Gamma conjecture via mirror symmetry

Sergey Galkin; Hiroshi Iritani

GAMMA CONJECTURE VIA MIRROR SYMMETRY

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class AF of F , called the principal asymptotic class. Gamma conjecture [26] of Vasily Golyshev and the present authors claims that the principal asymptotic class AF equals the Gamma class ΓF associated to Euler's Γ-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.

GAMMA CONJECTURE VIA MIRROR SYMMETRY arXiv:1508.00719v1 [math.AG] 4 Aug 2015 SERGEY GALKIN AND HIROSHI IRITANI Abstract. The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class AF of F , called the principal asymptotic class. Gamma conjecture [26] of Vasily Golyshev and the present authors claims that the b F associated to Euler’s Γ-function. principal asymptotic class AF equals the Gamma class Γ We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space. Contents 1. Introduction 1.1. Gamma conjecture 1.2. Riemann–Roch 1.3. Mirror symmetry 1.4. Plan of the paper 2. Quantum cohomology and quantum connection 3. Gamma conjecture I (and I’) 3.1. Property O 3.2. Gamma class 3.3. Principal asymptotic class and Gamma conjecture I 4. Gamma conjecture II 4.1. Formal fundamental solution 4.2. Lift to an analytic solution 4.3. Asymptotic basis and Gamma conjecture II 4.4. Stokes matrix 5. Mirror heuristics 5.1. Polynomial loop space 5.2. Solution as a path integral 5.3. Direct calculation 5.4. Calculation by localization 5.5. Comparison 6. Toric Manifold 7. Toric complete intersections 8. Quantum Lefschetz 9. Grassmannians 2 2 2 3 4 5 6 6 6 7 10 10 11 11 12 13 14 15 15 16 17 18 19 22 24 2010 Mathematics Subject Classification. 53D37 (primary), 14N35, 14J45, 14J33, 11G42 (secondary). Key words and phrases. Fano varieties; quantum cohomology; mirror symmetry; Dubrovin’s conjecture; Gamma class; Apery constant; derived category of coherent sheaves; exceptional collection; Landau–Ginzburg model. 1 2 S. GALKIN AND H. IRITANI 9.1. Abelian quotient and non-abelian quotient 9.2. Preliminary lemmas 9.3. Comparison of cohomology and K-groups 9.4. Quantum cohomology central charge 9.5. Integral representation of quantum cohomology central charges 9.6. Eguchi–Hori–Xiong mirror and quantum period 9.7. Apéry constants References 25 26 28 30 31 35 35 38 1. Introduction 1.1. Gamma conjecture. Gamma conjecture [26] is a conjecture which relates quantum cohomology of a Fano manifold with its topology. The small quantum cohomology of F defines a flat connection (quantum connection) over C× and its solution is given by a (multivalued) cohomology-valued function JF (t) called the J-function. Under a certain condition (Property O), the limit of the J-function: AF := lim t→+∞ JF (t) ∈ H (F ) h[pt], JF (t)i exists and defines the principal asymptotic class AF of F . Gamma conjecture I says that AF b F = Γ(T b F ) of the tangent bundle of F (see §3.2): equals the Gamma class Γ bF . AF = Γ Suppose that the quantum cohomology of F is semisimple. In this case, we can define higher asymptotic classes AF,i , 1 6 i 6 N = dim H (F ) from exponential asymptotics of flat sections of the quantum connection. Gamma conjecture II says that there exists a full exceptional b (F ) such that we have collection E1 , E2 , . . . , EN of Dcoh b F · Ch(Ei ) AF,i = Γ i = 1, . . . , N. √ √ √ P deg dim F p Here we write Ch(E) := (2π −1) 2 ch(E) = p=0 (2π −1) chp (E) for the (2π −1)modified version of the Chern character. The principal asymptotic class AF corresponds to the exceptional object E = OF . 1.2. Riemann–Roch. One may view Gamma conjecture as a square root of the index theorem. Recall the Hirzebruch–Riemann–Roch formula: Z ch(E1∨ ) · ch(E2 ) · tdF χ(E1 , E2 ) = F P F i i for vector bundles E1 , E2 on F , where χ(E1 , E2 ) = dim i=0 (−1) dim Ext (E1 , E2 ) is the Euler pairing and tdF = td(T F ) is the Todd class of F . The famous identity x x x x/2 =e Γ 1− √ Γ 1+ √ 1 − e−x 2π −1 2π −1 √ √ deg bF · Γ b ∗ of the Todd class, which in turn gives the factorization (2π −1) 2 tdF = eπ −1c1 (F ) · Γ F factorizes the Hirzebruch–Riemann–Roch formula as: h b F , Ch(E2 ) · Γ bF . (1) χ(E1 , E2 ) = Ch(E1 ) · Γ GAMMA CONJECTURE VIA MIRROR SYMMETRY 3 b F and [·, ·) is a non-symmetric pairing on H (F ) given by b ∗ = (−1) deg 2 Γ Here we set Γ F Z √ √ 1 π −1c1 (X) π −1µ (e e α) ∪ β (2) [α, β) = (2π)dim F F with µ ∈ End(H (F )) defined by µ(φ) = (p− dim2 F )φ for φ ∈ H 2p (F ). Via the factorization (1), Gamma conjecture II implies part of Dubrovin’s conjecture [18]: the Euler matrix χ(Ei , Ej ) of the exceptional collection equals the Stokes matrix Sij = [AF,i , AF,j ) of the quantum differential equation. 1.3. Mirror symmetry. In the B-side of mirror symmetry, solutions to Picard–Fuchs equations are often given by hypergeometric series whose coefficients are ratios of Γ-functions. Recall that a mirror of a quintic threefold Q ⊂ P4 is given by the pencil of hypersurfaces Yt = {f (x) = t−1 } in the torus (C× )4 which can be compactified to smooth Calabi–Yau threefolds Y t [12], where f is the Laurent polynomial given by f (x) = x1 + x2 + x3 + x4 + 1 . x1 x2 x3 x4 For the holomorphicR volume form Ωt = d log x1 ∧ · · · ∧ d log x4 /df on Y t and a real 3-cycle C ⊂ Y t , the period C Ωt satisfies the Picard–Fuchs equations Z θ 4 − 55 t5 (θ + 51 )(θ + 25 )(θ + 35 )(θ + 54 ) Ωt = 0 C with θ = ∂ . t ∂t The Frobenius method yields the following solution to this differential equation: Φ(t) = ∞ X Γ(1 + 5n + 5ǫ) n=0 Γ(1 + n + ǫ)5 t5n+5ǫ where ǫ is an infinitesimal parameter satisfying ǫ4 = 0. Regarding ǫ as a hyperplane class on the quintic Q, we may identify the leading term of the series Φ(t) with the inverse Gamma class of Q: Γ(1 + 5ǫ) 1 . = 5 bQ Γ(1 + ǫ) Γ This is how the Gamma class originally arose in the context of mirror symmetry [40, 51]. HosonoR [39] conjectured (more generally for a complete intersection Calabi–Yau) that the period C Ωt of an integral 3-cycle C ⊂ Y t should be written in the form Z Φ(t) · Ch(V ) · TdQ (3) Q √ deg for a vector bundle V → Q which is “mirror” to C, where TdQ = (2π −1) 2 tdQ . In R physics terminology, the period C Ωt (or the quantity (3)) is called the central charge of the D-branes C (resp. V ). Hosono’s conjecture has been answered affirmatively in [44] by showing b structure in quantum that the natural integral structure H 3 (Y t , Z) agrees with the Γ-integral cohomology of Q, see also [38, 10, 42, 33]. A main purpose of this article is to explain a relationship between Gamma conjecture and mirror symmetry. In fact, Hosono’s conjecture for a quintic Q is closely related to the truth of Gamma conjecture for the ambient Fano manifold P4 . A mirror of P4 is given by the 4 S. GALKIN AND H. IRITANI Landau–Ginzburg model f : (C× )4 → C and the quantum differential equation for P4 has a solution given by the oscillatory integral: Z dx4 dx1 ∧ ··· ∧ . exp(tf (x)) x1 x4 Γ When the cycle Γ is a Lefschetz thimble of f (x) and C is the associated vanishing cycle, R this oscillatory integral can be written as a Laplace transform of the period C Ωt . Correspondingly, by quantum Lefschetz principle [29, 48, 50, 14], the quantum differential equation for Q arises from a Laplace transform of the quantum differential equation for P4 [19, 45]. b (P4 ) Gamma conjecture relates a Lefschetz thimble of f with an exceptional object E in Dcoh via the exponential asymptotics of the corresponding oscillatory integral; then the vanishing cycle C associated with Γ corresponds to the spherical object V = i∗ E on Q under Hosono’s conjecture, see [44, Theorem 6.9]. See the comparison table below for quantum differential equations (QDE) of a Fano manifold F and its anti-canonical section Q. space X mirror singularities of QDE Fano F × f : (C )n → C irregular solutions (central charges) oscillatory integral of f cycles on the mirror b (X) Dcoh objects of monodromy data Lefschetz thimble exceptional object Ei Stokes Sij = χ(Ei , Ej ) Calabi–Yau Q ∈ | − KF | Y t = f −1 (1/t) regular Laplace ↔ fiber → i∗ → period integral of Y t vanishing cycle spherical object Vi reflection Sij − (−1)n Sji b (X) and Table 1. We expect a mirror correspondence between objects of Dcoh integration cycles on the mirror; when a vanishing cycle C arises as a fiber of a Lefschetz thimble Γ, the spherical object V on Q mirror to C should be the pull-back of the exceptional object E on F mirror to Γ. A Lefschetz thimble of f gives a solution to the quantum differential equation of F which has a specific exponential asymptotics as t → ∞. 