X) +

∑

k>2

(−1)k(k − 1)!ζ(k) chk(TX)

.

The Gamma class ΓX has a loop space interpretation [53, 46]. Let LX denote the free loopspace of X and consider the locus X ⊂ LX of constant loops. The normal bundle N of X inLX has a natural S1-action (by loop rotation) and splits into the sum N+ ⊕N− of positiveand negative representations. Using the ζ-function regularization one obtains:

1

eS1(N+)=

1∏∞k=1 eS1(TX ⊗ Lk)

∼ zdeg /2z−c1(X)ΓX

GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO MANIFOLDS 3

where L denotes the S1-representation of weight one and z denotes a generator of H2S1(pt)

such that ch(L) = ez . Therefore ΓX can be regarded as a localization contribution fromconstant maps in Floer theory, cf. Givental’s equivariant Floer theory [27, 28].

The Gamma class can be also regarded as a ‘square root’ of the Todd class (or A-class).The Gamma function identity:

Γ(1− z)Γ(1 + z) =πz

sinπz=

2πiz

1− e−2πize−πiz

implies that we can factorize the Todd class in the Hirzebruch-Riemann–Roch (HRR) formulaas follows:

χ(E1, E2) =

∫

Fch(E∨

1 ) ∪ ch(E2) ∪ tdX

=[ΓX Ch(E1), ΓX Ch(E2)

)(1.1.1)

where χ(E1, E2) =∑dimX

p=0 (−1)p dimExtp(E1, E2) is the Euler pairing of vector bundles E1,

E2, Ch(Ei) =∑dimX

p=0 (2πi)p chp(Ei) is the modified Chern character and

[A,B) :=1

(2π)dimX

∫

X(eπic1(X)eπiµA) ∪B

is a non-symmetric pairing on H (X), where µ ∈ End(H (X)) is the grading operator definedby µ(φ) = (p − dimX

2 )φ for φ ∈ H2p(X). Geometrically this factorization corresponds to thedecomposition N = N+⊕N− of the normal bundle. Recall that Witten and Atiyah [2] derived

heuristically the index theorem by identifying the A-class with 1/eS1(N ). In this sense, theGamma conjectures can be regarded as a square root of the index theorem.

1.2. Gamma conjectures. The Gamma conjectures relate the quantum cohomology of aFano manifold and the Gamma class in terms of differential equations. For a Fano manifoldF , the quantum cohomology algebra (H (F ), ⋆0) (at the origin τ = 0) defines the quantumconnection [18]:

∇z∂z = z∂

∂z− 1

z(c1(F )⋆0) + µ

acting on H (F )⊗C[z, z−1]. It has a regular singularity at z = ∞ and an irregular singularityat z = 0. Flat sections near z = ∞ are constructed by the so-called Frobenius methodand can be put into correspondence with cohomology classes in a natural way. Flat sectionsnear z = 0 are classified by their exponential growth order (along a sector). Our underlyingassumption is Conjecture O (Definition 3.1.1) which roughly says that c1(F )⋆0 has a simpleeigenvalue T > 0 of the biggest norm. Under Conjecture O, we can single out a flat sections0(z) with the smallest asymptotics ∼ e−T/z as z → +0 along R>0; then we transport the flatsection s0(z) to z = ∞ and identify the corresponding cohomology class AF . We call AF theprincipal asymptotic class of a Fano manifold. We say that F satisfies Gamma Conjecture I(Definition 3.3.3) if AF equals the Gamma class:

AF = ΓF .

More generally (under semisimplicity assumption), we can identify a cohomology class Aisuch that the corresponding flat section has an exponential asymptotics ∼ e−ui/z as z → 0along a fixed sector (of angle bigger than π) for each eigenvalue ui of (c1(F )⋆0), i = 1, . . . , N .


These classes A1, . . . , AN ∈ H (F ) form a basis which we call the asymptotic basis of F . Wesay that F satisfies Gamma Conjecture II (Definition 4.6.1) if the basis can be written as:

Ai = ΓF Ch(Ei)

for a certain exceptional collection E1, . . . , EN of the derived category Dbcoh(F ). Gamma

Conjecture I says that the exceptional object OF corresponds to the biggest real positiveeigenvalue T of (c1(F )⋆0).

The quantum connection has an isomonodromic deformation over the cohomology groupH (F ). By Dubrovin’s theory [18, 19], the asymptotic basis Ai changes by mutation

(A1, . . . , Ai, Ai+1, . . . , AN ) −→ (A1, . . . , Ai+1, Ai − [Ai, Ai+1)Ai+1, . . . , AN )

when the eigenvalues ui and ui+1 are interchanged (see Figure 6). Via the HRR formula(1.1.1) this corresponds to a mutation of the exceptional collection Ei. By mutation, thebraid group acts on the set of asymptotic bases: we formulate this in terms of a markedreflection system in §4.2. Note that Gamma Conjecture II implies (part (2) of) Dubrovin’sconjecture [20] (see §4.6): the Stokes matrix Sij = [Ai, Aj) of the quantum connection equalsthe Euler pairing χ(Ei, Ej). While we are writing this paper, we are informed that Dubrovin[21] gave a new formulation of his conjecture that includes Gamma Conjecture II above.

1.3. When the Gamma Conjectures hold. We can show the Gamma Conjectures forseveral cases. The Gamma Conjectures for Pn were implicit but essentially shown in the workof Dubrovin [19]; they also follow from mirror symmetry computations in [44, 45, 50]. In §6,we give a detailed proof of the following theorem:

Theorem 1.3.1 (Theorem 6.0.4). Gamma Conjectures I and II hold for the projective space

P = PN−1. An asymptotic basis of P is formed by mutations of the Gamma basis ΓPCh(O(i))

associated to Beilinson’s exceptional collection O(i) : 0 6 i 6 N − 1.

More generally one can deduce the Gamma Conjectures for toric manifolds or toric completeintersections from the mirror theorem [29] and the calculation of mirror periods [44, 45, 47].

Theorem 1.3.2 (see Theorems 5.1.2, 5.1.3, 5.2.5 for precise statements). When certain con-ditions (Conditions 5.1.1, 5.2.2) ensuring Conjecture O are satisfied, a Fano toric completeintersection satisfies Gamma Conjecture I. For a Fano toric manifold, a K-theoretic analogueof Gamma Conjecture II holds.

We also give a sketch of argument in §5.3 showing that Gamma conjecture I should becompatible with quantum Lefschetz principle [16], i.e. deducing Gamma Conjecture I for ahypersurface assuming that it holds for an ambient space.

In §7, we prove the Gamma Conjectures for Grassmannians G(r,N). The quantum Satakeprinciple [33] or the abelian/non-abelian correspondence [9, 15] says that the quantum con-nection of G(r,N) is the r-th wedge of the quantum connection of PN−1. This fact impliesthat the asymptotic basis of G(r,N) is the r-th wedge of that of PN−1. The same holds forthe Gamma basis, and we obtain the following result:

Theorem 1.3.3 (Theorem 7.1.1). Gamma Conjectures I and II hold for Grassmannians G =

G(r,N). An asymptotic basis of G is formed by mutations of the Gamma basis ΓGCh(SνV ∗)associated to Kapranov’s exceptional collection SνV ∗ : ν ⊂ r × (N − r)-box.


1.4. Limit formula and Apery’s irrationality. Let J(t) = J(−tKF ) denote Givental’sJ-function (3.5.7) restricted to the anti-canonical line Cc1(F ). This is a cohomology-valuedsolution to the quantum differential equation. We show the following (continuous and discrete)limit formulae:

Theorem 1.4.1 (Corollary 3.5.9). Suppose that a Fano manifold F satisfies Conjecture Oand Gamma Conjecture I. Then the Gamma class of F can be obtained as the limit of theratio of the J-function

ΓF = limt→+∞

J(t)

〈[pt], J(t)〉 .

Theorem 1.4.2 (see Theorem 3.6.1 for precise statements). Under the same assumptions asabove, the primitive part of the Gamma class can be obtained as the following discrete limit(assuming the limit exists):

(1.4.3) 〈γ, ΓF 〉 = limn→∞

〈γ, Jrn〉〈[pt], Jrn〉

for every γ ∈ H (F ) with c1(F ) ∩ γ = 0. Here we write J(t) = ec1(F ) log t∑∞

n=0 Jrntrn with r

the Fano index.

Discrete limits similar to (1.4.3) were studied by Almkvist-van Straten-Zudilin [1] in thecontext of Calabi–Yau differential equations and are called Apery limits (or Apery constants).Golyshev [32] and Galkin [25] studied the limits (1.4.3) for Fano manifolds. In fact, these limitsare related to famous Apery’s proof of the irrationality of ζ(2) and ζ(3). An Apery limit of theGrassmannian G(2, 5) gives a fast approximation of ζ(2) and an Apery limit of the orthogonalGrassmannian OGr(5, 10) gives a fast approximation of ζ(3); they prove the irrationality ofζ(2) and ζ(3). It would be extremely interesting to find a Fano manifold whose Apery limitsgive fast approximations of other zeta values.

1.5. Mirror symmetry. The ‘Gamma structure’ is closely related to mirror symmetry andhas been observed since its early days. The following references serve as motivation to theGamma conjectures.

• The Gamma class of a Calabi–Yau threefold X is given by ΓX = 1 − π2

6 c2(X) −ζ(3)c3(X); the number ζ(3)χ(X) appeared in the computation of mirror periods byCandelas et al. [13]; it also appeared in the conifold period formula of van Enckevort–van Straten [22].

• Libgober [52] found the (inverse) Gamma class from hypergeometric solutions to thePicard–Fuchs equation of the mirror, inspired by the work of Hosono et al. [43].

• Kontsevich’s homological mirror symmetry suggests that the monodromy of thePicard–Fuchs equation of mirrors should be related to Auteq(Db

coh(X)). In the relatedworks of Horja [41], Borisov–Horja [11] and Hosono [42], Gamma/hypergeometric se-ries play an important role.

• In the context of Fano/Landau–Ginzburg mirror symmetry, Iritani [44, 45] andKatzarkov-Kontsevich-Pantev [50] introduced a rational or integral structure of quan-tum connection in terms of the Gamma class, by shifting the natural integral structureof the Landau–Ginzburg model given by Lefschetz thimbles.

Although not directly related to mirror symmetry, we remark that the Gamma class alsoappears in a recent progress [39, 37] in physics on the sphere/hemisphere partition functions.


1.6. Plan of the paper. In §2, we review quantum cohomology and quantum differentialequation for Fano manifolds. We explain asymptotically exponential flat sections, their muta-tions and Stokes matrices. The exposition here is mostly based on Dubrovin’s work [18, 19],but we make the following technical refinement: we carefully deal with the case where thequantum cohomology is semisimple but the Euler multiplication (E⋆τ ) has repeated eigenval-ues (see Propositions 2.7.4 and 2.9.1). We need this case since it happens for Grassmannians.In §3, we formulate Conjecture O and Gamma Conjecture I. We also prove limit formulaefor the principal asymptotic class. In §4, we formulate Gamma Conjecture II and explain arelationship to (original) Dubrovin’s conjecture. In §5, we prove Gamma conjectures for toriccomplete intersections (under certain conditions) and explain the compatibility of GammaConjecture I with quantum Lefschetz principle. In §6, we give a detailed proof of the GammaConjectures for projective spaces. In §7, we deduce the Gamma Conjectures for Grassman-nians from the truth of the Gamma Conjectures for projective spaces. Main tools in theproof are isomonodromic deformation and quantum Satake principle. For this purpose weextend quantum Satake principle [33] to big quantum cohomology (Theorem 7.3.1) usingabelian/non-abelian correspondence [9, 15, 51].

2. Quantum cohomology, quantum connection and solutions.

In this section we discuss background material on quantum cohomology and quantum con-nection of a Fano manifold. Quantum connection is defined as a meromorphic flat connectionof the trivial cohomology bundle over the z-plane, with singularities at z = 0 and z = ∞.We discuss two fundamental solutions associated to the regular singularity (z = ∞) and theirregular singularity (z = 0). Under the semisimplicity assumption, we discuss mutations andStokes matrices, reviewing Dubrovin’s theory [18, 20, 19] of isomonodromic deformation.

2.1. Quantum cohomology. Let F be a Fano manifold, i.e. a smooth projective varietysuch that the anticanonical line bundle ω−1

F = det(TF ) is ample. Let H (F ) = Heven(F ;C)denote the even part of the Betti cohomology group over C. For α1, α2, . . . αn ∈ H (F ),

let 〈α1, α2, . . . , αn〉F0,n,d denote the genus-zero n points Gromov–Witten invariant of degree

d ∈ H2(F ;Z), see e.g. [54]. Informally speaking, this counts the (virtual) number of rationalcurves in F which intersect the Poincare dual cycles of α1, . . . , αn. It is a rational number whenαi’s are rational classes. The quantum product α1⋆τ α2 ∈ H (F ) of two classes α1, α2 ∈ H (F )with parameter τ ∈ H (F ) is given by

(2.1.1) (α1 ⋆τ α2, α3)F =∑

d∈Eff(F )

∞∑

n=0

1

n!〈α1, α2, α3, τ, . . . , τ〉F0,3+n,d

where (α, β)F =∫F α∪ β is the Poincare pairing and Eff(F ) ⊂ H2(F ;Z) is the set of effective

curve classes. The quantum product is associative and commutative, and recovers the cupproduct in the limit where Re

(∫d τ)→ −∞ for all non-zero effective curve classes d. It is not

known if the quantum product ⋆τ converges in general, however it does for all the examplesin this paper (see also Remark 2.1.2). The quantum product ⋆τ with τ ∈ H2(F ) is called thesmall quantum product; for general τ ∈ H (F ) it is called the big quantum product.

We are mainly interested in the quantum product ⋆0 specialized to τ = 0. An effectiveclass d contributing to the sum

∑

d∈Eff(F )

〈α1, α2, α3〉F0,3,d


has to satisfy 12

∑3i=1 degαi = dimF + c1(F ) · d and there are only finitely many such d when

F is Fano. Therefore the specialization at τ = 0 makes sense.

Remark 2.1.2. Writing τ = h + τ ′ with h ∈ H2(F ) and τ ′ ∈ ⊕p 6=1H2p(F ) and using the

divisor axiom in Gromov–Witten theory, we have:

(α1 ⋆τ α2, α3)F =∑

d∈Eff(F )

∞∑

n=0

1

n!

⟨α1, α2, α3, τ

′, . . . , τ ′⟩F0,n+3,d

e∫dh.

Therefore the quantum product (2.1.1) makes sense as a formal power series in τ ′ and theexponentiated H2-variables eh1 , . . . , ehr , where we write h = h1p1 + · · · + hrpr by choosing anef basis p1, . . . , pr of H2(X;Z).

2.2. Quantum connection. Following Dubrovin [18, 20, 19], we introduce a meromor-phic flat connection associated to the quantum product. Consider a trivial vector bundleH (F )×P

1 → P1 over P1 and fix an inhomogeneous co-ordinate z on P

1. Define the quantumconnection ∇ on the trivial bundle by the formula

(2.2.1) ∇z∂z = z∂

∂z− 1

z(c1(F )⋆0) + µ

where µ ∈ End(H (F )) is the grading operator defined by µ|H2p(F ) = (p− dimF2 ) idH2p(F ). The

connection is smooth away from 0,∞; the singularity at z = ∞ is regular (or more preciselylogarithmic) and the singularity at z = 0 is irregular. The quantum connection preserves thePoincare pairing in the following sense: we have

(2.2.2) z∂

∂z(s1(−z), s2(z)) = ((∇z∂zs1)(−z), s2(z)) + (s1(−z),∇z∂zs2(z))

for s1, s2 ∈ H (F )⊗ C[z, z−1]. Here we need to flip the sign of z for the first entry s1.What is important in Dubrovin’s theory is the fact that ∇ admits an isomonodromic de-

formation over H (F ). Suppose that ⋆τ converges on a region B ⊂ H (F ). Then the aboveconnection is extended to a meromorphic flat connection on H (F )× (B× P

1) → (B ×P1) as

follows:

∇α = ∂α +1

z(α⋆τ ) α ∈ H (F )

∇z∂z = z∂

∂z− 1

z(E⋆τ ) + µ

(2.2.3)

with (τ, z) a point on the base B × P1. Here ∂α denotes the directional derivative in the

direction of α and

E = c1(F ) +

N∑

i=1

(1− 1

2deg φi

)τ iφi

is the Euler vector field, where we write τ =∑N

i=1 τiφi by choosing a homogeneous

basis φ1, . . . , φN of H (F ). We refer to this extension as the big quantum connec-tion. The Poincare pairing (·, ·)F is flat with respect to the big quantum connection ∇,i.e. ∂α(s1(τ,−z), s2(τ, z)) = ((∇αs1)(τ,−z), s2(τ, z)) + (s1(τ,−z),∇αs2(τ, z)).

Remark 2.2.4. The connection in the z-direction can be identified with the connection inthe anticanonical direction after an appropriate rescaling. Consider the quantum product ⋆τrestricted to the anticanonical line τ = c1(F ) log t, t ∈ C

×:

(2.2.5) (α1 ⋆c1(F ) log t α2, α3)F =∑

d∈Eff(F )

〈α1, α2, α3〉0,3,d tc1(F )·d.


This is a polynomial in t since F is Fano, and coincides with ⋆0 when t = 1. It can berecovered from the product ⋆0 by the formula:

(2.2.6) (α⋆c1(F ) log t) = tdegα/2t−µ(α⋆0)tµ.

The quantum connection restricted to the anticanonical line and z = 1 is

∇c1(F )

∣∣∣z=1

= t∂

∂t+ (c1(F )⋆c1(F ) log t).

On the other hand we have

zµ[∇z∂z

]τ=0

z−µ = z∂

∂z− (c1(F )⋆−c1(F ) log z)

Therefore, the connections ∇c1(F )|z=1 and ∇z∂z |τ=0 are gauge equivalent via zµ under the

change of variables t = z−1.

2.3. Fundamental solution around the regular singular point z = ∞. We now restrictour attention to the connection ∇ (2.2.1) defined by the quantum product ⋆0 at τ = 0. Thefollowing proposition is well-known.

Proposition 2.3.1. There exists a unique holomorphic function S : P1 \ 0 → End(H (F ))with S(∞) = idH (F ) such that

∇(S(z)z−µzρα) = 0 for all α ∈ H (F );

T (z) = zµS(z)z−µ is regular at z = ∞ and T (∞) = idH (F ),

where ρ = (c1(F )∪) ∈ End(H (F )) and we define z−µ = exp(−µ log z), zρ = exp(ρ log z).Moreover we have

(S(−z)α, S(z)β)F = (α, β)F α, β ∈ H (F ).

Proof. Dubrovin studied a fundamental solution of this form for general Frobenius manifolds[19, Lemma 2.5, 2.6]. For the convenience of the reader, we explain the construction inour setting. We study the equivalent differential equation ∇(z−µT (z)zρα) = 0 for T (z) =zµS(z)z−µ with the initial condition T (∞) = id. The differential equation for T reads:

z∂

∂zT (z)− 1

zzµ(c1(F )⋆0)z

−µT (z) + T (z)ρ = 0.

Expand:

T (z) = id+T1z−1 + T2z

−2 + T3z−3 + · · ·

(c1(F )⋆0) = G0 +G1 +G2 + · · · (finite sum)

where Gk ∈ End(H (F )) is an endomorphism of degree 1 − k, i.e. zµGkz−µ = z1−kGk and

G0 = c1(F )∪ = ρ. The above equation is equivalent to the system of equations:

0 = ρ− ρ

0 = T1 +G1 + [ρ, T1]

...

0 = mTm +Gm +Gm−1T1 + · · ·+G1Tm−1 + [ρ, Tm].

These equations can be solved recursively for T1, T2, T3, . . . because the map X 7→ mX+[ρ,X]is invertible (since ρ is nilpotent). One can easily show that the power series T (z) converges.


By construction, Tk is an endomorphism of degree > (1−k) and hence z−kz−µTkzµ contains

only negative powers in z for k > 1. Therefore S(z) = z−µT (z)zµ is regular at z = ∞ andsatisfies S(∞) = id.

Finally we see (S(−z)α, S(z)β)F = (α, β)F . We claim that(S(−z)(eπiz)−µ(eπiz)ρα, S(z)z−µzρβ

)F= (e−πiµeπiρα, β)F .

Because ∇ preserves the Poincare pairing (2.2.2), the left-hand side is independent of z. Onthe other hand, the left-hand side equals

(e−πiµT (−z)(eπiz)ρα, T (z)zρβ

)F

and can be expanded in C[[z−1]][log z]. The constant term of the Taylor expansion in z−1 andlog z equals the right-hand side. The claim follows. Replacing α, β with (eπiz)−ρ(eπiz)µαand z−ρzµβ, we arrive at the conclusion.

Remark 2.3.2 ([18, 55, 45]). The fundamental solution S(z) is given by descendant Gromov–Witten invariants. Let ψ denote the first Chern class of the universal cotangent line bundleover M0,2(F, d) at the first marking. Then we have:

(S(z)α, β)F = (α, β)F +∑

m>0

1

zm+1

∑

d∈Eff(F )\0

(−1)m+1 〈αψm, β〉F0,2,d .

Here again the summation in d is finite (for a fixed m) because F is Fano. A similar funda-mental solution exists for the isomonodromic deformation of ∇ associated to the big quantumcohomology. The big quantum connection ∇ over H (F )× P

1 admits a fundamental solutionof the form S(τ, z)z−µzρ extending the one in Proposition 2.3.1 such that

∇(S(τ, z)z−µzρα) = 0, S(τ,∞) = id, (S(τ,−z)α, S(τ, z)β)F = (α, β)F .

