GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO

GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO MANIFOLDS: GAMMA CONJECTURES

arXiv:1404.6407v1 [math.AG] 25 Apr 2014

SERGEY GALKIN, VASILY GOLYSHEV, AND HIROSHI IRITANI Abstract. We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class AF to a Fano manifold F . We say that F satisfies Gamma Conjecture I if AF equals the Gamma class b F . When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Γ Conjecture II if the columns of the central connection matrix of the quantum cohomology b F Ch(Ei ) for an exceptional collection {Ei } in the derived category of coherare formed by Γ b ent sheaves Dcoh (F ). Gamma Conjecture II refines part (3) of Dubrovin’s conjecture [20]. We prove Gamma Conjectures for projective spaces, toric manifolds, certain toric complete intersections and Grassmannians.

Contents 1. Introduction 1.1. Gamma class 1.2. Gamma conjectures 1.3. When the Gamma Conjectures hold 1.4. Limit formula and Apery’s irrationality 1.5. Mirror symmetry 1.6. Plan of the paper 2. Quantum cohomology, quantum connection and solutions. 2.1. Quantum cohomology 2.2. Quantum connection 2.3. Fundamental solution around the regular singular point z = ∞ 2.4. Formal fundamental solution around the irregular singular point z = 0 2.5. Laplace transform of ∇ 2.6. Asymptotically exponential flat section 2.7. Changing integration paths: mutation and Stokes matrix 2.8. Isomonodromic deformation and marked reflection system 2.9. Deformation of repeated eigenvalues 3. Gamma Conjecture I 3.1. Conjecture O 3.2. Flat sections with the smallest asymptotics 3.3. Gamma Conjecture I: statement 3.4. The leading asymptotics of dual flat sections 3.5. A limit formula for the principal asymptotic class 3.6. Apery limit 3.7. Quantum cohomology central charges Date: April 28, 2014. 1

2 2 3 4 5 5 6 6 6 7 8 9 11 12 13 16 18 20 20 21 22 24 27 29 31

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GALKIN, GOLYSHEV, AND IRITANI

4. Gamma Conjecture II 4.1. Semiorthonormal basis and Gram matrix 4.2. Marked reflection system 4.3. Mutation of MRSs 4.4. Exceptional collections 4.5. Asymptotic basis and MRS of a Fano manifold 4.6. Dubrovin’s (original) conjecture and Gamma conjecture II 4.7. Quantum cohomology central charges 5. Remarks on toric case and quantum Lefschetz principle 5.1. Toric manifolds 5.2. Toric complete intersections 5.3. Compatibility with the quantum Lefschetz principle 6. Gamma conjectures for projective spaces 6.1. Quantum connection of projective spaces 6.2. Frobenius solutions and Mellin solutions 6.3. Monodromy transformation and mutation 7. Gamma conjectures for Grassmannians 7.1. Statement 7.2. Quantum Pieri and quantum Satake 7.3. The wedge product of the big quantum connection of P 7.4. The wedge product of MRS 7.5. MRS of Grassmannian 7.6. The wedge product of Gamma basis 7.7. The abelian/non-abelian correspondence and the Gamma basis 7.8. Gamma Conjecture I for Grassmannians References

32 32 32 33 34 35 35 36 37 37 38 40 41 41 42 44 46 46 46 49 52 53 54 55 56 57

1. Introduction 1.1. Gamma class. The Gamma class of a complex manifold X is the cohomology class bX = Γ

n Y i=1

Γ(1 + δi ) ∈ H (X, R)

where δ1 , . . . , δn are the Chern roots of the tangent bundle T X and Γ(x) is Euler’s Gamma function. A well-known Taylor expansion for the Gamma function implies that it is expanded in the Euler constant Ceu and the Riemann zeta values ζ(k), k = 2, 3, . . . :   X b X = exp −Ceu c1 (X) + Γ (−1)k (k − 1)!ζ(k) chk (T X) . k>2

bX has a loop space interpretation [53, 46]. Let LX denote the free loop The Gamma class Γ space of X and consider the locus X ⊂ LX of constant loops. The normal bundle N of X in LX has a natural S 1 -action (by loop rotation) and splits into the sum N+ ⊕ N− of positive and negative representations. Using the ζ-function regularization one obtains: 1 1 bX ∼ z deg /2 z −c1 (X) Γ = Q∞ k) e 1 (T X ⊗ L eS 1 (N+ ) k=1 S

GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO MANIFOLDS

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where L denotes the S 1 -representation of weight one and z denotes a generator of HS2 1 (pt) b X can be regarded as a localization contribution from such that ch(L) = ez . Therefore Γ constant maps in Floer theory, cf. Givental’s equivariant Floer theory [27, 28]. b 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) =

2πiz πz = e−πiz sin πz 1 − e−2πiz

implies that we can factorize the Todd class in the Hirzebruch-Riemann–Roch (HRR) formula as follows: Z ch(E1∨ ) ∪ ch(E2 ) ∪ tdX χ(E1 , E2 ) = F (1.1.1) h b X Ch(E1 ), Γ b X Ch(E2 ) = Γ

P X where χ(E1 , E2 ) = dim (−1)p dim Extp (E1 , E2 ) is the Euler pairing of vector bundles E1 , Pdim X p=0 p E2 , Ch(Ei ) = p=0 (2πi) chp (Ei ) is the modified Chern character and Z 1 [A, B) := (eπic1 (X) eπiµ A) ∪ B (2π)dim X X

is a non-symmetric pairing on H (X), where µ ∈ End(H (X)) is the grading operator defined by µ(φ) = (p − dim2 X )φ for φ ∈ H 2p (X). Geometrically this factorization corresponds to the decomposition N = N+ ⊕N− of the normal bundle. Recall that Witten and Atiyah [2] derived b heuristically the index theorem by identifying the A-class with 1/eS 1 (N ). In this sense, the Gamma conjectures can be regarded as a square root of the index theorem. 1.2. Gamma conjectures. The Gamma conjectures relate the quantum cohomology of a Fano manifold and the Gamma class in terms of differential equations. For a Fano manifold F , the quantum cohomology algebra (H (F ), ⋆0 ) (at the origin τ = 0) defines the quantum connection [18]: ∂ 1 ∇z∂z = z − (c1 (F )⋆0 ) + µ ∂z z −1 acting on H (F ) ⊗ C[z, z ]. It has a regular singularity at z = ∞ and an irregular singularity at z = 0. Flat sections near z = ∞ are constructed by the so-called Frobenius method and can be put into correspondence with cohomology classes in a natural way. Flat sections near z = 0 are classified by their exponential growth order (along a sector). Our underlying assumption is Conjecture O (Definition 3.1.1) which roughly says that c1 (F )⋆0 has a simple eigenvalue T > 0 of the biggest norm. Under Conjecture O, we can single out a flat section s0 (z) with the smallest asymptotics ∼ e−T /z as z → +0 along R>0 ; then we transport the flat section s0 (z) to z = ∞ and identify the corresponding cohomology class AF . We call AF the principal asymptotic class of a Fano manifold. We say that F satisfies Gamma Conjecture I (Definition 3.3.3) if AF equals the Gamma class: bF . AF = Γ

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

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These classes A1 , . . . , AN ∈ H (F ) form a basis which we call the asymptotic basis of F . We say that F satisfies Gamma Conjecture II (Definition 4.6.1) if the basis can be written as: bF Ch(Ei ) Ai = Γ

b (F ). Gamma for a certain exceptional collection {E1 , . . . , EN } of the derived category Dcoh Conjecture I says that the exceptional object OF corresponds to the biggest real positive eigenvalue T of (c1 (F )⋆0 ). The quantum connection has an isomonodromic deformation over the cohomology group H (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, the braid group acts on the set of asymptotic bases: we formulate this in terms of a marked reflection system in §4.2. Note that Gamma Conjecture II implies (part (2) of) Dubrovin’s conjecture [20] (see §4.6): the Stokes matrix Sij = [Ai , Aj ) of the quantum connection equals the 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 for several cases. The Gamma Conjectures for Pn were implicit but essentially shown in the work of 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 b P Ch(O(i)) P = PN −1 . An asymptotic basis of P is formed by mutations of the Gamma basis Γ 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 complete intersections 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 conditions (Conditions 5.1.1, 5.2.2) ensuring Conjecture O are satisfied, a Fano toric complete intersection satisfies Gamma Conjecture I. For a Fano toric manifold, a K-theoretic analogue of Gamma Conjecture II holds.

We also give a sketch of argument in §5.3 showing that Gamma conjecture I should be compatible with quantum Lefschetz principle [16], i.e. deducing Gamma Conjecture I for a hypersurface assuming that it holds for an ambient space. In §7, we prove the Gamma Conjectures for Grassmannians G(r, N ). The quantum Satake principle [33] or the abelian/non-abelian correspondence [9, 15] says that the quantum connection of G(r, N ) is the r-th wedge of the quantum connection of PN −1 . This fact implies that the asymptotic basis of G(r, N ) is the r-th wedge of that of PN −1 . The same holds for the 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 = b G Ch(S ν V ∗ ) G(r, N ). An asymptotic basis of G is formed by mutations of the Gamma basis Γ associated to Kapranov’s exceptional collection {S ν V ∗ : ν ⊂ r × (N − r)-box}.


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1.4. Limit formula and Apery’s irrationality. Let J(t) = J(−tKF ) denote Givental’s J-function (3.5.7) restricted to the anti-canonical line Cc1 (F ). This is a cohomology-valued solution 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 O and Gamma Conjecture I. Then the Gamma class of F can be obtained as the limit of the ratio of the J-function J(t) b F = lim . Γ t→+∞ h[pt], J(t)i

Theorem 1.4.2 (see Theorem 3.6.1 for precise statements). Under the same assumptions as above, the primitive part of the Gamma class can be obtained as the following discrete limit (assuming the limit exists): (1.4.3)

bF i = lim hγ, Γ

n→∞

hγ, Jrn i h[pt], Jrn i

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

P∞

n=0 Jrn t

rn

with r

Discrete limits similar to (1.4.3) were studied by Almkvist-van Straten-Zudilin [1] in the context 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 limits are related to famous Apery’s proof of the irrationality of ζ(2) and ζ(3). An Apery limit of the Grassmannian G(2, 5) gives a fast approximation of ζ(2) and an Apery limit of the orthogonal Grassmannian OGr(5, 10) gives a fast approximation of ζ(3); they prove the irrationality of ζ(2) and ζ(3). It would be extremely interesting to find a Fano manifold whose Apery limits give fast approximations of other zeta values. 1.5. Mirror symmetry. The ‘Gamma structure’ is closely related to mirror symmetry and has been observed since its early days. The following references serve as motivation to the Gamma conjectures. bX = 1 − π2 c2 (X) − • The Gamma class of a Calabi–Yau threefold X is given by Γ 6 ζ(3)c3 (X); the number ζ(3)χ(X) appeared in the computation of mirror periods by Candelas 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 the Picard–Fuchs equation of the mirror, inspired by the work of Hosono et al. [43]. • Kontsevich’s homological mirror symmetry suggests that the monodromy of the b (X)). In the related Picard–Fuchs equation of mirrors should be related to Auteq(Dcoh works of Horja [41], Borisov–Horja [11] and Hosono [42], Gamma/hypergeometric series play an important role. • In the context of Fano/Landau–Ginzburg mirror symmetry, Iritani [44, 45] and Katzarkov-Kontsevich-Pantev [50] introduced a rational or integral structure of quantum connection in terms of the Gamma class, by shifting the natural integral structure of the Landau–Ginzburg model given by Lefschetz thimbles. Although not directly related to mirror symmetry, we remark that the Gamma class also appears in a recent progress [39, 37] in physics on the sphere/hemisphere partition functions.

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1.6. Plan of the paper. In §2, we review quantum cohomology and quantum differential equation for Fano manifolds. We explain asymptotically exponential flat sections, their mutations 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 the quantum cohomology is semisimple but the Euler multiplication (E⋆τ ) has repeated eigenvalues (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 formulae for the principal asymptotic class. In §4, we formulate Gamma Conjecture II and explain a relationship to (original) Dubrovin’s conjecture. In §5, we prove Gamma conjectures for toric complete intersections (under certain conditions) and explain the compatibility of Gamma Conjecture I with quantum Lefschetz principle. In §6, we give a detailed proof of the Gamma Conjectures for projective spaces. In §7, we deduce the Gamma Conjectures for Grassmannians from the truth of the Gamma Conjectures for projective spaces. Main tools in the proof are isomonodromic deformation and quantum Satake principle. For this purpose we extend quantum Satake principle [33] to big quantum cohomology (Theorem 7.3.1) using abelian/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 connection of a Fano manifold. Quantum connection is defined as a meromorphic flat connection of 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 the irregular singularity (z = 0). Under the semisimplicity assumption, we discuss mutations and Stokes 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 variety such that the anticanonical line bundle ωF−1 = det(T F ) is ample. Let H (F ) = H even (F ; C) denote the even part of the Betti cohomology group over C. For α1 , α2 , . . . αn ∈ H (F ), let hα1 , α2 , . . . , αn iF0,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 rational curves in F which intersect the Poincaré dual cycles of α1 , . . . , αn . It is a rational number when αi ’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 =

X

d∈Eff(F )

R

∞ X 1 hα1 , α2 , α3 , τ, . . . , τ iF0,3+n,d n! n=0

where (α, β)F = F α ∪ β is the Poincaré pairing and Eff(F ) ⊂ H2 (F ; Z) is the set of effective curve classes. The quantum product R is associative and commutative, and recovers the cup product 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 examples in this paper (see also Remark 2.1.2). The quantum product ⋆τ with τ ∈ H 2 (F ) is called the small 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 effective class d contributing to the sum X hα1 , α2 , α3 iF0,3,d d∈Eff(F )


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P has to satisfy 21 3i=1 deg αi = dim F + c1 (F ) · d and there are only finitely many such d when F is Fano. Therefore the specialization at τ = 0 makes sense. L 2p Remark 2.1.2. Writing τ = h + τ ′ with h ∈ H 2 (F ) and τ ′ ∈ p6=1 H (F ) and using the divisor axiom in Gromov–Witten theory, we have: ∞ R X X F 1

(α1 ⋆τ α2 , α3 )F = α1 , α2 , α3 , τ ′ , . . . , τ ′ 0,n+3,d e d h . n! n=0 d∈Eff(F )

Therefore the quantum product (2.1.1) makes sense as a formal power series in τ ′ and the exponentiated H 2 -variables eh1 , . . . , ehr , where we write h = h1 p1 + · · · + hr pr by choosing a nef basis {p1 , . . . , pr } of H 2 (X; Z).