1.4. Plan of the paper. In §2, we review definitions and basic facts on quantum cohomology and quantum connection. In §3 and §4, we discuss Gamma conjecture I and II respectively. This part is a review of our previous paper [26] with Vasily Golyshev. In §5, we give a heuristic argument which gives mirror oscillatory integral and the Gamma class in terms of polynomial loop spaces. In §6 and §7, we discuss Gamma conjecture for toric varieties and toric complete intersections using Batyrev–Borisov/Givental/Hori–Vafa mirrors. In §8, we discuss compatibility of Gamma conjecture I with taking hyperplane sections (quantum Lefschetz). In §9, we discuss Gamma conjecture for Grassmannians Gr(r, n) using the Hori–Vafa mirror which is the rth alternate product of the mirrors of Pn−1 . Acknowledgments. We thank Vasily Golyshev for insightful discussions during the collaboration [26]. H.I. thanks Kentaro Hori, Mauricio Romo and Kazushi Ueda for the discussion on the papers [7, 16, 36]. This project was supported by JSPS and Russian Foundation for Basic Research under the Japan–Russia Research Cooperative Program “Categorical and Analytic GAMMA CONJECTURE VIA MIRROR SYMMETRY 5 Invariants in Algebraic Geometry”. S.G. was supported by RFBR, research project No.1551-50045 fa. H.I. was supported by JSPS Kakenhi Grant number 25400069, 23224002, 26610008. 2. Quantum cohomology and quantum connection Let H (F ) = H (F, C) denote the even degree part of the Betti cohomology group of F . The (small) quantum product ⋆0 on H (F ) is defined by the formula: X (α ⋆0 β, γ)F = hα, β, γi0,3,d d∈H2 (F,Z) where (·, ·)F is the Poincaré pairing on F and h· · ·i0,3,d is the genus-zero three point Gromov– Witten invariants, which roughly speaking counts the number of rational curves in F passing through the cycles Poincaré dual to α, β and γ (see e.g. [53]). By the dimension axiom, these Gromov–Witten invariants are non-zero only if 2c1 (F ) · d + 2 dim F = deg α + deg β + deg γ; since F is Fano, there are finitely many such curve classes d and the above sum is finite. The product ⋆0 is associative and commutative, and (H (F ), ⋆0 ) becomes a finite dimensional algebra. More generally we can define the big quantum product ⋆τ for τ ∈ H (F ): (α ⋆τ β, γ) = X d∈H2 (F,Z) ∞ X 1 hα, β, γ, τ, . . . , τ i0,3+n,d . n! n=0 This defines a formal1 deformation of the small quantum cohomology as a commutative ring. In this paper, we will restrict our attention to the small quantum product ⋆0 . The quantum connection is a meromorphic flat connection on the trivial H (F )-bundle over P1 . It is given by: ∇z∂z = z (4) 1 ∂ − (c1 (F )⋆0 ) + µ ∂z z where z is an inhomogeneous co-ordinate on P1 and µ ∈ End(H (F )) is defined by µ(φ) = (p − dim2 F )φ for φ ∈ H 2p (F ). This is regular singular (logarithmic) at z = ∞ and irregular singular at z = 0. We have a canonical fundamental solution around z = ∞ as follows: Proposition 2.1 ([17, 26]). There exists a unique End(H (F ))-valued power series S(z) = id +S1 z −1 + S2 z −2 + · · · which converges over the whole z −1 -plane such that ∇(S(z)z −µ z c1 (F ) φ) = 0 µ T (z) = z S(z)z −µ ∀φ ∈ H (F ) is regular at z = ∞ and T (∞) = id. Here we set z c1 (F ) = exp(c1 (F ) log z) and z −µ = exp(−µ log z). The fundamental solution S(z)z −µ z c1 (F ) identifies the space of flat sections with the cohomology group H (F ). In other words, a basis of the cohomology group H (F ) yields a basis of flat sections via the map S(z)z −µ z c1 (F ) — this amounts to solving the quantum differential equation by the Frobenius method around z = ∞. 1the convergence of the big quantum product is not known in general. 6 S. GALKIN AND H. IRITANI Remark 2.2. The quantum connection can be extended to a meromorphic flat connection on the trivial H (F )-bundle over H (F ) × P1 via the big quantum product: (5) where E = c1 (F ) + P i τ= N i=1 τ φi . PN 1 α ∈ H (F ), ∇α = ∂α + (α⋆τ ) z 1 ∇z∂z = z∂z − (E⋆τ ) + µ z i=1 (1 − deg φi i 2 )τ φi with {φi }N i=1 a homogeneous basis of H (F ) and Remark 2.3 ([26]). Via the gauge transformation z µ and the change z = t−1 of variables, we have ∂ ∂ −µ µ z ∇z∂z z = z − (c1 (F )⋆−c1 (F ) log z ) = − t + c1 (F )⋆c1 (F ) log t ∂z ∂t This gives the quantum connection along the “anticanonical line” Cc1 (F ). 3. Gamma conjecture I (and I’) 3.1. Property O. We start with Property O for a Fano manifold. This is a supplementary condition we need to formulate Gamma conjecture I. Definition 3.1 (Property O [26]). Let F be a Fano manifold and define a non-negative real number T as (6) T := max{|u| : u ∈ C is an eigenvalue of (c1 (F )⋆0 )} ∈ Q. We say that F satisfies Property O if the following conditions are satisfied: (1) T is an eigenvalue of (c1 (F )⋆0 ) of multiplicity one; (2) if u is an eigenvalue of (c1 (F )⋆0 ) with |u| = T , there exists an rth root ζ of unity such that u = ζT , where r is the Fano index of F , i.e. the maximal integer r > 0 such that c1 (F )/r is an integral class. Remark 3.2. ‘O’ means the structure sheaf of F . Under mirror symmetry, it is conjectured that the set of eigenvalues of (c1 (F )⋆0 ) agrees with the set of critical values of the mirror Landau–Ginzburg potential f . Under Property O, number T should be a critical value of f and the Lefschetz thimble corresponding to T should be mirror to the structure sheaf O. Conjecture O [26] says that every Fano manifold satisfies Property O. Some Fano orbifolds with non-trivial π1orb (F ) do not satisfy Property O [25, 26]. Remark 3.3. Perron–Frobenius theorem says that an irreducible square matrix with nonnegative entries has a positive eigenvalue with the biggest norm whose multiplicity is one. It is likely that c1 (F )⋆0 is represented by a non-negative matrix if we have a basis of H (F ) consisting of ‘positive’ (algebraic) cycles. This remark is due to Kaoru Ono. Remark 3.4. Property O for homogeneous spaces G/P was recently proved by Cheong [13]. 3.2. Gamma class. The Gamma class [51, 52, 42] is a characteristic class defined for an almost complex manifold F . Let δ1 , . . . , δn be the Chern roots of the tangent bundle T F of b F is defined to be F such that c(T F ) = (1 + δ1 )(1 + δ2 ) · · · (1 + δn ). The Gamma class Γ bF = Γ n Y i=1 Γ(1 + δi ) ∈ H (F, R) GAMMA CONJECTURE VIA MIRROR SYMMETRY 7 R∞ where Γ(z) = 0 e−t tz−1 dt is Euler’s Γ-function. Since Γ(z) is holomorphic at z = 1, via the Taylor expansion, the right-hand side makes sense as a symmetric power series in δ1 , . . . , δn , and therefore as a (real) cohomology class of F . This is a transcendental class and is given by ! ∞ X k−1 b F = exp −γc1 (F ) + Γ (−1) (k − 1)!ζ(k) chk (T F ) k=2 where ζ(z) is the Riemann zeta function. As explained in the Introduction (see §1.2), the √ √ deg bF Γ b ∗ shows that the Gamma class Γ bF can be regarded formula (2π −1) 2 tdF = eπ −1c1 (F ) Γ F as a square root of the Todd class tdF . There is also an interpretation of the Gamma class b F arises from the in terms of the free loop space LF of F due to Lu [52] (see also [43, 26]): Γ 1 ζ-function regularization of the S -equivariant Euler class eS 1 (N+ ), where N+ is the positive normal bundle of the locus F of constant loops in LF . 3.3. Principal asymptotic class and Gamma conjecture I. Consider the space of flat sections for the quantum connection ∇ (4) over the positive real line R>0 . We introduce the subspace A of flat sections having the smallest asymptotics ∼ e−T /z as z → +0. o n A := s : R>0 → H (F ) : ∇s(z) = 0, keT /z s(z)k = O(z −m ) as z → +0 (∃m) where T > 0 is the number in Definition 3.1. Proposition 3.5 ([26, Proposition 3.3.1]). Suppose that a Fano manifold F satisfies Property O. We have dimC A = 1. Moreover, for every element s(z) ∈ A, the limit limz→+0 eT /z s(z) exists and lies in the T -eigenspace E(T ) of (c1 (F )⋆0 ). The principal asymptotic class of F is defined to be the class corresponding to a generator of the one-dimensional space A. Definition 3.6 ([26]). Suppose that a Fano manifold F satisfies Property O. A cohomology class AF ∈ H (F ) satisfying i h A = C S(z)z −µ z c1 (F ) AF is called the principal asymptotic class. Here S(z)z −µ z c1 (F ) is the fundamental solution in Proposition 2.1. The principal asymptotic class is determined up to multiplication by a nonzero complex number; when h[pt], AF i 6= 0, we can normalize AF so that h[pt], AF i = 1. Since the space A is identified via the asymptotics near the irregular singular point z = 0 and the fundamental solution S(z)z −µ z c1 (F ) is normalized at the regular singular point z = ∞, the definition of the class AF involves analytic continuation along the positive real line R>0 on the z-plane. Note that S(z)z −µ z c1 (F ) has a standard determination for z ∈ R>0 given by z −µ z c1 (F ) = exp(−µ log z) exp(c1 (F ) log z) and log z ∈ R. Conjecture 3.7 (Gamma Conjecture I [26]). Let F be a Fano manifold satisfying Property bF of F . O. The principal asymptotic class AF of F is given by the Gamma class Γ There is another description of the principal asymptotic class AF in terms of solutions to the quantum differential equation. We introduce Givental’s J-function [29] by the formula: −1 dim F JF (t) = z 2 1 with t = z −1 S(z)z −µ z c1 (F )   N (7) X X φ i φi tc1 (F )·d  = ec1 (F ) log t 1 + 1 − ψ 0,1,d i=1 d∈H2 (F,Z),d6=0 8 S. GALKIN AND H. IRITANI where S(z)z −µ z c1 (F ) is the fundamental solution in Proposition 2.1 and ψ is the first Chern