Here zµS(τ, z)z−µ is not necessarily regular at z = ∞ for τ /∈ H2(F ) (cf. Lemma 7.5.3).

2.4. Formal fundamental solution around the irregular singular point z = 0. Herewe assume that the ring (H (F ), ⋆0) is semisimple, i.e. isomorphic to the direct sum of C asa ring. In view of Remark 2.2.4, we refer to this property as that the quantum cohomologyof F is anticanonically semisimple. Note that anticanonical semisimplicity is stronger1 thansemisimplicity of the big quantum cohomology for generic τ .

Let ψ1, . . . , ψN denote the idempotent basis of (H (F ), ⋆0) such that

ψi ⋆0 ψj = δi,jψi

whereN = dimH (F ). Let Ψi := ψi/√

(ψi, ψi)F , i = 1, . . . , N be the normalized idempotents.They form an orthonormal basis: (Ψi,Ψj)F = δij . We write

Ψ =(Ψ1, . . . ,ΨN

)

for the matrix with the column vectors Ψ1, . . . ,ΨN . We regard Ψ as an endomorphismCN → H (F ). Let u1, . . . , uN be the eigenvalues of (c1(F )⋆0) given by c1(F ) ⋆0 Ψi = uiΨi.

Define U to be the diagonal matrix with entries u1, . . . , uN :

(2.4.1) U =

u1u2

. . .

uN

.

1For example, del Pezzo surfaces of degree 6 4 have generically semisimple big quantum cohomology, butare not anticanonically semisimple.


Around the irregular singular point, the connection ∇ (2.2.1) has the following formal funda-mental solution.

Proposition 2.4.2 ([19, Lemma 4.3], [57, Theorem 8.15]). There exists a unique formalpower series of the form

R(z) = id+R1z +R2z2 + · · · ,

with Rk ∈ End(CN ) such that one has

∇(ΨR(z)e−U/zv) = 0, R(−z)TR(z) = id

for all v ∈ CN .

Dubrovin [19] discussed the case where u1, . . . , uN are distinct. Teleman [57] remarked thatthe proposition holds even when u1, . . . , uN are not distinct. Note that Ψ1, . . . ,ΨN are uniqueup to order and signs. Hence the formal fundamental solution ΨR(z)e−U/z is unique up tothe right multiplication by a signed permutation matrix.

We review the construction of R(z) for the convenience of the reader. By the constantgauge transformation by Ψ, we can transform the connection ∇ into the UV -form [19]:

Ψ∗∇z ∂∂z

= z∂

∂z− 1

zU + V

with U given by (2.4.1) and V = Ψ−1µΨ. We have the following lemma:

Lemma 2.4.3. The matrix V = (Vij) is anti-symmetric Vij = −Vji. Moreover, Vij = 0whenever ui = uj .

Proof. The anti-symmetricity of V follows from the fact that µ is skew-adjoint: (µα, β)F =−(α, µβ)F and that Ψ1, . . . ,ΨN are orthonormal. To see the latter statement, we use theisomonodromic deformation discussed in §2.2. In the case where the big quantum cohomologyis not known to be analytic, we work over the formal neighbourhood of τ = 0 in the followingdiscussion. Around the semisimple point τ = 0, the spectrum u1, . . . , uN of E⋆τ forms alocal co-ordinate system, and ψi =

∂τ∂ui

, i = 1, . . . , N give an idempotent basis of ⋆τ (see [19,

Lecture 3]). The matrix Ψ of normalized idempotents is also extended to a matrix-valuedfunction on a neighbourhood of τ = 0. With respect to this co-ordinate system and aftergauge transformation by Ψ, the connection (2.2.3) takes the form (see [19, Lemma 3.2]):

Ψ∗∇ ∂∂ui

=∂

∂ui+

1

zEi + Vi

Ψ∗∇z ∂∂z

= z∂

∂z− 1

zU + V

where Ei = diag[0, . . . , 0,i-th1 , 0, . . . , 0] and V = Ψ−1µΨ, Vi = Ψ−1∂uiΨ are matrix-valued

functions in (u1, . . . , uN ). The flatness of the equation implies:

[Ei, V ] = [Vi, U ]

and it follows that Vij = (uj − ui)(Vi)ij if i 6= j. Therefore Vij = 0 if i 6= j and ui = uj . Ifi = j, Vij = 0 by the anti-symmetricity.

Proof of Proposition 2.4.2. It suffices to solve the UV -system (z ∂∂z− 1

zU+V )(R(z)e−U/z) = 0.This is equivalent to the system of equations:

(n+ V )Rn + [Rn+1, U ] = 0


with R0 = id. Suppose we know R0, . . . , Rn. The above equation determines the (i, j)-entry(Rn+1)ij if ui 6= uj . For a pair (i, j) with ui = uj , the (i, j)-entry of the equation

(n + 1 + V )Rn+1 + [Rn+2, U ] = 0

determines (Rn+1)ij because [Rn+2, U ]ij = 0 and (V Rn+1)ij can be written in terms of theentries (Rn+1)kj with uk 6= uj in view of Lemma 2.4.3.

2.5. Laplace transform of ∇. Let λ denote the Laplace-dual variable of z−1. Under theformal substitution ∂z−1 → −λ, z → ∂λ, the differential equation ∇z∂z(z

ν−1y(z)) = 0 istransformed to the equation:

∇ ∂∂λy(λ) :=

(∂

∂λ+ (λ− c1(F )⋆0)

−1(µ+ ν)

)y(λ) = 0

where ν ∈ C is an arbitrary parameter. The connection ∇ is called the second structure

connection in [54]. Suppose again that ⋆0 is semisimple. Then ∇ has logarithmic singularities

at the eigenvalues u1, . . . , uN of c1(F )⋆0 and at ∞. The residue of ∇ at λ = ui is given by

Resλ=ui ∇ = Pui(µ+ ν)

where Pui is the projection to the ui-eigenspace of c1(F )⋆0. By Lemma 2.4.3, the residue isconjugate (via Ψ and reordering the basis) to the matrix of the form:

(2.5.1)

(νIr ∗0 0N−r

)

where Ir is the identity matrix of size r = ♯j : ui = uj and 0N−r is the zero matrix of sizeN − r. If the residue is non-resonant2, i.e. ν = 0 or ν /∈ Z, we have a fundamental solution

of the form U(λ − ui) exp(− log(λ − ui)Resλ=ui ∇) with U(x) holomorphic near x = 0 and

U(0) = id. In particular the local monodromy at λ = ui is conjugate to exp(−2πiResλ=ui ∇).From this we have the following:

Proposition 2.5.2. Suppose that ⋆0 is semisimple. Consider the second structure connection

∇ with parameter ν = 0.

(1) For an eigenvector v of (c1(F )⋆0) of eigenvalue ui, there exists a unique holomorphicflat section yv(λ) near λ = ui such that yv(ui) = v.

(2) Let Mui denote the monodromy transformation around λ = ui which acts in the space

of flat sections of ∇. For a ∇-flat section g, there exists an ui-eigenvector v such thatMui g = g + yv, where yv is the flat section in (1).

Proof. (1): yv is given by yv(λ) = U(λ− ui) exp(− log(λ− ui)Resλ=ui ∇)v = U(λ−ui)v. (2):this follows from the form (2.5.1) of the residue matrix.

Remark 2.5.3 ([19, Lecture 5]). When ν = 1/2, ∇ has an invariant metric given by ((λ −c1(F )⋆0)f(λ), g(λ))F .

2The residue is said to be non-resonant if the eigenvalues do not differ by positive integers.


•u1•u2

•u3 = u4 = u5

•u6 = u7

•u8

•u9

L1

L2

L3 = L4 = L5

L6 = L7

L8

L9

Figure 1. Half-lines Li in the admissible direction eiφ = 1.

2.6. Asymptotically exponential flat section. The formal fundamental solution inProposition 2.4.2 is typically a divergent formal power series. By choosing an angular sectorin the z-plane with vertex at z = 0, we can lift each of the formal flat sections to an analyticflat section which is asymptotic to it along the sector. When u1, . . . , uN are distinct, this isdiscussed in [3, Theorem A, Proposition 7], [4] and [19, Lectures 4, 5]. Following [12, §8], wediscuss the case where some of u1, . . . , un may coincide.

We assume the anticanonical semisimplicity and choose an ordering and signs of a normal-ized idempotent basis Ψ1, . . . ,ΨN . The choice determines an ordering u1, . . . , uN of the

spectrum of c1(F )⋆0 and a formal fundamental solution ΨR(z)e−U/z in Proposition 2.4.2. Fora real number φ, we say that the direction eiφ ∈ S1 (or the phase φ ∈ R) is admissible3 foru1, . . . , uN if Im(e−iφui) 6= Im(e−iφuj) for every pair (ui, uj) such that ui 6= uj . This means

that eiφ is not parallel to any non-zero difference ui − uj of eigenvalues (see Figure 1). Wedenote by Σ ⊂ S1 the finite set of non-admissible directions.

By Proposition 2.5.2, there exists a unique flat section yi(λ) of ∇ (with parameter ν =0) near λ = ui such that yi(λ) is regular at λ = ui and that yi(ui) = Ψi. Choose anadmissible direction eiφ and construct a flat section yi(z) of ∇ by the vector-valued Laplacetransformation:

(2.6.1) yi(z) =1

z

∫

ui+eiφR>0

yi(λ)e−λ/zdλ.

The integration path is a straight half-line Li starting from ui to ∞ in the direction of eiφ

and fi(λ) is analytically continued along the path (Figure 1). Since ∇ is regular singular atλ = ∞, the integrand is at most of polynomial growth. Therefore the integral converges if| arg z − φ| < π

2 . By bending the path Li continuously (but avoiding other singularities), wecan analytically continue yi(z) along any path in C

×.

Proposition 2.6.2 ([3, 4, 19, 12]). Take an admissible direction eiφ and let eiθ : φ1 < θ <φ2 be the connected component of S1 \Σ containing eiφ. Then the flat sections yi(z) (2.6.1)satisfy the asymptotic expansion

(2.6.3) Y (z) := (y1(z), . . . , yN (z)) ∼ ΨR(z)e−U/z

as |z| → 0 in the sector:

(2.6.4) φ1 −π

2< arg z < φ2 +

π

2.

Moreover, we have:

3Our admissible direction is perpendicular to the one in [19, Definition 4.2].


(1) A fundamental solution Y satisfying this asymptotic condition is unique: we call itthe (asymptotically exponential) fundamental solution associated to eiφ.

(2) Let Y −(z) = (y−1 (z), . . . , y−N (z)) be the fundamental solution associated to −eiφ. Then

we have (y−i (−z), yj(z))F = δij for z in the sector (2.6.4).

For the uniqueness of flat sections satisfying the asymptotics (2.6.3), it is important thatthe angle of the sector (2.6.4) is bigger than π. The order and signs of the asymptoticallyexponential flat sections (y1(z), . . . , yN (z)) associated to eiφ depend on the order and signsof normalized idempotents Ψ1, . . . ,ΨN .

Remark 2.6.5. The precise meaning of the asymptotic expansion (2.6.3) is as follows. Forevery small ǫ > 0 and non-negative integer n, there exists a positive constant C = C(n, ǫ)such that ∣∣∣∣∣e

ui/zyi(z)−n∑

k=1

ΨRkeizk

∣∣∣∣∣ 6 C|z|n+1

holds for all i and z with φ1 − π2 + ǫ 6 arg z 6 φ2 + π

2 − ǫ and |z| 6 1, where we write

R(z) = R0 +R1z +R2z2 + · · · .

Proof of Proposition 2.6.2. As shown in [12, §8.4] by an elementary calculation, the functionyi(z) is a solution to the differential equation ∇z∂zyi(z) = 0 and has the asymptotic expansion:

yi(z) ∼ e−ui/z(Ψi +O(z)).

The asymptotic expansion is valid whenever we can bend the path Li so that Re(λ/z) increasesmonotonically along the path: it follows that this holds in the sector φ1−π

2 < arg z < arg φ2+π2

(see Remark 2.6.6 and Figure 2).We then see the uniqueness of a fundamental solution Y [3, Remark 1.4]. Suppose that we

have two solutions Y1, Y2 subject to (2.6.3). Then there exists a constant matrix C such that

Y1 = Y2C. We have e−U/zCeU/z → id as z → 0 in the sector. Since the angle of the sector isbigger than π, it happens only when C = id.

Finally we show Part (2). Since y−i (z) and yj(z) are flat, the pairing (y−i (−z), yj(z))Fdoes not depend on z (see (2.2.2)). By the asymptotic condition (2.6.3), we havee(uj−ui)/z(y−i (−z), yj(z))F = (e−ui/zy−i (−z), euj/zyj(z))F → (Ψi,Ψj)F = δij as |z| → 0

in the sector (2.6.4). Thus (y−i (−z), yj(z))F = δij if ui = uj. If ui 6= uj, we have

(y−i (−z), yj(z))F = 0 since the angle of the sector is bigger than π.

Remark 2.6.6. Each flat section yi(z) in (2.6.1) can have the asymptotic expansion yi(z) ∼e−ui/zΨR(z)ei in a bigger sector. Set Σi = (uj − ui)/|uj − ui| : uj 6= ui and let eiθ : ξ1 <θ < ξ2 be the connected component of S1 \ Σi which contains the admissible direction eiφ.By construction, the flat section yi(z) has the asymptotic expansion in the sector ξ1 − π

2 <arg z < ξ2 +

π2 as in Figure 2. (The sector here can be bigger than 2π.)

2.7. Changing integration paths: mutation and Stokes matrix. In the previous sec-tion, we constructed an asymptotically exponential flat section yi(z) as the Laplace transform

(2.6.1) of the ∇-flat section yi(λ) over the straight half-line Li. We study the change of flatsections under a change of integration paths. To illustrate, we consider the change of pathsfrom L1 to L′

1 depicted in Figure 3. In the passage from L1 to L′1, the path is assumed to

cross only one eigenvalue u2 = · · · = ur of multiplicity r − 1. We use the straight paths

L1, . . . , LN as branch cuts for ∇-flat sections; yi(λ) defines a single-valued analytic functionover C\⋃j 6=i Lj. Let M denote the anti-clockwise monodromy transformation around λ = u2


•ui

•uk

•uj

•un

•uo

•ul•

um

②

❨

ξ1

ξ2

ξ2 +π2

ξ1 − π2

φadmissible

Figure 2. The flat section yi(z) has the asymptotics e−ui/zΨR(z)ei in thesector (ξ1 − π

2 , ξ2 +π2 ).

•u1

•u2 = · · · = ur

•ur+1

......uN• LN

L1

L2 = · · · = Lr

L′1

Lr+1

Figure 3. Right mutation of L1

acting on the space of ∇-flat sections. By Proposition 2.5.2, the monodromy transform of y1is of the form:

My1 = y1 − c12y2 − · · · − c1ryr

for some coefficients c12, . . . , c1r ∈ C. From this it follows that the flat section y′1(z) definedby the integral over L′

1 is given by:

(2.7.1) y′1(z) = y1(z)− c12y2(z)− · · · − c1ryr(z).

We call the flat section y′1(z) (or the path L′1) the right mutation of y1(z) (resp. L1) with

respect to u2 = · · · = ur. The left mutation of a flat section or a path is the inverse operation:see Figure 4.

•u1

•u2 = · · · = ur

•ur+1

L1

L2 = · · · = Lr

L′r+1

Lr+1

Figure 4. Left mutation of Lr+1


A left or right mutation occurs when the admissible direction eiφ varies and crosses a non-admissible direction. We now let the phase φ decrease by the angle π continuously. Thenthe asymptotically exponential flat sections y1, . . . , yN undergo a sequence of right mutations.Let y−1 , . . . , y

−N be the basis of asymptotically exponential flat sections associated to −eiφ as

in Proposition 2.6.2. In the situation of Figure 3, we have:

(2.7.2) y−1 (z) = y1(z)− c12y2(z) − · · · − c1ryr(z)−(

linear combinations of yj(z)

with Im(e−iφuj) < Im(e−iφu2)

).

In this way we can write y−i as a linear combination of yj’s and vice versa. The transitionmatrix is called the Stokes matrix.

•

②

③

Y (z)Y −(z) φ

admissible

Π+

Π−

Figure 5. Domains of the two solutions Y and Y −

Definition 2.7.3 ([19, Definition 4.3]). Let Y = (y1, . . . , yN ), Y− = (y−1 , . . . , y

−N ) be the

fundamental solution associated to an admissible direction eiφ and −eiφ respectively as inProposition 2.6.2. Let Π+∪Π− be the domain in Figure 5 where both Y and Y − are defined.The Stokes matrices (or Stokes multipliers) are the constant matrices4 S and S− satisfying

Y (z) = Y −(z)S for z ∈ Π+

Y (z) = Y −(z)S− for z ∈ Π−

Proposition 2.7.4. Let (y1, . . . , yN ) and (y−1 , . . . , y−N ) be the asymptotically exponential

fundamental solution associated to admissible directions eiφ and −eiφ respectively and letS = (Sij), S− = (S−,ij) be the Stokes matrices. We have

(1) S− = ST .(2) Sij = S−,ji = (yi(e

−πiz), yj(z))F for z ∈ Π+. Here we write e−πiz instead of −zto specify the path of analytic continuation. Similarly, (y−i (e

πiz), y−j (z))F gives the

coefficient (S−1)ij of the inverse matrix of S.(3) The Stokes matrix S is a triangular matrix with diagonal entries all equal to one.

More precisely, we have Sii = 1 for all i and

Sij = 0 if i 6= j and ui = uj ;

Sij = 0 if Im(e−iφui) < Im(e−iφuj).

4Our Stokes matrices are inverse to the ones in [19, Definition 4.3].


Proof. Dubrovin [19, Thorem 4.3] discussed the case where u1, . . . , uN are distinct. We have

yj(z) =∑N

k=1 y−k (z)Skj for z ∈ Π+ and yj(z) =

∑Nk=1 y

−k (z)S−,kj for z ∈ Π−. Taking the

pairing with yi(−z) and using the property (y−k (z), yi(−z))F = δki from Proposition 2.6.2, weobtain

(yi(−z), yj(z))F = Sij for z ∈ Π+

(yi(−z), yj(z))F = S−,ij for z ∈ Π−

Replacing z with −z in the second formula and taking into account the direction of analyticcontinuation, we see that (1) and (2) hold. (The discussion for (y−i (e

πiz), y−j (z))F is similar.)As already discussed, the coefficients of the Stokes matrix arise from a sequence of mutations.Part (3) is obvious from this.

Remark 2.7.5. As in Figure 1, we often choose an ordering of normalized idempotentsΨ1, . . . ,ΨN so that the corresponding eigenvalues satisfy Im(e−iφu1) > · · · > Im(e−iφuN ).Then the Stokes matrix becomes upper -triangular.

Let us go back to the situation of Figure 3. The coefficient c1i appearing in (2.7.2) coincideswith the coefficient S1i of the Stokes matrix, because by inverting (2.7.2) we obtain

y1(z) = y−1 (z) + c12y−2 (z) + · · ·+ c1ry

−r (z) +

(linear combinations of y−j (z)

with Im(e−iφuj) < Im(e−iφu2)

)

for z ∈ Π− and therefore c1i = S−,i1 = S1i. Recall that c1i arises as the coefficient of the rightmutation (2.7.1). Therefore we obtain the following corollary:

Corollary 2.7.6. Let eiφ be an admissible direction and let (y1(z), . . . , yN (z)) be the asymp-totically exponential fundamental solution associated to eiφ in Proposition 2.6.2. Let ua, ubbe distinct eigenvalues such that there are no eigenvalues uj with Im(e−iφua) > Im(e−iφuj) >

Im(e−iφub). Then the right mutation of the flat section ya(z) with respect to ub is given by

ya 7→ ya −∑

j:uj=ub

Sajyj.

Similarly, the left mutation of yb(z) with respect to ua is given by:

yb 7→ yb −∑

j:uj=ua

Sjbyj

where Sij = (yi(e−πiz), yj(z))F are the coefficients of the Stokes matrix associated to eiφ.

2.8. Isomonodromic deformation and marked reflection system. The contents in§§2.3–2.7 can be generalized to the isomonodromic deformation (2.2.3) of ∇. Suppose thatthe quantum product is anticanonically semisimple and analytic in a neighbourhood of τ = 0in H (F ). Then the eigenvalues u1, . . . , uN of E⋆τ form a local co-ordinate system on thebase H (F ) [19, Lecture 3] and we can make u1, . . . , uN pairwise distinct by a small de-formation of τ : we take a base point τ ∈ H (F ) such that the corresponding eigenvaluesu = u1, . . . , uN are pairwise distinct. The quantum connection ∇|τ then admits a uniqueisomonodromic deformation over the universal cover CN (C)

∼ of the configuration space

(2.8.1) CN (C) = (u1, . . . , uN ) ∈ CN : ui 6= uj for all i 6= j

/SN .


Proposition 2.8.2 ([19, Lemma 3.2, Exercise 3.3, Lemma 3.3]). We have a unique mero-morphic flat connection on the trivial H (F )-bundle over CN (C)

∼ × P1 of the form:

∇ ∂∂ui

=∂

∂ui+

1

zCi

∇z ∂∂z

= z∂

∂z− 1

zU + µ

where Ci and U are End(H (F ))-valued holomorphic functions on CN (C)∼, such that it re-

stricts to the big quantum connection (2.2.3) in a neighbourhood of the base point u. Herethe eigenvalues of U give the co-ordinates u1, . . . , uN on the base.

Remark 2.8.3 ([19, Lemma 3.3]). This isomonodromic deformation defines a Frobenius man-ifold structure on an open dense subset of CN (C)

∼.