2.2. Quantum connection. Following Dubrovin [18, 20, 19], we introduce a meromorphic flat connection associated to the quantum product. Consider a trivial vector bundle H (F ) × P1 → P1 over P1 and fix an inhomogeneous co-ordinate z on P1 . Define the quantum connection ∇ on the trivial bundle by the formula 1 ∂ − (c1 (F )⋆0 ) + µ (2.2.1) ∇z∂z = z ∂z z where µ ∈ End(H (F )) is the grading operator defined by µ|H 2p (F ) = (p − dim2 F ) idH 2p (F ) . The connection is smooth away from {0, ∞}; the singularity at z = ∞ is regular (or more precisely logarithmic) and the singularity at z = 0 is irregular. The quantum connection preserves the Poincaré pairing in the following sense: we have ∂ (2.2.2) z (s1 (−z), s2 (z)) = ((∇z∂z s1 )(−z), s2 (z)) + (s1 (−z), ∇z∂z s2 (z)) ∂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 deformation over H (F ). Suppose that ⋆τ converges on a region B ⊂ H (F ). Then the above connection is extended to a meromorphic flat connection on H (F ) × (B × P1 ) → (B × P1 ) as follows: 1 ∇α = ∂α + (α⋆τ ) α ∈ H (F ) z (2.2.3) 1 ∂ − (E⋆τ ) + µ ∇z∂z = z ∂z z with (τ, z) a point on the base B × P1 . Here ∂α denotes the directional derivative in the direction of α and N X 1 1 − deg φi τ i φi E = c1 (F ) + 2 i=1 PN i is the Euler vector field, where we write τ = i=1 τ φi by choosing a homogeneous basis {φ1 , . . . , φN } of H (F ). We refer to this extension as the big quantum connection. The Poincaré pairing (·, ·)F is flat with respect to the big quantum connection ∇, i.e. ∂α (s1 (τ, −z), s2 (τ, z)) = ((∇α s1 )(τ, −z), s2 (τ, z)) + (s1 (τ, −z), ∇α s2 (τ, z)). Remark 2.2.4. The connection in the z-direction can be identified with the connection in the anticanonical direction after an appropriate rescaling. Consider the quantum product ⋆τ restricted to the anticanonical line τ = c1 (F ) log t, t ∈ C× : X (2.2.5) (α1 ⋆c1 (F ) log t α2 , α3 )F = hα1 , α2 , α3 i0,3,d tc1 (F )·d . d∈Eff(F )

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This is a polynomial in t since F is Fano, and coincides with ⋆0 when t = 1. It can be recovered from the product ⋆0 by the formula: (α⋆c1 (F ) log t ) = tdeg α/2 t−µ (α⋆0 )tµ .

(2.2.6)

The quantum connection restricted to the anticanonical line and z = 1 is ∂ = t + (c1 (F )⋆c1 (F ) log t ). ∇c1 (F ) ∂t z=1 On the other hand we have i h ∂ − (c1 (F )⋆−c1 (F ) log z ) z −µ = z z µ ∇z∂z ∂z τ =0 Therefore, the connections ∇c1 (F ) |z=1 and ∇z∂z |τ =0 are gauge equivalent via z µ under the change of variables t = z −1 . 2.3. Fundamental solution around the regular singular point z = ∞. We now restrict our attention to the connection ∇ (2.2.1) defined by the quantum product ⋆0 at τ = 0. The following 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 −µ z ρ α) = 0 µ

T (z) = z S(z)z

−µ

for all α ∈ H (F );

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

where ρ = (c1 (F )∪) ∈ End(H (F )) and we define z −µ = exp(−µ log z), z ρ = 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 in our setting. We study the equivalent differential equation ∇(z −µ T (z)z ρ α) = 0 for T (z) = z µ S(z)z −µ with the initial condition T (∞) = id. The differential equation for T reads: z

1 ∂ T (z) − z µ (c1 (F )⋆0 )z −µ T (z) + T (z)ρ = 0. ∂z z

Expand: T (z) = id +T1 z −1 + T2 z −2 + T3 z −3 + · · ·

(c1 (F )⋆0 ) = G0 + G1 + G2 + · · ·

(finite sum)

where Gk ∈ End(H (F )) is an endomorphism of degree 1 − k, i.e. z µ Gk z −µ = z 1−k Gk 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−1 T1 + · · · + G1 Tm−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.


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By construction, Tk is an endomorphism of degree > (1−k) and hence z −k z −µ Tk z µ contains only negative powers in z for k > 1. Therefore S(z) = z −µ T (z)z µ is regular at z = ∞ and satisfies S(∞) = id. Finally we see (S(−z)α, S(z)β)F = (α, β)F . We claim that S(−z)(eπi z)−µ (eπi z)ρ α, S(z)z −µ z ρ β F = (e−πiµ eπiρ α, β)F . Because ∇ preserves the Poincaré pairing (2.2.2), the left-hand side is independent of z. On the other hand, the left-hand side equals e−πiµ T (−z)(eπi z)ρ α, T (z)z ρ β F

and can be expanded in C[[z −1 ]][log z]. The constant term of the Taylor expansion in z −1 and log z equals the right-hand side. The claim follows. Replacing α, β with (eπi z)−ρ (eπi z)µ α and z −ρ z µ β, 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 bundle over M 0,2 (F, d) at the first marking. Then we have: X X 1 (−1)m+1 hαψ m , βiF0,2,d . (S(z)α, β)F = (α, β)F + z m+1 m>0

d∈Eff(F )\{0}

Here again the summation in d is finite (for a fixed m) because F is Fano. A similar fundamental solution exists for the isomonodromic deformation of ∇ associated to the big quantum cohomology. The big quantum connection ∇ over H (F ) × P1 admits a fundamental solution of the form S(τ, z)z −µ z ρ extending the one in Proposition 2.3.1 such that ∇(S(τ, z)z −µ z ρ α) = 0,

S(τ, ∞) = id,

(S(τ, −z)α, S(τ, z)β)F = (α, β)F .

Here z µ S(τ, z)z −µ is not necessarily regular at z = ∞ for τ ∈ / H 2 (F ) (cf. Lemma 7.5.3).

2.4. Formal fundamental solution around the irregular singular point z = 0. Here we assume that the ring (H (F ), ⋆0 ) is semisimple, i.e. isomorphic to the direct sum of C as a ring. In view of Remark 2.2.4, we refer to this property as that the quantum cohomology of F is anticanonically semisimple. Note that anticanonical semisimplicity is stronger1 than semisimplicity 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,j ψi p where N = dim H (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 endomorphism CN → H (F ). Let u1 , . . . , uN be the eigenvalues of (c1 (F )⋆0 ) given by c1 (F ) ⋆0 Ψi = ui Ψi . Define U to be the diagonal matrix with entries u1 , . . . , uN :   u1   u2   (2.4.1) U = . ..   . uN

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

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Around the irregular singular point, the connection ∇ (2.2.1) has the following formal fundamental solution. Proposition 2.4.2 ([19, Lemma 4.3], [57, Theorem 8.15]). There exists a unique formal power series of the form R(z) = id +R1 z + R2 z 2 + · · · ,

with Rk ∈ End(CN ) such that one has for all v ∈ CN .

∇(ΨR(z)e−U/z v) = 0,

R(−z)T R(z) = id

Dubrovin [19] discussed the case where u1 , . . . , uN are distinct. Teleman [57] remarked that the proposition holds even when u1 , . . . , uN are not distinct. Note that Ψ1 , . . . , ΨN are unique up to order and signs. Hence the formal fundamental solution ΨR(z)e−U/z is unique up to the right multiplication by a signed permutation matrix. We review the construction of R(z) for the convenience of the reader. By the constant gauge transformation by Ψ, we can transform the connection ∇ into the U V -form [19]: ∂ 1 − U +V ∂z z with U given by (2.4.1) and V = Ψ−1 µΨ. We have the following lemma: Ψ ∗ ∇z

∂ ∂z

=z

Lemma 2.4.3. The matrix V = (Vij ) is anti-symmetric Vij = −Vji . Moreover, Vij = 0 whenever 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 the isomonodromic deformation discussed in §2.2. In the case where the big quantum cohomology is not known to be analytic, we work over the formal neighbourhood of τ = 0 in the following discussion. Around the semisimple point τ = 0, the spectrum {u1 , . . . , uN } of E⋆τ forms a ∂τ , i = 1, . . . , N give an idempotent basis of ⋆τ (see [19, local co-ordinate system, and ψi = ∂u i Lecture 3]). The matrix Ψ of normalized idempotents is also extended to a matrix-valued function on a neighbourhood of τ = 0. With respect to this co-ordinate system and after gauge transformation by Ψ, the connection (2.2.3) takes the form (see [19, Lemma 3.2]): Ψ∗ ∇

∂ ∂ui

Ψ ∗ ∇z

∂ ∂z

∂ 1 + Ei + Vi ∂ui z ∂ 1 =z − U +V ∂z z

=

i-th

where Ei = diag[0, . . . , 0, 1 , 0, . . . , 0] and V = Ψ−1 µΨ, Vi = Ψ−1 ∂ui Ψ 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 . If i = j, Vij = 0 by the anti-symmetricity. ∂ Proof of Proposition 2.4.2. It suffices to solve the U V -system (z ∂z − z1 U +V )(R(z)e−U/z ) = 0. This is equivalent to the system of equations:

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


11

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 the entries (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 the formal substitution ∂z −1 → −λ, z → ∂λ , the differential equation ∇z∂z (z ν−1 y(z)) = 0 is transformed to the equation: ∂ −1 b + (λ − c1 (F )⋆0 ) (µ + ν) yˆ(λ) = 0 ∇ ∂ yˆ(λ) := ∂λ ∂λ b is called the second structure where ν ∈ C is an arbitrary parameter. The connection ∇ b has logarithmic singularities connection in [54]. Suppose again that ⋆0 is semisimple. Then ∇ b at λ = ui is given by at the eigenvalues u1 , . . . , uN of c1 (F )⋆0 and at ∞. The residue of ∇ b = Pu (µ + ν) Resλ=ui ∇ i

where Pui is the projection to the ui -eigenspace of c1 (F )⋆0 . By Lemma 2.4.3, the residue is conjugate (via Ψ and reordering the basis) to the matrix of the form: ∗ νIr (2.5.1) 0 0N −r where Ir is the identity matrix of size r = ♯{j : ui = uj } and 0N −r is the zero matrix of size N − r. If the residue is non-resonant2, i.e. ν = 0 or ν ∈ / Z, we have a fundamental solution b with U (x) holomorphic near x = 0 and of the form U (λ − ui ) exp(− log(λ − ui ) Resλ=ui ∇) b U (0) = id. In particular the local monodromy at λ = ui is conjugate to exp(−2πi Resλ=ui ∇). From this we have the following: Proposition 2.5.2. Suppose that ⋆0 is semisimple. Consider the second structure connection b with parameter ν = 0. ∇

(1) For an eigenvector v of (c1 (F )⋆0 ) of eigenvalue ui , there exists a unique holomorphic flat section yˆv (λ) near λ = ui such that yˆv (ui ) = v. (2) Let Mui denote the monodromy transformation around λ = ui which acts in the space b For a ∇-flat b of flat sections of ∇. section gˆ, there exists an ui -eigenvector v such that Mui gˆ = gˆ + yˆv , where yˆv is the flat section in (1).

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

b has an invariant metric given by ((λ − Remark 2.5.3 ([19, Lecture 5]). When ν = 1/2, ∇ 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.

12


u1 • u3 = u4 = u5 •

u6 = u7 • u9 •

u2 • u8 •

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 in Proposition 2.4.2 is typically a divergent formal power series. By choosing an angular sector in the z-plane with vertex at z = 0, we can lift each of the formal flat sections to an analytic flat section which is asymptotic to it along the sector. When u1 , . . . , uN are distinct, this is discussed in [3, Theorem A, Proposition 7], [4] and [19, Lectures 4, 5]. Following [12, §8], we discuss the case where some of u1 , . . . , un may coincide. We assume the anticanonical semisimplicity and choose an ordering and signs of a normalized 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. For a real number φ, we say that the direction eiφ ∈ S 1 (or the phase φ ∈ R) is admissible 3 for {u1 , . . . , uN } if Im(e−iφ ui ) 6= Im(e−iφ uj ) 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). We denote by Σ ⊂ S 1 the finite set of non-admissible directions. b (with parameter ν = By Proposition 2.5.2, there exists a unique flat section yî (λ) of ∇ 0) near λ = ui such that yî (λ) is regular at λ = ui and that yî (ui ) = Ψi . Choose an admissible direction eiφ and construct a flat section yi (z) of ∇ by the vector-valued Laplace transformation: Z 1 yî (λ)e−λ/z dλ. (2.6.1) yi (z) = z ui +eiφ R>0 The integration path is a straight half-line Li starting from ui to ∞ in the direction of eiφ b is regular singular at and fi (λ) is analytically continued along the path (Figure 1). Since ∇ λ = ∞, the integrand is at most of polynomial growth. Therefore the integral converges if | arg z − φ| < π2 . By bending the path Li continuously (but avoiding other singularities), we can 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 S 1 \ Σ containing eiφ . Then the flat sections yi (z) (2.6.1) satisfy the asymptotic expansion (2.6.3) as |z| → 0 in the sector: (2.6.4)

Y (z) := (y1 (z), . . . , yN (z)) ∼ ΨR(z)e−U/z φ1 −

π π < arg z < φ2 + . 2 2

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


13

(1) A fundamental solution Y satisfying this asymptotic condition is unique: we call it the (asymptotically exponential) fundamental solution associated to eiφ . − (z)) be the fundamental solution associated to −eiφ . Then (2) Let Y − (z) = (y1− (z), . . . , yN − we have (yi (−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 that the angle of the sector (2.6.4) is bigger than π. The order and signs of the asymptotically exponential flat sections (y1 (z), . . . , yN (z)) associated to eiφ depend on the order and signs of normalized idempotents Ψ1 , . . . , ΨN . Remark 2.6.5. The precise meaning of the asymptotic expansion (2.6.3) is as follows. For every small ǫ > 0 and non-negative integer n, there exists a positive constant C = C(n, ǫ) such that n X ui /z k n+1 ΨR e z e y (z) − i k i 6 C|z| k=1

holds for all i and z with φ1 − R(z) = R0 + R1 z + R2 z 2 + · · · .

π 2

+ ǫ 6 arg z 6 φ2 +

π 2

− ǫ and |z| 6 1, where we write

Proof of Proposition 2.6.2. As shown in [12, §8.4] by an elementary calculation, the function yi (z) is a solution to the differential equation ∇z∂z yi (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) increases monotonically along the path: it follows that this holds in the sector φ1 − π2 < 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 = Y2 C. We have e−U/z CeU/z → id as z → 0 in the sector. Since the angle of the sector is bigger than π, it happens only when C = id. Finally we show Part (2). Since yi− (z) and yj (z) are flat, the pairing (yi− (−z), yj (z))F does not depend on z (see (2.2.2)). By the asymptotic condition (2.6.3), we have e(uj −ui )/z (yi− (−z), yj (z))F = (e−ui /z yi− (−z), euj /z yj (z))F → (Ψi , Ψj )F = δij as |z| → 0 in the sector (2.6.4). Thus (yi− (−z), yj (z))F = δij if ui = uj . If ui 6= uj , we have (yi− (−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 /z ΨR(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 S 1 \ Σi which contains the admissible direction eiφ . By construction, the flat section yi (z) has the asymptotic expansion in the sector ξ1 − π2 < 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 section, we constructed an asymptotically exponential flat section yi (z) as the Laplace transform b (2.6.1) of the ∇-flat section yî (λ) over the straight half-line Li . We study the change of flat sections under a change of integration paths. To illustrate, we consider the change of paths from 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 b L1 , . . . , LSN as branch cuts for ∇-flat sections; yî (λ) defines a single-valued analytic function over C \ j6=i Lj . Let M denote the anti-clockwise monodromy transformation around λ = u2

14


ξ2 +

②

π 2

✗

um •

un •

✙

• ui

✠

uo •

ξ1 −

❨

ul ✒ • • uk

✯ ξ2

✲ φ

uj •

admissible

❥ ξ1

π 2

Figure 2. The flat section yi (z) has the asymptotics e−ui /z ΨR(z)ei in the sector (ξ1 − π2 , ξ2 + π2 ). u1 •

u2 = · · · = ur • ur+1 • uN •

.. .