class of the universal cotangent line bundle over the moduli space of stable maps. This is a cohomology-valued (and multi-valued) function. Since S(z)z −µ z c1 (F ) is a fundamental solution of the quantum connection and by Remark 2.3, we have P (t, ∇c1 (F ) )1 = 0 ∂ )JF (t) = 0 P (t, t ∂t ⇐⇒ ∂ ∂ ∂ for any differential operator P (t, t ∂t ) ∈ Cht, t ∂t i, where ∇c1 (F ) = t ∂t + c1 (F )⋆c1 (F ) log t is the quantum connection along the anticanonical line. In other words, JF (t) satisfies all the differential relations satisfied by the identity class 1 with respect to the connection ∇c1 (F ) ; ∂ in this sense JF (t) is a solution of the quantum connection. Differential operators P (t, t ∂t ) annihilating JF (t) are called quantum differential operators. The principal asymptotic class AF can be computed by the t → +∞ asymptotics of the J-function: Proposition 3.8 ([26]). Suppose that a Fano manifold F satisfies Property O and let AF be the principal asymptotic class. Then we have an asymptotic expansion of the form: JF (t) = Ct− dim F 2 eT t (AF + α1 t−1 + α2 t−2 + · · · ). as t → +∞ on the positive real line, where C 6= 0 is a non-zero constant and αi ∈ H (F ). dim F Proof. It follows from [26, Proposition 3.6.2] that limt→∞ t 2 e−T t JF (t) exists and is proportional to AF . The fact that the remainder admits an asymptotic expansion of the form α1 t−1 + α2 t−2 + · · · follows from the proof there, in particular from [26, Proposition 3.2.1]. This proposition says that CJF (t) converges to CAF in the projective space P(H (F )) as bF i = 1, we obtain the following corollary. t → +∞. Since h[pt], Γ Corollary 3.9 ([26, Corollary 3.6.9]). Let F be a Fano manifold satisfying Property O. Gamma conjecture I holds for F if and only if we have JF (t) b F = lim . Γ t→∞ h[pt], JF (t)i We can replace the continuous limit in the above corollary with a discrete limit of ratios of the Taylor coefficients. Expand the J-function as: ∞ X c1 log t Jn tn . (8) JF (t) = e n=0 Note that Jn = 0 if n is not divisible by the Fano index r of F . We have the following: Proposition 3.10 ([26, Theorem 3.7.1]). Suppose that a Fano manifold F satisfies Property O and Gamma conjecture I. Let r be the Fano index of F . Then we have hα, Jrn i bF i = 0 lim inf − hα, Γ n→∞ h[pt], Jrn i for every α ∈ H (F ) with α ∩ c1 (F ) = 0. Define the (unregularized and regularized) quantum period of F [15] to be ∞ X Gn tn GF (t) = h[pt], JF (t)i = n=0 (9) bF (κ) = G ∞ X 1 n!Gn κ = κ n=0 n Z 0 ∞ GF (t)e−t/κ dt GAMMA CONJECTURE VIA MIRROR SYMMETRY 9 where Gn = h[pt], Jn i. It is shown in [26, Lemma 3.7.6] that if F satisfies Property O and if bF (κ) equals 1/T . In particular h[pt], AF i = 6 0, the convergence radius of G p (10) lim sup rn (rn)!|Grn | = T. n→∞ p Suppose that this limit sup (10) can be replaced with the limit, i.e. limn→∞ rn (rn)!|Grn | = T . Then the argument in the proof of [26, Theorem 3.7.1] shows, under the same assumption as in Proposition 3.10, that hα, Jrn i bF i = hα, Γ (11) lim n→∞ h[pt], Jrn i for a class α ∈ H (F ) such that α ∩ c1 (F ) = 0. Therefore we can consider the following variant of Gamma conjecture I: Conjecture 3.11 (Gamma conjecture I’). For a Fano manifold F and a class α ∈ H (F ) with α ∩ c1 (F ) = 0, the limit formula (11) holds. Remark 3.12. By the abovepdiscussion, Gamma conjecture I’ holds if Gamma conjecture I holds and one has limn→∞ rn (rn)!|Grn | = T . The discrete limit in the left-hand side of (11) is called the Apéry constant (or Apéry limit) and was studied by Almkvist–van-Straten–Zudilin [1] in the context of Calabi–Yau differential equations and by Golyshev [31] and Galkin [23] for Fano manifolds. Under Gamma conjecture I’, these limits are expressed in terms of the zeta values ζ(2), ζ(3), ζ(4), . . . . For some Fano manifolds, they are precisely the limits which Apéry used to prove the irrationality of ζ(2) and ζ(3): an Apéry limit for the Grassmannian G(2, 5) gives a fast approximation of ζ(2) and an Apéry limit for the orthogonal Grassmannian OG(5, 10) gives a fast approximation of ζ(3) [23, 31]. Most of the Apéry limits of Fano manifolds are not fast enough to prove irrationality (see [23]). It would be very interesting to find a Fano manifold which gives a fast approximation of ζ(5), for example. We give a sufficient condition that ensures that the limit sup in (10) can be replaced with ± the limit. When a Laurent polynomial f (x) ∈ C[x± 1 , . . . , xm ] is mirror to F , we expect that the quantum period GF (t) (9) for F should be given by the constant term series of f : Z ∞ X 1 1 tf (x) dx1 · · · dxm √ Const(f n )tn = e GF (t) = m x1 · · · xm n! (2π −1) (S 1 )m n=0 where Const(f n ) denotes the constant term of the Laurent polynomial f (x)n . When this holds, f (x) is said to be a weak Landau–Ginzburg model of F [58]. Lemma 3.13. Let F be a Fano manifold of index r. Suppose that F admits a weak Landau– ± Ginzburg model f (x) ∈ C[x± 1 , . . . , xm ] whose coefficients are non-negative real numbers. Suprn pose also that Const(f ) 6= 0 for all but finitely many n ∈ Z>0 . Then the coefficients Gn of the quantum period (9) are non-negative and the limit p p lim rn (rn)!|Grn | = lim rn Const(f rn ) n→∞ exists. n→∞ p Proof. Some of the techniques here are borrowed from [24]. We set αn = log( rn Const(f rn )). It suffices to show that limn→∞ αn exists. By assumption, there exists n0 ∈ Z>0 such that αn is well-defined for all n > n0 . Since Const(f r(n+m) ) > Const(f rn ) Const(f rm ), we have m n αn + αm αn+m > n+m n+m 10 S. GALKIN AND H. IRITANI Set α := lim supn→∞ αn . For any ǫ > 0, there exists n1 > 1 such that αn1 > α − ǫ. Then we have, for all k > 1 and 0 6 i < n1 , kn1 n0 + i αn0 +i + αn αkn1 +n0 +i > kn1 + n0 + i kn1 + n0 + i 1 The right-hand side converges to αn1 as k → ∞. This implies that there exists n2 > n0 such that for all n > n2 , we have αn > α − 2ǫ. Since ǫ > 0 was arbitrary, this implies lim inf n→∞ αn > α and the conclusion follows. Remark 3.14. When discussing the continuous limit, we only need to assume Part (1) of Property O (Definition 3.1). More precisely, Propositions 3.5, 3.8 and Corollary 3.9 hold for Fano manifolds satisfying only Part (1) of Property O, and Gamma conjecture I (Conjecture 3.7) makes sense for such Fano manifolds. On the other hand, we need Part (2) of Property O in the proof of Proposition 3.10. Remark 3.15. Golyshev and Zagier have announced the proof of Gamma conjecture I for Fano threefolds of Picard rank one. 4. Gamma conjecture II The small quantum cohomology (H (F ), ⋆0 ) of a Fano manifold F is said to be semisimple if it is isomorphic to the direct sum of C as a ring. This is equivalent to the condition that (H (F ), ⋆0 ) has no nilpotent elements. In this section, we give a refinement of Gamma conjecture I for a Fano manifold with semisimple quantum cohomology. 4.1. Formal fundamental solution. Suppose that (H (F ), ⋆0 ) is semisimple. Then we have an idempotent basis ψ1 , . . . , ψN of H (F ) suchpthat ψi ⋆0 ψj = δij ψi . Define the normalized idempotent basis Ψ1 , . . . , ΨN to be Ψi = ψi / (ψi , ψi ), where (ψi , ψi ) denotes the Poincaré pairing of ψi with itself. Note that (Ψ1 , . . . , ΨN ) is unique up to sign and ordering. We set   | | | Ψ = Ψ1 Ψ2 · · · ΨN  . | | | This is a matrix with column vectors Ψi . We may regard it as a linear map CN → H (F ). Let u1 , . . . , uN be the eigenvalues of (c1 (F )⋆0 ) such that c1 (F ) ⋆0 Ψi = ui Ψi . Let U be the diagonal matrix with entries u1 , . . . , uN :   u1   u2   U =  . .   . uN The following proposition is well-known in the context of Frobenius manifolds. Proposition 4.1 ([17, Lectures 4,5], [61, Theorem 8.15]). Suppose that the small quantum cohomology (H (F ), ⋆0 ) is semisimple. The quantum connection (4) near the irregular singular point z = 0 admits a formal fundamental matrix solution of the form: ΨR(z)e−U/z where R(z) = id +R1 z + R2 z 2 + · · · ∈ End(CN )[[z]] is a matrix-valued formal power series. The formal solution ΨR(z)e−U/z is unique up to multiplication by a signed permutation matrix from the right (which corresponds to the ambiguity of Ψ1 , . . . , ΨN ). GAMMA CONJECTURE VIA MIRROR SYMMETRY 11 Remark 4.2. Since R(z) is a formal power series in positive powers of z, the product R(z)e−U/z does not make sense as a power series. The meaning of the proposition is that the formal gauge transformation by ΨR(z) turns ∇z∂z into z∂z − U/z. Remark 4.3. We do not need to assume that u1 , . . . , uN are mutually distinct. 