The construction of an asymptotically exponential solution Y (z) = (y1(z), . . . , yN (z)) in§2.6 can be done in a parallel way for the isomonodromic deformation. We denote by u apoint of CN (C) and by u its lift to CN (C)

∼.

• We have a formal fundamental solution ΨuRueU/z depending analytically on u ∈

CN (C)∼ as in Proposition 2.4.2.

• For each u ∈ CN (C)∼ and an admissible phase φ for u, we have a fundamental solution

Yu,φ for ∇ which satisfies the asymptotic condition

Yu,φ(z) ∼ ΨuRueU/z

as |z| → 0 in a sector of the form φ − π2 − ǫ < arg z < φ + π

2 + ǫ for some ǫ > 0 asin Proposition 2.6.2. Note that the sector here is naturally identified with a subset ofthe universal cover of C× as we specify the phase φ.

• When (u, φ) varies, the fundamental solution Yu,φ changes discontinuously by a leftor right mutation as described in §2.7.

See also Proposition 2.9.1.The asymptotically exponential flat sections define a marked reflection system (MRS) on a

certain vector space H; see also §4.2, 4.3. We identify the space of flat sections of ∇ over theuniversal cover (CN (C)× C

×)∼ and write

H :=f : (CN (C)×C

×)∼ → H (F ) : ∇f = 0

for the finite dimensional vector space of flat sections. The vector space H can be identified5

with H (F ) via the fundamental solution S(τ, z)z−µzρ of Remark 2.3.2. This is equipped witha non-degenerate (but not necessarily symmetric or anti-symmetric) pairing

[f, g) := (f(u, e−πiz), g(u, z))F

for f, g ∈ H. Note that the right-hand side does not depend on (u, z) because of the flatness(cf. (2.2.2)). Let φ be an admissible phase for u ∈ CN (C). The asymptotically exponentialfundamental solution Yu,φ = (y1, . . . , yN ) associated to (u, φ) gives a basis of H. It is semi-orthonormal by the analogue of Proposition 2.7.4:

[yi, yj) =

0 if Im(e−iφui) < Im(e−iφuj)

1 if i = j

5We make this identification along u×R>0 ⊂ (CN(C)×C×)∼ where the connection ∇ is identified with

the quantum connection and the determination of z−µzρ = exp(−µ log z) exp(ρ log z) is canonical. See also§4.5.


The basis is unique up to signs and orderings. We call y1, . . . , yN the asymptotic basis andthe tuple

MRS(F, u, φ) = (H, y1, . . . , yN,u = u1, . . . , uN, eiφ)the marked reflection system of a Fano manifold F . Here each basis vector yi ∈ H is markedby a complex number ui ∈ u. Let us number u1, . . . , uN so that Im(e−iφu1) > Im(e−iφu2) >· · · > Im(e−iφuN ). By Corollary 2.7.6, the right and left mutation of (y1, . . . , yN ) is givenrespectively by:

(y1, . . . , yi, yi+1, . . . , yN ) 7→ (y1, . . . , yi−1, yi+1, yi − [yi, yi+1)yi+1, . . . , yN ) (right)

(y1, . . . , yi, yi+1, . . . , yN ) 7→ (y1, . . . , yi−1, yi+1 − [yi, yi+1)yi, yi, . . . , yN ) (left)

The right mutation occurs for example when u moves along the loop in CN (C) depicted inFigure 6. These loops generate the braid group BN , the fundamental group of CN (C). It iseasy to see that the right mutations σi satisfy the braid relations:

σiσi+1σi = σi+1σiσi+1

σiσj = σjσi if |i− j| > 2.(2.8.4)

•

•

•

•

φ

admissible direction❨

u1...

ui

ui+1 ...uN

Figure 6. Braid σi

2.9. Deformation of repeated eigenvalues. In the previous section §2.8, we considered ageneric deformation where u1, . . . , uN are pairwise distinct. Suppose now that the quantumproduct ⋆τ0 at τ0 is semisimple but (c1(F )⋆τ0) has repeated eigenvalues. In this case, we havethe isomonodromic deformation over the union

(CN (C)∼/π1(B

∗)) ∪Bwhere B is a small open ball in H (F ) centred at τ0 and B∗ ⊂ B is the locus where (E⋆τ )has pairwise distinct eigenvalues; π1(B

∗) acts on CN (C)∼ through the natural map π1(B

∗) →π1(CN (C)). We glue B∗ and CN (C)

∼/π1(B∗) by the map sending τ to the set of eigenvalues

of E⋆τ .We show that the asymptotically exponential flat sections yi(z) associated to a repeated

eigenvalue (at a semisimple point) deform holomorphically along the base of isomonodromicdeformation. The flat sections corresponding to the same repeated eigenvalue remain mutuallyorthogonal after small deformation.

Proposition 2.9.1. Suppose that the quantum product ⋆τ0 is convergent and semisimple forsome τ0 ∈ H (F ). Let B ⊂ H (F ) be an open ball centred at τ = τ0 such that ⋆τ is analyticand semisimple on B. Consider the isomonodromic deformation over B defined by the bigquantum connection ∇ in (2.2.3). Let eiφ be an admissible direction for the spectrum of(c1(F )⋆τ0). Then, shrinking B if necessary, we have:


(1) A formal fundamental solution ΨτRτ (z)e−Uτ /z for ∇ exists and depends analytically

on τ ∈ B (cf. Proposition 2.4.2). Here Ψτ = (Ψτ1 , . . . ,Ψ

τN ) is a basis of normalized

idempotents, Rτ (z) = id+R1(τ)z + R2(τ)z2 + · · · is a formal power series in z with

coefficients Ri(τ) ∈ End(CN ) and Uτ = diag[u1(τ), . . . , uN (τ)] is the diagonal matrixsuch that E ⋆τ Ψ

τi = ui(τ)Ψ

τi .

(2) We have a fundamental solution Yτ = (yτ1 , . . . , yτN ) for ∇ which depends analytically

on τ ∈ B and satisfies the asymptotic condition

Yτ ∼ ΨτRτ (z)e−Uτ/z

as |z| → 0 in a sector of the form φ − π2 − ǫ < arg z < φ + π

2 + ǫ for some ǫ > 0(cf. Proposition 2.6.2).

(3) The flat sections yτi (z) are semiorthonormal in the sense that

(yτi (e−iπz), yτj (z))F = Sij =

0 if Im(e−iφui,0) < Im(e−iφuj,0)

0 if ui,0 = uj,0 and i 6= j

1 if i = j

where (u1,0, . . . , uN,0) are the eigenvalues of (c1(F )⋆τ0) and Sij is the Stokes matrix

at τ = τ0 associated to eiφ (cf. Proposition 2.7.4).

Proof of Proposition 2.9.1. We generalize the construction of yi(z) by Laplace transformation(2.6.1) to the isomonodromic deformation over B. For each fixed τ , one can construct the for-mal fundamental solution ΨτRτe

−Uτ/z and the asymptotically exponential flat sections yτi (z)by the same method as in §§2.4–2.6. It only suffices to show that they depend analytically onτ . The matrix Ψτ of normalized idempotents depends analytically on τ , and hence Uτ also

depends analytically on τ . We use the second structure connection ∇ (with parameter ν = 0;see §2.5) extended to the base B × Cλ. The extension is given by the formula:

∇α = ∂α − (α⋆τ )(λ− E⋆τ )−1µ

∇ ∂∂λ

=∂

∂λ+ (λ− E⋆τ )

−1µ

where α ∈ H (F ) and (τ, λ) ∈ B × Cλ, see [54]. Under the gauge transformation by Ψτ , wehave

(Ψτ )∗∇ = d+

N∑

i=1

(EiV

λ− ui(τ)− Vi

)d(λ− ui(τ)) +

N∑

i=1

Vidλ

whereEi, V , Vi are as in the proof of Lemma 2.4.3. Therefore the connection ∇ has logarithmicsingularities along the normal crossing divisor

∏Ni=1(λ − ui(τ)) = 0 in B × Cλ. The residue

Ri = Ψτ (EiV )Ψ−1τ |τ=τ0 of ∇ at (τ, λ) = (τ0, ui,0) along the divisor λ− ui(τ) = 0 is nilpotent;

hence in a neighbourhood of (τ, λ) = (τ0, ui,0), we have a fundamental solution for ∇ of theform:

U(λ, τ) exp

−

∑

j:uj,0=ui,0

Rj log(λ− uj(τ))

where U(λ, τ) is an End(H (F ))-valued holomorphic function defined near (λ, τ) = (ui,0, τ0)

such that U(ui,0, τ0) = id. We define a ∇-flat section yi(τ, λ) by applying the above funda-mental solution to the eigenvector Ψτ=0

i . This is holomorphic near (τ, λ) = (τ0, ui,0). Here we


use the fact that RjΨτ=0i = 0 whenever uj,0 = ui,0 (this fact follows from Lemma 2.4.3). Now

we define yτi (z) as the Laplace transform (cf. (2.6.1))

yτi (z) =1

z

∫

ui(τ)+eiφR>0

yi(τ, λ)e−λ/zdλ.

This is flat for ∇ (in both z and τ) and depends analytically on τ as far as eiφ is admissiblefor u1(τ), . . . , uN (τ). It then follows that the asymptotic expansion ΨτRτe

−Uτ/z of Yτ =(yτ1 , . . . , y

τN ) depends analytically on τ , as the asymptotic series Rτ (z) is determined by the

Taylor coefficients of yi(τ, λ) at λ = ui(τ), see Watson’s lemma in [12, §8.4].The semi-orthonormality in Part (3) follows from Proposition 2.7.4 (3) because the pairing

(yτi (e−iπz), yτj (z))F does not depend on τ and z.

3. Gamma Conjecture I

In this section we formulate Gamma conjecture I for arbitrary Fano manifolds. We do notneed to assume the semisimplicity of the quantum cohomology algebra.

3.1. Conjecture O. Conjecture O is a property of a Fano variety. We study Gamma Con-jecture I assuming Conjecture O.

Definition 3.1.1. Let F be a Fano manifold and let c1(F )⋆0 ∈ End(H (F )) be the quantumproduct at τ = 0 (see §2.1). Set(3.1.2) T := max|u| : u is an eigenvalue of (c1(F )⋆0).We say that F satisfies Conjecture O if

(1) T is an eigenvalue of (c1(F )⋆0).(2) If u is an eigenvalue of (c1(F )⋆0) with |u| = T , we have u = Tζ for some r-th root of

unity ζ ∈ C, where r is the Fano index of F .(3) The multiplicity of the eigenvalue T is one.

The number T ∈ R>0 above is an algebraic number. This is an invariant of a monotonesymplectic manifold. Conjecture O says that T is non-zero unless F is a point.

Remark 3.1.3. Let r be the Fano index. The formula (2.2.5) shows that we have ⋆0 = ⋆τ withτ = c1(F )(2πi/r). This together with (2.2.6) gives the symmetry:

(3.1.4) (c1(F )⋆0) = e2πi/re−2πiµ/r(c1(F )⋆0)e2πiµ/r

Therefore the spectrum of (c1(F )⋆0) is invariant under multiplication by e2πi/r.

Remark 3.1.5 (O is the structure sheaf). Mirror symmetry for Fano manifolds claims thatF is mirror to a holomorphic function f : Y → C on a complex manifold Y . Under mirrorsymmetry it is expected that the eigenvalues of (c1(F )⋆0) should coincide with the criticalvalues of f . Each Morse critical point of f associates a vanishing cycle (or Lefschetz thimble)which gives an object of the Fukaya-Seidel category of f . Under homological mirror symmetry,the vanishing cycle associated to T should correspond to the structure sheaf O of F . Thisexplains the name ‘ConjectureO’. Our Gamma conjectures give a direct link between T andOin terms of differential equations (without referring to mirror symmetry): the asymptotically

exponential flat section associated to T should correspond to ΓF = ΓF Ch(O); more generally

a simple eigenvalue should correspond to ΓF Ch(E) for an exceptional object E ∈ Dbcoh(F )

(see Gamma Conjecture I, II in §3.3, §4.6).


Remark 3.1.6 (Conifold point). Let f : (C×)n → C be a convenient6 Laurent polynomialmirror to a Fano manifold F . Suppose that all the coefficients of f are positive real. Thenthe Hessian

∂2f

∂ log xi∂ log xj(x1, . . . , xn)

is positive definite on the subspace (R>0)n. Thus f |(R>0)n attains a global minimum at a

unique critical point xcon in the domain (R>0)n; also the critical point xcon is non-degenerate.

We call the point xcon the conifold point. Many examples suggest that T equals Tcon :=f(xcon).

Remark 3.1.7 (Perron-Frobenius eigenvalue). This remark is due to Kaoru Ono. We say thata square matrix M of size n is irreducible if the linear map C

n → Cn, v 7→ Mv admits

no invariant co-ordinate subspace. If (c1(F )⋆0) is represented by an irreducible matrix withnonnegative entries, T is a simple eigenvalue of (c1(F )⋆0) by Perron-Frobenius theorem.

Remark 3.1.8 (Fano orbifold). For a Fano orbifold, T may have multiplicity bigger than one.

Consider the Z/3Z action on P2 given by [x, y, z] 7→ [x, e2πi/3y, e4πi/3z] and let F be the

quotient Fano orbifold P2/(Z/3Z). The mirror is f = x−1

1 x−12 + x21x

−12 + x−1

1 x22 and T = 3has multiplicity three. Under homological mirror symmetry, three critical points in f−1(T )correspond to three flat line bundles which are mutually orthogonal in the derived category.One could guess that the multiplicity of T equals the number of irreducible representationsof πorb1 (F ) for a Fano orbifold F .

Remark 3.1.9. In the rest of the section we assume Conjecture O. However, the argumentin this section (i.e. §3) except for §3.6 works under the following weaker assumption: thereexists a complex number T such that (1) T is an eigenvalue of (c1(F )⋆0) of multiplicity one

and (2) if T is an eigenvalue of (c1(F )⋆0) with Re(T ) > Re(T ), then T = T . In §3.6, we usethe condition part (2) in Definition 3.1.1.

Under Conjecture O, we write

(3.1.10) T ′ := max Re(u) : u 6= T, u is an eigenvalue of (c1(F )⋆0) < T

for the second biggest real part for the eigenvalues of (c1(F )⋆0).

3.2. Flat sections with the smallest asymptotics. Under Conjecture O, one can find aone-dimensional vector space A of flat sections s(z) having the most rapid exponential decay

s(z) ∼ e−T/z as z → +0 on the positive real line. We refer to such sections as flat sectionswith the smallest asymptotics.

Proposition 3.2.1. Suppose that a Fano manifold F satisfies Conjecture O. Consider thequantum connection ∇ in (2.2.1) and define:

A :=s : R>0 → H (F ) : ∇s = 0, ‖eT/zs(z)‖ = O(z−m) as z → +0 (∃m)

(3.2.2)

Then:

(1) A is a one-dimensional complex vector space.(2) For s ∈ A, the limit limz→+0 e

T/zs(z) exists and lies in the T -eigenspace E(T ) of(c1(F )⋆0). The map A → E(T ) defined by this limit is an isomorphism.

(3) Let s : R>0 → H (F ) be a flat section which does not belong to A. Then we have

limz→+0 ‖eλ/zs(z)‖ = ∞ for all λ > T ′.

6A Laurent polynomial is said to be convenient if the Newton polytope of f contains the origin in its interior.


Proof. We postpone the proof of (1) and (3) till §3.4. Here we construct a ∇-flat section

s0(z) such that the limit limz→+0 eT/zs0(z) exists and is a non-zero eigenvector of (c1(F )⋆0)

with eigenvalue T . This together with (1) proves (2). The method here is similar to theconstruction of the asymptotically exponential flat section yi(z) in §2.6. Consider again the

second structure connection ∇ from §2.5 with parameter ν = 0. Since we do not assume the

semisimplicity, we do not know if ∇ is Fuchsian. However we know by Conjecture O that ∇ islogarithmic at λ = T . Following the argument in §2.5 and using Lemma 3.2.4 below instead of

Lemma 2.4.3, we can construct a ∇-flat section ϕ(λ) holomorphic near λ = T such that ϕ(T )is a non-zero T -eigenvector of (c1(F )⋆0). The ∇-flat section s0(z) is defined by (cf. (2.6.1)):

(3.2.3) s0(z) =1

z

∫ ∞

Tϕ(λ)e−λ/zdλ

Note that ∇ is smooth in the region Re(λ) > T and therefore ϕ(λ) is analytically continued

along the positive real line. The integral converges for z > 0 because ∇ is regular singular atλ = ∞. Watson’s lemma [12, §8.4] shows that limz→+0 e

T/zs0(z) = ϕ(T ).

Lemma 3.2.4. Suppose that Conjecture O holds. Let E(T ) ⊂ H (F ) denote the one-dimensional T -eigenspace of (c1(F )⋆0) and H ′ = Im(T − (c1(F )⋆0)) be the complementarysubspace. Then E(T ) and H ′ are orthogonal with respect to the Poincare pairing and theendomorphism µ satisfies µ(E(T )) ⊂ H ′.

Proof. Take α ∈ E(T ) and β ∈ H ′. There exists an element γ ∈ H (F ) such that β =(T − c1(F )⋆0)γ. Then we have

(α, β)F = (α, Tγ − c1(F ) ⋆0 γ)F = (Tα− c1(F ) ⋆0 α, γ)F = 0.

Thus E(T ) ⊥ H ′ with respect to (·, ·)F . The skew-adjointness of µ with respect to (·, ·)Fshows that (µα,α)F = 0; hence µ(α) ∈ H ′.

3.3. Gamma Conjecture I: statement. The Gamma class [52, 53, 44, 50, 45] of a smoothprojective variety F is the class

ΓF :=

dimF∏

i=1

Γ(1 + δi) ∈ H (F )

where δ1, . . . , δdimF are the Chern roots of the tangent bundle TF so that

c(TF ) =∏dimFi=1 (1 + δi). The Gamma function Γ(1 + δi) in the right-hand side should be

expanded in the Taylor series in δi. We have

ΓF = exp

(−Ceuc1(F ) +

∞∑

k=2

(−1)k(k − 1)!ζ(k) chk(TF )

)

where Ceu = 0.57721... is Euler’s constant and ζ(k) is the special value of Riemann’s zetafunction.

The fundamental solution S(z)z−µzρ from Proposition 2.3.1 identifies the space of flatsections over the positive real line R>0 with the cohomology group:

Φ: H (F ) −→ s : R>0 → H (F ) : ∇s = 0α 7−→ (2π)−

dimF2 S(z)z−µzρα

(3.3.1)

where we use the standard determination for z−µzρ = exp(−µ log z) exp(ρ log z) such thatlog z ∈ R for z ∈ R>0.


Definition 3.3.2. Suppose that a Fano manifold F satisfies Conjecture O. The principalasymptotic class of F is a cohomology class AF ∈ H (F ) such that A = CΦ(AF ), where A isthe space (3.2.2) of flat sections with the smallest asymptotics (recall that A is of dimensionone by Proposition 3.2.1). The class AF is defined up to a constant; when 〈[pt], AF 〉 6= 0, wecan normalize AF so that 〈[pt], AF 〉 = 1.

Definition 3.3.3. Suppose that a Fano manifold F satisfies Conjecture O. We say that F

satisfies Gamma Conjecture I if the principal asymptotic class AF is given by ΓF .

We explain that the Gamma class should be viewed as a square root of the Todd class (or

A-class) in the Riemann–Roch theorem. Let H denote the vector space of ∇-flat sections overthe positive real line (cf. §2.8):

H := s : R>0 → H (F ) : ∇s = 0and let [·, ·) be a non-symmetric pairing on H defined by

(3.3.4) [s1, s2) := (s1(e−iπz), s2(z))F

for s1, s2 ∈ H, where s1(e−iπz) denotes the analytic continuation of s1(z) along the semicircle

[0, 1] ∋ θ 7→ e−iπθz. The right-hand side does not depend on z because of the flatness (2.2.2).The non-symmetric pairing [·, ·) induces a pairing on H (F ) via the map Φ (3.3.1); we denoteby the same symbol [·, ·) the corresponding pairing on H (F ). Since S(z) preserves the pairing(Proposition 2.3.1), we have

(3.3.5) [α, β) =1

(2π)dimF(eπiµe−πiρα, β)F =

1

(2π)dimF(eπiρeπiµα, β)F

for α, β ∈ H (F ). We write Ch(·), Td(·) for the following characteristic classes:

Ch(V ) = (2πi)deg2 ch(V ) =

rankV∑

j=1

e2πiδj

Td(V ) = (2πi)deg2 td(V ) =

rankV∏

j=1

2πiδj1− e−2πiδj

where δ1, . . . , δrank V are the Chern roots of a vector bundle V . We also write TdF = Td(TF ).

Lemma 3.3.6 ([44, 45]). Let V1, V2 be vector bundles over F . Then[ΓF Ch(V1), ΓF Ch(V2)

)= χ(V1, V2)

where χ(V1, V2) =∑dimF

i=0 (−1)i dimExti(V1, V2).

Proof. Recall the Gamma function identity:

(3.3.7) Γ(1 + z)Γ(1− z) =2πiz

eiπz − e−iπz.

This immediately implies that

eπiρ(eπi

deg2 ΓF

)· ΓF = TdF .

Therefore[ΓF Ch(V1), ΓF Ch(V2)

)=

1

(2πi)dimF

∫

F

(eπi

deg2 Ch(V1)

)Ch(V2)TdF .

This equals χ(V1, V2) by the Hirzebruch-Riemann–Roch formula.


Under Gamma Conjecture I, the canonical generator Φ(ΓF ) of A satisfies the followingnormalization.