L1 L2 = · · · = Lr L′1 Lr+1 .. . LN

Figure 3. Right mutation of L1 b acting on the space of ∇-flat sections. By Proposition 2.5.2, the monodromy transform of yˆ1 is of the form: M yˆ1 = yˆ1 − c12 yˆ2 − · · · − c1r yˆr

for some coefficients c12 , . . . , c1r ∈ C. From this it follows that the flat section y1′ (z) defined by the integral over L′1 is given by: (2.7.1)

y1′ (z) = y1 (z) − c12 y2 (z) − · · · − c1r yr (z).

We call the flat section y1′ (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 •

L1 L′r+1 • ur+1

• u2 = · · · = ur

Figure 4. Left mutation of Lr+1

L2 = · · · = Lr Lr+1


15

A left or right mutation occurs when the admissible direction eiφ varies and crosses a nonadmissible direction. We now let the phase φ decrease by the angle π continuously. Then the asymptotically exponential flat sections y1 , . . . , yN undergo a sequence of right mutations. − be the basis of asymptotically exponential flat sections associated to −eiφ as Let y1− , . . . , yN in Proposition 2.6.2. In the situation of Figure 3, we have: linear combinations of yj (z) (2.7.2) y1− (z) = y1 (z) − c12 y2 (z) − · · · − c1r yr (z) − with Im(e−iφ uj ) < Im(e−iφ . u2 )

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

Π+ ③ ✾

Y − (z)

•

Y (z)

② ✿

✲

φ admissible

Π−

Figure 5. Domains of the two solutions Y and Y − − ) be the Definition 2.7.3 ([19, Definition 4.3]). Let Y = (y1 , . . . , yN ), Y − = (y1− , . . . , yN fundamental solution associated to an admissible direction eiφ and −eiφ respectively as in Proposition 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− − ) be the asymptotically exponential Proposition 2.7.4. Let (y1 , . . . , yN ) and (y1− , . . . , yN fundamental solution associated to admissible directions eiφ and −eiφ respectively and let S = (Sij ), S− = (S−,ij ) be the Stokes matrices. We have

(1) S− = S T . (2) Sij = S−,ji = (yi (e−πi z), yj (z))F for z ∈ Π+ . Here we write e−πi z instead of −z to specify the path of analytic continuation. Similarly, (yi− (eπi z), yj− (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 Sij = 0

if i 6= j and ui = uj ;

if Im(e−iφ ui ) < Im(e−iφ uj ).

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

16


Proof. Dubrovin [19, Thorem 4.3] discussed the case where u1 , . . . , uN are distinct. We have PN P − − yj (z) = N k=1 yk (z)S−,kj for z ∈ Π− . Taking the k=1 yk (z)Skj for z ∈ Π+ and yj (z) = − pairing with yi (−z) and using the property (yk (z), yi (−z))F = δki from Proposition 2.6.2, we obtain (yi (−z), yj (z))F = Sij (yi (−z), yj (z))F = S−,ij

for z ∈ Π+

for z ∈ Π−

Replacing z with −z in the second formula and taking into account the direction of analytic continuation, we see that (1) and (2) hold. (The discussion for (yi− (eπi z), yj− (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 idempotents Ψ1 , . . . , ΨN so that the corresponding eigenvalues satisfy Im(e−iφ u1 ) > · · · > Im(e−iφ uN ). 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) coincides with the coefficient S1i of the Stokes matrix, because by inverting (2.7.2) we obtain linear combinations of yj− (z) y1 (z) = y1− (z) + c12 y2− (z) + · · · + c1r yr− (z) + −iφ −iφ with Im(e

uj ) < Im(e

u2 )

for z ∈ Π− and therefore c1i = S−,i1 = S1i . Recall that c1i arises as the coefficient of the right mutation (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 asymptotically exponential fundamental solution associated to eiφ in Proposition 2.6.2. Let ua , ub be distinct eigenvalues such that there are no eigenvalues uj with Im(e−iφ ua ) > Im(e−iφ uj ) > Im(e−iφ ub ). Then the right mutation of the flat section ya (z) with respect to ub is given by X ya 7→ ya − Saj yj . j:uj =ub

Similarly, the left mutation of yb (z) with respect to ua is given by: X yb 7→ yb − Sjb yj j:uj =ua

where Sij = (yi (e−πi z), 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 that the quantum product is anticanonically semisimple and analytic in a neighbourhood of τ = 0 in H (F ). Then the eigenvalues u1 , . . . , uN of E⋆τ form a local co-ordinate system on the base H (F ) [19, Lecture 3] and we can make u1 , . . . , uN pairwise distinct by a small deformation of τ : we take a base point τ ◦ ∈ H (F ) such that the corresponding eigenvalues u◦ = {u◦1 , . . . , u◦N } are pairwise distinct. The quantum connection ∇|τ ◦ then admits a unique isomonodromic 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 .


17

Proposition 2.8.2 ([19, Lemma 3.2, Exercise 3.3, Lemma 3.3]). We have a unique meromorphic flat connection on the trivial H (F )-bundle over CN (C)∼ × P1 of the form: ∂ 1 + Ci ∇ ∂ = ∂ui ∂ui z ∂ 1 ∇z ∂ = z − U +µ ∂z ∂z z where Ci and U are End(H (F ))-valued holomorphic functions on CN (C)∼ , such that it restricts to the big quantum connection (2.2.3) in a neighbourhood of the base point u◦ . Here the 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 manifold 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 a ˜ its lift to CN (C)∼ . point of CN (C) and by u ˜ ∈ • We have a formal fundamental solution Ψu˜ Ru˜ eU/z depending analytically on u CN (C)∼ as in Proposition 2.4.2. ˜ ∈ CN (C)∼ and an admissible phase φ for u, we have a fundamental solution • For each u Yu˜ ,φ for ∇ which satisfies the asymptotic condition Yu˜ ,φ (z) ∼ Ψu˜ Ru˜ eU/z

as |z| → 0 in a sector of the form φ − π2 − ǫ < arg z < φ + π2 + ǫ for some ǫ > 0 as in Proposition 2.6.2. Note that the sector here is naturally identified with a subset of the universal cover of C× as we specify the phase φ. • When (˜ u, φ) varies, the fundamental solution Yu˜ ,φ changes discontinuously by a left or 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 the universal 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 −µ z ρ of Remark 2.3.2. This is equipped with a non-degenerate (but not necessarily symmetric or anti-symmetric) pairing [f, g) := (f (˜ u, e−πi z), 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 exponential fundamental solution Yu˜ ,φ = (y1 , . . . , yN ) associated to (˜ u, φ) gives a basis of H. It is semiorthonormal by the analogue of Proposition 2.7.4: ( 0 if Im(e−iφ ui ) < Im(e−iφ uj ) [yi , yj ) = 1 if i = j × ∼ 5We make this identification along {u◦ } × R where the connection ∇ is identified with >0 ⊂ (CN (C) × C ) the quantum connection and the determination of z −µ z ρ = exp(−µ log z) exp(ρ log z) is canonical. See also §4.5.

18


The basis is unique up to signs and orderings. We call {y1 , . . . , yN } the asymptotic basis and the tuple ˜ , φ) = (H, {y1 , . . . , yN }, u = {u1 , . . . , uN }, eiφ ) MRS(F, u the marked reflection system of a Fano manifold F . Here each basis vector yi ∈ H is marked by a complex number ui ∈ u. Let us number u1 , . . . , uN so that Im(e−iφ u1 ) > Im(e−iφ u2 ) > · · · > Im(e−iφ uN ). By Corollary 2.7.6, the right and left mutation of (y1 , . . . , yN ) is given respectively by: (y1 , . . . , yi , yi+1 , . . . , yN ) 7→ (y1 , . . . , yi−1 , yi+1 , yi − [yi , yi+1 )yi+1 , . . . , yN ) (y1 , . . . , yi , yi+1 , . . . , yN ) 7→ (y1 , . . . , yi−1 , yi+1 − [yi , yi+1 )yi , yi , . . . , yN )

(right) (left)

The right mutation occurs for example when u moves along the loop in CN (C) depicted in Figure 6. These loops generate the braid group BN , the fundamental group of CN (C). It is easy to see that the right mutations σi satisfy the braid relations: σi σi+1 σi = σi+1 σi σi+1 (2.8.4) σi σj = σj σi if |i − j| > 2. u1 • .. . • ui ❨ ❥•

ui+1

admissible direction

✲

φ

.. . uN • Figure 6. Braid σi

2.9. Deformation of repeated eigenvalues. In the previous section §2.8, we considered a generic deformation where u1 , . . . , uN are pairwise distinct. Suppose now that the quantum product ⋆τ0 at τ0 is semisimple but (c1 (F )⋆τ0 ) has repeated eigenvalues. In this case, we have the isomonodromic deformation over the union (CN (C)∼ /π1 (B ∗ )) ∪ B

where 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 isomonodromic deformation. The flat sections corresponding to the same repeated eigenvalue remain mutually orthogonal after small deformation. Proposition 2.9.1. Suppose that the quantum product ⋆τ0 is convergent and semisimple for some τ0 ∈ H (F ). Let B ⊂ H (F ) be an open ball centred at τ = τ0 such that ⋆τ is analytic and semisimple on B. Consider the isomonodromic deformation over B defined by the big quantum 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:


19

(1) A formal fundamental solution Ψτ Rτ (z)e−Uτ /z for ∇ exists and depends analytically on τ ∈ B (cf. Proposition 2.4.2). Here Ψτ = (Ψτ1 , . . . , ΨτN ) is a basis of normalized idempotents, Rτ (z) = id +R1 (τ )z + R2 (τ )z 2 + · · · is a formal power series in z with coefficients Ri (τ ) ∈ End(CN ) and Uτ = diag[u1 (τ ), . . . , uN (τ )] is the diagonal matrix such that E ⋆τ Ψτi = ui (τ )Ψτi . τ ) for ∇ which depends analytically (2) We have a fundamental solution Yτ = (y1τ , . . . , yN on τ ∈ B and satisfies the asymptotic condition Yτ ∼ Ψτ Rτ (z)e−Uτ /z

as |z| → 0 in a sector of the form φ − π2 − ǫ < arg z < φ + π2 + ǫ for some ǫ > 0 (cf. Proposition 2.6.2). (3) The flat sections yiτ (z) are semiorthonormal in the sense that  −iφ u ) < Im(e−iφ u )  i,0 j,0 0 if Im(e (yiτ (e−iπ z), yjτ (z))F = Sij = 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 formal fundamental solution Ψτ Rτ e−Uτ /z and the asymptotically exponential flat sections yiτ (z) by the same method as in §§2.4–2.6. It only suffices to show that they depend analytically on τ . The matrix Ψτ of normalized idempotents depends analytically on τ , and hence Uτ also b (with parameter ν = 0; depends analytically on τ . We use the second structure connection ∇ see §2.5) extended to the base B × Cλ . The extension is given by the formula: b α = ∂α − (α⋆τ )(λ − E⋆τ )−1 µ ∇ b ∂ = ∂ + (λ − E⋆τ )−1 µ ∇ ∂λ ∂λ

where α ∈ H (F ) and (τ, λ) ∈ B × Cλ , see [54]. Under the gauge transformation by Ψτ , we have N N X X Ei V ∗b − Vi d(λ − ui (τ )) + Vi dλ (Ψτ ) ∇ = d + λ − ui (τ ) i=1

i=1

b where Ei , V , Vi are as in the proof of Lemma 2.4.3. QNTherefore the connection ∇ has logarithmic singularities along the normal crossing divisor i=1 (λ − ui (τ )) = 0 in B × Cλ . The residue b Ri = Ψτ (Ei V )Ψ−1 τ |τ =τ0 of ∇ at (τ, λ) = (τ0 , ui,0 ) along the divisor λ − ui (τ ) = 0 is nilpotent; b of the hence in a neighbourhood of (τ, λ) = (τ0 , ui,0 ), we have a fundamental solution for ∇ form:   X Rj log(λ − uj (τ )) U (λ, τ ) exp − j:uj,0 =ui,0

where U (λ, τ ) is an End(H (F ))-valued holomorphic function defined near (λ, τ ) = (ui,0 , τ0 ) b such that U (ui,0 , τ0 ) = id. We define a ∇-flat section yî (τ, λ) by applying the above fundaτ =0 mental solution to the eigenvector Ψi . This is holomorphic near (τ, λ) = (τ0 , ui,0 ). Here we

20


use the fact that Rj Ψτi =0 = 0 whenever uj,0 = ui,0 (this fact follows from Lemma 2.4.3). Now we define yiτ (z) as the Laplace transform (cf. (2.6.1)) Z 1 τ yî (τ, λ)e−λ/z dλ. yi (z) = z ui (τ )+eiφ R>0 This is flat for ∇ (in both z and τ ) and depends analytically on τ as far as eiφ is admissible for {u1 (τ ), . . . , uN (τ )}. It then follows that the asymptotic expansion Ψτ Rτ e−Uτ /z of Yτ = τ ) depends analytically on τ , as the asymptotic series R (z) is determined by the (y1τ , . . . , yN τ Taylor coefficients of yî (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 (yiτ (e−iπ z), yjτ (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 not need 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 Conjecture I assuming Conjecture O. Definition 3.1.1. Let F be a Fano manifold and let c1 (F )⋆0 ∈ End(H (F )) be the quantum product 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 = T ζ 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 monotone symplectic 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 with τ = c1 (F )(2πi/r). This together with (2.2.6) gives the symmetry: (3.1.4)

(c1 (F )⋆0 ) = e2πi/r e−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 that F is mirror to a holomorphic function f : Y → C on a complex manifold Y . Under mirror symmetry it is expected that the eigenvalues of (c1 (F )⋆0 ) should coincide with the critical values 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 . This explains the name ‘Conjecture O’. Our Gamma conjectures give a direct link between T and O in terms of differential equations (without referring to mirror symmetry): the asymptotically bF = Γ b F Ch(O); more generally exponential flat section associated to T should correspond to Γ b F Ch(E) for an exceptional object E ∈ D b (F ) a simple eigenvalue should correspond to Γ coh (see Gamma Conjecture I, II in §3.3, §4.6).