4.2. Lift to an analytic solution. By choosing an angular sector, the above formal solution can be lifted to an actual analytic solution. This is an instance of the Hukuhara– Turrittin theorem for irregular connections (see e.g. [64, Theorem 19.1], [60, II, 5.d]). We √ −1φ ∈ S 1 ) is admissible for a multiset {u1 ,√u2 , . . . , uN } ⊂ C if say that√a phase φ ∈ R (or e √ −1φ −1φ − − Im(ui e ) 6= Im(uj e ) for every pair (ui , uj ) with ui 6= uj , i.e. e −1φ is not parallel to any non-zero difference ui − uj . Proposition 4.4 ([64, Theorem 12.2], [4, Theorem A], [17, Lectures 4,5], [11, §8], [26, Proposition 2.5.1]). Let φ ∈ R be an admissible phase for the spectrum {u1 , u2 , . . . , uN } of (c1 (F )⋆0 ). φ There exist ǫ > 0 and an analytic fundamental solution Yφ (z) = (y1φ (z), . . . , yN (z)) for the π quantum connection (4) on the angular sector | arg(z) − φ| < 2 + ǫ around z = 0 such that one has the asymptotic expansion Yφ (z)eU/z ∼ ΨR(z) (12) as z → 0 in the sector | arg(z) − φ| < π2 + ǫ, where ΨR(z)e−U/z is the formal fundamental solution in Proposition 4.1. Such an analytic solution Yφ (z) is unique when we fix the sign and the ordering of Ψ1 , . . . , ΨN . Remark 4.5. Notice that each flat section yiφ (z) has the exponential asymptotics ∼ e−ui /z Ψi as z → 0 in the sector | arg(z) − φ| < π2 + ǫ. Remark 4.6. The precise meaning of the asymptotic expansion (12) is as follows. For any 0 < ǫ′ < ǫ and n ∈ Z>0 , there exists a constant C = C(ǫ′ , n) such that −U/z Yφ (z)e for all z with | arg z − φ| 6 π 2 + ǫ′ − n X ΨRk z k 6 C|z|n+1 k=0 and |z| 6 1, where we write R(z) = P∞ k=0 Rk z k. φ 4.3. Asymptotic basis and Gamma conjecture II. Let Yφ (z) = (y1φ (z), . . . , yN (z)) be the analytic fundamental solution in Proposition 4.4. We regard Yφ (z) as a function defined on the universal cover of C× ; initially it is defined on the angular sector | arg(z) − φ| < π + ǫ with |z| ≪ 1, but can be analytically continued to the whole universal cover since it is a solution to a linear differential equation. For an admissible phase φ for {u1 , . . . , uN }, we define the higher asymptotic classes AφF,i ∈ H (F ), i = 1, . . . , N by yiφ (z) parallel translate to arg(z) = 0 = 1 S(z)z −µ z c1 (F ) AφF,i (2π)dim F/2 where S(z)z −µ z c1 (F ) is the fundamental solution for the quantum connection in Proposition 2.1. We call {AφF,1 , AφF,2 , . . . , AφF,N } the asymptotic basis at the phase φ. The asymptotic basis is the same as what Dubrovin [17] called the central connection matrix. Remark 4.7. Suppose that F satisfies Property O. When we take an admissible phase φ from the interval (− π2 , π2 ), the class AφF,i corresponding to the eigenvalue ui = T is proportional to the principal asymptotic class AF . 12 S. GALKIN AND H. IRITANI Remark 4.8. The asymptotic basis at a phase φ is unique up to sign and ordering. The sign and the ordering depend on those of the normalized idempotents Ψ1 , . . . , ΨN . Each asymptotic class AφF,i is marked by the eigenvalue ui of (c1 (F )⋆0 ). With respect to the pairing [·, ·) in (2), these data {(AφF,i , ui )}N i=1 form a marked reflection system [26], see also Remark 4.13. Conjecture 4.9 (Gamma conjecture II [26]). Suppose that a Fano manifold F has a semisimb (F ) has a full exceptional collection. Let φ be ple small quantum cohomology and that Dcoh an admissible phase for the spectrum {u1 , . √ . . , uN } of (c1 (F )⋆0 ). We number the eigenvalues √ √ −1φ − −1φ − −1φ − u1 ) > Im(e u2 ) > · · · > Im(e uN ). There exists a u1 , . . . , uN so that Im(e φ φ φ φ b full exceptional collection E1 , . . . , EN such that AF,i = ΓF · Ch(Ei ). √ P F p Recall that Ch(E) = dim p=0 (2π −1) chp (E) is the modified Chern character. Remark 4.10. Part (3) of Dubrovin’s conjecture [18, Conjecture 4.2.2] says that the columns of the central connection matrix are given by C ′ (Ch(Ei )) for some linear operator C ′ ∈ b F ∪. Recently End(H (F )) commuting with c1 (F )∪. Gamma conjecture II says that C ′ = Γ Dubrovin [20] also proposed the same conjecture as Gamma conjecture II. 4.4. Stokes matrix. Let Yφ (z) and Yφ− (z) be the analytic fundamental solutions from Propo√ √ sition 4.4 associated respectively to admissible directions e −1φ and −e −1φ . The domains of definitions of Yφ and Yφ− are shown in Figure 1. Let Π± be the angular regions as in Figure 1 which are components of the intersection of the domains of Yφ and Yφ− . The Stokes matrices φ are the constant matrices S φ and S− satisfying Yφ (z) = Yφ− (z)S φ Yφ (z) = for z ∈ Π+ ; φ Yφ− (z)S− for z ∈ Π− . Π+ ✾ q Yφ− (z) • ✶ ② Yφ (z) ✲ φ admissible direction Π− Figure 1. φ Proposition 4.11 ([17, Theorem 4.3], [26, Proposition 2.6.4]). Let Yφ (z) = (y1φ (z), . . . , yN (z)) be the fundamental solution √ Stokes matrices at phase φ are given √ from Proposition 4.4. The φ φ = (yiφ (e−π −1 z), yjφ (z)), where yiφ (e−π −1 z) denotes the analytic continuation = S−,ji by Sij GAMMA CONJECTURE VIA MIRROR SYMMETRY 13 √ φ are semi-orthogonal of yi (z) along the path [0, π] ∋ θ → 7 e− −1θ z. The flat sections y1φ , . . . , yN in the following sense: ( √ √ 0 if (i 6= j and ui = uj ) or Im(e− −1φ ui ) < Im(e− −1φ uj ); φ Sij = 1 if i = j. Using the non-symmetric pairing [·, ·) given in (2), we have √ φ Sij = (yiφ (e− −1π z), yjφ (z)) √ √ 1 −µ c1 (F ) φ −µ π −1µ c1 (F ) −π −1c1 (F ) φ = z e A , S(z)z z A S(−z)z e F,i F,j (2π)dim F = [AφF,i , AφF,j ) where we used the fact that (S(−z)α, S(z)β) = (α, β) and (z −µ α, z −µ β) = (α, β) (see [17]). Therefore, the factorization (1) of the Hirzebruch–Riemann–Roch formula implies the following corollary. Corollary 4.12. Suppose that a Fano manifold F satisfies Gamma conjecture II. Then there b (F ) such that χ(E , E ) equals the Stokes exists a full exceptional collection E1 , . . . , EN of Dcoh i j matrix Sij . (This conclusion is part (2) of Dubrovin’s conjecture [18, Conjecture 4.2.2]). Remark 4.13. The asymptotic basis can be defined similarly for the big quantum product ⋆τ with τ ∈ H (F ) as far as ⋆τ is semisimple. We have an asymptotic basis Aφ,τ F,i depending on both φ and τ , and Gamma conjecture II makes sense at general τ ∈ H (F ). The truth of the Gamma conjecture II is, however, independent of the choice of (τ, φ). This is because the asymptotic basis changes (discontinuously) by mutation as (τ, φ) varies, and we can consider the corresponding mutation for exceptional collections. The right mutation of asymptotic bases takes the form i i+1 i i+1 (A1 , A2 , . . . , Ai , Ai+1 , . . . AN ) 7→ (A1 , A2 , . . . , Ai+1 , Ai − [Ai , Ai+1 )Ai+1 , · · · , AN ) √ √ if we √order the asymptotic basis so that Im(e− −1φ u1 ) > Im(e− −1φ u2 ) > · · · > Im(e− −1φ√uN ). The right mutation happens when the eigenvalue ui+1 crosses the ray ui + R>0 e −1φ . (The left mutation is the inverse of the right mutation). The braid group action on asymptotic bases and Stokes matrices has been studied by Dubrovin, see [17, 26] for more details. Remark 4.14. In general, the quantum connection of a smooth projective variety (or a projective orbifold) X is underlain by the integral local system consisting of flat sections dim F b X Ch(E) with E ∈ K 0 (X). This is called the Γ-integral b structure (2π)− 2 S(z)z −µ z c1 (F ) Γ [42, 47]. In this language, Gamma conjecture implies that this integral structure should be compatible with the Stokes structure at the irregular singular point z = 0 of the quantum connection. Katzarkov-Kontsevich-Pantev [47] imposed the compatibility of rational structure with Stokes structure as part of conditions for nc-Hodge structure. Our Gamma conjecture makes their compatibility condition explicit for Fano manifolds. Note that Gamma conjecture II implies the integrality of the Stokes matrix Sij . 5. Mirror heuristics In this section we give a heuristic argument which gives the mirror oscillatory integral and the Gamma class from the polynomial loop space (quasi map space) for PN −1 . This is 14 S. GALKIN AND H. IRITANI motivated by Givental’s equivariant Floer theory heuristics [27, 28]. The argument in this section can be applied more generally to toric varieties to yield their mirrors and Gamma classes. 5.1. Polynomial loop space. Givental [27, 28] conjectured that the quantum D-module (i.e. quantum connection) should be identified with the S 1 -equivariant Floer theory for the free loop space (see also [63, 41, 2]). Following Givental, we consider an algebraic version of the loop space instead of the actual free loop space. The (Laurent) polynomial loop space of PN −1 is defined to be Lpoly PN −1 = C[ζ, ζ −1 ]N \ {0} /C× . where C× acts on C[ζ, ζ −1 ]N by scalar multiplication. The polynomial loop space Lpoly PN −1 ∞ by the diagonal S 1 can be also described as a symplectic reduction of C[ζ, ζ −1 ]N ∼ = CP −1 N action. For a point (a1 (ζ), . . . , aN (ζ)) ∈ C[ζ, ζ ] , we write ai (ζ) = n∈Z ai,n ζ n and regard (ai,n : 1 6 i 6 N, n ∈ Z) as a co-ordinate system on C[ζ, ζ −1 ]N . With respect to the standard Kähler form √ N −1 X X dai,n ∧ dai,n (13) ω= 2 i=1 n∈Z C[ζ, ζ −1 ]N , on given by the diagonal S 1 -action admits a moment map2 µ : C[ζ, ζ −1 ]N → Lie(S 1 )∗ ∼ =R µ(a1 (ζ), . . . , aN (ζ)) = π N X X i=1 n∈Z |ai,n |2 . √ P P ∂ ∂ This satisfies ιX ω + dµ = 0 for the vector field X = 2π −1 N i=1 n∈Z (ai,n ∂ai,n − ai,n ∂ai,n ) generating the S 1 -action. Then we have Lpoly PN −1 ∼ = µ−1 (u)/S 1 for every u ∈ R>0 . Via the symplectic reduction, Lpoly PN −1 is equipped with the reduced symplectic form ωu such that the pull-back of ωu to µ−1 (u) equals the restriction ω|µ−1 (u) . The class ωu represents the cohomology class uc1 (O(1)) on Lpoly PN −1 . The loop rotation defines the S 1 -action on Lpoly PN −1 given by [a1 (ζ), . . . , aN (ζ)] 7→ [a1 (λζ), . . . , aN (λζ)] with λ ∈ S 1 . With respect to the reduced symplectic form ωu on Lpoly PN −1 , this S 1 -action admits a moment map Hu given by Hu ([a1 (ζ), . . . , aN (ζ)]) = π N X X i=1 n∈Z n|ai,n |2 with (a1 (ζ), . . . , aN (ζ)) ∈ µ−1 (u). The function Hu is an analogue of the action functional on the free loop space. Remark 5.1. More precisely, the polynomial loop space should be regarded as an analogue of the universal cover of the free loop space LPN −1 . In fact, we have an analogue of the deck transformation on Lpoly PN −1 given by [a1 (ζ), . . . , aN (ζ)] 7→ [ζa1 (ζ), . . . , ζaN (ζ)]; this corresponds to a generator of π1 (LPN −1 ) ∼ = π2 (PN −1 ) = Z. Recall that the action functional N −1 is defined on the universal cover of LP . 