Proposition 3.3.8. Suppose that a Fano manifold F satisfies Gamma Conjecture I,

i.e. Φ(ΓF )(z) = (2π)− dimF/2S(z)z−µzρΓF is a generator of A. Then the limit

limz→+0

eT/zΦ(ΓF )(z) ∈ E(T )

has length one with respect to the Poincare pairing (·, ·)F . In other words, Φ(ΓF )(z) is given

by the Laplace transform (3.2.3) along [T,∞] of a ∇-flat section ϕ(λ) such that ϕ(λ) is

holomorphic near λ = T and that ϕ(T ) is a T -eigenvector of unit length, where ∇ is thesecond structure connection in §2.5 with parameter ν = 0.

Proof. By Proposition 3.2.1, the limit limz→+0 eT/zΦ(ΓF )(z) exists and lies in E(T ). By the

construction there, there exists a ∇-flat section ϕ(λ) such that

• ϕ(λ) is holomorphic near λ = T ;

• Φ(ΓF )(z) is given as the Laplace transform (3.2.3) of ϕ(λ);

• limz→+0 eT/zΦ(ΓF )(z) = ϕ(T ) ∈ E(T ).

By bending the integration path of (3.2.3) as in Figure 7, we obtain the analytic continuation

Φ(ΓF )(e−πiz). Again by Watson’s lemma, we have limz→+0 e

−T/zΦ(ΓF )(e−πiz) = ϕ(T ).

Therefore we have[ΓF , ΓF

)=(Φ(ΓF )(e

−πiz),Φ(ΓF )(z))F→ (ϕ(T ), ϕ(T ))F

as z → +0. By Lemma 3.3.6, the left-hand side equals χ(OF ,OF ) = 1 since F is Fano. Thuswe have (ϕ(T ), ϕ(T ))F = 1 and the conclusion follows.

•T

•

•

••

••

Φ(ΓF )(z)

Φ(ΓF )(e−πiz)

Figure 7. Spectrum of (c1(F )⋆0) and the paths of integration (on the λ-plane).

3.4. The leading asymptotics of dual flat sections. We now study flat sections ofthe dual quantum connection, which we call dual flat sections. We show that the limitlimz→+0 e

T/zf(z) exists for every dual flat section f(z).Let H (F ) = Heven(F ;C) denote the even part of the homology group with complex coeffi-

cients. The dual quantum connection is a meromorphic flat connection on the trivial bundleH (F )× P

1 → P1 given by (cf. (2.2.1))

(3.4.1) ∇∨z ∂∂z

= z∂

∂z+

1

z(c1(F )⋆0)

t − µt

where the superscript “t” denotes the transpose. This is dual to the quantum connection ∇in the sense that we have

d〈f(z), s(z)〉 = 〈∇∨f(z), s(z)〉 + 〈f(z),∇s(z)〉


where 〈·, ·〉 is the natural pairing between homology and cohomology. Because of the property(2.2.2), we can identify the dual flat connection ∇∨ with the quantum connection ∇ withthe opposite sign of z. In order to avoid possible confusion about signs, we shall not use thisidentification. We write E(T )∗ ⊂ H (F ) for the one-dimensional eigenspace of (c1(F )⋆0)

t witheigenvalue T and H ′∗ = Im(T − (c1(F )⋆0)

t) for the complementary subspace. They are dualto E(T ), H ′ from Lemma 3.2.4.

Proposition 3.4.2. Suppose that a Fano manifold F satisfies Conjecture O. Then:

(1) The limit

LA(f) := limz→+0

e−T/zf(z)

exists for every ∇∨-flat section f(z) and lies in E(T )∗. In other words, LA defines alinear functional:

LA:f : R>0 → H (F ) : ∇∨f(z) = 0

−→ E(T )∗.

(2) If LA(f) = 0, we have limz→+0 e−λ/zf(z) = 0 for every λ > T ′.

Proof. As usual we make use of the Laplace transformation. Let λ be the Laplace dual variableof z−1. The Laplace dual of the dual quantum connection ∇∨ is given by (cf. §2.5):

(3.4.3) ∇(ν)∂∂λ

=∂

∂λ+ (λ+ (c1(F )⋆0)

t)−1(−µt + ν).

Here ν ∈ R is a parameter. This is smooth away from the spectrum of (−c1(F )⋆0) and is

logarithmic at λ = −T and λ = ∞. By Lemma 3.2.4, the residue of ∇(ν) at λ = −T isrepresented by the matrix:

ν ∗

0 0

under the decomposition H (F ) = E(T )∗⊕H ′∗. When the residue is non-resonant, we have a

fundamental solution near λ = −T of the form U(λ+ T ) exp(− log(λ+ T )Resλ=−T ∇(ν)) forsome holomorphic U(x) with U(0) = id. Therefore we have:

Claim 3.4.4. (1) Suppose ν = 0. Then we have a ∇(0)-flat section ϕ0(λ) which is holomorphicnear λ = −T such that ϕ0(−T ) is a non-zero eigenvector in E(T )∗.

(2) Suppose ν /∈ Z. The monodromy M of ∇(ν) around λ = −T is a quasi-reflection,

i.e. rank(id−M) = 1. Moreover, a monodromy-invariant ∇(ν)-flat section around λ = −T isholomorphic at λ = −T .

First we construct a ∇∨-flat section f0(z) such that limz→+0 eT/zf0(z) exists and is a non-

zero eigenvector in E(T )∗. Take a ∇(0)-flat section ϕ0(λ) in Claim 3.4.4 (1). By an argumentsimilar to §2.6 and §3.2, we can check that the Laplace transform

(3.4.5) f0(z) :=1

z

∫

−T+eiφR>0

e−λ/zϕ0(λ)dλ

satisfies the differential equation ∇∨f0(z) = 0 and we have limz→+0 eT/zf0(z) = ϕ0(−T ) ∈

E(T )∗. Here we choose the phase φ ∈ (−π/2, π/2) so that the ray −T + eiφR>0 does not

contain the singularities of ∇(0) other than −T .


Let f(z) be an arbitrary ∇∨-flat section. Changing co-ordinates t = z−1, we have

∂f

∂t= (c1(F )⋆0)

tf − 1

tµtf.

An argument using Gronwall’s inequality shows that there exist constants C0, C1 > 0 suchthat ‖f(t−1)‖ 6 C0e

C1t for all t ∈ [1,∞). Therefore the following Laplace transformationconverges for λ with Re(λ) ≪ 0 and ν ≫ 0, ν /∈ Z.

(3.4.6) ϕ(λ) :=

∫ ∞

0tν−1eλtf(t−1)dt.

The condition ν ≫ 0 ensures that the integral converges near t = 0 (since ∇∨ is regularsingular at t = 0, f(t−1) is of at most polynomial growth near t = 0). By the standard

argument using integration by parts, we can show that ϕ(λ) is flat for the connection ∇(ν) in

(3.4.3). Since the connection ∇(ν) is smooth on Re(λ) < −T, ϕ(λ) is analytically continuedthere.

We now consider the Laplace transform (3.4.6) of the ∇∨-flat section f(z) = f0(z) defined

in (3.4.5). Let ϕν(λ) denote the resulting ∇(ν)-flat section. Suppose ν ≫ 0 and ν /∈ Z.

Claim 3.4.7. The flat section ϕν(λ) has a non-trivial monodromy around λ = −T .Proof of Claim 3.4.7. By (3.4.5) and the definition of ϕν , we have

ϕν(λ) =

∫ ∞

0tν−1eλtdt

∫

−T+eiφR>0

e−ξtϕ0(ξ)dξ

=

∫

−T+eiφR>0

(∫ ∞

0tν−1et(λ−ξ)dt

)ϕ0(ξ)dξ

= Γ(ν)

∫

−T+eiφR>0

(ξ − λ)−νϕ0(ξ)dξ

for Re(λ) < −T . The right-hand side is analytically continued to the complement of the ray−T + eiφR>0. We can directly evaluate the monodromy around λ = −T using this formulaand see that it is non-trivial. Alternatively we prove this by contradiction. Suppose ϕν(λ) hastrivial monodromy around λ = −T . The above formula shows that it is analytically continued

to the whole λ-plane without singularities by Part (2) of Claim 3.4.4. Since ∇(ν) is regularsingular at λ = ∞, this implies that ϕν(λ) is a polynomial. Expand ϕν(λ) = φ0 +λφ1 + · · ·+λnφn with φn 6= 0. Then the differential equation ∇(ν)ϕν = 0 gives (−µt+n+ ν)φn = 0. Butfor ν ≫ 0, −µt + n+ ν has no kernel as n > 0. This is a contradiction.

Now consider the ∇(ν)-flat section ϕ(λ) (3.4.6) associated to a general ∇∨-flat sectionf(z). Because the monodromy around λ = −T is a quasi-reflection (Claim 3.4.4) and themonodromy of ϕν(λ) around λ = −T is non-trivial (Claim 3.4.7), we can find a linear combi-nation ϕ(λ) = ϕ(λ) + cϕν(λ), c ∈ C such that it has trivial monodromy around λ = −T . ByClaim 3.4.4 again, ϕ(λ) defines a single-valued analytic function on the region Re(λ) < −T ′.Note that ϕ(λ) is a Laplace transform of f(z) = f(z) + cf0(z):

ϕ(λ) =

∫ ∞

0tν−1etλf(t−1)dt

and we have the Laplace inversion formula:

f(z) =1

2πizν−1

∫ −a+i∞

−a−i∞ϕ(λ)e−λ/zdλ


which is valid for a > T ′. The differential equation7 ∇(ν)ϕ(λ) = 0 near λ = ∞ together withthe condition ν ≫ 0 ensures that ϕ(λ) = O(|λ|−2) as λ → ∞ in the region Re(λ) < −T ′and the above integral converges. For any a > T ′, we have

limz→+0

e−a/z f(z) = limz→+0

1

2πizν−1

∫i∞

−i∞ϕ(−a+ x)e−x/zdx = 0.

In particular, limz→+0 e−T/zf(z) = limz→+0 e

−T/z(f(z)− cf0(z)) = −cϕ0(−T ) lies in E(T )∗.We proved the existence of the map LA.

Suppose now that LA(f) = 0. Then we have c = 0 and thus limz→+0 e−a/zf(z) =

limz→+0 e−a/z f(z) = 0 for a > T ′. The proof of Proposition 3.4.2 is completed.

Proof of (1) and (3) of Proposition 3.2.1. We have already shown that dimA > 1 in §3.2. Tosee that dimA = 1, it suffices to show that A ⊂ ImLA∗, where LA∗ is the dual map of LA

LA∗ : E(T ) → s : R>0 → H (F ) : ∇s(z) = 0.In order to prove (1) and (3) simultaneously, we show that A ⊂ Im(LA∗), where A is definedby:

A =

s : R>0 → H (F ) :

∇s(z) = 0, ∃λ > T ′ such that

‖eλ/zs(z)‖ 6→ ∞ as z → +0

.

Obviously we have A ⊂ A. Thus A ⊂ Im(LA∗) implies A = A = Im(LA∗); (1) and (3)

follow from this. Let s(z) be a flat section from A. Then there exist λ > T ′ and a sequence

zn such that zn ∈ R>0, zn → +0 as n → ∞ and limn→∞ eλ/zns(zn) exists. Then for everyf ∈ Ker(LA), we have

〈f, s〉 =⟨e−λ/znf(zn), e

λ/zns(zn)⟩.

By Proposition 3.4.2 (2), the right hand side converges to zero as n → ∞. Therefore s is in

the annihilator of Ker(LA). This means that s ∈ Im(LA∗). We have shown A ⊂ Im(LA∗).The proof of Proposition 3.2.1 is now complete.

In the above proof, we showed the following.

Proposition 3.4.8. The space A in Proposition 3.2.1 coincides with Im(LA∗).

3.5. A limit formula for the principal asymptotic class. The leading asymptotic be-haviour of dual flat sections as z → +0 is given by the pairing with a flat section with thesmallest asymptotics. From this we obtain a limit formula for AF .

The fundamental solution for the dual quantum connection ∇∨ (3.4.1) is given by theinverse transpose (S(z)z−µzρ)−t of the fundamental solution for ∇ in Proposition 2.3.1. Thisyields the following identification (cf. (3.3.1))

Φ∨ : H (F ) −→ f : R>0 → H (F ) : ∇∨f(z) = 0α 7−→ (2π)

dimF2 S(z)−tzµ

tz−ρ

tα

(3.5.1)

between dual flat sections and homology classes.

Proposition 3.5.2. Suppose that F satisfies Conjecture O. Let AF be a principal as-ymptotic class (Definition 3.3.2). Let ψ∗

0 be an element of E(T )∗ which is dual to ψ0 :=

limz→+0 eT/zΦ(AF )(z) ∈ E(T ) in the sense that 〈ψ∗

0 , ψ0〉 = 1. Then we have:

LA(Φ∨(α)) := limz→+0

e−T/zΦ∨(α)(z) = 〈α,AF 〉ψ∗0 .

7We have κ∂κϕ = (1 + κ(c1(F )⋆0)t)−1(−µt + ν)ϕ for κ = λ−1.


Proof. By the definition of Φ and Φ∨ (see (3.3.1), (3.5.1)), we have:

〈α,AF 〉 = 〈Φ∨(α)(z),Φ(AF )(z)〉 = 〈e−T/zΦ∨(α)(z), eT/zΦ(AF )(z)〉.This converges to 〈LA(Φ∨(α)), ψ0〉 as z → +0 by Propositions 3.2.1, 3.4.2.

Definition 3.5.3 ([30]). Let S(z)z−µzρ be the fundamental solution from Proposition 2.3.1and set t := z−1. The J-function J(t) of F is a cohomology-valued function defined by:

J(t) := zdimF

2 z−ρzµS(z)−11

where 1 ∈ H (F ) is the identity class. Alternatively we can define J(t) by the requirementthat

(3.5.4) 〈α, J(t)〉 =( z2π

) dimF2 〈Φ∨(α)(z), 1〉

holds for all α ∈ H (F ), using the dual flat sections Φ∨(α) in (3.5.1).

Remark 3.5.5. The J-function is a solution to the scalar differential equation associated tothe quantum connection on the anticanonical line τ = c1(F ) log t. More precisely, we have

P(t, [∇c1(F )]z=1

)1 = 0 ⇐⇒ P

(t, t ∂∂t

)J(t) = 0

for any differential operators P ∈ C〈t, t ∂∂t〉, where [∇c1(F )]z=1 = t ∂∂t + (c1(F )⋆c1(F ) log t). Dif-ferential equations satisfied by the J-function are called the quantum differential equations.See also Remark 2.2.4.

Remark 3.5.6. Using the fact that S(z)−1 equals the adjoint of S(−z) (Proposition 2.3.1) andRemark 2.3.2, we have

(3.5.7) J(t) = ec1(F ) log t

1 +

N∑

i=1

∑

d∈Eff(F )\0

⟨φi

1− ψ

⟩F

0,1,d

tc1(F )·dφi

where φiNi=1 and φiNi=1 are bases of H (F ) which are dual with respect to the Poincarepairing (·, ·)F . This gives a well-known form of Givental’s J-function [30] restricted to theanticanonical line τ = c1(F ) log t.

Theorem 3.5.8. Suppose that a Fano manifold F satisfies Conjecture O. Let AF be aprincipal asymptotic class (Definition 3.3.2) and let J(t) be the J-function. We have

limt→+∞

J(t)

〈[pt], J(t)〉 =AF

〈[pt], AF 〉where the limit is taken over the positive real line and exists if and only if 〈[pt], AF 〉 6= 0.

Proof. By Proposition 3.5.2, we have

limz→+0

e−T/zΦ∨(α)(z) = 〈α,AF 〉ψ∗0

limz→+0

e−T/zΦ∨([pt])(z) = 〈[pt], AF 〉ψ∗0

for some non-zero ψ∗0 ∈ E(T )∗. Therefore by (3.5.4) we have

limt→+∞

〈α, J(t)〉〈[pt], J(t)〉 = lim

z→+0

e−T/z〈Φ∨(α)(z), 1〉e−T/z〈Φ∨([pt])(z), 1〉 =

〈α,AF 〉〈[pt], AF 〉

for all α ∈ H (F ). Here we used the fact 〈ψ∗0 , 1〉 6= 0 which is proved in Lemma 3.5.10 below.

The limit exists if and only if 〈[pt], AF 〉 6= 0. The Theorem is proved.


Corollary 3.5.9. Suppose that a Fano manifold F satisfies Conjecture O. The followingstatements are equivalent:

(1) F satisfies Gamma Conjecture I (Definition 3.3.3).

(2) ‖Φ(ΓF )(z)‖ 6 Ce−λ/z on (0, 1] for some C > 0 and λ > T ′, where T ′ is given in(3.1.10) and Φ is given in (3.3.1).

(3) We have

limt→+∞

J(t)

〈[pt], J(t)〉 = ΓF

where J(t) is the J-function of F given in Definition 3.5.3 and the limit is taken overthe positive real line.

Proof. The equivalence of (1) and (2) is clear from Proposition 3.2.1. The equivalence of (1)and (3) follows from Theorem 3.5.8.

Lemma 3.5.10. Suppose that Conjecture O holds. For a non-zero eigenvector ψ∗0 ∈ E(T )∗,

we have 〈ψ∗0 , 1〉 6= 0. In other words, the E(T )-component of 1 with respect to the decomposi-

tion H (F ) = E(T )⊕H ′ in Lemma 3.2.4 does not vanish.

Proof. Let ψ0 ∈ E(T ) be a non-zero eigenvector of (c1(F )⋆0) with eigenvalue T . We claimthat ψ0 ⋆0 H

′ = 0. Take α ∈ H ′ and write α = Tγ − c1(F ) ⋆0 γ. Then we have

ψ0 ⋆ α = Tψ0 ⋆0 γ − ψ0 ⋆0 c1(F ) ⋆0 γ = 0.

The claim follows. Write 1 = cψ0 + ψ′ with ψ′ ∈ H ′ and c ∈ C. Applying (ψ0⋆0), we obtainψ0 = cψ0 ⋆0 ψ0. Thus c 6= 0 and Lemma follows.

Remark 3.5.11. See [17, §7] for a discussion of the limit formula in the classification problemof Fano manifolds.

3.6. Apery limit. We can replace the continuous limit of the ratio of the J-function appear-ing in Theorem 3.5.8 with the (discrete) limit of the ratio of the Taylor coefficients. Golyshev[32] considered such limits and called them Apery constants of Fano manifolds. A general-ization to Apery class was studied by Galkin [25]. This limit sees the primitive part of theGamma class.

Expand the J-function as

J(t) = ec1(F ) log t∑

n>0

Jntn.

The coefficient Jn ∈ H (F ) is given by:

Jn :=

N∑

i=1

∑

d∈Eff(F ):c1(F )·d=n

⟨φiψ

n−2−dimφi⟩F0,1,d

φi

where we set dimφi = dimF − 12 deg φi. See Remark 3.5.6. The main theorem in this section

is stated as follows:

Theorem 3.6.1. Suppose that F satisfies Conjecture O and suppose also that the principalasymptotic class AF satisfies 〈[pt], AF 〉 6= 0. Let r be the Fano index. For every α ∈ H (F )such that c1(F ) ∩ α = 0, we have

lim infn→∞

∣∣∣∣〈α, Jrn〉〈[pt], Jrn〉

− 〈α,AF 〉〈[pt], AF 〉

∣∣∣∣ = 0.


Remark 3.6.2. Note that Jn = 0 unless r divides n. In the left-hand side of the above formula,we set 〈α, Jrn〉/〈[pt], Jrn〉 = ∞ if 〈[pt], Jrn〉 = 0. It is however expected that the followingproperties will hold for Fano manifolds:

• the Gromov–Witten invariants 〈[pt], Jrn〉 are all positive for n > 0;• lim infn→∞ in Theorem 3.6.1 can be replaced with limn→∞.

Remark 3.6.3. More generally, if α, β ∈ H (F ) are homology classes such that α ∩ c1(F ) =β ∩ c1(F ) = 0 and if we have 〈β,AF 〉 6= 0, we have the limit formula

lim infn→∞

∣∣∣∣〈α, Jrn〉〈β, Jrn〉

− 〈α,AF 〉〈β,AF 〉

∣∣∣∣ = 0.

This can be shown by the same argument as below. We shall restrict to the case β = [pt] asGamma Conjecture I implies 〈[pt], AF 〉 6= 0.

Define the functions G and G as:

G(t) := 〈[pt], J(t)〉 =∞∑

n=0

Gntn, where Gn := 〈[pt], Jn〉,

G(κ) :=

∞∑

n=0

n!Gnκn =

1

κ

∫ ∞

0G(t)e−t/κdt.

These functions G, G are called (unregularized or regularized) quantum periods of Fano man-ifolds [17].

Lemma 3.6.4. Let F be an arbitrary Fano manifold. The convergence radius of G(κ) isbigger than or equal to 1/T .

Proof. Recall that we have G(t) = (2πt)− dimF/2〈Φ∨([pt])(z), 1〉 (see (3.5.4)) and Φ∨([pt])(z)is a ∇∨-flat section (where z = t−1). Therefore the Laplace transform

ϕ(λ) = (2π)−dimF

2

∫ ∞

0tν−1eλtΦ∨([pt])(t−1)dt

with ν ≫ 0 is ∇(ν)-flat as we discussed around (3.4.6). Then the convergence radius of

〈ϕ(−κ−1), 1〉 =∫ ∞

0tν+

dimF2

−1G(t)e−t/κdt

= κν+dimF

2−1

∞∑

n=0

Γ(n+ ν + dimF2 )Gnκ

n(3.6.5)

is bigger than or equal to 1/T as ∇(ν) has no singularities in |λ| > T. The function G(κ)has the same radius of convergence as 〈ϕ(−κ−1), 1〉 and the conclusion follows.