21

Remark 3.1.6 (Conifold point). Let f : (C× )n → C be a convenient6 Laurent polynomial mirror to a Fano manifold F . Suppose that all the coefficients of f are positive real. Then the Hessian ∂2f (x1 , . . . , xn ) ∂ log xi ∂ log xj 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 that a square matrix M of size n is irreducible if the linear map Cn → Cn , v 7→ M v admits no invariant co-ordinate subspace. If (c1 (F )⋆0 ) is represented by an irreducible matrix with nonnegative 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/3 y, e4πi/3 z] and let F be the −1 2 −1 2 −1 quotient Fano orbifold P2 /(Z/3Z). The mirror is f = x−1 1 x2 + x1 x2 + x1 x2 and T = 3 has 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 representations of π1orb (F ) for a Fano orbifold F . Remark 3.1.9. In the rest of the section we assume Conjecture O. However, the argument in this section (i.e. §3) except for §3.6 works under the following weaker assumption: there exists 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 use the 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 a one-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 sections with the smallest asymptotics. Proposition 3.2.1. Suppose that a Fano manifold F satisfies Conjecture O. Consider the quantum connection ∇ in (2.2.1) and define: o n (3.2.2) A := s : R>0 → H (F ) : ∇s = 0, keT /z s(z)k = O(z −m ) as z → +0 (∃m)

Then: (1) A is a one-dimensional complex vector space. (2) For s ∈ A, the limit limz→+0 eT /z s(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 keλ/z s(z)k = ∞ for all λ > T ′ . 6A Laurent polynomial is said to be convenient if the Newton polytope of f contains the origin in its interior.

22


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 /z s0 (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 the construction of the asymptotically exponential flat section yi (z) in §2.6. Consider again the b from §2.5 with parameter ν = 0. Since we do not assume the second structure connection ∇ b is Fuchsian. However we know by Conjecture O that ∇ b is semisimplicity, we do not know if ∇ logarithmic at λ = T . Following the argument in §2.5 and using Lemma 3.2.4 below instead of b 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)): Z 1 ∞ ϕ(λ)e−λ/z dλ (3.2.3) s0 (z) = z T

b is smooth in the region Re(λ) > T and therefore ϕ(λ) is analytically continued Note that ∇ b is regular singular at along the positive real line. The integral converges for z > 0 because ∇ T /z λ = ∞. Watson’s lemma [12, §8.4] shows that limz→+0 e s0 (z) = ϕ(T ).

Lemma 3.2.4. Suppose that Conjecture O holds. Let E(T ) ⊂ H (F ) denote the onedimensional T -eigenspace of (c1 (F )⋆0 ) and H ′ = Im(T − (c1 (F )⋆0 )) be the complementary subspace. Then E(T ) and H ′ are orthogonal with respect to the Poincaré pairing and the endomorphism 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, T γ − c1 (F ) ⋆0 γ)F = (T α − c1 (F ) ⋆0 α, γ)F = 0.

Thus E(T ) ⊥ H ′ with respect to (·, ·)F . The skew-adjointness of µ with respect to (·, ·)F shows 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 smooth projective variety F is the class bF := Γ

dim YF i=1

Γ(1 + δi ) ∈ H (F )

where δ1 , . . . , δdim F are the Chern roots of the tangent bundle T F so that Q F c(T F ) = dim i=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 ! ∞ X b F = exp −Ceu c1 (F ) + (−1)k (k − 1)!ζ(k) chk (T F ) Γ k=2

where Ceu = 0.57721... is Euler’s constant and ζ(k) is the special value of Riemann’s zeta function. The fundamental solution S(z)z −µ z ρ from Proposition 2.3.1 identifies the space of flat sections over the positive real line R>0 with the cohomology group: (3.3.1)

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

dim F 2

S(z)z −µ z ρ α

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


23

Definition 3.3.2. Suppose that a Fano manifold F satisfies Conjecture O. The principal asymptotic class of F is a cohomology class AF ∈ H (F ) such that A = CΦ(AF ), where A is the space (3.2.2) of flat sections with the smallest asymptotics (recall that A is of dimension one by Proposition 3.2.1). The class AF is defined up to a constant; when h[pt], AF i = 6 0, we can normalize AF so that h[pt], AF i = 1. Definition 3.3.3. Suppose that a Fano manifold F satisfies Conjecture O. We say that F bF . satisfies Gamma Conjecture I if the principal asymptotic class AF is given by Γ

We explain that the Gamma class should be viewed as a square root of the Todd class (or b A-class) in the Riemann–Roch theorem. Let H denote the vector space of ∇-flat sections over the positive real line (cf. §2.8): H := {s : R>0 → H (F ) : ∇s = 0}

and let [·, ·) be a non-symmetric pairing on H defined by

[s1 , s2 ) := (s1 (e−iπ z), s2 (z))F

(3.3.4)

for s1 , s2 ∈ H, where s1 (e−iπ z) denotes the analytic continuation of s1 (z) along the semicircle [0, 1] ∋ θ 7→ e−iπθ z. 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 denote by the same symbol [·, ·) the corresponding pairing on H (F ). Since S(z) preserves the pairing (Proposition 2.3.1), we have 1 1 (eπiµ e−πiρ α, β)F = (eπiρ eπiµ α, β)F (3.3.5) [α, β) = (2π)dim F (2π)dim F for α, β ∈ H (F ). We write Ch(·), Td(·) for the following characteristic classes: Ch(V ) = (2πi)

deg 2

ch(V ) =

rank XV

e2πiδj

j=1

Td(V ) = (2πi)

deg 2

td(V ) =

rank YV j=1

2πiδj 1 − e−2πiδj

where δ1 , . . . , δrank V are the Chern roots of a vector bundle V . We also write TdF = Td(T F ). Lemma 3.3.6 ([44, 45]). Let V1 , V2 be vector bundles over F . Then h b F Ch(V1 ), Γ bF Ch(V2 ) = χ(V1 , V2 ) Γ

where χ(V1 , V2 ) =

Pdim F i=0

(−1)i dim Exti (V1 , V2 ).

Proof. Recall the Gamma function identity: Γ(1 + z)Γ(1 − z) =

(3.3.7)

This immediately implies that

Therefore

h

2πiz . eiπz − e−iπz

deg b F = TdF . bF · Γ eπiρ eπi 2 Γ

b F Ch(V1 ), Γ bF Ch(V2 ) = Γ

1 (2πi)dim F

Z deg eπi 2 Ch(V1 ) Ch(V2 ) TdF . F

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

24


b F ) of A satisfies the following Under Gamma Conjecture I, the canonical generator Φ(Γ normalization.

Proposition 3.3.8. Suppose that a Fano manifold F satisfies Gamma Conjecture I, b F )(z) = (2π)− dim F/2 S(z)z −µ z ρ Γ bF is a generator of A. Then the limit i.e. Φ(Γ b F )(z) ∈ E(T ) lim eT /z Φ(Γ

z→+0

b F )(z) is given has length one with respect to the Poincaré pairing (·, ·)F . In other words, Φ(Γ b by the Laplace transform (3.2.3) along [T, ∞] of a ∇-flat section ϕ(λ) such that ϕ(λ) is b is the holomorphic near λ = T and that ϕ(T ) is a T -eigenvector of unit length, where ∇ second structure connection in §2.5 with parameter ν = 0. b F )(z) exists and lies in E(T ). By the Proof. By Proposition 3.2.1, the limit limz→+0 eT /z Φ(Γ b construction there, there exists a ∇-flat section ϕ(λ) such that • ϕ(λ) is holomorphic near λ = T ; b F )(z) is given as the Laplace transform (3.2.3) of ϕ(λ); • Φ(Γ b F )(z) = ϕ(T ) ∈ E(T ). • limz→+0 eT /z Φ(Γ By bending the integration path of (3.2.3) as in Figure 7, we obtain the analytic continuation b F )(e−πi z). Again by Watson’s lemma, we have limz→+0 e−T /z Φ(Γ b F )(e−πi z) = ϕ(T ). Φ(Γ Therefore we have h b F )(e−πi z), Φ(Γ b F )(z) bF , Γ b F = Φ(Γ Γ → (ϕ(T ), ϕ(T ))F F

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

−πi

z) ✛

•

•

• •

•

T

b )(z) Φ(Γ ✲ F

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 of the dual quantum connection, which we call dual flat sections. We show that the limit limz→+0 eT /z f (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 coefficients. The dual quantum connection is a meromorphic flat connection on the trivial bundle H (F ) × P1 → P1 given by (cf. (2.2.1)) 1 ∂ + (c1 (F )⋆0 )t − µt (3.4.1) ∇∨ ∂ = z z ∂z ∂z z where the superscript “t” denotes the transpose. This is dual to the quantum connection ∇ in the sense that we have dhf (z), s(z)i = h∇∨ f (z), s(z)i + hf (z), ∇s(z)i


25

where h·, ·i 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 ∇ with the opposite sign of z. In order to avoid possible confusion about signs, we shall not use this identification. We write E(T )∗ ⊂ H (F ) for the one-dimensional eigenspace of (c1 (F )⋆0 )t with eigenvalue T and H ′ ∗ = Im(T − (c1 (F )⋆0 )t ) for the complementary subspace. They are dual to 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 ) := lim e−T /z f (z) z→+0

exists for every ∇∨ -flat section f (z) and lies in E(T )∗ . In other words, LA defines a linear functional: LA : f : R>0 → H (F ) : ∇∨ f (z) = 0 −→ E(T )∗ .

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

Proof. As usual we make use of the Laplace transformation. Let λ be the Laplace dual variable of z −1 . The Laplace dual of the dual quantum connection ∇∨ is given by (cf. §2.5): (3.4.3)

∂ e (ν) + (λ + (c1 (F )⋆0 )t )−1 (−µt + ν). ∇ ∂ = ∂λ ∂λ

Here ν ∈ R is a parameter. This is smooth away logarithmic at λ = −T and λ = ∞. By Lemma represented by the matrix:  ∗ ν    0 0

from the spectrum of (−c1 (F )⋆0 ) and is e (ν) at λ = −T is 3.2.4, the residue of ∇    

under the decomposition H (F ) = E(T )∗ ⊕ H ′ ∗ . When the residue is non-resonant, we have a e (ν) ) for fundamental solution near λ = −T of the form U (λ + T ) exp(− log(λ + T ) Resλ=−T ∇ some holomorphic U (x) with U (0) = id. Therefore we have:

e (0) -flat section ϕ0 (λ) which is holomorphic Claim 3.4.4. (1) Suppose ν = 0. Then we have a ∇ near λ = −T such that ϕ0 (−T ) is a non-zero eigenvector in E(T )∗ . e (ν) around λ = −T is a quasi-reflection, (2) Suppose ν ∈ / Z. The monodromy M of ∇ e (ν) -flat section around λ = −T is i.e. rank(id −M ) = 1. Moreover, a monodromy-invariant ∇ holomorphic at λ = −T .

First we construct a ∇∨ -flat section f0 (z) such that limz→+0 eT /z f0 (z) exists and is a none (0) -flat section ϕ0 (λ) in Claim 3.4.4 (1). By an argument zero eigenvector in E(T )∗ . Take a ∇ similar to §2.6 and §3.2, we can check that the Laplace transform Z 1 e−λ/z ϕ0 (λ)dλ (3.4.5) f0 (z) := z −T +eiφ R>0

satisfies the differential equation ∇∨ f0 (z) = 0 and we have limz→+0 eT /z f0 (z) = ϕ0 (−T ) ∈ E(T )∗ . Here we choose the phase φ ∈ (−π/2, π/2) so that the ray −T + eiφ R>0 does not e (0) other than −T . contain the singularities of ∇

26


Let f (z) be an arbitrary ∇∨ -flat section. Changing co-ordinates t = z −1 , we have 1 ∂f = (c1 (F )⋆0 )t f − µt f. ∂t t An argument using Gronwall’s inequality shows that there exist constants C0 , C1 > 0 such that kf (t−1 )k 6 C0 eC1 t for all t ∈ [1, ∞). Therefore the following Laplace transformation converges for λ with Re(λ) ≪ 0 and ν ≫ 0, ν ∈ / Z. Z ∞ tν−1 eλt f (t−1 )dt. (3.4.6) ϕ(λ) := 0

The condition ν ≫ 0 ensures that the integral converges near t = 0 (since ∇∨ is regular singular at t = 0, f (t−1 ) is of at most polynomial growth near t = 0). By the standard e (ν) in argument using integration by parts, we can show that ϕ(λ) is flat for the connection ∇ e (ν) is smooth on {Re(λ) < −T }, ϕ(λ) is analytically continued (3.4.3). Since the connection ∇ there. We now consider the Laplace transform (3.4.6) of the ∇∨ -flat section f (z) = f0 (z) defined e (ν) -flat section. Suppose ν ≫ 0 and ν ∈ in (3.4.5). Let ϕν (λ) denote the resulting ∇ / 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 Z Z ∞ ν−1 λt e−ξt ϕ0 (ξ)dξ t e dt ϕν (λ) = =

Z

0

−T +eiφ R>0

= Γ(ν)

Z

Z

−T +eiφ R>0 ∞ ν−1 t(λ−ξ)

t

0

−T +eiφ R>0

e

dt ϕ0 (ξ)dξ

(ξ − λ)−ν ϕ0 (ξ)dξ

for Re(λ) < −T . The right-hand side is analytically continued to the complement of the ray −T + eiφ R>0 . We can directly evaluate the monodromy around λ = −T using this formula and see that it is non-trivial. Alternatively we prove this by contradiction. Suppose ϕν (λ) has trivial monodromy around λ = −T . The above formula shows that it is analytically continued e (ν) is regular to the whole λ-plane without singularities by Part (2) of Claim 3.4.4. Since ∇ singular at λ = ∞, this implies that ϕν (λ) is a polynomial. Expand ϕν (λ) = φ0 + λφ1 + · · · + e (ν) ϕν = 0 gives (−µt + n + ν)φn = 0. But λn φn with φn 6= 0. Then the differential equation ∇ t for ν ≫ 0, −µ + n + ν has no kernel as n > 0. This is a contradiction.

e (ν) -flat section ϕ(λ) (3.4.6) associated to a general ∇∨ -flat section Now consider the ∇ f (z). Because the monodromy around λ = −T is a quasi-reflection (Claim 3.4.4) and the monodromy of ϕν (λ) around λ = −T is non-trivial (Claim 3.4.7), we can find a linear combination ϕ(λ) ˜ = ϕ(λ) + cϕν (λ), c ∈ C such that it has trivial monodromy around λ = −T . By Claim 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): Z ∞ tν−1 etλ f˜(t−1 )dt ϕ(λ) ˜ = 0

and we have the Laplace inversion formula: 1 ν−1 z f˜(z) = 2πi

Z

−a+i∞

−a−i∞

−λ/z ϕ(λ)e ˜ dλ


27

e (ν) ϕ(λ) which is valid for a > T ′ . The differential equation7 ∇ ˜ = 0 near λ = ∞ together with −2 the condition ν ≫ 0 ensures that ϕ(λ) ˜ = O(|λ| ) as λ → ∞ in the region {Re(λ) < −T ′ } and the above integral converges. For any a > T ′ , we have Z 1 ν−1 i∞ −a/z ˜ z ϕ(−a ˜ + x)e−x/z dx = 0. lim e f (z) = lim z→+0 z→+0 2πi −i∞ In particular, limz→+0 e−T /z f (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/z f (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 dim A > 1 in §3.2. To see that dim A = 1, it suffices to show that A ⊂ Im LA∗ , 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 defined by: ∇s(z) = 0, ∃λ > T ′ such that A˜ = s : R>0 → H (F ) : . keλ/z s(z)k 6→ ∞ as z → +0 ˜ Thus A˜ ⊂ Im(LA∗ ) implies A = A˜ = Im(LA∗ ); (1) and (3) Obviously we have A ⊂ A. ˜ Then there exist λ > T ′ and a sequence follow from this. Let s(z) be a flat section from A. {zn } such that zn ∈ R>0 , zn → +0 as n → ∞ and limn→∞ eλ/zn s(zn ) exists. Then for every f ∈ Ker(LA), we have D E hf, si = e−λ/zn f (zn ), eλ/zn s(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 behaviour of dual flat sections as z → +0 is given by the pairing with a flat section with the smallest asymptotics. From this we obtain a limit formula for AF . The fundamental solution for the dual quantum connection ∇∨ (3.4.1) is given by the inverse transpose (S(z)z −µ z ρ )−t of the fundamental solution for ∇ in Proposition 2.3.1. This yields the following identification (cf. (3.3.1)) (3.5.1)

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

dim F 2

t

t

S(z)−t z µ z −ρ α

between dual flat sections and homology classes. Proposition 3.5.2. Suppose that F satisfies Conjecture O. Let AF be a principal asymptotic class (Definition 3.3.2). Let ψ0∗ be an element of E(T )∗ which is dual to ψ0 := limz→+0 eT /z Φ(AF )(z) ∈ E(T ) in the sense that hψ0∗ , ψ0 i = 1. Then we have: LA(Φ∨ (α)) := lim e−T /z Φ∨ (α)(z) = hα, AF iψ0∗ . z→+0

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

28


Proof. By the definition of Φ and Φ∨ (see (3.3.1), (3.5.1)), we have: hα, AF i = hΦ∨ (α)(z), Φ(AF )(z)i = he−T /z Φ∨ (α)(z), eT /z Φ(AF )(z)i.