2We identify Lie(S 1 )∗ with R so that the radian angular form dθ on S 1 = {e √ −1θ : θ ∈ R} corresponds to 2π. With this choice, the reduced symplectic form ωu on µ−1 (u)/S 1 represents an integral cohomology class precisely when u ∈ Z. GAMMA CONJECTURE VIA MIRROR SYMMETRY 15 5.2. Solution as a path integral. Recall that symplectic Floer theory is an infinitedimensional analogue of the Morse theory with respect to the action functional on the loop space. We consider the Morse theory on Lpoly PN −1 with respect to the Bott–Morse function Hu . Note that the critical set of Hu is a disjoint union of infinite copies of PN −1 given by (PN −1 )n = {[a1 ζ n , . . . , aN ζ n ]} ⊂ Lpoly PN −1 for each n ∈ Z. The Floer fundamental cycle ∆ is defined to be the closure of the stable manifold associated to the critical component (PN −1 )0 . This is given by ∆ = [a1 (ζ), . . . , aN (ζ)] ∈ Lpoly PN −1 : ai (ζ) ∈ C[ζ] . Under the isomorphism between the Floer homology and the quantum cohomology [57], the Floer fundamental cycle corresponds to the identity class 1 ∈ H (PN −1 ). Consider the following equivariant 2-form on Lpoly PN −1 (in the Cartan model) Ωu = ωu − zHu where z ∈ HS2 1 (pt, Z) is a positive generator. Note that Ω is equivariantly closed since Hu is a Hamiltonian for the S 1 -action. Givental [27, 28] proposed that the infinite-dimensional integral (which could be viewed as a Feynman path integral) Z eΩu /z (14) ∆ should give a solution to the quantum differential equation for PN −1 . This mayR be viewed as the image of the Floer fundamental cycle ∆ under the homomorphism C 7→ C eΩu /z . The integral does not have a rigorous definition in mathematics, but we can heuristically compute this quantity in two different ways — one is by a direct computation and the other is by localization. The former method yields a mirror oscillatory integral and the latter (due to Givental [27, 28]) yields the J-function of PN −1 . In Givental’s original calculation, however, an infinite (constant) factor corresponding to the Gamma class has been ignored. From bPN−1 -component of the this calculation we obtain a (mirror) integral representation of the Γ J-function. 5.3. Direct calculation. We compute the infinite dimensional integral (14) directly. We regard z as a positive real parameter. Since ∆ = (µ−1 (u) ∩ C[ζ]N )/S 1 , we have Z Z dθ −Hu ωu /z ∧ eω/z e−Hu e e = (14) = 2π −1 N −1 N 1 µ (u)∩C[ζ] (µ (u)∩C[ζ] )/S where dθ is the angular form (connection form) on the principal S 1 -bundle µ−1 (u) → P P µ−1 (u)/S 1 given by dθ = 2µ√π −1 N i=1 k∈Z (ai,n dai,n − ai,n dai,n ) (satisfying dθ(X) = 2π) √ and ω is the Kähler form (13). Changing co-ordinates ai,n → zai,n , we find that this equals Z dθ e−zHu ∧ eω . 2π µ−1 (u/z)∩C[ζ]N If C[ζ]N were a finite dimensional vector space, the top-degree component of the differential dθ form dµ ∧ 2π ∧ eω would equal the Liouville volume form on C[ζ]N associated to ω. Therefore we may write this as Z δ(µ − u/z)e−zHu d vol C[ζ]N 16 S. GALKIN AND H. IRITANI V V∞ √−1 and δ(x) is the Dirac delta-function. Using da ∧ da where d vol = N i,n i,n i=1 n=0 2 √ R ∞ 1 −1xξ dξ, we can compute this as: δ(x) = 2π −∞ e Z ∞ Z √ 1 dξe −1ξ(µ−u/z) e−zHu d vol 2π C[ζ]N −∞ Z ∞ Z ∞ N Y √ √ Y 1 2 = e−π|ai,n | (nz− −1ξ) d vol dξe− −1ξu/z 2π −∞ C[ζ]N i=1 n=0 Z ∞ ∞ N Y √ Y 1 1 √ (15) = dξe− −1ξu/z 2π −∞ nz − −1ξ i=1 n=0 By the ζ-function regularization, we can regularize the infinite product to get: √ √ 1 √ Q∞ ∼ (2πz)−1/2 z − −1ξ/z Γ(− −1ξ/z). n=0 (nz − −1ξ) Therefore (15) should equal, after the change ξ → zξ of co-ordinates, Z ∞ √ √ z − −1(u+N log z)ξ −1ξ)N . dξe (16) Γ(− N/2 2π(2πz) −∞ This integral makes sense if we perturb the integration contour so that√it avoids the singu√ larity at ξ = 0. We will consider the perturbed contour from −∞ + −1ǫ to ∞ + −1ǫ with ǫ > 0. Then the integral (16) is just a (finite-dimensional) Fourier transform and the following R ∞ discussion can be made completely rigorous. Using the integral representation Γ(z) = 0 e−x z x−1 dx of the Γ-function, we find that (16) equals Z ∞ Z dx1 dxN −√−1ξ(u+N log z+PN z i=1 log xi ) e−(x1 +···+xN ) dξ ··· e N/2 x x N 2π(2πz) 1 N [0,∞) −∞ Z P dx1 dxN z −(x1 +···+xN ) ··· δ(u + N log z + N = i=1 log xi )e N/2 xN (2πz) [0,∞)N x1 Z e−u /z z dx1 dxN −1 − x1 +···+xN−1 + x1 ···x N−1 = (17) ··· e N/2 xN −1 (2πz) [0,∞)N−1 x1 where in the last line we considered the co-ordinate change xi → xi /z. The Landau–Ginzburg mirror of PN −1 is given by the Laurent polynomial function f (x1 , . . . , xN −1 ) = x1 +· · ·+xN −1 + e−u x1 ···xN−1 and this is the associated oscillatory integral [27, 37]. 5.4. Calculation by localization. Next we calculate the quantity (14) using the localization formula of equivariant cohomology (or Duistermaat–Heckman formula) [21, 8, 3]. The S 1 fixed set in ∆ is the disjoint union of (PN −1 )n with n > 0 and we have [Ωu ]|(PN−1 )n = uh−znu, where h := c1 (O(1)) ∈ H 2 (PN −1 ) is the hyperplane class. Therefore Z ∞ Z X euh/z−nu Ωu /z e = (PN−1 )n eS 1 (Nn ) ∆ n=0 where Nn is the (infinite-rank) normal bundle of (PN −1 )n in ∆ and eS 1 (Nn ) = Y (h + kz)N = k>−n,k6=0 ∞ Y (h + kz)N k=1 n Y (h − kz)N . k=1 GAMMA CONJECTURE VIA MIRROR SYMMETRY 17 Q N The infinite factor ∞ k=1 (h + kz) was discarded in the original calculation of Givental [28]. Using again the ζ-function regularization, we find that this factor yields the Gamma class: z 1/2 1 Q∞ ∼ z h/z Γ(1 + h/z). 2π k=1 (h + kz) Thus we should have Z ∞ Z X Ωu /z e = ∆ z N/2 euh/z−nu z N h/z Γ(1 + h/z)N Qn N 2π N−1 k=1 (h − kz) n=0 P Z ∞ X z (eu z N )h−n N Qn . Γ(1 + h) = N (2πz)N/2 PN−1 k=1 (h − k) n=0 Recall that the J-function (7) of PN −1 is given by [29] (18) JPN−1 (t) = ∞ X n=0 tN (h+n) . N k=1 (h + k) Qn b PN−1 = Γ(1 + h)N , we obtain Thus, using Γ Z h √ √ z π −1 N −1 b eΩu /z = (19) t), Γ (e J −1) (2π N−1 N−1 P P (2πz)N/2 ∆ under the identification t = e−u/N z −1 , where [·, ·) is the non-symmetric pairing defined in (2). Remark 5.2. The quantity (19) coincides, up to a factor, with the quantum cohomology central charge of OPN−1 [42, 26]. For a vector bundle E on a Fano manifold F , the quantum cohomology central charge Z(E) is defined to be: h √ √ bF Ch(E) Z(E) = (2π −1)dim F JF (eπ −1 t), Γ dim F b F Ch(E) . 1, S(z)z −µ z c1 (F ) Γ =z 2 where t = z −1 . 5.5. Comparison. We computed the infinite dimensional integral (14) in two ways. Comparing (17) and (19), we should have the equality (20) Z h e−u √ √ − x1 +···+xN−1 + x ···x /z dx1 dxN −1 1 N−1 bPN−1 e ··· = (2π −1)N −1 JPN−1 (eπ −1 t), Γ x1 xN −1 [0,∞)N−1 with t = e−u/N z −1 . This oscillatory integral representation yields the asymptotic expansion (as t → +∞) h √ √ −N t b PN−1 ∼ const × t− N−1 2 e Z(OPN−1 ) = (2π −1)N −1 JPN−1 (eπ −1 t), Γ which can be used to prove the Gamma conjecture for PN −1 . See §6 and [26, §3.8]. Remark 5.3. We have a rigorous independent proof of the equality (20) (see √ [42, 47]). Recall that we can write the left-hand side as the Fourier transform (16) of Γ(− −1ξ)N ; by closing the integration √ in the√lower half ξ-plane and writing the integral as the sum of residues √ contour at ξ = 0, − −1, −2 −1, −3 −1, . . . , we arrive at the expression in the right-hand side. 18 S. GALKIN AND H. IRITANI Remark 5.4. A similar regularization of an infinite dimensional integral appears in the computation of (sphere or hemisphere) partition functions of (2,2) supersymmetric gauge theories, see Benini-Cremonesi [7], Doroud–Gomis–Le-Floch–Lee [16] and Hori-Romo [36]. It appears that the computations in §5.3 and §5.4 correspond, in the terminology of [7, 16], to the localization on the Coulomb branch and on the Higgs branch respectively. 