Lemma 3.6.6. Suppose that F satisfies Conjecture O and 〈[pt], AF 〉 6= 0. Then the conver-

gence radius of G(κ) is exactly 1/T .

Proof. In view of the discussion in the previous lemma, it suffices to show that the func-tion 〈ϕ(−κ−1), 1〉 in (3.6.5) has a singularity at κ = 1/T . By (3.5.4) we have G(t) =

(2πt)− dimF/2〈Φ∨([pt])(z), 1〉. Thus by Proposition 3.5.2 and Lemma 3.5.10, we have

limt→+∞

e−tT (2πt)dimF

2 G(t) = 〈[pt], AF 〉〈ψ∗0 , 1〉 6= 0.

Therefore the Laplace transform (3.6.5) diverges as κ approaches 1/T from the left for ν ≫0.


Proof of Theorem 3.6.1. Set β := α− 〈α,AF 〉〈[pt],AF 〉 [pt]. Then 〈β,AF 〉 = 0. It suffices to show that

(3.6.7) lim infn→∞

∣∣∣∣〈β, Jrn〉〈[pt], Jrn〉

∣∣∣∣ = 0.

Set bn := 〈β, Jn〉. Then we have∑∞

n=0 bntn = 〈β, J(t)〉 = (2πt)− dimF/2〈Φ∨(β)(t−1), 1〉 by

(3.5.4). By Proposition 3.5.2, we have LA(Φ∨(β)) = 〈β,AF 〉ψ∗0 = 0. Thus by Proposition

3.4.2 we have limt→+∞ e−tλΦ∨(β)(t−1) = 0 for every λ > T ′. Consider again the Laplacetransform

ϕβ(λ) = (2π)−dimF

2

∫ ∞

0tν−1etλΦ∨(β)(t−1)dt

for ν ≫ 0 (as in the proof of Lemma 3.6.4). This is a ∇(ν)-flat section which is regular onλ : Re(λ) < −T ′. Then its component

〈ϕβ(−κ−1), 1〉 = κν+dimF

2−1

∞∑

n=0

Γ(n+ ν + dimF2 )bnκ

n

has the radius of convergence strictly bigger than 1/T . This is because the only singularities of

∇(ν) on |λ| > T are −e2πik/rT : k = 0, . . . , r− 1 and 〈ϕβ(λ), 1〉 is regular at λ = −T and

satisfies 〈ϕβ(e2πi/rλ), 1〉 = e−(ν+ dimF2

−1)2πi/r〈ϕβ(λ), 1〉. (Here we used Part (2) of ConjectureO.) This implies that there exist numbers C,R > 0 such that R > 1/T and

|bnn!| 6 CR−n.

On the other hand, Lemma 3.6.6 implies that

lim supn→∞

|Gnn!|1/n = T.

Let ǫ > 0 be such that (1− ǫ)RT > 1. Then we have a number n0 such that for every n > n0we have supm>n |Gmm!|1/m > T (1 − ǫ/2). Namely we can find a sequence n0 6 n1 < n2 <n3 < · · · such that |Gni

|ni! > T ni(1− ǫ)ni . Therefore∣∣∣∣bni

Gni

∣∣∣∣ =∣∣∣∣bnini!

Gnini!

∣∣∣∣ 6CR−ni

T ni(1− ǫ)ni=

C

(RT (1− ǫ))ni.

This implies lim infn→∞ |bn/Gn| = 0, which proves (3.6.7).

3.7. Quantum cohomology central charges. For a vector bundle V on F , we define

(3.7.1) Z(V ) := (2πz)dimF

2

(1,Φ(ΓF Ch(V ))(z)

)F.

This quantity is called the quantum cohomology central charge8 of V in [45]. Using theproperty S(z)∗ = S(−z)−1, (3.3.1) and Definition 3.5.3, we can write Z(V ) in terms of theJ-function:

(3.7.2) Z(V ) = (−1)dimF(eπiρJ(e−πit), Γ∗

F Ch(V ∗))F

where t = z−1 and Γ∗F := (−1)

deg2 ΓF =

∏dimFi=1 Γ(1 − δi) is the dual Gamma class of F .

Therefore, as a component of the J-function, Z(V ) gives a scalar-valued solution to thequantum differential equations (see Remark 3.5.5). If F satisfies Gamma Conjecture I, thecentral charge Z(O) satisfies the following smallest asymptotics:

Z(O) ∼√(ψ0, ψ0)F (2πz)

dim F2 e−T/z as z → +0

8The central charge Z(V ) here is denoted by (2πi)dimFZ(V ) in [45].


where ψ0 ∈ E(T ) is the idempotent for ⋆0. This follows from Proposition 3.3.8 and the proofof Lemma 3.5.10.

4. Gamma Conjecture II

In this section we restrict to a Fano manifold F with semisimple quantum cohomology andstate a refinement of Gamma Conjecture I for such F . We represent the irregular monodromyof quantum connection by a marked reflection system (§2.8) given by a basis of asymptoti-cally exponential flat sections. Gamma Conjecture II says that the marked reflection system

coincides with a certain Gamma-basis ΓF Ch(Ei) for some exceptional collection Ei ofthe derived category Db

coh(F ). This also refines Dubrovin’s conjecture [20].

4.1. Semiorthonormal basis and Gram matrix. Let V = (V ; [·, ·); v1, . . . , vN ) be a tripleof a vector space V , a bilinear form [·, ·) = [·, ·)V , and a collection of vectors v1, . . . , vN ∈ V .The bilinear form [·, ·) is not necessarily symmetric nor skew-symmetric. We say that Vis a semiorthonormal collection if the Gram matrix Gij = [vi, vj) is uni-uppertriangular:[vi, vi) = 1 for all i, and [vi, vj) = 0 for all i > j. This implies that G is non-degenerate, hencevectors vi are linearly independent so N 6 dimV . We say that V is a semiorthonormal basis(SOB) if N = dimV i.e. vi form a basis of vector space V , this implies that pairing [·, ·) isnon-degenerate.

Set of SOBs admits an obvious action of (Z/2Z)N : if v1, . . . , vN is SOB then so is±v1, . . . ,±vN , in this case we say that SOBs are the same up to sign. For u, v ∈ V , de-fine the right mutation Ruv and the left mutation Luv by

Ruv := v − [v, u)u, Luv := v − [u, v)u.

The braid group BN on N strands acts on the set of SOBs of N vectors by:

σi(v1, . . . , vN ) = (v1, . . . , vi−1, vi+1, Rvi+1vi, vi+2, . . . , vN )

σ−1i (v1, . . . , vN ) = (v1, . . . , vi−1, Lvivi+1, vi, vi+2, . . . , vN )

(4.1.1)

where σ1, . . . , σN are generators of BN and satisfy the braid relation (2.8.4).

4.2. Marked reflection system. A marked reflection system (MRS) of phase φ ∈ R is atuple (V, [·, ·), v1 , . . . , vN,m, eiφ) consisting of

• a complex vector space V ;• a bilinear form [·, ·) on V ;• an unordered basis v1, . . . , vN of V ;• a marking m : v1, . . . , vN → C; we write ui = m(vi) and u = u1, . . . , uN;• eiφ ∈ S1

such that the vectors vi are semiorthonormal in the sense that:

(4.2.1) [vi, vj) =

1 if i = j

0 if i 6= j and Im(e−iφui) 6 Im(e−iφuj)

cf. Proposition 2.9.1. Under an appropriate ordering of the basis, (V, [·, ·), v1, . . . , vN ) givesan SOB. An MRS is said to be admissible if the phase φ is admissible for u, i.e. eiφ is notparallel to any non-zero difference ui − uj. By a small perturbation of the marking, one canalways make an MRS admissible. The circle S1 acts on the space of all marked reflectionsystems by rotating simultaneously the phase and all markings: φ 7→ α+ φ, ui 7→ eiαui.

We write hφ : C → R for the R-linear function hφ(z) = Im(e−iφz).


Remark 4.2.2 ([31]). The data of admissible marked reflection system may be used to producea polarized local system on C\u (cf. the second structure connection): by identifying the fiberwith V , endowing it with a bilinear form ·, · (either with the symmetric form v1, v2 =[v1, v2) + [v2, v1) or with the skew-symmetric form v1, v2 = [v1, v2) − [v2, v1)), choosing“eiφ∞” for the base point, joining it with the points ui with the level rays hφ(z) = hφ(ui)as paths, trivializing the local system outside the union of the paths, and requiring that theturn around ui act in the monodromy representation by the reflection with respect to vi :rvi(v) = v−v, vivi. In case ui is a multiple marking all the respective reflections commute,so the monodromy of the local system is well-defined as a product of such reflections. The form·, · could be degenerate, however construction above also provides local systems with fiberthe quotient-space V/Ker·, · (use reflections with respect to images of vi in the quotient-spaces).

When u1, . . . , uN are pairwise distinct, this construction could be reversed with a littleambiguity: if there is such a local system and the paths are given, we take V to be a fiber,define vi ∈ V as a reflection vector of the monodromy at ui (which could be determined upto sign if ·, · is non-degenerate), and define [vi, vi) = 1 for all i, for i 6= j put [vi, vj) = 0 ifhφ(ui) < hφ(uj) and [vi, vj) = vi, vj otherwise.

4.3. Mutation of MRSs. We describe the change of an MRS under deformation of param-eters (u, φ). The description here is analogous to the one in §§2.7–2.9. By perturbing themarking (or the phase) a little, we may start with an admissible MRS (see Remark 4.3.1below). We draw a straight ray Li emanating from each marking ui in the direction of φ andobtain a picture as in Figure 1. We let the marking u and the phase φ vary. We allow thata multiple marking separates into distinct ones, but do not allow that two different markingsui 6= uj collide unless the corresponding vectors vi, vj are orthogonal [vi, vj) = [vj , vi) = 0.When a multiple marking separates into distinct ones, the vectors v1, . . . , vN remain the same(under sufficiently small deformation). When the parameter (u, φ) crosses the codimension-one wall consisting of non-admissible configurations (u, φ), we transform the vectors v1, . . . , vNby a left/right mutation. The situation is illustrated in Figure 8; see also Figures 3 and 4. Inthis Figure 8, a vector vl equipped with the marking ui is transformed to:

∏

k:uj=uk

Rvkvl (right mutation) or∏

k:uj=uk

Lvkvl (left mutation).

cf. Corollary 2.7.6. The vectors that are not assigned the marking ui remain the same. Notethat the vectors corresponding to the same marking uj are orthogonal to each other and theabove product of mutations is well-defined.

•uj

•

•

ui

ui

right left

phase φ

Figure 8. Right and left mutation

Remark 4.3.1. Since we have [vi, vj) = [vj , vi) = 0 if hφ(ui) = hφ(uj) and i 6= j (see (4.2.1)),all admissible MRSs obtained from a non-admissible MRS by a small perturbation of themarking or the phase are connected by mutation.


Remark 4.3.2. Let CN (C) denote the configuration space (2.8.1) of distinct N points on C.Formally speaking the above procedure defines a local system over CN (C)×S1 such that thefiber at an admissible parameter (u, eiφ) is the set of all MRSs on the vector space V withmarking u and phase φ. The base space CN (C)× S1 has a wall and chamber structure givenby the wall W of non-admissible parameters:

W = (u, eiφ) ∈ CN (C)× S1 : hφ(ui) = hφ(uj) for some i 6= j.

The locus of self-intersection of W gives a codimension-two stratum:

W ′ = (u, eiφ) ∈ CN (C)× S1 : hφ(ui) = hφ(uj) = hφ(uk) for some i 6= j 6= k ⊂W.

Note that W is invariant under the S1-action on CN (C) × S1 discussed above. To definea local system, it suffices to define a parallel transportation along a path connecting twopoints from (CN (C) × S1) \W . For a given path, we can perturb the path so that it meetsonly the smooth part W \W ′ of the wall. Then we define the parallel transportation as thecomposition of right/left mutations corresponding to crossings of the wall. To check that theparallel transportation depends only on the homotopy class of the path, it suffices to recallthat the mutations satisfy the braid relation (2.8.4). Note that the S1 symmetry for MRSsreduces the local system to the quotient space (CN (C)× S1)/S1 ∼= CN (C).

4.4. Exceptional collections. In this section we sketch only the necessary definitions of(full) exceptional collections in triangulated categories and braid group action on them. Fordetails we refer the reader to [7] or to the survey [35].

Recall that an object E in a triangulated category T over C is called exceptional ifHom(E,E) = C and Hom(E,E[j]) = 0 for all j 6= 0. An ordered tuple of exceptionalobjects E1, . . . , EN is called an exceptional collection if Hom(Ei, Ej) = 0 for all i, j withi > j. If E1, . . . , EN is an exceptional collection and k1, . . . , kN is a set of integers thenE1[k1], . . . , EN [kN ] is also an exceptional collection, in this case we say that two collectionscoincide up to shift. An exceptional object E gives a pair of functors called left mutation LEand right mutation RE which fit into the distinguished triangles9:

LEF [−1] → Hom•(E,F ) ⊗ E → F → LEF,

REF → F → Hom•(F,E)∗ ⊗ E → REF [1].

The set of exceptional collections of N objects in T admits an action of the braid group BNin a way parallel to the BN -action (4.1.1) on SOBs, i.e. the generator σi replaces Ei, Ei+1with Ei+1, REi+1Ei and σ−1

i replaces Ei, Ei+1 with LEiEi+1, Ei.

The subcategory in T generated by Ei (minimal full triangulated subcategory that con-tains all Ei) stays invariant under mutations. An exceptional collection is called full if itgenerates the whole category T ; clearly braid group and shift action respects fullness. Whena triangulated category T has a full exceptional collection E1, . . . , EN , the GrothendieckK-group K0(T ) is the abelian group freely generated by the classes [Ei]. The Gram ma-trix Gij = χ(Ei, Ej) of the Euler pairing χ(E,F ) =

∑(−1)i dimHom(E,F [i]) is uni-

uppertriangular in the basis [Ei]. Thus any exceptional collection E1, . . . , EN producesan SOB (K0(T ) ⊗ C;χ(·, ·); [E1], . . . , [En]). The action of the braid group BN and shiftsEi → Ei[ki] described above descends to the action of BN and change of signs on the K-group described in §4.1.

9What we write as LEF , REF here are denoted respectively by LEF [1], REF [−1] in [7, 35].


4.5. Asymptotic basis and MRS of a Fano manifold. Here we resume the discussionin §2.8 and §2.9. Let F be a Fano manifold such that ⋆τ is convergent and semisimple forsome τ ∈ H (F ). Choose a phase φ ∈ R that is admissible for the spectrum of (c1(F )⋆τ ). ByProposition 2.9.1, we have a basis yτ1 (z), . . . , y

τN (z) of asymptotically exponential flat sections

for the quantum connection ∇|τ defined over the sector φ − π2 − ǫ < arg z < φ + π

2 + ǫ forsome ǫ > 0. They are canonically defined up to sign and ordering. The basis is marked bythe spectrum u = u1, . . . , uN of (c1(F )⋆τ ). We can naturally identify yτi (z) on the universalcover of the z-plane C×. By ∇-parallel transportation of yτi (z) to τ = 0, arg z = 0 and |z| ≫ 1,we can identify the basis yτ1 , . . . , y

τN of flat sections with a basis A1, . . . , AN of H (F ) via the

fundamental solution S(z)z−µzρ in Proposition 2.3.1, i.e.

yτi (z)∣∣∣parallel transportto τ = 0, arg z = 0

= Φ(Ai)(z) := (2π)−dimF

2 S(z)z−µzρAi

where Φ was given in (3.3.1). The basis A1, . . . , AN is defined up to sign and ordering andis semiorthonormal with respect to the pairing [·, ·) in (3.3.4)–(3.3.5) by Proposition 2.9.1.We call it the asymptotic basis of F at τ with respect to the phase φ.

The marked reflection system (MRS) of F is defined to be the tuple:

MRS(F, τ, φ) := (H (F ), [·, ·), A1 , . . . , AN,m : Ai 7→ ui, eiφ).

The marking m assigns to Ai the corresponding eigenvalue ui of (c1(F )⋆τ ). It is defined upto sign. Note that MRS(F, τ, φ) and MRS(F, τ, φ + 2π) are not necessarily the same: theydiffer by the monodromy around z = 0. By the discussion in §§2.7–2.9, under isomonodromicdeformation of the quantum connection, the corresponding MRSs change by mutation asdescribed in the previous section §4.3.

Remark 4.5.1. When ⋆0 is semisimple, we have the two MRSs MRS(F, 0,±ǫ) defined at thecanonical position τ = 0 for a sufficiently small ǫ > 0. When φ = 0 is admissible for thespectrum of (c1(F )⋆0), the two MRSs coincide.

4.6. Dubrovin’s (original) conjecture and Gamma conjecture II. We recall Dubrovin’sConjecture (see also Remark 4.6.5). Let F be a Fano manifold. Dubrovin [20, Conjecture4.2.2] conjectured:

(1) the quantum cohomology of F is semisimple if and only if Dbcoh(F ) admits a full

exceptional collection E1, . . . , EN ;

and if the quantum cohomology of F is semisimple, there exists a full exceptional collectionE1, . . . , EN of Db

coh(F ) such that:

(2) the Stokes matrix S = (Sij) is given by Sij = χ(Ei, Ej), i, j = 1, . . . , N ;(3) the central connection matrix has the form C = C ′C ′′ when the columns of C ′′ are the

components of Ch(Ej) ∈ H (F ) and C ′ : H (F ) → H (F ) is some operator satisfyingC ′(c1(F )a) = c1(F )C

′(a) for any a ∈ H (F ).

The central connection matrix C above is given by

C =1

(2π)dimF/2

| |A1 · · · AN| |

where the column vectors A1, . . . , AN ∈ H (F ) are an asymptotic basis in §4.5. GammaConjecture II makes precise the operator C ′ in Part (3) of Dubrovin’s conjecture.


Definition 4.6.1. Let F be a Fano manifold such that the quantum product ⋆τ is convergentand semisimple for some τ ∈ H (F ) and that Db

coh(F ) has a full exceptional collection. We saythat F satisfies Gamma Conjecture II if there exists a full exceptional collection E1, . . . , ENsuch that the asymptotic basis A1, . . . , AN of F at τ with respect to an admissible phase φ(see §4.5) is given by:

(4.6.2) Ai = ΓF Ch(Ei).

In other words, the operator C ′ in Part (3) of Dubrovin’s conjecture is given by C ′(α) =

(2π)− dimF/2ΓFα.

Remark 4.6.3. The validity of Gamma Conjecture II does not depend on the choice of τ andφ. Under deformation of τ and φ, the marked reflection system MRS(F, τ, φ) changes bymutations as described in §§2.7–2.9 and §4.3. On the other hand, the braid group acts on fullexceptional collections by mutations (§4.4). These mutations are compatible in the K-groupbecause the Euler pairing χ(·, ·) is identified with the pairing [·, ·) (3.3.5) by Lemma 3.3.6.

Remark 4.6.4. If F satisfies Gamma Conjecture II, the quantum differential equations ofF specify a distinguished mutation-equivalence class of (the images in the K-group of) fullexceptional collections of F given by an asymptotic basis. It is not known in general whetherany two given full exceptional collections are connected by mutations and shifts.

Remark 4.6.5. While preparing this paper, we are informed that Dubrovin [21] himself for-mulated the same conjecture as Gamma Conjecture II.

Proposition 4.6.6. Gamma Conjecture II implies Parts (2) and (3) of Dubrovin’s conjecture.

Proof. Recall from Proposition 2.9.1 that the Stokes matrix is given by the pairing [yτi , yτj )

of asymptotically exponential flat sections. Under Gamma Conjecture II, we have [yτi , yτj ) =

[Ai, Aj) = χ(Ei, Ej), where we used the fact that the pairing [·, ·) is identified with the Eulerpairing (Lemma 3.3.6).

A relationship between Gamma Conjectures I and II is explained as follows.

Proposition 4.6.7. Suppose that a Fano manifold F satisfies Conjecture O and that thequantum product ⋆0 is semisimple. Suppose also that F satisfies Gamma Conjecture II. If theexceptional object corresponding to the eigenvalue T and an admissible phase φ with |φ| < π/2is the structure sheaf O (or its shift), then F satisfies Gamma Conjecture I.

Proof. Under the assumptions, the flat section Φ(ΓF ) corresponding to O satisfies the asymp-

totics Φ(ΓF )(z) ∼ e−T/zψ0 for some ψ0 ∈ E(T ), as z → 0 in the sector φ − π2 − ǫ < arg z <

φ + π2 + ǫ (see Proposition 2.6.2). Therefore Φ(ΓF ) belongs to the space A (3.2.2). Gamma

Conjecture I holds (see Definition 3.3.3, Theorem 3.5.8).

4.7. Quantum cohomology central charges. Suppose that F satisfies Gamma ConjectureII and that ⋆0 is semisimple. Let ψ1, . . . , ψN be the idempotent basis for ⋆0 and u1, . . . , uN bethe corresponding eigenvalues of (c1(F )⋆0) i.e. c1(F )⋆0ψi = uiψi. Let φ ∈ R be an admissiblephase for u1, . . . , uN and let E1, . . . , EN be an exceptional collection corresponding (via(4.6.2)) to the asymptotic basis A1, . . . , AN at τ = 0 with respect to the phase φ. Then thequantum cohomology central charges of E1, . . . , EN (see (3.7.1), (3.7.2)) have the followingasymptotics:

Z(Ei) ∼√

(ψi, ψi)F (2πz)dimF/2e−ui/z


as z → 0 in the sector φ− π2 − ǫ < arg z < φ+ π

2 + ǫ for a sufficiently small ǫ > 0. This follows

from the definition (3.7.1) of Z(Ei) and the asymptotics Φ(Ai)(z) ∼ e−ui/zψi/√

(ψi, ψi)F .See also §3.7.