This converges to hLA(Φ∨ (α)), ψ0 i as z → +0 by Propositions 3.2.1, 3.4.2.

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

dim F 2

z −ρ z µ S(z)−1 1

where 1 ∈ H (F ) is the identity class. Alternatively we can define J(t) by the requirement that z dim F 2 (3.5.4) hα, J(t)i = hΦ∨ (α)(z), 1i 2π 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 to the 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

∂ ∂ i, where [∇c1 (F ) ]z=1 = t ∂t + (c1 (F )⋆c1 (F ) log t ). Diffor any differential operators P ∈ Cht, t ∂t 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) and Remark 2.3.2, we have   F N X X φ i tc1 (F )·d φi  (3.5.7) J(t) = ec1 (F ) log t 1 + 1 − ψ 0,1,d i=1 d∈Eff(F )\{0}

i N e where {φi }N i=1 and {φ }i=1 are bases of H (F ) which are dual with respect to the Poincar´ pairing (·, ·)F . This gives a well-known form of Givental’s J-function [30] restricted to the anticanonical line τ = c1 (F ) log t.

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

t→+∞

J(t) AF = h[pt], J(t)i h[pt], AF i

where the limit is taken over the positive real line and exists if and only if h[pt], AF i = 6 0. Proof. By Proposition 3.5.2, we have lim e−T /z Φ∨ (α)(z) = hα, AF iψ0∗

z→+0

for some non-zero ψ0∗ ∈

lim e−T /z Φ∨ ([pt])(z) = h[pt], AF iψ0∗ z→+0 E(T )∗ . Therefore by (3.5.4) we have

hα, J(t)i hα, AF i e−T /z hΦ∨ (α)(z), 1i = = lim −T /z ∨ t→+∞ h[pt], J(t)i z→+0 e h[pt], AF i hΦ ([pt])(z), 1i lim

for all α ∈ H (F ). Here we used the fact hψ0∗ , 1i 6= 0 which is proved in Lemma 3.5.10 below. The limit exists if and only if h[pt], AF i 6= 0. The Theorem is proved.


29

Corollary 3.5.9. Suppose that a Fano manifold F satisfies Conjecture O. The following statements are equivalent: (1) F satisfies Gamma Conjecture I (Definition 3.3.3). b F )(z)k 6 Ce−λ/z on (0, 1] for some C > 0 and λ > T ′ , where T ′ is given in (2) kΦ(Γ (3.1.10) and Φ is given in (3.3.1). (3) We have J(t) bF =Γ lim t→+∞ h[pt], J(t)i where J(t) is the J-function of F given in Definition 3.5.3 and the limit is taken over the 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 ψ0∗ ∈ E(T )∗ , 6 0. In other words, the E(T )-component of 1 with respect to the decomposiwe have hψ0∗ , 1i = 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 claim that ψ0 ⋆0 H ′ = 0. Take α ∈ H ′ and write α = T γ − c1 (F ) ⋆0 γ. Then we have ψ0 ⋆ α = T ψ0 ⋆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 obtain ψ0 = 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 problem of Fano manifolds. 3.6. Apery limit. We can replace the continuous limit of the ratio of the J-function appearing 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 generalization to Apery class was studied by Galkin [25]. This limit sees the primitive part of the Gamma class. Expand the J-function as X Jn tn . J(t) = ec1 (F ) log t n>0

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

N X

X

i=1 d∈Eff(F ):c1 (F )·d=n

D

φi ψ n−2−dim φi

EF

0,1,d

φi

where we set dim φi = dim F − 21 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 principal asymptotic class AF satisfies h[pt], AF i 6= 0. Let r be the Fano index. For every α ∈ H (F ) such that c1 (F ) ∩ α = 0, we have hα, Jrn i hα, AF i = 0. − lim inf n→∞ h[pt], Jrn i h[pt], AF i

30


Remark 3.6.2. Note that Jn = 0 unless r divides n. In the left-hand side of the above formula, we set hα, Jrn i/h[pt], Jrn i = ∞ if h[pt], Jrn i = 0. It is however expected that the following properties will hold for Fano manifolds: • the Gromov–Witten invariants h[pt], Jrn i are all positive for n > 0; • lim inf n→∞ 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 hβ, AF i 6= 0, we have the limit formula hα, Jrn i hα, AF i = 0. lim inf − n→∞ hβ, Jrn i hβ, AF i This can be shown by the same argument as below. We shall restrict to the case β = [pt] as Gamma Conjecture I implies h[pt], AF i 6= 0. b as: Define the functions G and G

G(t) := h[pt], J(t)i =

b G(κ) :=

∞ X

n=0

∞ X

Gn tn ,

n=0

1 n!Gn κ = κ n

Z

∞

where Gn := h[pt], Jn i,

G(t)e−t/κ dt.

0

b are called (unregularized or regularized) quantum periods of Fano manThese functions G, G ifolds [17].

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

Proof. Recall that we have G(t) = (2πt)− dim F/2 hΦ∨ ([pt])(z), 1i (see (3.5.4)) and Φ∨ ([pt])(z) is a ∇∨ -flat section (where z = t−1 ). Therefore the Laplace transform Z ∞ F − dim tν−1 eλt Φ∨ ([pt])(t−1 )dt ϕ(λ) = (2π) 2 0

e (ν) -flat as we discussed around (3.4.6). Then the convergence radius of with ν ≫ 0 is ∇ Z ∞ dim F tν+ 2 −1 G(t)e−t/κ dt hϕ(−κ−1 ), 1i = 0

(3.6.5)

= κν+

dim F 2

−1

∞ X

Γ(n + ν +

dim F 2

)Gn κn

n=0

e (ν) has no singularities in {|λ| > T }. The function G(κ) b is bigger than or equal to 1/T as ∇ −1 has the same radius of convergence as hϕ(−κ ), 1i and the conclusion follows. Lemma 3.6.6. Suppose that F satisfies Conjecture O and h[pt], AF i = 6 0. Then the converb 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 function hϕ(−κ−1 ), 1i in (3.6.5) has a singularity at κ = 1/T . By (3.5.4) we have G(t) = (2πt)− dim F/2 hΦ∨ ([pt])(z), 1i. Thus by Proposition 3.5.2 and Lemma 3.5.10, we have lim e−tT (2πt)

t→+∞

dim F 2

G(t) = h[pt], AF ihψ0∗ , 1i 6= 0.

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


31

hα,AF i [pt]. Then hβ, AF i = 0. It suffices to show that Proof of Theorem 3.6.1. Set β := α − h[pt],A Fi hβ, Jrn i = 0. (3.6.7) lim inf n→∞ h[pt], Jrn i P n − dim F/2 hΦ∨ (β)(t−1 ), 1i by Set bn := hβ, Jn i. Then we have ∞ n=0 bn t = hβ, J(t)i = (2πt) ∨ ∗ (3.5.4). By Proposition 3.5.2, we have LA(Φ (β)) = hβ, AF iψ0 = 0. Thus by Proposition 3.4.2 we have limt→+∞ e−tλ Φ∨ (β)(t−1 ) = 0 for every λ > T ′ . Consider again the Laplace transform Z ∞ F − dim ϕβ (λ) = (2π) 2 tν−1 etλ Φ∨ (β)(t−1 )dt 0

e (ν) -flat section which is regular on for ν ≫ 0 (as in the proof of Lemma 3.6.4). This is a ∇ ′ {λ : Re(λ) < −T }. Then its component ∞ X F −1 −1 ν+ dim 2 Γ(n + ν + dim2 F )bn κn hϕβ (−κ ), 1i = κ n=0

has the radius of convergence strictly bigger than 1/T . This is because the only singularities of e (ν) on {|λ| > T } are {−e2πik/r T : k = 0, . . . , r − 1} and hϕβ (λ), 1i is regular at λ = −T and ∇ dim F

satisfies hϕβ (e2πi/r λ), 1i = e−(ν+ 2 −1)2πi/r hϕβ (λ), 1i. (Here we used Part (2) of Conjecture O.) This implies that there exist numbers C, R > 0 such that R > 1/T and |bn n!| 6 CR−n .

On the other hand, Lemma 3.6.6 implies that

lim sup |Gn n!|1/n = T. n→∞

Let ǫ > 0 be such that (1 − ǫ)RT > 1. Then we have a number n0 such that for every n > n0 we have supm>n |Gm m!|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 bni ni ! CR−ni C = 6 Gn Gn ni ! T ni (1 − ǫ)ni = (RT (1 − ǫ))ni . i

i

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

3.7. Quantum cohomology central charges. For a vector bundle V on F , we define dim F b F Ch(V ))(z) . (3.7.1) Z(V ) := (2πz) 2 1, Φ(Γ F 8

This quantity is called the quantum cohomology central charge of V in [45]. Using the property S(z)∗ = S(−z)−1 , (3.3.1) and Definition 3.5.3, we can write Z(V ) in terms of the J-function: b ∗F Ch(V ∗ ) (3.7.2) Z(V ) = (−1)dim F eπiρ J(e−πi t), Γ deg 2

Qdim F

F

bF = b ∗ := (−1) Γ where t = z −1 and Γ i=1 Γ(1 − δi ) is the dual Gamma class of F . F Therefore, as a component of the J-function, Z(V ) gives a scalar-valued solution to the quantum differential equations (see Remark 3.5.5). If F satisfies Gamma Conjecture I, the central charge Z(O) satisfies the following smallest asymptotics: p dim F Z(O) ∼ (ψ0 , ψ0 )F (2πz) 2 e−T /z as z → +0 8The central charge Z(V ) here is denoted by (2πi)dim F Z(V ) in [45].

32


where ψ0 ∈ E(T ) is the idempotent for ⋆0 . This follows from Proposition 3.3.8 and the proof of Lemma 3.5.10. 4. Gamma Conjecture II In this section we restrict to a Fano manifold F with semisimple quantum cohomology and state a refinement of Gamma Conjecture I for such F . We represent the irregular monodromy of quantum connection by a marked reflection system (§2.8) given by a basis of asymptotically exponential flat sections. Gamma Conjecture II says that the marked reflection system b F Ch(Ei )} for some exceptional collection {Ei } of coincides with a certain Gamma-basis {Γ b the derived category Dcoh (F ). This also refines Dubrovin’s conjecture [20].

4.1. Semiorthonormal basis and Gram matrix. Let V = (V ; [·, ·); v1 , . . . , vN ) be a triple of 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 V is 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, hence vectors vi are linearly independent so N 6 dim V . We say that V is a semiorthonormal basis (SOB) if N = dim V i.e. vi form a basis of vector space V , this implies that pairing [·, ·) is non-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 , define the right mutation Ru v and the left mutation Lu v by Ru v := v − [v, u)u,

Lu v := v − [u, v)u.

The braid group BN on N strands acts on the set of SOBs of N vectors by: (4.1.1)

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

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 a tuple (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 ∈ S 1 such that the vectors vi are semiorthonormal in the sense that: ( 1 if i = j (4.2.1) [vi , vj ) = 0 if i 6= j and Im(e−iφ ui ) 6 Im(e−iφ uj ) cf. Proposition 2.9.1. Under an appropriate ordering of the basis, (V, [·, ·), v1 , . . . , vN ) gives an SOB. An MRS is said to be admissible if the phase φ is admissible for u, i.e. eiφ is not parallel to any non-zero difference ui − uj . By a small perturbation of the marking, one can always make an MRS admissible. The circle S 1 acts on the space of all marked reflection systems by rotating simultaneously the phase and all markings: φ 7→ α + φ, ui 7→ eiα ui . We write hφ : C → R for the R-linear function hφ (z) = Im(e−iφ z).


33

Remark 4.2.2 ([31]). The data of admissible marked reflection system may be used to produce a polarized local system on C\u (cf. the second structure connection): by identifying the fiber with 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 the turn around ui act in the monodromy representation by the reflection with respect to vi : rvi (v) = v − {v, vi }vi . 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 fiber the quotient-space V / Ker{·, ·} (use reflections with respect to images of vi in the quotientspaces). When u1 , . . . , uN are pairwise distinct, this construction could be reversed with a little ambiguity: 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 up to sign if {·, ·} is non-degenerate), and define [vi , vi ) = 1 for all i, for i 6= j put [vi , vj ) = 0 if hφ (ui ) < hφ (uj ) and [vi , vj ) = {vi , vj } otherwise. 4.3. Mutation of MRSs. We describe the change of an MRS under deformation of parameters (u, φ). The description here is analogous to the one in §§2.7–2.9. By perturbing the marking (or the phase) a little, we may start with an admissible MRS (see Remark 4.3.1 below). We draw a straight ray Li emanating from each marking ui in the direction of φ and obtain a picture as in Figure 1. We let the marking u and the phase φ vary. We allow that a multiple marking separates into distinct ones, but do not allow that two different markings ui 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 codimensionone wall consisting of non-admissible configurations (u, φ), we transform the vectors v1 , . . . , vN by a left/right mutation. The situation is illustrated in Figure 8; see also Figures 3 and 4. In this Figure 8, a vector vl equipped with the marking ui is transformed to: Y Y Rvk vl (right mutation) or Lvk vl (left mutation). k:uj =uk

k:uj =uk

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

❄

• ui

✻ left

• uj

✲ 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 the marking or the phase are connected by mutation.