6. Toric Manifold In this section, we discuss Gamma conjecture for Fano toric manifolds. We prove Gamma conjecture I by assuming a certain condition for the mirror Laurent polynomial f which is analogous to Property O. Let X be an n-dimensional Fano toric manifold. A mirror of X is given by the Laurent polynomial [27, 37, 30]: 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 for bi = (bi1 , . . . , bin ). By mirror symmetry [30, 42], the spectrum of (c1 (X)⋆0 ) is the set of critical values of f . The restriction of f to the real locus (R>0 )n is strictly convex since the logarithmic Hessian m X ∂2f (x) = bkj bki xbk ∂ log xi ∂ log xj k=1 )n . is positive definite for any x ∈ (R>0 One can also show that f |(R>0 )n is proper and bounded from below since the convex hull of b1 , . . . , bm contains the origin in its interior (cf. Remark 7.4). Therefore f |(R>0 )n admits a global minimum at a unique point xcon ∈ (R>0 )n . We call xcon the conifold point [25, 26]. Consider the following condition: Condition 6.1 (analogue of Property O for toric manifolds). Let X be a Fano toric manifold and f be its mirror Laurent polynomial. Let Tcon = f (xcon ) be the value of f at the conifold point xcon . One has (a) every critical value u of f satisfies |u| 6 Tcon ; (b) the conifold point is the unique critical point of f contained in f −1 (Tcon ). Note that Condition 6.1 implies Part (1) of Property O (Definition 3.1) which is sufficient to make sense of Gamma Conjecture I (see Remark 3.14). Remark 6.2. One can define the conifold point for every Laurent polynomial such that the Newton polytope contains the origin in its interior and that all the coefficients are positive, but Condition 6.1 does not always hold. For instance, the one-dimensional Laurent polynomial f (x) = x−1 + x + tx2 with a sufficiently small t > 0 does not satisfy Part (a) of the condition. Also, if the lattice generated by b1 , . . . , bm is not equal to Zn , then f (x) = xb1 + xb2 + · · · + xbm has non-trivial diagonal symmetry {ζ ∈ (C× )n : f (x) = f (ζx)} = 6 {1} and Part (b) of the condition fails. Theorem 6.3. Suppose that a Fano toric manifold X satisfies Condition 6.1. Then X satisfies Gamma Conjecture I. Proof. It follows from the argument in [42, §4.3.1] that Z dx1 · · · dxn bX e−f (x)/z ϕ(x, z) = z n/2 φ, S(z)z −µ z c1 (X) Γ x1 · · · xn X (R>0 )n GAMMA CONJECTURE VIA MIRROR SYMMETRY 19 ± for z > 0, where φ ∈ H (X) and ϕ(x, z) ∈ C[x± 1 , . . . , xn , z] is such that the class [ϕ(x, z)e−f (x)/z dx1 · · · dxn /(x1 · · · xn )] corresponds to φ under the mirror isomorphism in [42, Proposition 4.8] and n = dim X. When ϕ = 1, one has φ = 1 and this gives the integral representation Z dx1 · · · dxn e−f (x)/z Z(OX ) = x1 · · · xn (R>0 )n of the quantum cohomology central charge Z(OX ) from Remark 5.2. We already saw this in bX (20) for X = PN −1 . This integral representation implies that the flat section S(z)z −µ z c1 (X) Γ has the smallest asymptotics as z → +0, i.e. and the conclusion follows. b X = O(1) eTcon /z S(z)z −µ z c1 (X) Γ A toric manifold has a generically semisimple quantum cohomology. We note that the following weaker version of Gamma Conjecture II (Conjecture 4.9) can be shown for a toric manifold. Theorem 6.4 ([42]). 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 . (see §4.3) of X at τ with respect to φ is given by Γ Proof. Recall from Remark 4.13 that the asymptotic basis makes sense also for the big quantum product ⋆τ and that it changes by mutation as τ varies. By the theory of mutation, 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 [42, Theorem 4.11]. Remark 6.5 ([42]). Results similar to Theorems 6.3, 6.4 hold for a weak-Fano toric orbifold. In the weak-Fano case, however, we need to take into consideration the effect of the mirror map. Remark 6.6. Kontsevich’s homological mirror symmetry suggests that the basis [E1 ], . . . , [EN ] b (X) because the correin the K-group should be lifted to an exceptional collection in Dcoh sponding Lefschetz thimbles form an exceptional collection in the Fukaya–Seidel category of the mirror. 7. Toric complete intersections In this section we discuss Gamma conjecture I for a Fano complete intersection Y in a toric manifold X. Again the problem comes down to the truth of a mirror analogue of Property O. We begin with the remark that the number T can be evaluated on the ambient part. Let Hamb (Y ) := Im(i∗ : H (X) → H (Y )) denote the ambient part of the cohomology group, where i : Y → X is the inclusion. It is shown in [56, Proposition 4], [44, Corollary 2.5] that the ambient part Hamb (Y ) is closed under quantum multiplication ⋆τ when τ ∈ Hamb (Y ). Note that Givental’s mirror theorem [30] determines this ambient quantum cohomology (Hamb (Y ), ⋆τ ). 20 S. GALKIN AND H. IRITANI Proposition 7.1. 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 (6)) 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 assume that Y is a complete intersection in X obtained from the following nef partition [5, 30]. 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 theP fan 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 L 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 xbi fk (x) = −c0 δ0,k + i∈Ik where c0 is the constant arising from Givental’s mirror map [30]: 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) A mirror of Y [37, 30] 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 := dx1 x1 ∧ ··· ∧ dxn xn df1 ∧ · · · ∧ dfl . Consider the following condition, which implies Property O for Y . Condition 7.2 (analogue of Property 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: (a) Z is a smooth complete intersection of dimension n − l; (b) f0 |Zreal : Zreal → R attains global minimum at a unique critical point xcon ∈ Zreal ; we call it the conifold point; (c) the conifold point xcon is a non-degenerate critical point of f0 ; (d) the number T (6) for Y coincides with Tcon := f0 (xcon ); (e) Tcon is an eigenvalue of (c1 (Y )⋆0 ) with multiplicity one. Variant. We can consider the weaker version of Condition 7.2 where part (e) is replaced with (e′ ) Tcon is an eigenvalue of (c1 (Y )⋆0 )|Hamb (Y ) with multiplicity one. When (a), (b), (c), (d), (e′ ) hold, we say that Y satisfies Condition 7.2 on the ambient part. Example 7.3. 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 GAMMA CONJECTURE VIA MIRROR SYMMETRY 21 on the affine variety Z = {x ∈ (C× )n : f1 (x) = xn−d+1 + xn−d+2 + · · · + xn = 1}. Following Przyjalkowski [59], introduce homogeneous co-ordinates [yn−d+1 : · · · : yn ] of Pn−d−1 and consider the change of variables: yi for n − d + 1 6 i 6 n. xi = yn−d+1 + · · · + yn Then we have (yn−d+1 + · · · + yn )d f0 (x) = Qn−d + x1 + · · · + xn−d − δn,d 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}. As in the case of mirrors of toric manifolds (see §6), the Laurent polynomial expression of f0 shows the existence of a conifold point in Condition 7.2. (Similarly, the mirrors of complete intersections in weighted projective spaces have Laurent polynomial expressions [59].) We have Tcon = (n + 1 − d)dd/(n+1−d) − δn,d d!. It is easy to check that Tcon 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 [30]). Therefore Y satisfies Condition 7.2 on the ambient part. Furthermore, if 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 Y satisfies Condition 7.2 if d < n. RemarkP 7.4 ([44]). Using an inequality of the form β1 u1 + · · · + βm um > uβ1 1 · · · uβmm for ui > 0, βi > 0, i βi = 1, we find that f0 |Zreal is bounded (from below) by a convex function: f0 (x) + l = f0 (x) + f1 (x) + · · · + fl (x) > −c0 + ǫ n X 1/k −1/k (xi + xi ) i=1 ∀x ∈ Zreal where ǫ > 0 and k > 0 are constants. In particular f0 |Zreal : Zreal → R is proper and attains global minimum at some point. It also follows that the oscillatory integral (21) below converges. Theorem 7.5. Let Y be a Fano complete intersection in a toric manifold constructed from a nef partition as above. If Y satisfies Condition 7.2, Y satisfies Gamma Conjecture I. If Y satisfies Condition 7.2 on the ambient part, the ambient quantum cohomology of Y satisfies Gamma conjecture I. Proof. In [44, Theorem 5.7], it is proved that the quantum cohomology central charge (see Remark 5.2) of OY has an integral representation: Z dim Y −µ c1 (Y ) b 2 (21) Z(OY ) = z e−f0 /z ωZ . 1, S(z)z z ΓY = X Zreal We note that the constant term c0 in f0 comes from the value of the mirror map at τ = 0. In [44], a slightly more general statement was proved: we have mirrors (Zτ , f0,τ , ωZ,τ ) 2 (Y ) and parametrized by τ ∈ Hamb Z dim Y −µ c (Y ) 1 b ef0,τ /z ωZ,τ z 2 1, S(τ, z)z z ΓY = Y Zτ,real 22 S. GALKIN AND H. IRITANI where S(τ, z)z −µ z c1 (Y ) is the fundamental solution for the big quantum connection (5) which restricts to the one in Proposition 2.1 at τ = 0. By differentiating this in the τ -direction, we obtain Z dim Y −µ c (Y ) 1 b e−f0,τ /z ωZ,τ z∇α1 · · · z∇αk 1, S(τ, z)z z ΓY = z∂α1 · · · z∂αk z 2 Y Zτ,real H 2 (Y 2 (Y Hamb for α1 , . . . , αk ∈ ). Since Hamb (Y ) is generated by ), we obtain an integral representation of the form: Z dim Y bY ϕ(x, z)e−f0 /z ωZ φ, S(z)z −µ z c1 (Y ) Γ = z 2 Y Zreal bY for every φ ∈ Hamb (Y ) (for some function ϕ(x, z)). Since the flat section S(z)z −µ z c1 (Y ) Γ takes values in the ambient part Hamb (Y ), we obtain integral representations for all comb Y . These integral representations and Condition 7.2 show that ponents of S(z)z −µ z c1 (Y ) Γ T /z −µ c (Y ) 1 b ke S(z)z z ΓY k grows at most polynomially as z → +0. 