5. Remarks on toric case and quantum Lefschetz principle

The relevance of the “Gamma structure” in toric mirror symmetry has been observed bymany people [43, 52, 41, 42, 11, 44, 45, 34, 47]. In this section, we give several remarkson a possible proof of the Gamma conjectures for toric complete intersections. A remaining(subtle) question is related to the truth of Conjecture O. We also observe in general thatGamma Conjecture I is ‘compatible’ with the quantum Lefschetz principle [16].

5.1. Toric manifolds. Let X be an n-dimensional Fano toric manifold. A mirror of X isgiven by the Laurent polynomial [27, 40, 29]:

f(x) = xb1 + xb2 + · · ·+ xbm

where x = (x1, . . . , xn) ∈ (C×)n, b1, . . . , bm ∈ Zn are primitive generators of the 1-dimensional

cones of the fan of X and xbi = xbi11 · · · xbinn . By mirror symmetry [29, 45], the spectrum of(c1(X)⋆0) is the set of critical values of f . Consider the following condition:

Condition 5.1.1 (analogue of Conjecture O for toric manifolds). Let X be a Fano toricmanifold and f be its mirror Laurent polynomial. Let Tcon be the value of f at the conifoldpoint as defined in Remark 3.1.6. One has

• every critical value u of f satisfies |u| 6 Tcon;• the conifold point is the unique critical point of f contained in f−1(Tcon).

Note that Condition 5.1.1 implies Parts (1) and (3) of Conjecture O (Definition 3.1.1),which are enough to make sense of Gamma Conjecture I.

Theorem 5.1.2. Suppose that a Fano toric manifold X satisfies Condition 5.1.1. Then Xsatisfies Gamma Conjecture I.

Proof. It follows from the argument in [45, §4.3.1] that(φ,Φ(ΓX)(z)

)X

=1

(2πz)n/2

∫

(R>0)ne−f(x)/zϕ(x, z)

dx1 · · · dxnx1 · · · xn

for z > 0, where Φ is the map in (3.3.1), φ ∈ H (X) and ϕ(x, z) ∈ C[x±1 , . . . , x±n , z] is such

that the class [ϕ(x, z)e−f(x)/zdx1 · · · dxn/(x1 · · · xn)] corresponds to φ under the mirror iso-morphism in [45, Proposition 4.8] and n = dimX. When ϕ = 1, one has ϕ = 1 and this is the

quantum cohomology central charge (2πz)−n/2Z(OX); see §3.7. This integral representation

implies that the flat section Φ(ΓX)(z) has the smallest asymptotics, i.e. ‖Φ(ΓX)(z)eTcon/z‖grows at most polynomially as z → +0.

A toric manifold has a generically semisimple quantum cohomology. We note that thefollowing weaker version of Gamma Conjecture II can be shown for a toric manifold.

Theorem 5.1.3 ([44, 45]). Let X be a Fano toric manifold. We choose a semisimple pointτ ∈ H (X) and an admissible phase φ. There exists a Z-basis [E1], . . . , [EN ] of the K-groupsuch that the matrix (χ(Ei, Ej))16i,j6N is uni-uppertriangular and that the asymptotic basis

(§4.5) of X at τ with respect to φ is given by ΓX Ch(Ei), i = 1, . . . , N .


Proof. It suffices to prove the theorem at a single semisimple point τ . If τ is in the imageof the mirror map, we can take [Ei] to be the mirror images of the Lefschetz thimbles (withphase φ) under the isomorphism between the integral structures of the A-model and of theB-model, given in [45, Theorem 4.11].

Remark 5.1.4 ([44, 45]). Results similar to Theorems 5.1.2, 5.1.3 hold for a weak-Fano toricorbifold (under mild technical assumptions).

Remark 5.1.5. Kontsevich’s homological mirror symmetry suggests that the basis[E1], . . . , [EN ] in the K-group should be lifted to an exceptional collection in Db

coh(X) be-cause the corresponding Lefschetz thimbles form an exceptional collection in the Fukaya-Seidelcategory of the mirror.

5.2. Toric complete intersections. In this section Y denotes a Fano complete intersectionin a toric manifold X. Define

Hamb(Y ) := Im(i∗ : H (X) → H (Y ))

to be the ambient part of the cohomology group, where i : Y → X is the inclusion. It isshown in [55, Proposition 4], [47, Corollary 2.5] that the ambient part Hamb(Y ) is closedunder quantum multiplication ⋆τ when τ ∈ Hamb(Y ). Therefore it makes sense to ask if theambient part (Hamb(Y ), ⋆τ ) satisfies Conjecture O and Gamma Conjectures I, II. Note also

that ΓY belongs to the ambient part Hamb(Y ). A mirror theorem of Givental [29] gives theinformation of the ambient part of quantum cohomology of Y ; hence one can first try to proveGamma conjectures for the ambient part. We remark the following easy fact.

Proposition 5.2.1. The spectrum of (c1(Y )⋆0) on H (Y ) is the same as the spectrum of(c1(Y )⋆0) on Hamb(Y ) as a set (when we ignore the multiplicities). In particular the numberT for Y (see (3.1.2)) can be evaluated on the ambient part.

Proof. It follows from the fact that c1(Y ) belongs to Hamb(Y ) and the fact that H (Y ) is amodule over the algebra (Hamb(Y ), ⋆0).

We now assume that Y is a complete intersection in X obtained from the following nefpartition [5, 29]. Let D1, . . . ,Dm denote prime toric divisors of X. Each prime toric divisorDi corresponds to a primitive generator bi ∈ Z

n of a 1-dimensional cone of the fan of X. Weassume that there exists a partition 1, . . . ,m = I0 ⊔ I1 ⊔ · · · ⊔ Il such that

∑i∈Ik

Di is nef

for k = 1, . . . , l and∑

i∈I0Di is ample. Let Lk := O(

∑i∈Ik

Di) be the corresponding nef line

bundle. We assume that Y is the zero-locus of a transverse section of ⊕lk=1Lk over X. Then

c1(Y ) = c1(X) −∑lk=1 c1(Lk) =

∑i∈I0

[Di] is ample and Y is a Fano manifold. Define the

Laurent polynomial functions f0, f1, . . . , fl : (C×)n → C as follows:

fk(x) = −c0δ0,k +∑

i∈Ik

xbi

where c0 is the constant arising from Givental’s mirror map [29]:

c0 :=∑

d∈Eff(X):c1(Y )·d=1[Di]·d>0 (∀i)

∏lk=1(c1(Lk) · d)!∏mi=1([Di] · d)!

.


A mirror of Y [40, 29] is the affine variety Z := x ∈ (C×)n : f1(x) = · · · = fl(x) = 1equipped with a function f0 : Z → C and a holomorphic volume form

ωZ :=dx1x1

∧ · · · ∧ dxnxn

df1 ∧ · · · ∧ dfl.

Consider the following condition.

Condition 5.2.2 (analogue of Conjecture O for Y ). Let Y , Z and f0 : Z → C be as above.Let Zreal := (x1, . . . , xn) ∈ Z : xi ∈ R>0 (∀i) be the positive real locus of Z. We have:

• Z is a smooth complete intersection of dimension n− l;• f0|Zreal

: Zreal → R attains global minimum at a unique critical point xcon ∈ Zreal; wecall it the conifold point ;

• the conifold point xcon is a non-degenerate critical point of f0;• the number T (3.1.2) for Y coincides with Tcon := f0(xcon);• Tcon is an eigenvalue of (c1(Y )⋆0) with multiplicity one.

Example 5.2.3. Let Y be a degree d hypersurface in Pn with d < n+1. The mirror is given

by the function

f0(x) =1

x1x2 · · · xn+ x1 + · · ·+ xn−d − δd,nd!

on the affine variety Z = x ∈ (C×)n : f1(x) = xn−d+1 + xn−d+2 + · · · + xn = 1. FollowingPrzyjalkowski [56], introduce homogeneous co-ordinates [yn−d+1 : · · · : yn] of P

n−d−1 andconsider the change of variables:

xi =yi

yn−d+1 + · · · + ynfor n− d+ 1 6 i 6 n.

Then we have

f0(x) =(yn−d+1 + · · · + yn)

d

∏n−di=1 xi ·

∏nj=n−d+1 yj

+ x1 + · · ·+ xn−d − δn,dd!

Setting yn = 1 we obtain a Laurent polynomial expression (with positive coefficients) for f0in the variables x1, . . . , xn−d, yn−d+1, . . . , yn−1. Note that the change of variables maps Zreal

isomorphically onto the real locus xi > 0, yj > 0 : 1 6 i 6 n − d, n − d + 1 6 j 6 n − 1.This fact together with Remark 3.1.6 shows the existence of a conifold point in Condition5.2.2. (Similarly, one can obtain Laurent polynomial expressions for the mirrors of completeintersections in weighted projective spaces [56].)

One can also check that Tcon = (n + 1 − d)dd/(n+1−d) − δn,dd! coincides with T for Y andgives a simple eigenvalue of (c1(Y )⋆0)|Hamb(Y ) restricted to the ambient part (using Givental’s

mirror theorem [29]). When the Fano index n+1−d is greater than one, one has c1(Y )⋆0θ = 0for every primitive class θ ∈ Hn−1(Y ): this follows from the fact that (c1(Y )⋆0) preserves theprimitive part in Hn−1(Y ) and for degree reasons. Therefore Condition 5.2.2 holds for Y ifn+ 1− d > 1.

Remark 5.2.4 ([47]). Using an inequality of the form β1u1 + · · · + βmum > uβ11 · · · uβmm forui > 0, βi > 0,

∑i βi = 1, we find that f0|Zreal

is bounded (from below) by a convex function:

f0(x) + l = f0(x) + f1(x) + · · ·+ fl(x) > −c0 + ǫn∑

i=1

(x1/ki + x

−1/ki ) ∀x ∈ Zreal


where ǫ > 0 and k > 0 are constants. In particular f0|Zreal: Zreal → R is proper and attains

global minimum at some point. It also follows that the oscillatory integral (5.2.6) belowconverges.

Theorem 5.2.5. Let Y be a Fano complete intersection in a toric manifold constructed froma nef partition as above. Suppose that Y satisfies Condition 5.2.2. Then Gamma ConjectureI holds for Y .

Proof. In [47, Theorem 5.7], it is proved that the quantum cohomology central charge (see§3.7) of OY has an integral representation:

(5.2.6) Z(OY ) = (2πz)dim Y

2

(1,Φ(ΓY )(z)

)Y=

∫

Zreal

e−f0/zωZ .

The constant c0 in f0 comes from the value of the mirror map at τ = 0. In [47], the mirror(Z, f0) is parameterized by τ ∈ H2

amb(Y ) and the same integral representation is shown tohold for general τ ∈ H2

amb(Y,R). Since Hamb(Y ) is generated by H2amb(Y ), by differentiating

this integral representation in τ , we obtain an integral representation of the form:

(2πz)dim Y

2

(φ,Φ(ΓY )(z)

)Y=

∫

Zreal

ϕ(x, z)e−f0/zωZ

for every φ ∈ Hamb(Y ) (for some function ϕ(x, z)). Since the flat section Φ(ΓY ) takes values

in the ambient part Hamb(Y ), we obtain integral representations for all components of Φ(ΓY ).

These integral representations and Condition 5.2.2 show that ‖Φ(ΓY )(z)eT/z‖ grows at mostpolynomially as z → +0.

5.3. Compatibility with the quantum Lefschetz principle. We give an argument toshow that our Gamma conjecture is compatible with the quantum Lefschetz principle ingeneral. The discussion in this section is only sketchy and is not (analytically) rigorous.

Let X be a Fano manifold of index r > 2. We write −KX = rh for an ample class h. LetY ⊂ X be a degree-a Fano hypersurface in the linear system |ah| with 0 < a < r. Assumingthe truth of Gamma Conjecture I for X, we study Gamma Conjecture I for Y .

We write the J-function of X (Definition 3.5.3) as

JX(t) = erh log t∞∑

n=0

Jrntrn

with Jd ∈ H (X). By quantum Lefschetz theorem [16] of Coates-Givental, the J-function ofY is

JY (t) = e(r−a)h log t−c0t∞∑

n=0

(ah+ 1) · · · (ah+ an)(i∗Jrn)t(r−a)n,

where i : Y → X is the natural inclusion and c0 is the constant:

c0 =

a!∑

h·d=1〈[pt]ψr−2〉X0,1,d if r − a = 1;

0 if r − a > 1.

The quantum Lefschetz principle can be rephrased in terms of the Laplace transformation:

Lemma 5.3.1. We have

JY (t = ua/(r−a)) =e−c0t

Γ(1 + ah)u

∫ ∞

0i∗JX(t = qa/r)e−q/udq.


Suppose that X satisfies Gamma Conjecture I. By Proposition 3.5.2 and (3.5.4), we findthe following asymptotics of JX :

JX(t) ∼ Ct−dimX

2 eTX tΓX(1 +O(t−1)

)

as t → +∞ along the real line. Here C 6= 0 is a constant and TX is the number T in (3.1.2)for X. The method of stationary phase applied to Lemma 5.3.1 suggests that we have theasymptotics

JY (t) ∼ C ′t−dimY

2 e(T0−c0)tΓY(1 +O(t−1)

)as t→ +∞.

This would imply Gamma Conjecture I for Y . Here we used ΓY = i∗ΓX/Γ(1 + ah) and thenumber T0 > 0 is determined by the relation:

(T0r − a

)r−a= aa

(TXr

)r.

Problem 5.3.2. Check that T0 − c0 is the number T (3.1.2) for the hypersurface Y . Alsoprove Conjecture O for Y .

Remark 5.3.3 ([25]). It is easier to see the compatibility with quantum Lefschetz in terms ofthe Apery limit (§3.6). Suppose we have the Apery limit formula for X (cf. Theorem 3.6.1):

limn→∞

〈γ, Jrn〉〈[pt], Jrn〉

= 〈γ, ΓX〉

for γ ∈ H (X) with c1(X) ∩ γ = 0. Both sides do not change when we pass to a Fanohypersurface Y in | − (a/r)KX | (when γ is the push-forward of a class in H (Y )). Thereforethe same limit formula holds for Y .

6. Gamma conjectures for projective spaces

In this section we prove the Gamma Conjectures for projective spaces following the methodof Dubrovin [19, Example 4.4]. The Gamma conjectures for projective spaces also follow fromthe computations in [44, 45, 50]. A more general result for toric varieties was explained in theprevious section, but we include a detailed proof for the projective spaces for the convenienceof the reader.

Theorem 6.0.4. Gamma Conjectures I and II hold for the projective space P = PN−1.

An asymptotic basis is formed by mutations of the Gamma basis ΓPCh(O(i)) associated toBeilinson’s exceptional collection O(i) : 0 6 i 6 N − 1.6.1. Quantum connection of projective spaces. Let P denote the projective space PN−1.Let h = c1(O(1)) ∈ H2(P) denote the hyperplane class. The quantum product by h is givenby:

h ⋆0 hi =

hi+1 if 0 6 i 6 N − 2;

1 if i = N − 1.

It follows that the quantum multiplication (c1(P)⋆0) = N(h⋆0) is a semisimple operator withpairwise distinct eigenvalues. Also Conjecture O (Definition 3.1.1) holds for P with T = N .Let ∇ be the quantum connection (2.2.1) at τ = 0. The differential equation ∇f(z) = 0 for

a cohomology-valued function f(z) =∑N−1

i=0 fi(z)zi−N−1

2 hN−1−i reads:

fi(z) = N−i(z∂z)if0(z) 1 6 i 6 N − 1,

f0(z) = zNN−N (z∂z)Nf0(z).


Therefore a flat section f(z) for ∇ is determined by its top-degree component z−N−1

2 f0(z)which satisfies the scalar differential equation:

(6.1.1)(DN −NN (−t)N

)f0 = 0

where we set t := z−1 and write D := t∂t = −z∂z.Remark 6.1.2. Equation (6.1.1) is the quantum differential equation of P with t replaced with−t (see Remark 3.5.5).

6.2. Frobenius solutions and Mellin solutions. Solving the differential equation (6.1.1)by the Frobenius method, we obtain a series solution

(6.2.1) Π(t;h) := e−Nh log t∞∑

n=0

Γ(h− n)N

Γ(h)NtNn

taking values in the ring C[h]/(hN ). Expanding Π(t;h) in the nilpotent element h, we obtaina basis Πk(t) : 0 6 k 6 N − 1 of solutions as follows:

Π(t; ǫ) = Π0(t) + Π1(t)h+ · · ·+ΠN−1(t)hN−1.

Since hN = 0, we may identify h with the hyperplane class of P. The basis Πk(t) yieldsthe fundamental solution S(z)z−µzρ of ∇ near the regular singular point t = z−1 = 0 inProposition 2.3.1. More precisely, we have:

(6.2.2) S(z)z−µzρhN−1−k =

t−N−1

2 (− 1ND)N−1Πk(t)...

tN−3

2 (− 1ND)Πk(t)

tN−1

2 Πk(t)

where the right-hand side is presented in the basis 1, h, . . . , hN−1.Another solution to equation (6.1.1) is given by the Mellin transform of Γ(s)N :

(6.2.3) Ψ(t) :=1

2πi

∫ c+i∞

c−i∞Γ(s)N t−Nsds

with c > 0. The integral does not depend on c > 0. One can easily check that Ψ(t) satisfiesequation (6.1.1) by using the identity sΓ(s) = Γ(s+1). Comparing (6.2.1) and (6.2.3), we canthink of Ψ(t) as being obtained from Π(t;h) by multiplying Γ(h)N and replacing the discretesum over n with a “continuous” sum (i.e. integral) over s.

Note that we have a standard determination for Πk(t) and Ψ(t) along the positive real lineby requiring that log t ∈ R and t−Ns = e−Ns log t for t ∈ R>0. We see that Ψ(t) gives rise to aflat section with the smallest asymptotics along the positive real line (see §3.2).Proposition 6.2.4. The solution Ψ(t) (6.2.3) of (6.1.1) satisfies the asymptotics

Ψ(t) ∼ Ct−N−1

2 e−Nt(1 +O(t−1))

as t → ∞ in the sector −π2 < arg t < π

2 , where C = N−1/2(2π)(N−1)/2. Write the Gamma

class of P = PN−1 as ΓP = Γ(1 + h)N =

∑N−1k=0 ckh

N−1−k. Then we have the followingconnection formula:

Ψ(t) =

N−1∑

k=0

ckΠk(t) =

∫

P

ΓP ∪Π(t;h)

under the analytic continuation along the positive real line.


The connection formula in this proposition and equation (6.2.2) show that:

(6.2.5) S(z)z−µzρΓP =

t−N−1

2 (− 1ND)N−1Ψ(t)...

tN−3

2 (− 1ND)Ψ(t)

tN−1

2 Ψ(t)

.

This flat section has the smallest asymptotics ∼ e−Nt as t→ +∞ (see (3.2.2)). Therefore weconclude (recall Definition 3.3.3):

Corollary 6.2.6. Gamma Conjecture I holds for P.

Proof of Proposition 6.2.4. We follow Dubrovin [19, Example 4.4] and use the method ofstationary phase to obtain the asymptotics of Ψ(t). We write

Ψ(t) =1

2πi

∫ c+i∞

c−i∞eft(s)ds

with ft(s) = N(log Γ(s)− s log t). The integral is approximated by a contribution around thecritical point of ft(s). Using Stirling’s formula

log Γ(s) ∼(s− 1

2

)log s− s+

1

2log(2π) +O(s−1)

we find a critical point s0 of ft(s) such that s0 ∼ t+ 12 +O(t−1) as t→ ∞. Then we have

ft(s0) ∼ −Nt+ N

2log(2π/t) +O(t−1),

f ′′t (s0) ∼N

t+O(t−2).

From these we obtain the asymptotics:

Ψ(t) ∼ 1√2πf ′′t (s0)

eft(s0) =1√N

(2π

t

)N−12

e−Nt.

To show the connection formula, we close the integration contour in (6.2.3) to the left andexpress Ψ(t) as the sum of residues at s = 0,−1,−2,−3, . . . . We have for n ∈ Z>0

Ress=−n Γ(s)N t−Nsds = Resh=0 Γ(h− n)N tNn−Nhdh

=

∫

P

Γ(1 + h)NΓ(h− n)N

Γ(h)NtNn−Nh.

In the second line we used the fact that Γ(1+h)/Γ(h) = h and∫Pg(h) = Resh=0(g(h)/h

N )dhfor any g(h) ∈ C[[h]]. Therefore we have

Ψ(t) =∞∑

n=0

Ress=−n Γ(s)N t−Nsds =

∫

P

ΓP ∪Π(t;h)

as required.


6.3. Monodromy transformation and mutation. We use monodromy transformationand mutation to deduce Gamma Conjecture II for P from the truth of Gamma Conjecture Ifor P.

The differential equation (6.1.1) is invariant under the rotation t→ e2πi/N t. Therefore the

functions Ψ(j)(t) = Ψ(e−2πij/N t), j ∈ Z are also solutions to (6.1.1). By changing co-ordinates

t→ e−2πij/N t in Proposition 6.2.4, we find that Ψ(j) satisfies the asymptotic condition

(6.3.1) Ψ(j)(t) ∼ Cjt−N−1

2 e−Nζ−j

in the sector −π2 + 2πj

N < arg t < π2 + 2πj

N where ζ = exp(2πi/N) and Cj = Ce(N−1)πij/N .