34


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) × S 1 such that the fiber at an admissible parameter (u, eiφ ) is the set of all MRSs on the vector space V with marking u and phase φ. The base space CN (C) × S 1 has a wall and chamber structure given by the wall W of non-admissible parameters: W = {(u, eiφ ) ∈ CN (C) × S 1 : 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) × S 1 : hφ (ui ) = hφ (uj ) = hφ (uk ) for some i 6= j 6= k} ⊂ W. Note that W is invariant under the S 1 -action on CN (C) × S 1 discussed above. To define a local system, it suffices to define a parallel transportation along a path connecting two points from (CN (C) × S 1 ) \ W . For a given path, we can perturb the path so that it meets only the smooth part W \ W ′ of the wall. Then we define the parallel transportation as the composition of right/left mutations corresponding to crossings of the wall. To check that the parallel transportation depends only on the homotopy class of the path, it suffices to recall that the mutations satisfy the braid relation (2.8.4). Note that the S 1 symmetry for MRSs reduces the local system to the quotient space (CN (C) × S 1 )/S 1 ∼ = 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. For details 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 if Hom(E, E) = C and Hom(E, E[j]) = 0 for all j 6= 0. An ordered tuple of exceptional objects E1 , . . . , EN is called an exceptional collection if Hom(Ei , Ej ) = 0 for all i, j with i > j. If E1 , . . . , EN is an exceptional collection and k1 , . . . , kN is a set of integers then E1 [k1 ], . . . , EN [kN ] is also an exceptional collection, in this case we say that two collections coincide up to shift. An exceptional object E gives a pair of functors called left mutation LE and right mutation RE which fit into the distinguished triangles9: LE F [−1] → Hom• (E, F ) ⊗ E → F → LE F, RE F → F → Hom• (F, E)∗ ⊗ E → RE F [1].

The set of exceptional collections of N objects in T admits an action of the braid group BN in a way parallel to the BN -action (4.1.1) on SOBs, i.e. the generator σi replaces {Ei , Ei+1 } with {Ei+1 , REi+1 Ei } and σi−1 replaces {Ei , Ei+1 } with {LEi Ei+1 , Ei }. The subcategory in T generated by Ei (minimal full triangulated subcategory that contains all Ei ) stays invariant under mutations. An exceptional collection is called full if it generates the whole category T ; clearly braid group and shift action respects fullness. When a triangulated category T has a full exceptional collection E1 , . . . , EN , the Grothendieck K-group K 0 (T ) is the abelian group freely generated by P the classes [Ei ]. The Gram matrix Gij = χ(Ei , Ej ) of the Euler pairing χ(E, F ) = (−1)i dim Hom(E, F [i]) is uniuppertriangular in the basis {[Ei ]}. Thus any exceptional collection E1 , . . . , EN produces an SOB (K 0 (T ) ⊗ C; χ(·, ·); [E1 ], . . . , [En ]). The action of the braid group BN and shifts Ei → Ei [ki ] described above descends to the action of BN and change of signs on the Kgroup described in §4.1. 9What we write as L F , R F here are denoted respectively by L F [1], R F [−1] in [7, 35]. E E E E


35

4.5. Asymptotic basis and MRS of a Fano manifold. Here we resume the discussion in §2.8 and §2.9. Let F be a Fano manifold such that ⋆τ is convergent and semisimple for some τ ∈ H (F ). Choose a phase φ ∈ R that is admissible for the spectrum of (c1 (F )⋆τ ). By τ (z) of asymptotically exponential flat sections Proposition 2.9.1, we have a basis y1τ (z), . . . , yN for the quantum connection ∇|τ defined over the sector φ − π2 − ǫ < arg z < φ + π2 + ǫ for some ǫ > 0. They are canonically defined up to sign and ordering. The basis is marked by the spectrum u = {u1 , . . . , uN } of (c1 (F )⋆τ ). We can naturally identify yiτ (z) on the universal cover of the z-plane C× . By ∇-parallel transportation of yiτ (z) to τ = 0, arg z = 0 and |z| ≫ 1, τ of flat sections with a basis A , . . . , A of H (F ) via the we can identify the basis y1τ , . . . , yN 1 N −µ ρ fundamental solution S(z)z z in Proposition 2.3.1, i.e. dim F yiτ (z) parallel transport = Φ(Ai )(z) := (2π)− 2 S(z)z −µ z ρ Ai to τ = 0, arg z = 0

where Φ was given in (3.3.1). The basis {A1 , . . . , AN } is defined up to sign and ordering and is 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 up to sign. Note that MRS(F, τ, φ) and MRS(F, τ, φ + 2π) are not necessarily the same: they differ by the monodromy around z = 0. By the discussion in §§2.7–2.9, under isomonodromic deformation of the quantum connection, the corresponding MRSs change by mutation as described 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 the canonical position τ = 0 for a sufficiently small ǫ > 0. When φ = 0 is admissible for the spectrum of (c1 (F )⋆0 ), the two MRSs coincide. 4.6. Dubrovin’s (original) conjecture and Gamma conjecture II. We recall Dubrovin’s Conjecture (see also Remark 4.6.5). Let F be a Fano manifold. Dubrovin [20, Conjecture 4.2.2] conjectured: b (F ) admits a full (1) the quantum cohomology of F is semisimple if and only if Dcoh exceptional collection E1 , . . . , EN ;

and if the quantum cohomology of F is semisimple, there exists a full exceptional collection b (F ) such that: E1 , . . . , EN of Dcoh

(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 satisfying C ′ (c1 (F )a) = c1 (F )C ′ (a) for any a ∈ H (F ).

The central connection matrix C above is given by  | 1 A1 · · · C= (2π)dim F/2 |

 | AN  |

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

36


Definition 4.6.1. Let F be a Fano manifold such that the quantum product ⋆τ is convergent b (F ) has a full exceptional collection. We say and semisimple for some τ ∈ H (F ) and that Dcoh that F satisfies Gamma Conjecture II if there exists a full exceptional collection {E1 , . . . , EN } such 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)

b F Ch(Ei ). Ai = Γ

In other words, the operator C ′ in Part (3) of Dubrovin’s conjecture is given by C ′ (α) = b F α. (2π)− dim F/2 Γ

Remark 4.6.3. The validity of Gamma Conjecture II does not depend on the choice of τ and φ. Under deformation of τ and φ, the marked reflection system MRS(F, τ, φ) changes by mutations as described in §§2.7–2.9 and §4.3. On the other hand, the braid group acts on full exceptional collections by mutations (§4.4). These mutations are compatible in the K-group because 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 of F specify a distinguished mutation-equivalence class of (the images in the K-group of) full exceptional collections of F given by an asymptotic basis. It is not known in general whether any 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 formulated 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 [yiτ , yjτ ) of asymptotically exponential flat sections. Under Gamma Conjecture II, we have [yiτ , yjτ ) = [Ai , Aj ) = χ(Ei , Ej ), where we used the fact that the pairing [·, ·) is identified with the Euler pairing (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 the quantum product ⋆0 is semisimple. Suppose also that F satisfies Gamma Conjecture II. If the exceptional object corresponding to the eigenvalue T and an admissible phase φ with |φ| < π/2 is the structure sheaf O (or its shift), then F satisfies Gamma Conjecture I.

b F ) corresponding to O satisfies the asympProof. Under the assumptions, the flat section Φ(Γ −T /z b F )(z) ∼ e totics Φ(Γ ψ0 for some ψ0 ∈ E(T ), as z → 0 in the sector φ − π2 − ǫ < arg z < b F ) belongs to the space A (3.2.2). Gamma φ + π2 + ǫ (see Proposition 2.6.2). Therefore Φ(Γ Conjecture I holds (see Definition 3.3.3, Theorem 3.5.8).

4.7. Quantum cohomology central charges. Suppose that F satisfies Gamma Conjecture II and that ⋆0 is semisimple. Let ψ1 , . . . , ψN be the idempotent basis for ⋆0 and u1 , . . . , uN be the corresponding eigenvalues of (c1 (F )⋆0 ) i.e. c1 (F ) ⋆0 ψi = ui ψi . Let φ ∈ R be an admissible phase 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 the quantum cohomology central charges of E1 , . . . , EN (see (3.7.1), (3.7.2)) have the following asymptotics: p Z(Ei ) ∼ (ψi , ψi )F (2πz)dim F/2 e−ui /z


37

as z → 0 in the sector φ − π2 − ǫ < arg z < φ + π2 + ǫ for a sufficiently small ǫ > 0. This follows p from the definition (3.7.1) of Z(Ei ) and the asymptotics Φ(Ai )(z) ∼ e−ui /z ψi / (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 by many people [43, 52, 41, 42, 11, 44, 45, 34, 47]. In this section, we give several remarks on 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 that Gamma 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 is given 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 = xb1i1 · · · xbnin . 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 toric manifold and f be its mirror Laurent polynomial. Let Tcon be the value of f at the conifold point 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 X satisfies Gamma Conjecture I. Proof. It follows from the argument in [45, §4.3.1] that Z dx1 · · · dxn 1 b e−f (x)/z ϕ(x, z) φ, Φ(ΓX )(z) = x1 · · · xn X (2πz)n/2 (R>0 )n

± for z > 0, where Φ is the map in (3.3.1), φ ∈ H (X) and ϕ(x, z) ∈ C[x± 1 , . . . , xn , z] is such −f (x)/z that the class [ϕ(x, z)e dx1 · · · dxn /(x1 · · · xn )] corresponds to φ under the mirror isomorphism in [45, Proposition 4.8] and n = dim X. When ϕ = 1, one has ϕˇ = 1 and this is the quantum cohomology central charge (2πz)−n/2 Z(OX ); see §3.7. This integral representation b X )(z) has the smallest asymptotics, i.e. kΦ(Γ b X )(z)eTcon /z k implies that the flat section Φ(Γ grows at most polynomially as z → +0.

A toric manifold has a generically semisimple quantum cohomology. We note that the following 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 point τ ∈ H (X) and an admissible phase φ. There exists a Z-basis {[E1 ], . . . , [EN ]} of the K-group such that the matrix (χ(Ei , Ej ))16i,j6N is uni-uppertriangular and that the asymptotic basis b X Ch(Ei ), i = 1, . . . , N . (§4.5) of X at τ with respect to φ is given by Γ

38


Proof. It suffices to prove the theorem at a single semisimple point τ . If τ is in the image of the mirror map, we can take [Ei ] to be the mirror images of the Lefschetz thimbles (with phase φ) under the isomorphism between the integral structures of the A-model and of the B-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 toric orbifold (under mild technical assumptions). Remark 5.1.5. Kontsevich’s homological mirror symmetry suggests that the basis b (X) be[E1 ], . . . , [EN ] in the K-group should be lifted to an exceptional collection in Dcoh cause the corresponding Lefschetz thimbles form an exceptional collection in the Fukaya-Seidel category of the mirror. 5.2. Toric complete intersections. In this section Y denotes a Fano complete intersection in 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 is shown in [55, Proposition 4], [47, Corollary 2.5] that the ambient part Hamb (Y ) is closed under quantum multiplication ⋆τ when τ ∈ Hamb (Y ). Therefore it makes sense to ask if the ambient part (Hamb (Y ), ⋆τ ) satisfies Conjecture O and Gamma Conjectures I, II. Note also b Y belongs to the ambient part H (Y ). A mirror theorem of Givental [29] gives the that Γ amb information of the ambient part of quantum cohomology of Y ; hence one can first try to prove Gamma 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 number T 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 a module over the algebra (Hamb (Y ), ⋆0 ). We now assume that Y is a complete intersection in X obtained from the following nef partition [5, 29]. Let D1 , . . . , Dm denote prime toric divisors of X. Each prime toric divisor Di corresponds to a primitive generator bi ∈ Zn of a 1-dimensional cone of thePfan of X. We assume that there exists a partition {1, . . . , m} = I0 ⊔ I1 ⊔ · · · ⊔ Il such that i∈Ik Di is nef P P for k = 1, . . . , l and i∈I0 Di 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=1 Lk over X. Then P P 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: X fk (x) = −c0 δ0,k + xbi i∈Ik

where c0 is the constant arising from Givental’s mirror map [29]: Ql X k=1 (c1 (Lk ) · d)! Q c0 := . m i=1 ([Di ] · d)! d∈Eff(X):c1 (Y )·d=1 [Di ]·d>0 (∀i)


39

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

dx1 x1

∧ ··· ∧

dxn xn

df1 ∧ · · · ∧ dfl

.

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 ; we call 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 1 + x1 + · · · + xn−d − δd,n d! f0 (x) = x1 x2 · · · xn

on the affine variety Z = {x ∈ (C× )n : f1 (x) = xn−d+1 + xn−d+2 + · · · + xn = 1}. Following Przyjalkowski [56], introduce homogeneous co-ordinates [yn−d+1 : · · · : yn ] of Pn−d−1 and consider the change of variables: yi xi = for n − d + 1 6 i 6 n. yn−d+1 + · · · + yn Then we have (yn−d+1 + · · · + yn )d + x1 + · · · + xn−d − δn,d d! f0 (x) = Qn−d Qn i=1 xi · j=n−d+1 yj

Setting yn = 1 we obtain a Laurent polynomial expression (with positive coefficients) for f0 in 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 Condition 5.2.2. (Similarly, one can obtain Laurent polynomial expressions for the mirrors of complete intersections in weighted projective spaces [56].) One can also check that Tcon = (n + 1 − d)dd/(n+1−d) − δn,d d! coincides with T for Y and gives 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 )⋆0 θ = 0 for every primitive class θ ∈ H n−1 (Y ): this follows from the fact that (c1 (Y )⋆0 ) preserves the primitive part in H n−1 (Y ) and for degree reasons. Therefore Condition 5.2.2 holds for Y if n + 1 − d > 1.

Remark 5.2.4 ([47]). Using an inequality of the form β1 u1 + · · · + βm um > uβ1 1 · · · uβmm for P ui > 0, βi > 0, i βi = 1, we find that f0 |Zreal is bounded (from below) by a convex function: n X −1/k 1/k ) (xi + xi f0 (x) + l = f0 (x) + f1 (x) + · · · + fl (x) > −c0 + ǫ i=1

∀x ∈ Zreal

40


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) below converges. Theorem 5.2.5. Let Y be a Fano complete intersection in a toric manifold constructed from a nef partition as above. Suppose that Y satisfies Condition 5.2.2. Then Gamma Conjecture I 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: Z dim Y b Y )(z) e−f0 /z ωZ . 1, Φ(Γ = (5.2.6) Z(OY ) = (2πz) 2 Y

Zreal

The constant c0 in f0 comes from the value of the mirror map at τ = 0. In [47], the mirror 2 (Y ) and the same integral representation is shown to (Z, f0 ) is parameterized by τ ∈ Hamb 2 (Y, R). Since H 2 hold for general τ ∈ Hamb amb (Y ) is generated by Hamb (Y ), by differentiating this integral representation in τ , we obtain an integral representation of the form: Z dim Y b 2 ϕ(x, z)e−f0 /z ωZ φ, Φ(ΓY )(z) = (2πz) Y

Zreal

b Y ) takes values for every φ ∈ Hamb (Y ) (for some function ϕ(x, z)). Since the flat section Φ(Γ b Y ). in the ambient part Hamb (Y ), we obtain integral representations for all components of Φ(Γ T /z b Y )(z)e k grows at most These integral representations and Condition 5.2.2 show that kΦ(Γ polynomially as z → +0.