8. Quantum Lefschetz In this section we show that Gamma conjecture I is compatible with the quantum Lefschetz principle. 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 (7) as JX (t) = erh log t ∞ X Jrn trn n=0 with Jd ∈ H (X). By the quantum Lefschetz theorem [50, 14], the J-function of Y is (22) JY (t) = e(r−a)h log t−c0 t ∞ X (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[pt], Jr i = a! h·d=1 h[pt]ψ r−2 iX 0,1,d if r − a = 1; (23) c0 = 0 if r − a > 1. The quantum Lefschetz principle can be rephrased in terms of the Laplace transformation: Lemma 8.1. We have a/(r−a) JY (t = u e−c0 t )= Γ(1 + ah)u Z ∞ i∗ JX (q a/r )e−q/u dq. 0 Remark 8.2. The Laplace transformation in the above lemma converges for sufficiently small u > 0 because of the exponential asymptotics as t → +∞ in Proposition 3.8 and the growth estimate kJX (t)k 6 C| log t|dim X as t → +0. Suppose that X satisfies Gamma Conjecture I. Recall from Proposition 3.8 that we have the following limit formula for JX : (24) b X ∝ lim t dim2 X e−TX t JX (t). Γ t→+∞ GAMMA CONJECTURE VIA MIRROR SYMMETRY 23 Here TX is the number T in (6) for X. Using the stationary phase approximation in Lemma 8.1, we obtain the following result. Theorem 8.3. Suppose that the J-function of X satisfies the limit formula (24) and let Y be a Fano hypersurface in the linear system |−(a/r)KX |, where r is the Fano index of X and 0 < a < r. Then the J-function of Y satisfies the limit formula: bY ∝ lim t dim2 Y e−(T0 −c0 )t JY (t) Γ t→+∞ where c0 is given in (23) and the positive number T0 > 0 is determined by the relation: r−a r T0 a TX =a . r−a r In particular, if Y satisfies Property O (Definition 3.1), then Y satisfies Gamma conjecture I by Corollary 3.9. n Proof. We write n = dim X and T = TX for simplicity. We set Je(t) := t 2 e−T t i∗ JX (t). The e = C1 i∗ Γ b X for some C1 6= 0. By Lemma 8.1, we have limit formula (24) gives limt→+∞ J(t) n−1 Z ∞ n−1 t 2 e−T0 t a/r −(T −c )t a/(r−a) 0 0 e a/r )dq q −an/(2r) eT q −q/u J(q JY (t = u )= t 2 e Γ(1 + ah)u 0 √ Z ∞ t a/r e a/r )dq = q −an/(2r) e−(q−T q +T0 )t J(tq Γ(1 + ah) 0 where in the second line we performed the change of variables q → ur/(r−a) q and used t = ua/(r−a) . Consider the function θ(q) = q − T q a/r + T0 on [0, ∞). This function has a unique r critical point at q0 := ( ar T ) r−a and attains a global minimum at q = q0 ; we have θ(q0 ) = 0 by the definition of T0 . By the stationary phase approximation, the t → +∞ asymptotics of this integral is determined by the behaviour of the integrand near q = q0 . To establish the asymptotics rigorously, we divide the interval [0, ∞) of integration into [0, q0 /2] and [q0 /2, ∞). We first estimate the integral over [0, q0 /2]. (25) √ t Z q0 /2 q −an/(2r) −θ(q)t e 0 √ e a/r )dq 6 e−tθ(q0 /2) t J(tq −tθ(q0 /2) 6e t n+1 − ar 2 r a Z (r/a)−1 e y −n/2 kJ(y)ky tq a/r . Z 0 0 q0 /2 q −an/(2r) kJe(tq a/r )kdq t(q0 /2)a/r (r/a)−1 e y −n/2 kJ(y)ky dy. where we set y = Note that is integrable near y = 0 (by the e definition of J) and is of polynomial growth as y → +∞. Therefore the integral Z t(q0 /2)a/r y −n/2 kJe(y)ky (r/a)−1 dy. 0 is of polynomial growth as t → +∞. Since θ(q0 /2) > 0, the integral (25) over [0, q0 /2] goes to zero (exponentially) as t → +∞. Next we consider the integral over [q0 /2, ∞). By the change √ t x/ , the integral over [q2 /2, ∞) can be written as of variables q = q0 e an Z ∞ 1− √ √ √ q0 2r a/r a x/ t (1− an )x/ t −tθ(q0 ex/ t ) e 2r r dx e e J tq e (26) 0 Γ(1 + ah) −√t log 2 24 S. GALKIN AND H. IRITANI P n Since we have an expansion of the form θ(q0 ey ) = a2 y 2 + ∞ n=3 an y , we have for a fixed x ∈ R, √ √ √ an 2 x/ t ) a/r a bX Je tq0 e r x/ t → C1 i∗ Γ → e−a2 x , e−tθ(q0 e e(1− 2r )x/ t → 1, as t → +∞. On the other hand, since θ(q0 ey ) grows exponentially as y → +∞, we have an estimate of the form θ(q0 ey ) > C2 y 2 on y ∈ [− log 2, ∞) for some C2 > 0. Therefore, when t > 1, the integrand of (26) can be estimated as √ √ √ an an 2 x/ t ) a/r a Je tq0 e r x/ t 6 C3 e|1− 2r ||x|e−C2 x e(1− 2r )x/ t e−tθ(q0 e √ for all x ∈ [− t log 2, ∞) for some C3 > 0. Thus the integrand is uniformly bounded by an integrable function and we can apply Lebesgue’s dominated convergence theorem to (26) to bX /Γ(1 + ah). The conclusion see that the limit of (26) as t → +∞ is proportional to i∗ Γ follows by noting that ∗b b Y = i ΓX . Γ Γ(1 + ah) It is natural to ask the following questions: Problem 8.4. Check that T0 − c0 is the number T (6) for the hypersurface Y . Also prove that Y satisfies Property O (assuming that X satisfies Property O). Remark 8.5 ([23]). It is easy to see that Apéry limits (11) (or Gamma conjecture I’) are compatible with quantum Lefschetz. Suppose that X is a Fano manifold of index r satisfying Gamma conjecture I’ and let Y ∈ |−(a/r)KX | be a Fano hypersurface of index r − a > 1. Then for any α ∈ H (Y ) with c1 (Y ) ∩ α = 0, we have the limit formula: lim n→∞ hi∗ α, JX,rn i bX i = hα, Γ bY i = hi∗ α, Γ h[pt], JX,rn i where JX,n is the Taylor coefficients of the J-function of X (as in (8)) and we used h ∩ i∗ α = 0 in the second equality. Since the index r − a is greater than one, the quantum Lefschetz (22) gives: α, JY,(r−a)n = (an)! hi∗ α, JX,rn i , [pt], JY,(r−a)n = (an)! h[pt], JX,rn i and thus Y also satisfies the Gamma conjecture I’. 9. Grassmannians In this section, we prove Gamma conjecture I’ (Conjecture 3.11) for Grassmannians Gr(r, n) using the Hori–Vafa mirror [37] and abelian/non-abelian correspondence [9]. The discussion in this section extends the proof of Dubrovin conjecture for Grassmannians by Ueda [62]. In the course of the proof, we obtain a formula for quantum cohomology central charges of Gr(r, n) in terms of mirror oscillatory integrals. We note that Gamma conjecture I’ for Gr(2, n) was proved by Golyshev [31]. Gamma conjectures I and II were proved for general Gr(r, n) in [26] by a different (but closely related) method based on the quantum Satake principle [32]. GAMMA CONJECTURE VIA MIRROR SYMMETRY 25 9.1. Abelian quotient and non-abelian quotient. Let G = Gr(r, n) denote the Grassmann variety of r-dimensional subspaces in Cn and let P = Pn−1 × · · · × Pn−1 (r times) denote the product of r copies of the projective space Pn−1 . We relate these two spaces in the framework of Martin [55]: G and P arise as non-abelian and abelian quotients of the same vector space Hom(Cr , Cn ): G = Hom(Cr , Cn )//GL(r), P = Hom(Cr , Cn )//(C× )r . We have a rational map P 99K G sending a collection of lines l1 , . . . , lr to the subspace spanned by l1 , . . . , lr . We can also relate P and G by the following diagram [55]: p F −−−−→ G   ιy (27) P where F := Fl(1, 2, . . . , r, n) is the partial flag variety, p : F → G is the natural projection and ι is a real-analytic embedding which sends a flag 0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vr ⊂ Cn to the lines L1 , . . . , Lr such that Li is the orthogonal complement of Vi−1 in Vi . Here we need to choose a Hermitian metric on Cn for the definition of ι. The diagram (27) naturally comes from the following description of G and P as symplectic reductions of Hom(Cr , Cn ): (28) G∼ = µ−1 G (η)/G, P∼ = µ−1 T (η0 )/T. F∼ = µ−1 G (η)/T, Here µG and µT are the moment maps of G = U (r) and T = (S 1 )r -actions on Hom(Cr , Cn ) respectively and η ∈ Lie(G)⋆ and η0 ∈ Lie(T )⋆ are non-zero central elements (i.e. scalar multiples of the identify matrix) such that π(η) = η0 (where π : Lie(G)⋆ → Lie(T )⋆ is the natural projection): µG (A) = A† A, r µT (A) = (A† A)i,i i=1 , µG / Lie(G)⋆ ◆◆◆ ◆◆◆ ◆ π µT ◆◆◆◆ ' Hom(Cr , Cn ) Lie(T )⋆ −1 The map ι in (27) is then induced from the inclusion µ−1 G (η) ⊂ µT (η0 ). We describe the cohomology groups of G and P. Let V1 ⊂ V2 ⊂ · · · ⊂ Vr denote the tautological bundles (with rank Vi = i) over F and set Li := (Vi /Vi−1 )∨ . The line bundle Li is the pull-back of the line bundle O(1) over the ith factor ∼ = Pn−1 of P under the non-algebraic map ι. We denote the corresponding line bundle over P also by Li , i.e. ι⋆ Li = Li . We set xi := c1 (Li ) ∈ H (P) or H (F). A symmetric polynomial in x1 , . . . , xr can be written in terms of Chern classes of Vr and thus makes sense as a cohomology class on G. The cohomology rings of P, G and F are described 26 S. GALKIN AND H. IRITANI in terms of xi as follows: . H (P) ∼ = C[x1 , . . . , xr ] hxn1 , . . . , xnr i, . H (G) ∼ = C[x1 , . . . , xr ]Sr hhn−r+1 , . . . , hn i, . H (F) ∼ = C[x1 , . . . , xr ] hhn−r+1 , . . . , hn i. Here hi = ci (Cn /Vr ) is the ith complete symmetric function of x1 , . . . , xr . An additive basis of H (G) is given by the Schubert classes: µ +r−j det xi j 16i,j6r (29) σµ := r−j det xi 16i,j6r where µ ranges over partitions such that n − r > µ1 > µ2 > · · · > µr > 0. This is the Poincaré dual of the Schubert cycle Ωµ = {V ∈ Gr(r, n) : dim(V ∩ Cn−r+i−µi ) > i, 1 6 i 6 r} P and deg σλ = 2|λ| = 2 ri=1 λi (degree as a cohomology class), see [34, §6, Chapter 1]. Let JPn−1 (t) be the J-function (see (7), (18)) of Pn−1 . We define the multivariable Jfunction of P by JP (t1 , . . . , tr ) = JPn−1 (t1 ) ⊗ JPn−1 (t2 ) ⊗ · · · ⊗ JPn−1 (tr ). ∼ H (Pn−1 )⊗r . Bertram–Ciocan-Fontanine–Kim [9] proved the This takes values in H (P) = following abelian/non-abelian correspondence between the J-functions of G and P. Theorem 9.1 ([9]). The J-function JG (t) of G is given by   Q ⋆ J (t , . . . , t ) (θ − θ ) ι √ i j P 1 r 16i<j6r  p⋆ JG (t) = e−σ1 π −1(r−1)  n n( r ) ∆ where ∆ := Q i<j (xi − xj ), θi := ti ∂t∂ i , ξ := eπ √ t1 =···=tr =ξt −1(r−1)/n and σ1 := x1 + · · · + xr . 