Using Π(e−2πij/N t;h) = Ch(O(i)) ∪Π(t;h), we also find the connection formula

Ψ(j)(t) =

∫

P

ΓPCh(O(j)) ∪Π(t;h).

Let yj(z) be the flat section corresponding to Ψ(j) (cf. (6.2.5)):

yj(z) := (2π)−N−1

2

t−N−1

2 (− 1ND)N−1Ψ(j)(t)

...

tN−3

2 (− 1ND)Ψ(j)(t)

tN−1

2 Ψ(j)(t)

.

The above connection formula and (6.2.2) show that

(6.3.2) yj(z) = (2π)−N−1

2 S(z)z−µzρ(ΓPCh(O(i))

)= Φ

(ΓPCh(O(i))

)

where we recall that Φ was defined in (3.3.1). Since the sectors where the asymptotics (6.3.1) of

Ψ(j) hold depend on j, the flat sections yj(z) do not quite form the asymptotically exponentialfundamental solution in the sense of Proposition 2.6.2. We will see however that they give itafter a sequence of mutations.

Recall from the discussion in §3.2 that the flat section y0(z) with the smallest asymptotics(along R>0) can be written as the Laplace integral:

y0(z) =1

z

∫ ∞

Tϕ(λ)e−λ/zdλ

for some ∇-flat section ϕ(λ) holomorphic near λ = T (= N). Recall that ∇ is the secondstructure connection (§2.5) with ν = 0. This integral representation is valid when Re(z) > 0.We have

yj(z) = e2πijµ/Ny0(e2πij/Nz) =

ζ−j

z

∫ ∞

Te2πijµ/Nϕ(λ)e−ζ

−jλ/zdλ

=

∫

Tζ−j+R>0ζ−j

e2πijµ/Nϕ(ζjλ)e−λ/zdλ.

Set ϕj(λ) = e2πijµ/Nϕ(ζjλ). Then ϕj(λ) is holomorphic near λ = ζ−jT and is flat for ∇. Thelatter fact follows easily from (3.1.4). Thus we obtain an integral representation of yj(z):

yj(z) =1

z

∫

Tζ−j+R>0ζ−j

ϕj(λ)e−λ/zdλ

which is valid when −π2 − 2πj

N < arg z < π2 − 2πj

N . By bending the radial integration path, wecan analytically continue yj(z) for arbitrary arg z. When arg z is close to zero, for example,we bend the paths as shown in Figure 9. Here we choose a range [j0, j0 +N − 1] of length N


and consider a system of integration paths for yj(z), j ∈ [j0, j0 + N − 1] when arg z is closeto zero.

Figure 9. Bent paths starting from Tζ−j, j = −3, · · · , 4 (N = 8).

Figure 10. Straightened paths in the admissible direction eiφ.

We can construct an asymptotically exponential fundamental solution in the sense of Propo-sition 2.6.2 by straightening the paths (see Figure 10). Note that limz→+0 e

T/zy0(z) = ϕ(T )is a T -eigenvector of (c1(P)⋆0) of unit length by Proposition 3.3.8. Therefore ϕj(ζ

−jT ) =

e2πijµ/Nϕ(T ) is also of unit length. It is a ζ−jT -eigenvector of (c1(P)⋆0) by (3.1.4). There-fore ϕj(ζ

−jT ), j ∈ [j0, j0 + N − 1] form a normalized idempotent basis for ⋆0. Let φ be anadmissible phase for the spectrum of (c1(P)⋆0) which is close to zero. By the discussion in§2.6, the ∇-flat sections

xj(z) =1

z

∫

Tζ−j+R>0eiφϕj(λ)e

−λ/zdλ ∼ e−ζ−jT/zϕj(ζ

−jT )

with j ∈ [j0, j0 +N − 1] give the asymptotically exponential fundamental solution associatedto eiφ. By the argument in §2.7, the two bases yj and xj are related by a sequence ofmutations. Because yj corresponds to the Beilinson collection O(j) (see (6.3.2)), xjcorresponds to a mutation of O(j) (see §§2.7–2.9 and Remark 4.6.3). This shows that the

asymptotic basis at τ = 0 with respect to phase φ is given by a mutation of ΓPCh(O(j)),j0 6 j 6 j0 +N − 1.

The proof of Theorem 6.0.4 is now complete.


Figure 11. A system of paths after isomonodromic deformation.

Remark 6.3.3. In §2.7–§2.9 and §4.3, we considered mutations for straight paths. Althoughour paths in Figure 9 are not straight, we can make them straight after isomonodromicdeformation (see Figure 11). We can therefore apply our mutation argument to such straightpaths. When the eigenvalues of E⋆τ align as in Figure 11, the corresponding flat sections yjform an asymptotically exponential fundamental solution and the Beilinson collection gives

the asymptotic basis ΓPCh(O(j)) at such a point τ .

7. Gamma conjectures for Grassmannians

We derive Gamma Conjectures for Grassmannians of type A from the truth of GammaConjectures for projective spaces. Let G = G(r,N) denote the Grassmannian of r-dimensionalsubspaces of CN and let P = P

N−1 = G(1, N) denote the projective space of dimension N−1.

7.1. Statement. Let Sν denote the Schur functor for a partition ν = (ν1 > ν2 > · · · > νr).Denote by V the tautological bundle10 over G = G(r,N). Kapranov [48, 49] showed thatthe vector bundles SνV ∗ form a full exceptional collection of Db

coh(G), where ν ranges overall partitions such that the corresponding Young diagrams are contained in an r × (N − r)rectangle, i.e. νi 6 N − r for all i.

Theorem 7.1.1. Gamma Conjectures I and II hold for Grassmannians G. An asymptotic

basis of G is formed by mutations of the Gamma basis ΓGCh(SνV ∗) associated to Kapranov’sexceptional collection SνV ∗ : ν is contained in an r × (N − r) rectangle .

Corollary 7.1.2 (Ueda [58]). Dubrovin’s Conjecture holds for G.

The proof of Gamma Conjecture II for G will be completed in §7.6; the proof of GammaConjecture I for G will be completed in §7.8.

7.2. Quantum Pieri and quantum Satake. For background material on classical coho-mology rings of Grassmannians, we refer the reader to [23]. Let x1, . . . , xr be the Chern rootsof the dual V ∗ of the tautological bundle over G = G(r,N). Every cohomology class of G canbe written as a symmetric polynomial in x1, . . . , xr. We have:

H (G) ∼= C[x1, . . . , xr]Sr/〈hN−r+1, . . . , hN 〉

10The dual V ∗ can be identified with the universal quotient bundle on the dual Grassmannian G(N − r,N).


where hi = hi(x1, . . . , xr) is the ith complete symmetric polynomial of x1, . . . , xr. An additivebasis of H (G) is given by the Schur polynomials11

(7.2.1) σλ = σλ(x1, . . . , xr) =det(x

λj+r−ji )16i,j6r

det(xr−ji )16i,j6r

with partition λ in an r×(N−r) rectangle. The class σλ is Poincare dual to the Schubert cycleΩλ and degσλ = 2|λ| = 2

∑ri=1 λi. For 1 6 k 6 N − r, we write σk = hk(x1, . . . , xr) = ck(Q)

for the Schubert class corresponding to the partition (k > 0 > · · · > 0). Here Q is theuniversal quotient bundle. They are called the special Schubert classes. The first Chern classis given by c1(G) = Nσ1. The classical Pieri formula describes the multiplication by thespecial Schubert classes:

(7.2.2) σk ∪ σλ =∑

σν

where the sum ranges over all partitions ν = (ν1 > · · · > νr) in an r× (N − r) rectangle suchthat |ν| = |λ|+ k and ν1 > λ1 > ν2 > λ2 > · · · > νr > λr. Bertram’s quantum Pieri formulagives the quantum multiplication by special Schubert classes:

Proposition 7.2.3 (Quantum Pieri [8], see also [10, 24]). We have

σk ⋆σ1 log q σλ =∑

σν + q∑

σµ

where the first sum is the same as the classical Pieri formula (7.2.2) and the second sumranges over all µ in an r × (N − r) rectangles such that |µ| = |λ|+ k −N and λ1 − 1 > µ1 >λ2 − 1 > µ2 > · · · > λr − 1 > µr > 0.

From the quantum Pieri formula, one can deduce that the quantum connection of G is thewedge power of the quantum connection of P. This is an instance of the quantum Satake prin-ciple of Golyshev-Manivel [33]. The geometric Satake correspondence of Ginzburg [26] impliesthat the intersection cohomology of an affine Schubert variety Xλ in the affine Grassmannianof a complex Lie group G becomes an irreducible representation of the Langlands dual Liegroup GL. Our target spaces P, G arise as certain minuscule Schubert varieties in the affineGrassmannian of GL(N,C). Therefore the cohomology groups of P and G are representationsof GL(N,C): H (P) is the standard representation of GL(H (P)) ∼= GL(N,C) and H (G) isthe r-th wedge power of the standard representation. In fact, by the ‘take the span’ rationalmap P× · · · × P (r factors) 99K G, we obtain the Satake identification

Sat : ∧r H (P)∼=−→ H (G), σλ1+r−1 ∧ σλ2+r−2 ∧ · · · ∧ σλr 7−→ σλ(7.2.4)

where σλi+r−i = (σ1)λi+r−i is the special Schubert class of P = G(1, N). The Satake identifi-

cation endows H (G) with the structure of a gl(H (P))-module.

Proposition 7.2.5 (Quantum Satake [33]). The quantum product (c1(G)⋆0) on H (G) coin-cides, upon the Satake identification Sat, with the action of (c1(P)⋆πi(r−1)σ1) ∈ gl(H (P)) onthe r-th wedge representation ∧rH (P).

Proof. Recall that c1(P) = Nσ1 and c1(G) = Nσ1. Thus it suffices to examine the quantumproduct by σ1. The action of (σ1⋆πi(r−1)σ1) on the basis element σλ1+r−1∧σλ2+r−2∧ · · ·∧σλr

11The Satake identification H (G) ∼= ∧rH (P) is also indicated by this expression.


of the r-th wedge ∧rH (P) is given by

r∑

i=1

σλ1+r−1 ∧ · · · ∧ (σ1 ⋆πi(r−1)σ1 σλi+r−i) ∧ · · · ∧ σλr

=

(r∑

i=1

σλ1+r−1 ∧ · · · ∧ith

σλi+r−i+1 ∧ · · · ∧ σλr

)+(−1)r−1δN−1,λ1+r−1σ0∧σλ2+r−2∧· · ·∧σλr

Under the Satake identification, the first term corresponds to the classical Pieri formula andthe second term corresponds to the quantum correction. The sign (−1)r−1 cancels the signcoming from the permutation of σ0 and σλ2+r−2 ∧ · · · ∧ σλr .

Remark 7.2.6 ([33]). More generally, the action of the power sum (xk1 + · · ·+ xkr )⋆0 on H (G)coincides, upon the Satake identification with the action of (σk⋆(r−1)πiσ1) ∈ gl(H (P)) on ther-th wedge representation ∧rH (P), cf. Theorem 7.3.1.

We have a similar result for the grading operator. The following lemma together withProposition 7.2.5 implies that the quantum connection of G is the wedge product of thequantum connection of P.

Lemma 7.2.7. Let µP and µG be the grading operators of P and G respectively (see §2.2).The action of µG coincides, upon the Satake identification with the action of µP ∈ gl(H (P))on the r-th wedge representation ∧rH (P).

Proof. This is immediate from the definition: we use dimG+ r(r − 1) = r dimP.

The Satake identification also respects the Poincare pairing up to sign. The Poincarepairing on H (P) induces the pairing on the r-th wedge ∧rH (P):

(α1 ∧ · · · ∧ αr, β1 ∧ · · · ∧ βr)∧P := det((αi, βj)P)16i,j6r.

Lemma 7.2.8. We have (α, β)∧P = (−1)r(r−1)/2(Sat(α),Sat(β))G for α, β ∈ ∧rH (P).

Proof. For two partitions λ, µ in an r×(N−r) rectangle, we have (σλ, σµ)G = 1 if λi+µr−i+1 =N − r for all i, and (σλ, σµ)G = 0 otherwise. On the other hand, if λi + µr−i+1 = N − r forall i, we have

(σλ1+r−1 ∧ · · · ∧ σλr , σµ1+r−1 ∧ · · · ∧ σµr)∧P= (σλ1+r−1 ∧ · · · ∧ σλr , σN−1−λr ∧ · · · ∧ σN−r−λ1)∧P = (−1)r(r−1)/2.

Otherwise, the pairing can be easily seen to be zero.

Remark 7.2.9. Proposition 7.2.5 implies the well-known formula that the spectrum of(c1(G)⋆0) consists of sums of r distinct eigenvalues of (c1(P)⋆(r−1)πiσ1), i.e.

Spec(c1(G)⋆0) =Ne(r−1)πi/N (ζ i1N + · · · + ζ irN ) : 0 6 i1 < i2 < · · · < ir 6 N − 1

where ζN = e2πi/N . In particular G satisfies Conjecture O (Definition 3.1.1) with T =N sin(πr/N)/ sin(π/N). The quantum product (c1(G)⋆0) has pairwise distinct eigenvalues ifk! and N are coprime, where k = min(r,N − r).


7.3. The wedge product of the big quantum connection of P. The quantum Satakeprinciple implies that the quantum connection for G (at τ = 0) is the r-th wedge product ofthe quantum connection for P (at τ = (r − 1)πi). By using the abelian/non-abelian corre-spondence of Bertram–Ciocan-Fontanine–Kim–Sabbah [9, 15, 51], we observe that this is alsotrue for the isomonodromic deformation corresponding to the big quantum cohomology of Pand G. When the eigenvalues of (c1(G)⋆0) are pairwise distinct, this can be also deduced fromthe quantum Pieri (Proposition 7.2.3) and Dubrovin’s reconstruction theorem [18]; howeverit is not always true that (c1(G)⋆0) has pairwise distinct eigenvalues (see Remark 7.2.9).

The quantum product ⋆(r−1)πiσ1 of P = PN−1 is semisimple and (c1(P)⋆(r−1)πiσ1) has

pairwise distinct eigenvalues u = Neπi(r−1)/N e2πik/N : 0 6 k 6 N − 1. As explainedin §2.8, the quantum connection of P has an isomonodromic deformation over the universalcover CN (C)

∼ of the configuration space (2.8.1) of distinct N points in C. Let ∇P be theconnection on the trivial bundle H (P) × (CN (C)

∼ × P1) → (CN (C)

∼ × P1) which gives the

isomonodromic deformation (see Proposition 2.8.2). Here the germ (CN (C)∼,u) at u is

identified with the germ (H (P), (r− 1)πiσ1) by the eigenvalues of (E⋆τ ) and ∇P is identifiedwith the big quantum connection (2.2.3) of P near u. Consider the r-th wedge product ofthis bundle:

(∧rH (P))× (CN (C)∼ × P

1) → (CN (C)∼ × P

1)

equipped with the meromorphic flat connection ∇∧P:

∇∧P(ξ1 ∧ ξ2 ∧ · · · ∧ ξr) :=r∑

i=1

ξ1 ∧ · · · ∧ (∇Pξi) ∧ · · · ∧ ξr.

Theorem 7.3.1. There exists an embedding f : (H (P), (r − 1)πiσ) ∼= (CN (C)∼,u) →

(H (G), 0) between germs of complex manifolds such that the big quantum connection ∇G

of G pulls back (via f) to the r-th wedge ∇∧P of the big quantum connection of P under theSatake identification Sat (7.2.4). More precisely, the bundle map

∧rH (P)× (CN (C)∼ × P

1) −→ H (G)× (CN (C)∼ × P

1)

(α, (u, z)) 7−→ (ir(r−1)/2 Sat(α), (u, z))

intertwines the connections ∇∧P, f∗∇G and the pairings (·, ·)∧P, (·, ·)G.Proof. The proposition follows by unpacking the definition of the “alternate product of Frobe-nius manifolds” in [51]. Let us first review the construction of the alternate product for thebig quantum cohomology Frobenius manifold of P. A pre-Saito structure [51, §1.1] with baseM is a meromorphic flat connection ∇ on a trivial vector bundle E0 × (M × P

1) → M × P1

of the form:

∇ = d+1

zC + (−1

zU + V )

dz

zfor some C ∈ End(E0) ⊗ Ω1

M and U, V ∈ End(E0) ⊗ OM (it follows from the flatness of ∇that V is constant). Here E0 is a finite dimensional complex vector space and the base spaceM is a complex manifold. The big quantum connection (2.2.3) is an example of a pre-Saitostructure. Consider the quantum connection of P restricted to H2(P) × P

1. This gives apre-Saito structure. The external tensor product of this pre-Saito structure yields a pre-Saitostructure ∇×r on the bundle:

⊗rH (P)× ((H2(P))r × P1) → (H2(P))r × P

1.

By Hertling-Manin’s reconstruction theorem [38], [51, Corollary 1.7] and [51, Lemma 2.1], thepre-Saito structure ∇×r admits a universal deformation over the base ⊗rH (P). Here the base


(H2(P))r is embedded into ⊗rH (P) by the map

χ1 : (H2(P))r → ⊗rH (P), χ1(τ1, . . . , τr) =

r∑

i=1

1⊗ · · · ⊗ 1⊗ ithτi ⊗ 1⊗ · · · ⊗ 1.

The universal deformation here is a pre-Saito structure ∇⊗r on the bundle

⊗rH (P)× (⊗rH (P)× P1) → ⊗rH (P)× P

1

which is pulled back by χ1 to the pre-Saito structure ∇×r. More precisely, the base of theuniversal deformation is a germ along the diagonal H2(P) ⊂ (H2(P))r ⊂ ⊗rH (P). Geomet-rically this is the big quantum connection of P× · · · × P (r factors). The pre-Saito structure∇⊗r is equivariant with respect to the natural W := Sr-action. We restrict the pre-Saitostructure ∇⊗r to the W -invariant base SymrH (P) ⊂ ⊗rH (P); fibers of the restriction arerepresentations of W . By taking the anti-symmetric part, we obtain a pre-Saito structure∇Sym on the bundle

∧rH (P)× (SymrH (P)× P1) → SymrH (P)× P

1.

Again the base is a germ along the diagonal H2(P) → Symr(H (P)). We further restrictthis pre-Saito structure to the subspace ElemrH (P) ⊂ SymrH (P) spanned by “elementarysymmetric” vectors

ElemrH (P) =r⊕

k=1

C

∑

i1,...,ir∈0,1∑ra=1 ia=k

σi1 ⊗ · · · ⊗ σir ⊂ SymrH (P)

which contains the diagonal H2(P). By Hertling-Manin’s reconstruction theorem and [51,Lemma 2.9], we obtain a universal deformation of the pre-Saito structure ∇Sym|ElemrH (P)

over the base ∧rH (P). More precisely, we have an embedding12

χ2 : ElemrH (P) → ∧rH (P) such that χ2(χ1(τ, . . . , τ)) = τσr−1 ∧ σr−2 ∧ · · · ∧ σ0and a pre-Saito structure ∇∧r on the bundle

(7.3.2) ∧r H (P)× (∧rH (P)× P1) → ∧rH (P)× P

1

which is a universal deformation of ∇Sym|Elemr H (P) via the embedding χ2. The pre-Saitostructure ∇∧r together with a primitive section cσr−1 ∧ · · · ∧ σ0 (for some c ∈ C) and themetric (·, ·)∧P endows the base space ∧rH (P) with a Frobenius manifold structure.

The main result of Ciocan-Fontanine–Kim–Sabbah [51, Theorem 2.13], [15, Theorem 4.1.1]implies that the germ of the pre-Saito structure∇∧r at χ2(χ1(t

, . . . , t)) with t = (r−1)πiσ1is isomorphic to the pre-Saito structure on the bundle

(7.3.3) H (G)× ((H (G), 0) × P1) → (H (G), 0) × P

1

defined by the big quantum connection of G. By the construction in [15, §3], the isomorphismbetween the above two pre-Saito bundles (7.3.2), (7.3.3) is induced by the Satake identificationSat between the fibers (up to a scalar multiple).

12The embedding χ2 is a highly non-trivial map except along the diagonal H2(P).


By the universal property of the pre-Saito structures ∇⊗r and ∇∧r, we obtain maps χ1, χ2

extending χ1 and χ2 which fit into the following commutative diagram.

(H2(P))r χ1 //

_

⊗rH (P) SymrH (P)? _oo

χ2 ''PPPP

PPPP

PPPP

ElemrH (P)? _oo _

χ2

(H (P))r

χ1

88rrrrrrrrrr∧rH (P)

H (P) ?

∆

OO

+

ψ

88qqqqqqqqqqqqqqqqqqqqqqqqqqqq f

22

The maps χ1, χ2 are such that

• the pre-Saito structure over (H (P))r defined by the r-fold external tensor product ofthe big quantum connection of P is isomorphic to the pull-back of ∇⊗r by χ1; theisomorphism here restricts to the given one along (H2(P))r (i.e. the fibers of the twopre-Saito bundles are identified by the identity of ⊗rH (P));

• the pre-Saito structure ∇Sym over SymrH (P) is isomorphic to the pull-back of ∇∧r

by χ2; the isomorphism here restricts to the given one along ElemrH (P) (i.e. thefibers of the two pre-Saito bundles are identified by the identity of ∧rH (P)).

The (non-injective) map χ1 can be given explicitly:

χ1(τ1, . . . , τr) =

r∑

i=1

1⊗ · · · ⊗ ithτi ⊗ · · · ⊗ 1.