5.3. Compatibility with the quantum Lefschetz principle. We give an argument to show that our Gamma conjecture is compatible with the quantum Lefschetz principle in general. 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. Let Y ⊂ X be a degree-a Fano hypersurface in the linear system |ah| with 0 < a < r. Assuming the 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

∞ X

Jrn trn

n=0

with Jd ∈ H (X). By quantum Lefschetz theorem [16] of Coates-Givental, the J-function of Y is ∞ X (r−a)h log t−c0 t JY (t) = e (ah + 1) · · · (ah + an)(i∗ Jrn )t(r−a)n , n=0

where i : Y → X is the natural inclusion and c0 is the constant: ( P a! h·d=1 h[pt]ψ r−2 iX 0,1,d if r − a = 1; c0 = 0 if r − a > 1.

The quantum Lefschetz principle can be rephrased in terms of the Laplace transformation: Lemma 5.3.1. We have a/(r−a)

JY (t = u

e−c0 t )= Γ(1 + ah)u

Z

∞ 0

i∗ JX (t = q a/r )e−q/u dq.


41

Suppose that X satisfies Gamma Conjecture I. By Proposition 3.5.2 and (3.5.4), we find the following asymptotics of JX : dim X b X 1 + O(t−1 ) JX (t) ∼ Ct− 2 eTX t Γ

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 the asymptotics dim Y b Y 1 + O(t−1 ) JY (t) ∼ C ′ t− 2 e(T0 −c0 )t Γ as t → +∞. b Y = i∗ Γ b X /Γ(1 + ah) and the This would imply Gamma Conjecture I for Y . Here we used Γ number T0 > 0 is determined by the relation: r−a TX r T0 = aa . r−a r

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

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

n→∞

hγ, Jrn i bX i = hγ, Γ h[pt], Jrn i

for γ ∈ H (X) with c1 (X) ∩ γ = 0. Both sides do not change when we pass to a Fano hypersurface Y in | − (a/r)KX | (when γ is the push-forward of a class in H (Y )). Therefore the 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 method of Dubrovin [19, Example 4.4]. The Gamma conjectures for projective spaces also follow from the computations in [44, 45, 50]. A more general result for toric varieties was explained in the previous section, but we include a detailed proof for the projective spaces for the convenience of the reader. Theorem 6.0.4. Gamma Conjectures I and II hold for the projective space P = PN −1 . b P Ch(O(i)) associated to An asymptotic basis is formed by mutations of the Gamma basis Γ Beilinson’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)) ∈ H 2 (P) denote the hyperplane class. The quantum product by h is given by: ( hi+1 if 0 6 i 6 N − 2; h ⋆0 hi = 1 if i = N − 1.

It follows that the quantum multiplication (c1 (P)⋆0 ) = N (h⋆0 ) is a semisimple operator with pairwise 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 PN −1 N−1 a cohomology-valued function f (z) = i=0 fi (z)z i− 2 hN −1−i reads: fi (z) = N −i (z∂z )i f0 (z) N

f0 (z) = z N

−N

N

1 6 i 6 N − 1,

(z∂z ) f0 (z).

42


Therefore a flat section f (z) for ∇ is determined by its top-degree component z − which satisfies the scalar differential equation: (6.1.1) D N − N N (−t)N f0 = 0

N−1 2

f0 (z)

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 ∞ X Γ(h − n)N N n −N h log t t (6.2.1) Π(t; h) := e Γ(h)N n=0

C[h]/(hN ).

taking values in the ring Expanding Π(t; h) in the nilpotent element h, we obtain a 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)} yields the fundamental solution S(z)z −µ z ρ of ∇ near the regular singular point t = z −1 = 0 in Proposition 2.3.1. More precisely, we have:   N−1 t− 2 (− N1 D)N −1 Πk (t)   ..   . −µ ρ N −1−k   (6.2.2) S(z)z z h = N−3  1 2 (− N D)Πk (t)   t N−1 t 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 : Z c+i∞ 1 (6.2.3) Ψ(t) := Γ(s)N t−N s ds 2πi c−i∞

with c > 0. The integral does not depend on c > 0. One can easily check that Ψ(t) satisfies equation (6.1.1) by using the identity sΓ(s) = Γ(s + 1). Comparing (6.2.1) and (6.2.3), we can think of Ψ(t) as being obtained from Π(t; h) by multiplying Γ(h)N and replacing the discrete sum 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 line by requiring that log t ∈ R and t−N s = e−N s log t for t ∈ R>0 . We see that Ψ(t) gives rise to a flat 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−N t (1 + O(t−1 ))

as t → ∞ in the sector − π2 < arg t < π2 , where C = N −1/2 (2π)(N −1)/2 . Write the Gamma b P = Γ(1 + h)N = PN −1 ck hN −1−k . Then we have the following class of P = PN −1 as Γ k=0 connection formula: Z N −1 X bP ∪ Π(t; h) Ψ(t) = ck Πk (t) = Γ k=0

P

under the analytic continuation along the positive real line.


43

The connection formula in this proposition and equation (6.2.2) show that:   N−1 t− 2 (− N1 D)N −1 Ψ(t)   ..   . −µ ρ b . (6.2.5) S(z)z z ΓP =  N−3    t 2 (− N1 D)Ψ(t)  N−1 t 2 Ψ(t)

This flat section has the smallest asymptotics ∼ e−N t as t → +∞ (see (3.2.2)). Therefore we conclude (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 of stationary phase to obtain the asymptotics of Ψ(t). We write Z c+i∞ 1 eft (s) ds Ψ(t) = 2πi c−i∞ with ft (s) = N (log Γ(s) − s log t). The integral is approximated by a contribution around the critical point of ft (s). Using Stirling’s formula 1 1 log s − s + log(2π) + O(s−1 ) log Γ(s) ∼ s − 2 2 we find a critical point s0 of ft (s) such that s0 ∼ t + ft (s0 ) ∼ −N t + ft′′ (s0 ) ∼

1 2

+ O(t−1 ) as t → ∞. Then we have

N log(2π/t) + O(t−1 ), 2

N + O(t−2 ). t

From these we obtain the asymptotics: 1 1 ft (s0 ) √ Ψ(t) ∼ p e = 2πft′′ (s0 ) N

2π t

N−1 2

e−N t .

To show the connection formula, we close the integration contour in (6.2.3) to the left and express 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−N s ds = Resh=0 Γ(h − n)N tN n−N h dh Z Γ(h − n)N N n−N h = Γ(1 + h)N t . Γ(h)N P R In the second line we used the fact that Γ(1 + h)/Γ(h) = h and P g(h) = Resh=0 (g(h)/hN )dh for any g(h) ∈ C[[h]]. Therefore we have Z ∞ X N −N s bP ∪ Π(t; h) Ress=−n Γ(s) t ds = Γ Ψ(t) = n=0

as required.

P

44


6.3. Monodromy transformation and mutation. We use monodromy transformation and mutation to deduce Gamma Conjecture II for P from the truth of Gamma Conjecture I for 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) ∼ Cj t−

N−1 2

e−N ζ

−j

2πj π (N −1)πij/N . in the sector − π2 + 2πj N < arg t < 2 + N where ζ = exp(2πi/N ) and Cj = Ce −2πij/N Using Π(e t; h) = Ch(O(i)) ∪ Π(t; h), we also find the connection formula Z (j) b P Ch(O(j)) ∪ Π(t; h). Ψ (t) = Γ P

Let yj (z) be the flat section corresponding to Ψ(j) (cf. (6.2.5)):   N−1 t− 2 (− N1 D)N −1 Ψ(j) (t)   ..   . − N−1 .  2 yj (z) := (2π) N−3    t 2 (− N1 D)Ψ(j) (t)  N−1 t 2 Ψ(j) (t)

The above connection formula and (6.2.2) show that N−1 b P Ch(O(i)) = Φ Γ b P Ch(O(i)) (6.3.2) yj (z) = (2π)− 2 S(z)z −µ z ρ Γ

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 exponential fundamental solution in the sense of Proposition 2.6.2. We will see however that they give it after 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: Z 1 ∞ ϕ(λ)e−λ/z dλ y0 (z) = z T b b is the second for some ∇-flat section ϕ(λ) holomorphic near λ = T (= N ). Recall that ∇ structure connection (§2.5) with ν = 0. This integral representation is valid when Re(z) > 0. We have Z ζ −j ∞ 2πijµ/N −j 2πijµ/N 2πij/N e ϕ(λ)e−ζ λ/z dλ yj (z) = e y0 (e z) = z T Z e2πijµ/N ϕ(ζ j λ)e−λ/z dλ. = T ζ −j +R>0 ζ −j

b The Set ϕj (λ) = e2πijµ/N ϕ(ζ j λ). Then ϕj (λ) is holomorphic near λ = ζ −j T and is flat for ∇. latter fact follows easily from (3.1.4). Thus we obtain an integral representation of yj (z): Z 1 yj (z) = ϕj (λ)e−λ/z dλ z T ζ −j +R>0 ζ −j 2πj π which is valid when − π2 − 2πj N < arg z < 2 − N . By bending the radial integration path, we can 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


45

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

Figure 9. Bent paths starting from T ζ −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 Proposition 2.6.2 by straightening the paths (see Figure 10). Note that limz→+0 eT /z y0 (z) = ϕ(T ) is a T -eigenvector of (c1 (P)⋆0 ) of unit length by Proposition 3.3.8. Therefore ϕj (ζ −j T ) = e2πijµ/N ϕ(T ) is also of unit length. It is a ζ −j T -eigenvector of (c1 (P)⋆0 ) by (3.1.4). Therefore ϕj (ζ −j T ), j ∈ [j0 , j0 + N − 1] form a normalized idempotent basis for ⋆0 . Let φ be an admissible phase for the spectrum of (c1 (P)⋆0 ) which is close to zero. By the discussion in §2.6, the ∇-flat sections Z −j 1 ϕj (λ)e−λ/z dλ ∼ e−ζ T /z ϕj (ζ −j T ) xj (z) = z T ζ −j +R>0 eiφ with j ∈ [j0 , j0 + N − 1] give the asymptotically exponential fundamental solution associated to eiφ . By the argument in §2.7, the two bases {yj } and {xj } are related by a sequence of mutations. Because {yj } corresponds to the Beilinson collection {O(j)} (see (6.3.2)), {xj } corresponds to a mutation of {O(j)} (see §§2.7–2.9 and Remark 4.6.3). This shows that the b P Ch(O(j)), asymptotic basis at τ = 0 with respect to phase φ is given by a mutation of Γ j0 6 j 6 j0 + N − 1. The proof of Theorem 6.0.4 is now complete.

46


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. Although our paths in Figure 9 are not straight, we can make them straight after isomonodromic deformation (see Figure 11). We can therefore apply our mutation argument to such straight paths. When the eigenvalues of E⋆τ align as in Figure 11, the corresponding flat sections yj form an asymptotically exponential fundamental solution and the Beilinson collection gives b P Ch(O(j))} at such a point τ . the asymptotic basis {Γ 7. Gamma conjectures for Grassmannians

We derive Gamma Conjectures for Grassmannians of type A from the truth of Gamma Conjectures for projective spaces. Let G = G(r, N ) denote the Grassmannian of r-dimensional subspaces of CN and let P = PN −1 = G(1, N ) denote the projective space of dimension N − 1. 7.1. Statement. Let S ν 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 that b (G), where ν ranges over the vector bundles S ν V ∗ form a full exceptional collection of Dcoh all 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 b G Ch(S ν V ∗ ) associated to Kapranov’s basis of G is formed by mutations of the Gamma basis Γ exceptional collection {S ν V ∗ : 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 Gamma Conjecture I for G will be completed in §7.8. 7.2. Quantum Pieri and quantum Satake. For background material on classical cohomology rings of Grassmannians, we refer the reader to [23]. Let x1 , . . . , xr be the Chern roots of the dual V ∗ of the tautological bundle over G = G(r, N ). Every cohomology class of G can be written as a symmetric polynomial in x1 , . . . , xr . We have: H (G) ∼ = C[x1 , . . . , xr ]Sr hhN −r+1 , . . . , hN i

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


47

where hi = hi (x1 , . . . , xr ) is the ith complete symmetric polynomial of x1 , . . . , xr . An additive basis of H (G) is given by the Schur polynomials11 λ +r−j

(7.2.1)

σλ = σλ (x1 , . . . , xr ) =

det(xi j

)16i,j6r r−j det(xi )16i,j6r

with partition λ in an r×(N P−r) rectangle. The class σλ is Poincaré dual to the Schubert cycle Ωλ and deg σλ = 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 the universal quotient bundle. They are called the special Schubert classes. The first Chern class is given by c1 (G) = N σ1 . The classical Pieri formula describes the multiplication by the special Schubert classes: X (7.2.2) σk ∪ σλ = σν

where the sum ranges over all partitions ν = (ν1 > · · · > νr ) in an r × (N − r) rectangle such that |ν| = |λ| + k and ν1 > λ1 > ν2 > λ2 > · · · > νr > λr . Bertram’s quantum Pieri formula gives the quantum multiplication by special Schubert classes: Proposition 7.2.3 (Quantum Pieri [8], see also [10, 24]). We have X X 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 sum ranges 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 the wedge power of the quantum connection of P. This is an instance of the quantum Satake principle of Golyshev-Manivel [33]. The geometric Satake correspondence of Ginzburg [26] implies that the intersection cohomology of an affine Schubert variety Xλ in the affine Grassmannian of a complex Lie group G becomes an irreducible representation of the Langlands dual Lie group GL . Our target spaces P, G arise as certain minuscule Schubert varieties in the affine Grassmannian of GL(N, C). Therefore the cohomology groups of P and G are representations of GL(N, C): H (P) is the standard representation of GL(H (P)) ∼ = GL(N, C) and H (G) is the r-th wedge power of the standard representation. In fact, by the ‘take the span’ rational map P × · · · × P (r factors) 99K G, we obtain the Satake identification (7.2.4)

∼ =

Sat : ∧r H (P) −→ H (G),

σλ1 +r−1 ∧ σλ2 +r−2 ∧ · · · ∧ σλr 7−→ σλ

where σλi +r−i = (σ1 )λi +r−i is the special Schubert class of P = G(1, N ). The Satake identification 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) coincides, upon the Satake identification Sat, with the action of (c1 (P)⋆πi(r−1)σ1 ) ∈ gl(H (P)) on the r-th wedge representation ∧r H (P). Proof. Recall that c1 (P) = N σ1 and c1 (G) = N σ1 . Thus it suffices to examine the quantum product 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) ∼ = ∧r H (P) is also indicated by this expression.

48


of the r-th wedge ∧r H (P) is given by r X i=1

=

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

r X i=1

ith

!

σλ1 +r−1 ∧ · · · ∧ σλi +r−i+1 ∧ · · · ∧ σλr + (−1)r−1 δN −1,λ1 +r−1 σ0 ∧ σλ2 +r−2 ∧ · · ·∧ σλr

Under the Satake identification, the first term corresponds to the classical Pieri formula and the second term corresponds to the quantum correction. The sign (−1)r−1 cancels the sign coming 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 the r-th wedge representation ∧r H (P), cf. Theorem 7.3.1. We have a similar result for the grading operator. The following lemma together with Proposition 7.2.5 implies that the quantum connection of G is the wedge product of the quantum 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 ∧r H (P). Proof. This is immediate from the definition: we use dim G + r(r − 1) = r dim P.

The Satake identification also respects the Poincaré pairing up to sign. The Poincaré pairing on H (P) induces the pairing on the r-th wedge ∧r H (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 ∈ ∧r H (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 for all 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. o n i1 ir ) : 0 6 i1 < i2 < · · · < ir 6 N − 1 Spec(c1 (G)⋆0 ) = N e(r−1)πi/N (ζN + · · · + ζN

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 if k! and N are coprime, where k = min(r, N − r).


49

7.3. The wedge product of the big quantum connection of P. The quantum Satake principle implies that the quantum connection for G (at τ = 0) is the r-th wedge product of the quantum connection for P (at τ = (r − 1)πi). By using the abelian/non-abelian correspondence of Bertram–Ciocan-Fontanine–Kim–Sabbah [9, 15, 51], we observe that this is also true for the isomonodromic deformation corresponding to the big quantum cohomology of P and G. When the eigenvalues of (c1 (G)⋆0 ) are pairwise distinct, this can be also deduced from the quantum Pieri (Proposition 7.2.3) and Dubrovin’s reconstruction theorem [18]; however it 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◦ = {N eπi(r−1)/N e2πik/N : 0 6 k 6 N − 1}. As explained in §2.8, the quantum connection of P has an isomonodromic deformation over the universal cover CN (C)∼ of the configuration space (2.8.1) of distinct N points in C. Let ∇P be the connection 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 identified with the big quantum connection (2.2.3) of P near u◦ . Consider the r-th wedge product of this bundle: (∧r H (P)) × (CN (C)∼ × P1 ) → (CN (C)∼ × P1 ) equipped with the meromorphic flat connection ∇∧P : r X ξ1 ∧ · · · ∧ (∇P ξi ) ∧ · · · ∧ ξr . ∇∧P (ξ1 ∧ ξ2 ∧ · · · ∧ ξr ) := i=1

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 the Satake identification Sat (7.2.4). More precisely, the bundle map ∧r H (P) × (CN (C)∼ × P1 ) −→ H (G) × (CN (C)∼ × P1 ) (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 Frobenius manifolds” in [51]. Let us first review the construction of the alternate product for the big quantum cohomology Frobenius manifold of P. A pre-Saito structure [51, §1.1] with base M is a meromorphic flat connection ∇ on a trivial vector bundle E 0 × (M × P1 ) → M × P1 of the form: 1 dz 1 ∇ = d + C + (− U + V ) z z z for some C ∈ End(E 0 ) ⊗ Ω1M and U, V ∈ End(E 0 ) ⊗ OM (it follows from the flatness of ∇ that V is constant). Here E 0 is a finite dimensional complex vector space and the base space M is a complex manifold. The big quantum connection (2.2.3) is an example of a pre-Saito structure. Consider the quantum connection of P restricted to H 2 (P) × P1 . This gives a pre-Saito structure. The external tensor product of this pre-Saito structure yields a pre-Saito structure ∇×r on the bundle: ⊗r H (P) × ((H 2 (P))r × P1 ) → (H 2 (P))r × P1 .

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

50


(H 2 (P))r is embedded into ⊗r H (P) by the map χ1 : (H 2 (P))r ֒→ ⊗r H (P),

χ1 (τ1 , . . . , τr ) =

r X i=1

ith

1 ⊗ · · · ⊗ 1 ⊗ τi ⊗ 1 ⊗ · · · ⊗ 1.

The universal deformation here is a pre-Saito structure ∇⊗r on the bundle ⊗r H (P) × (⊗r H (P) × P1 ) → ⊗r H (P) × P1 which is pulled back by χ1 to the pre-Saito structure ∇×r . More precisely, the base of the universal deformation is a germ along the diagonal H 2 (P) ⊂ (H 2 (P))r ⊂ ⊗r H (P). Geometrically 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-Saito structure ∇⊗r to the W -invariant base Symr H (P) ⊂ ⊗r H (P); fibers of the restriction are representations of W . By taking the anti-symmetric part, we obtain a pre-Saito structure ∇Sym on the bundle ∧r H (P) × (Symr H (P) × P1 ) → Symr H (P) × P1 . Again the base is a germ along the diagonal H 2 (P) ֒→ Symr (H (P)). We further restrict this pre-Saito structure to the subspace Elemr H (P) ⊂ Symr H (P) spanned by “elementary symmetric” vectors Elemr H (P) =

r M k=1

C

X

i1P ,...,ir ∈{0,1} r a=1 ia =k

σi1 ⊗ · · · ⊗ σir ⊂ Symr H (P)

which contains the diagonal H 2 (P). By Hertling-Manin’s reconstruction theorem and [51, Lemma 2.9], we obtain a universal deformation of the pre-Saito structure ∇Sym |Elemr H (P) over the base ∧r H (P). More precisely, we have an embedding12 χ2 : Elemr H (P) ֒→ ∧r H (P)

such that χ2 (χ1 (τ, . . . , τ )) = τ σr−1 ∧ σr−2 ∧ · · · ∧ σ0

and a pre-Saito structure ∇∧r on the bundle (7.3.2)

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

which is a universal deformation of ∇Sym |Elemr H (P) via the embedding χ2 . The pre-Saito structure ∇∧r together with a primitive section cσr−1 ∧ · · · ∧ σ0 (for some c ∈ C) and the metric (·, ·)∧P endows the base space ∧r H (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)πiσ1 is isomorphic to the pre-Saito structure on the bundle (7.3.3)

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

defined by the big quantum connection of G. By the construction in [15, §3], the isomorphism between the above two pre-Saito bundles (7.3.2), (7.3.3) is induced by the Satake identification Sat between the fibers (up to a scalar multiple). 12The embedding χ is a highly non-trivial map except along the diagonal H 2 (P). 2


51

By the universal property of the pre-Saito structures ∇⊗r and ∇∧r , we obtain maps χ ˜1 , χ ˜2 extending χ1 and χ2 which fit into the following commutative diagram. χ1

? _ Symr H (P) o ? _ Elemr H (P) / ⊗r H (P) o _ P 8 8 P PPP qq rr q r q r P PPP χ2 qq rrr PPP χ ˜2 qqq q ' rrr χ˜1 q q qqq (H (P))r ∧r H (P) q q ❡❡❡❡❡2 ψ O ❡ q ❡ ❡ q ❡ ❡ ❡ ❡❡ qq ❡❡❡❡❡❡ qqq ∆ ❡❡❡❡❡f❡ q ❡ ❡ ❡ q ❡ ❡ ? + qq❡❡❡❡❡❡❡❡❡❡ ❡

r (H 2 (P))

_

H (P)

The maps χ ˜1 , χ ˜2 are such that • the pre-Saito structure over (H (P))r defined by the r-fold external tensor product of the big quantum connection of P is isomorphic to the pull-back of ∇⊗r by χ ˜1 ; the isomorphism here restricts to the given one along (H 2 (P))r (i.e. the fibers of the two pre-Saito bundles are identified by the identity of ⊗r H (P)); • the pre-Saito structure ∇Sym over Symr H (P) is isomorphic to the pull-back of ∇∧r by χ ˜2 ; the isomorphism here restricts to the given one along Elemr H (P) (i.e. the fibers of the two pre-Saito bundles are identified by the identity of ∧r H (P)). The (non-injective) map χ ˜1 can be given explicitly: χ ˜1 (τ1 , . . . , τr ) =

r X i=1

ith

1 ⊗ · · · ⊗ τi ⊗ · · · ⊗ 1.

Pre-composed with the diagonal map ∆ : H (P) → (H (P))r , χ ˜1 induces a map ψ : H (P) → Symr H (P). We have that the pre-Saito structure on H (P) defined as the r-fold tensor product of the big quantum connection of P is isomorphic to the pull-back of ∇⊗r |Symr H (P) by ψ, where the identification of fibers is the identity of ⊗r H (P). Therefore the pull-back of the pre-Saito structure ∇Sym (defined as the anti-symmetric part of ∇⊗r |Symr H (P) ) by ψ is naturally identified with the r-th wedge power ∇∧P of the big quantum connection of P. Hence the composition f = χ ˜2 ◦ ψ pulls back the pre-Saito structure ∇∧r to the pre-Saito ∧P structure ∇ . 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 G are linearly independent, as they correspond to z∇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 at u ∈ CN (C)∼ near u◦ and write ∆i = (ψi , ψi )−1 P . 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 Q ( ra=1 ∆ia ) det ((ψia , σr−b )P )16a,b6r Sat(ψi1 ∧ · · · ∧ ψir ) with 1 6 i1 < i2 < · · · < ir 6 N ;

52


(3) the eigenvalues of the Euler multiplication (E G ⋆τ ) of G at τ = f (u) are given by ui1 + · · · + 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 this case, 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 ∧r H (P) is semisimple with pairwise distinct eigenvalues. Theorem 7.3.1 implies that w⋆u is conjugate to dfu (w)⋆f (u) under Sat. This proves Part (1). Moreover the eigenspace C Sat(ψi1 ∧ · · · ∧ ψir ), i1 < i2 < · · · < ir of dfu (w)⋆f (u) contains a unique idempotent basis vector ψi1 ,...,ir ∈ H (G). Set ψi1 ,...,ir = c Sat(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 (E P ⋆τ ) on ∧r H (P) is conjugate to (E G ⋆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 ∧r M is defined by the data: • • • • •

the the the the the

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

Note that the basis {vi1 ∧ · · · ∧ vir } is determined up to sign because {v1 , . . . , vN } is an unordered basis. In other words, ∧r M is defined up to sign. The following lemma shows that the above data is indeed an MRS. Recall that hφ (u) = Im(e−iφ u). Lemma 7.4.1. Let 1 6 i1 < · · · < ir 6 N , 1 6 j1 < · · · < jr 6 N be increasing sequences of integers. If hφ (ui1 + · · · + uir ) = hφ (uj1 + · · · + ujr ), we have ( 1 if ia = ja for all a = 1, . . . , r; [vi1 ∧ · · · ∧ vir , vj1 ∧ · · · ∧ vjr ) = 0 otherwise. Q 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), ∧r M1 and ∧r M2 are also related by mutations. We omit a proof of this fact since we do not use it in this paper; the details are left to the reader.


53

7.5. MRS of Grassmannian. Using the result from §7.3, we show that the MRS of G is isomorphic to the wedge product of the MRS of P. By the results in §7.3, the flat connections ∇P and ∇∧P on the base CN (C)∼ give isomonodromic deformations of the quantum connections of P and G respectively. Therefore we can define the MRSs of P or G at a point u ∈ 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 , eiφ ) 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 wedge {ui1 + · · · + uir : i1 < i2 < · · · < ir } of the spectrum of (E⋆u ). Then the MRS of G at u with respect to φ is given by the asymptotic basis 1 e−(r−1)πiσ1 Sat(Ai1 ∧ · · · ∧ Air ) r(r−1)/2 (2πi) 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) ∼ = ∧r MRS(P, u, φ) (see §7.4). Proof. Let y1 (z), . . . , yN (z) be the basis of asymptotically exponential flat sections for ∇P |u corresponding to A1 , . . . , AN . They are characterized by the asymptotic condition yi (z) ∼ e−ui /z (Ψi + O(z)) in the sector | arg z − φ| < π2 + 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 the asymptotically exponential flat sections ir(r−1)/2 Sat(yi1 ,...,ir (z)) give rise to the asymptotic basis of G for u and φ. P P Let S P (τ, z)z −µ z ρ with τ ∈ H (P) denote the fundamental solution for the big quantum P P connection of P as in Remark 2.3.2. The group elements S P (τ, z), z −µ , z ρ ∈ GL(H (P)) naturally act on the r-th wedge representation ∧r H (P); we denote these actions by the same symbols. Define the End(H (G))-valued function S G (z) by S G (z) Sat(α) = e(t

◦ ∪)/z

Sat(S P (t◦ , z)α)

with t◦ := (r − 1)πiσ1

for all α ∈ ∧r H (P). This satisfies S G (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: P

G

P

G

z µ Sat(α) = Sat(z µ α)

z ρ Sat(α) = Sat(z ρ α),

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

G

G

S G (z)z −µ z ρ Sat(α) = e(t

◦ ∪)/z

P

P

Sat(S P (t◦ , z)z −µ z ρ α).

These sections are flat for ∇G |τ =0 by the ‘quantum’ Satake (or Theorem 7.3.1). Note that we have: P ◦ P G G z µ S G (z)z −µ Sat(α) = e(t ∪) Sat(z µ S P (t◦ , z)z −µ α). G

G

G

G

By Lemma 7.5.3 below, we have [z µ S G (z)z −µ ]z=∞ = idH (G) . Hence S G (z)z −µ z ρ coincides with 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) 1 P P S P (t◦ , z)z −µ z ρ Ai . yi (z) parallel transport = dim P/2 (2π) to u◦ = {τP = t◦ }

54


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 1 G −µG ρG −(t◦ ∪) S (z)z z e Sat(Ai1 ∧ · · · ∧ Air ) . = (2π)dim G/2 (−2πi)r(r−1)/2

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 only up to sign.) Lemma 7.5.3. The fundamental solution S(τ, z)z −µ z ρ in Remark 2.3.2 satisfies [z µ S(τ, z)z −µ ]z=∞ = e−(τ ∪) for τ ∈ H 2 (F ).

Proof. The differential equation for T (τ, z) = z µ S(τ, z)z −µ in the τ -direction reads ∂α T (τ, z)+ z −1 (z µ (α⋆τ )z −µ )T (τ, z) = 0 for α ∈ H (F ). If τ, α ∈ H 2 (F ), we have that z −1 (z µ (α⋆τ )z −µ ) is regular at z = ∞ and equals (α∪) there. Here we use the fact that F is Fano and the divisor axiom (see Remark 2.1.2). The conclusion follows by solving the differential equation along H 2 (F ). Remark 7.5.4. When we identify the MRS of G with the r-th wedge of the MRS of P, we should use the identification (2πi)−r(r−1)/2 e−(r−1)πiσ1 Sat : ∧r H (P) ∼ = H (G) that respects the pairing [·, ·) in (3.3.5).

7.6. The wedge product of Gamma basis. We show that the r-th wedge of the Gamma b P Ch(O(i))} given by Beilinson’s exceptional collection for P matches up with the basis {Γ b G Ch(S ν V ∗ )} given by Kapranov’s exceptional collection for G. In view of Gamma basis {Γ the truth of Gamma Conjecture II for P (Theorem 6.0.4) and Proposition 7.5.1, the following proposition 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 r b G Ch(S ν V ∗ ) = (2πi)−(2) e−(r−1)πiσ1 Sat Γ b P Ch(O(ν1 + r − 1)) ∧ · · · ∧ Γ b P Ch(O(νr )) . Γ

We give an elementary algebraic proof of Proposition 7.6.1 in this section. A more geometric proof 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 the hyperplane class on P = PN −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 Q . i

GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO

GAMMA CLASSES AND QUANTUM COHOMOLOGY OF FANO

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