Pre-composed with the diagonal map ∆: H (P) → (H (P))r, χ1 induces a map ψ : H (P) →SymrH (P). We have that the pre-Saito structure on H (P) defined as the r-fold tensorproduct of the big quantum connection of P is isomorphic to the pull-back of ∇⊗r|SymrH (P)

by ψ, where the identification of fibers is the identity of ⊗rH (P). Therefore the pull-backof the pre-Saito structure ∇Sym (defined as the anti-symmetric part of ∇⊗r|SymrH (P)) by ψ

is naturally identified with the r-th wedge power ∇∧P of the big quantum connection of P.Hence the composition f = χ2 ψ pulls back the pre-Saito structure ∇∧r to the pre-Saitostructure ∇∧P. Since the pre-Saito structure ∇∧r is identified with the quantum connection∇G of G near f(t), we have f∗∇G ∼= ∇∧r under the Satake identification. The scalar factor

ir(r−1)/2 is put to make the pairings match (see Lemma 7.2.8).Finally we show that the map f is an embedding of germs. It suffices to show that the

differential of f at the base point t = (r − 1)πiσ1 is injective. Since we already know that∇G is pulled back to ∇∧P, it suffices to check that z∇∧P

σk(σr−1 ∧ · · · ∧ σ0)|t , k = 0, . . . , N − 1

are linearly independent, as they correspond to z∇G

df(σk)1 = df(σk). This follows from a

straightforward computation.

Corollary 7.3.4. Let ψ1, . . . , ψN be the idempotent basis of the quantum cohomology of P atu ∈ CN (C)

∼ near u and write ∆i = (ψi, ψi)−1P

. Let f be the embedding in Theorem 7.3.1.Then we have:

(1) the quantum product ⋆τ of G is semisimple near τ = 0;(2) the idempotent basis of G at τ = f(u) is given by

(∏ra=1 ∆ia) det ((ψia , σr−b)P)16a,b6r Sat(ψi1 ∧ · · · ∧ ψir)

with 1 6 i1 < i2 < · · · < ir 6 N ;


(3) the eigenvalues of the Euler multiplication (EG⋆τ ) of G at τ = f(u) are given byui1 + · · · + uir with 1 6 i1 < i2 < · · · < ir 6 N .

Proof. The quantum product ⋆τ for G is semisimple if and only if there exists a class v ∈ H (G)such that the endomorphism (v⋆τ ) is semisimple with pairwise distinct eigenvalues. In thiscase, each eigenspace of (v⋆τ ) contains a unique idempotent basis vector. On the other hand,since the quantum product of P is semisimple, we can find w ∈ H (P) such that the action of(w⋆u) on ∧rH (P) is semisimple with pairwise distinct eigenvalues. Theorem 7.3.1 implies thatw⋆u is conjugate to dfu(w)⋆f(u) under Sat. This proves Part (1). Moreover the eigenspaceC Sat(ψi1 ∧ · · · ∧ ψir), i1 < i2 < · · · < ir of dfu(w)⋆f(u) contains a unique idempotent basisvector ψi1,...,ir ∈ H (G). Set ψi1,...,ir = cSat(ψi1 ∧ · · · ∧ ψir). Then we have

(ψi1,...,ir , ψi1,...,ir)G = (ψi1,...,ir ⋆f(u) ψi1,...,ir , 1)G = (ψi1,...,ir , 1)G.

This implies by Lemma 7.2.8 that:

c2(ψi1 ∧ · · · ∧ ψir , ψi1 ∧ · · · ∧ ψir)∧P = c(ψi1 ∧ · · · ∧ ψir , σr−1 ∧ · · · ∧ σ0)∧P.

Part (2) follows from this. Part (3) follows from the fact that the Euler multiplication (EP⋆τ )on ∧rH (P) is conjugate to (EG⋆f(u)) on H (G) by Sat.

7.4. The wedge product of MRS. Let M = (V, [·, ·), v1, . . . , vN,m : vi 7→ ui, eiφ) be an

MRS (see §4.2). The r-th wedge product ∧rM is defined by the data:

• the vector space ∧rV ;• the pairing [α1 ∧ · · · ∧ αr, β1 ∧ · · · ∧ βr) := det([αi, βj))16i,j6r;• the basis vi1 ∧ · · · ∧ vir : 1 6 i1 < · · · < ir 6 N;• the marking m : vi1 ∧ · · · ∧ vir 7→ ui1 + · · ·+ uir ;• the same phase eiφ.

Note that the basis vi1 ∧ · · · ∧ vir is determined up to sign because v1, . . . , vN is anunordered basis. In other words, ∧rM is defined up to sign. The following lemma shows thatthe above data is indeed an MRS. Recall that hφ(u) = Im(e−iφu).

Lemma 7.4.1. Let 1 6 i1 < · · · < ir 6 N , 1 6 j1 < · · · < jr 6 N be increasing sequences ofintegers. If hφ(ui1 + · · ·+ uir) = hφ(uj1 + · · ·+ ujr), we have

[vi1 ∧ · · · ∧ vir , vj1 ∧ · · · ∧ vjr) =1 if ia = ja for all a = 1, . . . , r;

0 otherwise.

Proof. Suppose that∏ra=1[via , vjσ(a)

) 6= 0 for some permutation σ ∈ Sr. Then for each a,

we have either ia = jσ(a) or hφ(uia) > hφ(ujσ(a)) by (4.2.1). The assumption implies that

ia = jσ(a) for all a; this happens only when σ = id. The lemma follows.

The wedge product of an admissible MRS is not necessarily admissible.

Remark 7.4.2. We can define the tensor product of two MRSs similarly.

Remark 7.4.3. We can show that, when two MRSs M1 and M2 are related by mutations (see§4.3), ∧rM1 and ∧rM2 are also related by mutations. We omit a proof of this fact since wedo not use it in this paper; the details are left to the reader.


7.5. MRS of Grassmannian. Using the result from §7.3, we show that the MRS of G isisomorphic to the wedge product of the MRS of P. By the results in §7.3, the flat connec-tions ∇P and ∇∧P on the base CN (C)

∼ give isomonodromic deformations of the quantumconnections of P and G respectively. Therefore we can define the MRSs of P or G at a pointu ∈ CN (C)

∼ with respect to an admissible phase φ following §2.8 (and §4.5).Proposition 7.5.1. Let (H (P), [·, ·), A1 , . . . , AN, Ai 7→ ui, e

iφ) be the MRS of P at u ∈CN (C)

∼ with respect to phase φ. Suppose that the phase φ is admissible for the r-th wedgeui1 + · · ·+ uir : i1 < i2 < · · · < ir of the spectrum of (E⋆u). Then the MRS of G at u withrespect to φ is given by the asymptotic basis

1

(2πi)r(r−1)/2e−(r−1)πiσ1 Sat(Ai1 ∧ · · · ∧Air)

marked by ui1+· · ·+uir with 1 6 i1 < i2 < · · · < ir 6 N , where Sat is the Satake identification(7.2.4). In other words, one has MRS(G,u, φ) ∼= ∧rMRS(P,u, φ) (see §7.4).Proof. Let y1(z), . . . , yN (z) be the basis of asymptotically exponential flat sections for ∇P|ucorresponding to A1, . . . , AN . They are characterized by the asymptotic condition yi(z) ∼e−ui/z(Ψi + O(z)) in the sector | arg z − φ| < π

2 + ǫ for some ǫ > 0, where Ψ1, . . . ,ΨN

are normalized idempotent basis for P. For 1 6 i1 < i2 < · · · < ir 6 N , yi1,...,ir(z) :=yi1(z) ∧ · · · ∧ yir(z) gives a flat section for ∇∧P|u and satisfies the asymptotic condition

yi1,...,ir(z) ∼ e−(ui1+···+uir )/z(Ψi1 ∧ · · · ∧Ψir +O(z))

in the same sector. Note that ir(r−1)/2 Sat(Ψi1 ∧ · · · ∧ Ψir) give (analytic continuation of)

the normalized idempotent basis for G by Theorem 7.3.1 and Corollary 7.3.4. Therefore theasymptotically exponential flat sections i

r(r−1)/2 Sat(yi1,...,ir(z)) give rise to the asymptoticbasis of G for u and φ.

Let SP(τ, z)z−µP

zρP

with τ ∈ H (P) denote the fundamental solution for the big quantum

connection of P as in Remark 2.3.2. The group elements SP(τ, z), z−µP

, zρP ∈ GL(H (P))

naturally act on the r-th wedge representation ∧rH (P); we denote these actions by the samesymbols. Define the End(H (G))-valued function SG(z) by

SG(z) Sat(α) = e(t∪)/z Sat(SP(t, z)α) with t := (r − 1)πiσ1

for all α ∈ ∧rH (P). This satisfies SG(z = ∞) = idH (G). Using Lemma 7.2.7 and the‘classical’ Satake for the cup product by c1 (cf. Proposition 7.2.5), we find:

zρG

Sat(α) = Sat(zρP

α), zµG

Sat(α) = Sat(zµP

α)

where µG is the grading operator of G and ρG = (c1(G)∪). Therefore:(7.5.2) SG(z)z−µ

G

zρG

Sat(α) = e(t∪)/z Sat(SP(t, z)z−µ

P

zρP

α).

These sections are flat for ∇G|τ=0 by the ‘quantum’ Satake (or Theorem 7.3.1). Note that wehave:

zµG

SG(z)z−µG

Sat(α) = e(t∪) Sat(zµ

P

SP(t, z)z−µP

α).

By Lemma 7.5.3 below, we have [zµG

SG(z)z−µG

]z=∞ = idH (G). Hence SG(z)z−µ

G

zρG

coincideswith the fundamental solution of G from Proposition 2.3.1.

The asymptotic basis A1, . . . , AN of P are related to y1(z), . . . , yN (z) as (see §4.5)

yi(z)∣∣∣parallel transportto u

= τP = t

=1

(2π)dim P/2SP(t, z)z−µ

P

zρP

Ai.


This together with (7.5.2) and the definition of yi1,...,ir(z) implies that

ir(r−1)/2 Sat(yi1,...,ir(z))

∣∣∣parallel transport to u

=1

(2π)dimG/2SG(z)z−µ

G

zρG

[1

(−2πi)r(r−1)/2e−(t∪) Sat(Ai1 ∧ · · · ∧Air)

].

Recall that the base point u ∈ CN (C)∼ corresponds to 0 ∈ H (G) for G and to t ∈ H (P)

for P. The conclusion follows from this. (Note also that the asymptotic basis is defined onlyup to sign.)

Lemma 7.5.3. The fundamental solution S(τ, z)z−µzρ in Remark 2.3.2 satisfies

[zµS(τ, z)z−µ]z=∞ = e−(τ∪) for τ ∈ H2(F ).

Proof. The differential equation for T (τ, z) = zµS(τ, z)z−µ in the τ -direction reads ∂αT (τ, z)+z−1(zµ(α⋆τ )z

−µ)T (τ, z) = 0 for α ∈ H (F ). If τ, α ∈ H2(F ), we have that z−1(zµ(α⋆τ )z−µ) is

regular at z = ∞ and equals (α∪) there. Here we use the fact that F is Fano and the divisoraxiom (see Remark 2.1.2). The conclusion follows by solving the differential equation alongH2(F ).

Remark 7.5.4. When we identify the MRS of G with the r-th wedge of the MRS of P, weshould use the identification (2πi)−r(r−1)/2e−(r−1)πiσ1 Sat: ∧r H (P) ∼= H (G) that respectsthe pairing [·, ·) in (3.3.5).

7.6. The wedge product of Gamma basis. We show that the r-th wedge of the Gamma

basis ΓPCh(O(i)) given by Beilinson’s exceptional collection for P matches up with the

Gamma basis ΓG Ch(SνV ∗) given by Kapranov’s exceptional collection for G. In view ofthe truth of Gamma Conjecture II for P (Theorem 6.0.4) and Proposition 7.5.1, the followingproposition completes the proof of Gamma Conjecture II for G.

Proposition 7.6.1. Let N − r > ν1 > ν2 > · · · > νr > 0 be a partition in an r × (N − r)rectangle. We have

ΓGCh(SνV ∗) = (2πi)−(r2)e−(r−1)πiσ1 Sat

(ΓPCh(O(ν1 + r − 1)) ∧ · · · ∧ ΓPCh(O(νr))

).

We give an elementary algebraic proof of Proposition 7.6.1 in this section. A more geometricproof will be discussed in the next section §7.7. Let x1, . . . , xr denote the Chern roots of V ∗

as before. (Recall that V is the tautological bundle on G.) Let h = c1(O(1)) denote thehyperplane class on P = P

N−1.

Lemma 7.6.2. Let f1(z), . . . , fr(z) be power series in C[[z]]. One has

Sat(f1(h) ∧ f2(h) ∧ · · · ∧ fr(h)) =det(fj(xi))16i,j6r∏

i<j(xi − xj).

Proof. For monomial fi(z)’s this follows by the definition of the Schur polynomials (7.2.1)and the Satake identification Sat (7.2.4). The general case follows by linearity.

Lemma 7.6.3. One has

Ch(SνV ∗) =det(e2πixi(νj+r−j))16i,j6r∏

i<j(e2πixi − e2πixj )

Proof. Let L1, . . . , Lr be the K-theoretic Chern roots of V ∗ so that [V ∗] = L1+ · · ·+Lr. TheK-class [SνV ∗] can be expressed as the Schur polynomial σν(L1, . . . , Lr) in L1, . . . , Lr. Thelemma follows from the definition of the Schur polynomial (7.2.1) and Ch(Li) = e2πixi .


Lemma 7.6.4. The Gamma class of G is given by

ΓG = (2πi)−(r2)e−(r−1)πiσ1

∏

i<j

e2πixi − e2πixj

xi − xj

r∏

i=1

Γ(1 + xi)N

Proof. The tangent bundle TG of G is isomorphic to Hom(V,Q), where Q is the universal

quotient bundle. The exact sequence 0 → V → O⊕NG

→ Q → 0 implies that [TG] =

[Hom(V,O)⊕N ]− [Hom(V, V )] = N [V ∗]− [V ∗ ⊗ V ] in the K-group. Thus we have:

ΓG =

∏ri=1 Γ(1 + xi)

N

∏16i,j6r Γ(1 + xi − xj)

.

The denominator of the right-hand side equals∏

i<j

(Γ(1 + xi − xj)Γ(1 − xi + xj)

)=∏

i<j

2πi(xi − xj)

eπi(xi−xj) − e−πi(xi−xj)

= (2πi)(r2)∏

i<j

(xi − xj)e(r−1)πiσ1

∏i<j(e

2πixi − e2πixj )

where we used the Gamma function identity (3.3.7) and eπi(xi−xj) − e−πi(xi−xj) = (e2πixi −e2πixj )e−πi(xi+xj). The Lemma follows.

Proof of Proposition 7.6.1. By Lemmata 7.6.3 and 7.6.4, we have

ΓGCh(SνV ∗) = (2πi)−(r2)e−(r−1)πiσ1

det(e2πixi(νj+r−j))16i,j6r∏i<j(xi − xj)

r∏

i=1

Γ(1 + xi)N .

By Lemma 7.6.2, we have

Sat(ΓPCh(O(ν1 + r − 1)) ∧ · · · ∧ ΓPCh(O(νr))

)=

det(Γ(1 + xi)

Ne2πixi(νj+r−j))16i,j6r∏

i<j(xi − xj).

The conclusion follows from these formulas.

We have now completed the proof of Gamma Conjecture II for G.

7.7. The abelian/non-abelian correspondence and the Gamma basis. We give analternative proof of Proposition 7.6.1 in the spirit of the abelian/non-abelian correspondence[9, 15]. The product P

×r = PN−1 × · · · × P

N−1 (r factors) of projective spaces and theGrassmannian G = G(r,N) arise respectively as the GIT quotients Hom(Cr,CN )//(C×)r

and Hom(Cr,CN )//GL(r,C) of the same vector space Hom(Cr,CN ). We relate them by thefollowing diagram:

F := Fl(1, 2, . . . , r,N)p−−−−→ G

ι

y

P×r

where F is the partial flag variety parameterizing flags 0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vr ⊂ CN

with dimVi = i, p is the projection forgetting the intermediate flags V1, . . . , Vr−1 and ι is thereal-analytic inclusion sending a flag V1 ⊂ · · · ⊂ Vr to a collection (V1, V2⊖V1, . . . , Vr⊖Vr−1)of lines, where Vi ⊖ Vi−1 is the orthogonal complement of Vi−1 in Vi. The projection p is afiber bundle with fiber Fl(1, 2, . . . , r). The normal bundle Nι of the inclusion ι is isomorphicto the conjugate Tp of the relative tangent bundle Tp of p, as a topological vector bundle.


Let Li = (Vi/Vi−1)∗ be the line bundle on F and set xi := c1(Li). We have Euler(Tp) =∏

i<j(xi − xj) = (−1)(r2) Euler(Nι) and:

H (G) ∼= C[x1, . . . , xr]Sr/〈hN−r+1, . . . , hN 〉;

H (F) ∼= C[x1, . . . , xr]/〈hN−r+1, . . . , hN 〉;H (P×r) ∼= C[x1, . . . , xr]/〈xN1 , . . . , xNr 〉.

The injective map

ι∗p∗ : H (G) −→ H (P×r) ∼= ⊗rH (P)

identifies H (G) with the anti-symmetric part ∧rH (P) ⊂ ⊗rH (P), where we embed ∧rH (P)into ⊗rH (P) by α1∧· · ·∧αr 7→

∑σ∈Sr

sgn(σ)ασ(1)⊗· · ·⊗ασ(r). This is inverse to the Satake

identification (−1)(r2) Sat (7.2.4). The Kapranov exceptional bundle SνV ∗ is given by

SνV ∗ = p∗(Lν)

with Lν = L⊗ν11 ⊗ · · · ⊗L⊗νr

r . Because ι∗T (P×r) ∼= p∗TG⊕Tp⊕ Tp as topological bundles, wefind

(7.7.1) ι∗ΓP×r = p∗ΓG ∪ Γ(Tp) ∪ Γ(Tp) = p∗ΓG ∪ Td(Tp)e−πic1(Tp).

By the Grothendieck-Riemann–Roch theorem, we have

(7.7.2) Ch(SνV ∗) = Ch(p∗(Lν)) = (2πi)−(

r2)p∗(Ch(L

ν)Td(Tp)).

Combining (7.7.1) and (7.7.2), we have:

ι∗p∗(ΓGCh(SνV ∗)) = (2πi)−(

r2)ι∗p

∗p∗

(Ch(Lν)Td(Tp) ∪ p∗ΓG

)

= (2πi)−(r2)ι∗p

∗p∗

(Ch(Lν)eπic1(Tp)ι∗ΓP×r

)

= (2πi)−(r2)ι∗p

∗p∗ι∗(Ch (O(ν1 + r − 1)⊠ · · · ⊠O(νr)) e

−(r−1)πiσ1 ΓP×r

).

In the last line we used πic1(Tp) = πi((r − 1)x1 + (r − 3)x2 + · · · − (r − 1)xr) = −(r −1)πiσ1 +2πi((r− 1)x1 + (r− 2)x2 + · · ·+ xr−1). The map ι∗p

∗p∗ι∗ : ⊗rH (P) → ⊗rH (P) is

the anti-symmetrization map and the quantity in the last line can be identified an element

(−2πi)−(r2)e−(r−1)πiσ1

(ΓPCh(O(ν1 + r − 1)) ∧ · · · ∧ ΓPCh(O(νr))

).

of the wedge product ∧rH (P). Since ι∗p∗ is inverse to (−1)(

r2) Sat, the conclusion of Propo-

sition 7.6.1 follows.

Remark 7.7.3. Most of the above discussions can be applied to a general abelian/non-abeliancorrespondence. The bundle Tp corresponds to the sum of positive roots.

7.8. Gamma Conjecture I for Grassmannians. Here we prove that G satisfies GammaConjecture I. We may assume that r 6 N/2 by replacing r with N−r if necessary. Recall thatG satisfies Conjecture O (Remark 7.2.9) and that the quantum product ⋆τ of G is semisimplenear τ = 0 (Corollary 7.3.4). Let φ be an admissible phase for the spectrum of (c1(G)⋆0),sufficiently close to zero. Let TG = N sin(πr/N)/ sin(π/N) denote the biggest eigenvalueof (c1(G)⋆0). In view of Proposition 4.6.7, it suffices to show that the member AG of the

asymptotic basis (at τ = 0 with respect to φ) corresponding to TG is ±ΓG. By Proposition7.5.1, we have

AG = (2πi)−(r2)e−(r−1)πiσ1 Sat(A0 ∧ · · · ∧Ar−1)


where Ak ∈ H (P) is the member of the asymptotic basis for P at τ = (r−1)πiσ1 with respect

to φ, corresponding to the eigenvalue Neπi(r−1)/N e−2πik/N . By the discussion in §6, we knowthat Ak = ΓPCh(O(k)). Here we use the condition that φ is close to zero and r 6 N/2. Thus

Proposition 7.6.1 implies the equality AG = ±ΓG. Gamma Conjecture I for G is proved.The proof of Theorem 7.1.1 is now complete.

Acknowledgments. We thank Boris Dubrovin, Ionut Ciocan-Fontanine, Kohei Iwaki, Bum-sig Kim, Anton Mellit, Kaoru Ono for many useful suggestions and simplifications.

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National Research University Higher School of Economics, Faculty of Mathematics and

Laboratory of Algebraic Geometry, Vavilova str. 7, Moscow 117312, Russia

E-mail address: [email protected]

Algebra and Number Theory Sector, Institute for Information Transmission Problems, Bol-

shoy Karetny per.19, Moscow 127994, Russia


Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-

Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan







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