9.2. Preliminary lemmas. We discuss elementary topological properties of the maps in (27). Martin [55] has shown similar results for general abelian/non-abelian quotients. Lemma 9.2. Let Nι → F denote the normal bundle of ι and let Tp = Ker(dp) ⊂ T F denote the relative tangent bundle of p. Then the complex structure I on P induces an isomorphism: I : Tp ∼ = Nι . In particular, we have an isomorphism √ ∼ p⋆ T G ⊕ Tp ⊗R C, −1 ι⋆ T P = of topological complex vector bundles, where the complex structure from that on the C-factor. √ −1 on Tp ⊗R C is induced Proof. Recall that P, G, F are described as symplectic reductions (28). Note that rank Nι = rank Tp = dim G − dim T . Let X denote the fundamental vector field associated with X ∈ Lie(G). For x ∈ µ−1 G (η), it follows from the property of the moment map that IX x is perpendicular to Tx µ−1 G (η) for X ∈ Lie(G). Thus we have an orthogonal decomposition: Tx Hom(Cr , Cn ) = Tx µ−1 G (η) ⊕ ITx (G · x). GAMMA CONJECTURE VIA MIRROR SYMMETRY 27 Similarly we have an orthogonal decomposition Tx Hom(Cr , Cn ) = Tx µ−1 T (η0 ) ⊕ ITx (T · x). Note that the tangent space T[x] P at [x] ∈ P = µ−1 T (µ0 )/T is identified with the orthogonal −1 complement of Tx (T · x) in Tx µT (η0 ), which is preserved by the complex structure I. Under this identification, the orthogonal complement of Tx (T · x) in Tx (G · x) is identified with the fiber (Tp )[x] ; by the above decompositions, the complex structure I sends this to the −1 orthogonal complement of Tx µ−1 G (η) in Tx µT (η0 ). This shows I((Tp )[x] ) = (Nι )[x] and the conclusion follows. Remark 9.3. Note that as complex vector bundles √ (Tp ⊗R C, −1) ∼ = Tp ⊕ T p where T p denotes the conjugate of the complex vector bundle Tp . Because F, P are compact oriented manifolds, the push-forward map ι∗ : H (F) → H (P) is well-defined (even though ι is not algebraic). Here we need to be a little careful about the orientation of the normal bundle of ι. Lemma 9.4. The push-forward map ι⋆ : H (F) → H (P) is given by ι⋆ (f (x)) = f (x) ∪ ∆ Q r for any polynomial f (x) = f (x1 , . . . , xr ). Here ∆ = i<j (xj − xi ) = (−1)(2) ∆. Proof. The normal bundle Nι of ι is not a complex vector bundle, but can be written as the formal difference of complex vector bundles: Nι = ι⋆ T P ⊖ T F and thus defines an element of the K-group K 0 (P) of topological complex vector bundles. Using Lemma 9.2, Remark 9.3 and a topological isomorphism T F ∼ = Tp ⊕ p⋆ T G, we have: Nι = Tp ⊗ C ⊖ Tp = T p . On the other hand, we have a topological isomorphism: M ∨ Hom(L∨ Tp ∼ = i , Lj ). i<j Thus T p → F is isomorphic to the restriction of the following complex vector bundle over P: M Li ⊗ Lj −→ P i<j and F is the zero locus of the section of this bundle defined by the given Hermitian metric on Cn . Therefore, ι⋆ (f (x)) = f (x) ∪ ∆. Lemma 9.5. For a polynomial f (x) = f (x1 , . . . , xr ), we have 1 p⋆ (f (x)) = ∆ X σ∈Sr ! sgn(σ)f (xσ(1) , . . . , xσ(r) ) . Note that the right-hand side is a symmetric polynomial in x1 , . . . , xr . 28 S. GALKIN AND H. IRITANI Proof. The Sr -action on P induces a non-algebraic Sr action on F which preserves the fibration p : F → G. The action of σ ∈ Sr on F changes the orientation of each fiber of p by sgn(σ). Therefore p⋆ (f (x)) = sgn(σ)p⋆ (f (xσ(1) , . . . , xσ(r) )). Thus we have ! X 1 sgn(σ)f (xσ(1) , . . . , xσ(r) ) p⋆ (f (x)) = p⋆ r! σ∈Sr P 1 σ∈Sr sgn(σ)f (xσ(1) , . . . , xσ(r) ) = p⋆ (∆). r! ∆ The conclusion follows from p⋆ (∆) = p⋆ (Euler(Tp )) = r!. 9.3. Comparison of cohomology and K-groups. Using the diagram (27), we identify the cohomology (or the K-group) of G with the anti-symmetric part of the cohomology (resp. the K-group) of P. This identification was proved by Martin [55] for cohomology groups of general abelian/non-abelian quotients and has been generalized to K-groups by Harada– Landweber [35]. We compare the cohomological and K-theoretic identifications via the map b F Ch(E). E 7→ Γ We identify the rth wedge product ∧r H (Pn−1 ) with the anti-symmetric part of H (P) (with respect to the Sr -action) via the map: X ∧r H (Pn−1 ) ∋ α1 ∧ α2 ∧ · · · ∧ αr 7−→ sgn(σ)ασ(1) ⊗ ασ(2) ⊗ · · · ⊗ ασ(r) ∈ H (P). σ∈Sr Similarly we identify ∧r K 0 (P) with the anti-symmetric part of K 0 (P), where K 0 (·) denotes the topological K-group. For a non-increasing sequence µ1 > µ2 > · · · > µr of integers, we set Eµ := p⋆ (Lµ1 1 ⊗ Lµ2 2 ⊗ · · · ⊗ Lµr r ) . By the Borel–Weil theory, Eµ is the vector bundle on G associated to the irreducible GL(r)representation of highest weight µ. Shifting µi by some number µi 7→ µi + ℓ simultaneously corresponds to twisting the GL(r)-representation by det⊗ℓ , where det : GL(r) → C× corresponds to O(1) on G. These vector bundles span the K-group of G. We define a line bundle R on F by −(r−1) R = L1 −(r−2) ⊗ L2 ⊗ · · · ⊗ L−1 r−1 . This restricts to a half-canonical bundle3 on each fiber of p. One can easily check that: (30) det(Tp∨ ) = p⋆ (O(r − 1)) ⊗ R⊗2 where p⋆ O(1) = L1 L2 · · · Lr . We have the following result: Proposition 9.6. Let µ1 > µ2 > · · · > µr be a non-increasing sequence and let k1 > k2 > · · · > kr be the strictly decreasing sequence given by ki = µi + r − i. Then we have: (1) The map (r!)−1 p⋆ ι⋆ : ∧r H (P) → H (G) sends the class xk11 ∧ xk22 ∧ · · · ∧ xkr r to the Schubert class σµ (29) and is an isomorphism. (2) The map (r!)−1 p⋆ (R⊗ι⋆ (·)) : ∧r K 0 (Pn−1 ) → K 0 (G) sends the class Lk11 ∧Lk22 ∧· · ·∧Lkr r to the class Eµ and is an isomorphism (over Z). 3corresponding to the half-sum ρ of positive roots GAMMA CONJECTURE VIA MIRROR SYMMETRY 29 (3) We have the commutative diagram: 1 p (R⊗ι⋆ (·)) r! ⋆ ∧r K 0 (Pn−1 ) (31) b P Ch(·) Γ ǫr −π e r! √ −1(r−1)σ1 p / K 0 (G) ⋆ ⋆ι b G Ch(·) Γ / H (G) ∧r H (Pn−1 ) √ √ r P where ǫr = (2π −1)−(2) and Ch(E) = p>0 (2π −1)p chp (E) is the modified Chern character. Proof. (1): This follows from Lemma 9.5 and the definition of the Schubert class σµ (29). (2): Using the Grothendieck–Riemann–Roch theorem, we have ch p⋆ (R ⊗ ι⋆ (Lk11 ⊗ · · · ⊗ Lkr r )) = p⋆ (ch(R ⊗ Lk11 ⊗ · · · ⊗ Lkr r ⊗ ch(Tp ))   Y xi − xj  = p⋆ eµ1 x1 +···+µr xr 1 − exi −xj i<j   Y xi − xj . = p⋆ ek1 x1 +···+kr xr exj − exi i<j By Lemma 9.5, it follows that this is anti-symmetric in k1 , . . . , kr . Therefore 1 ch p⋆ (R ⊗ ι⋆ (Lk11 ∧ · · · ∧ Lkr r )) = ch p⋆ (R ⊗ ι⋆ (Lk11 ⊗ · · · ⊗ Lkr r )) r! = ch (p⋆ (Lµ1 1 ⊗ · · · ⊗ Lµ1 r )) = ch(Eµ ). This shows that (r!)−1 p⋆ (R ⊗ ι⋆ (Lk11 ∧ · · · ∧ Lkr r )) = Eµ . (3): It suffices to prove the commutativity of the following two diagrams: ∧r K 0 (Pn−1 ) bP Ch(·) Γ ∧r H 0 (Pr−1 ) ι⋆ ι⋆ / K 0 (F) p⋆ (R⊗(·)) K 0 (F) bP ) Ch(·) (ι⋆ Γ / H 0 (G) b P ) Ch(·) (ι⋆ Γ H (F) ǫr e−π √ / K 0 (G) −1(r−1)σ1 p ⋆ b G Ch(·) Γ / H (G) The commutativity of the left diagram is obvious. The commutativity of the right diagram follows from Grothendieck–Riemann–Roch. For α ∈ K 0 (F), we have Ch(p⋆ (R ⊗ α)) = ǫr p⋆ (Ch(R ⊗ α) Td(Tp )). From ι⋆ T P ∼ = p⋆ T G ⊕ Tp ⊕ Tp (Lemma 9.2 and Remark 9.3), we have b P = (p⋆ Γ b P )Γ(T b p )Γ(T b p∨ ) = (p⋆ Γ b P ) Td(Tp )e−π ι⋆ Γ √ −1c1 (Tp ) where we used the fact that the Gamma class is a square root of the Todd class (see §1.2). Therefore b G ) Ch(R ⊗ α) Td(Tp ) b G Ch(p⋆ (R ⊗ α)) = ǫr p⋆ (p⋆ Γ Γ √ b P ) Ch(α) Ch(R)e−π −1c1 (Tp ) . = ǫr p⋆ (ι⋆ Γ The relationship (30) gives Ch(R)e−π lows. √ −1c1 (Tp ) = e−π √ −1(r−1)σ1 and the commutativity fol 30 S. GALKIN AND H. IRITANI We define the Euler pairing on ∧r K 0 (Pn−1 ) as χ(α1 ∧ α2 ∧ · · · ∧ αr , β1 ∧ β2 ∧ · · · ∧ βr ) := det(χPn−1 (αi , βj ))16i,j6r . This is 1/r! of the Euler pairing induced from K 0 (P). Recall the non-symmetric pairing [·, ·) defined in (2). Similarly we define the pairing [·, ·) on ∧r H (Pn−1 ) by (32) [α1 ∧ α2 ∧ · · · ∧ αr , β1 ∧ β2 ∧ · · · ∧ βr ) := det([αi , βj ))16i,j6r . This is again 1/r! of the pairing [·, ·) on H (P). Proposition 9.7. The horizontal maps in the diagram (31) preserves the pairings. More precisely, the map (r!)−1 p⋆ (R ⊗ ι⋆ (·)) : ∧r K 0 (Pn−1 ) → K 0 (G) preserves the Euler pairing χ and the map (r!)−1 ǫr e−π √ −1(r−1)σ1 preserves the pairing [·, ·). p⋆ ι⋆ : ∧r H (Pn−1 ) → H (G) Proof. Since the vertical maps in (31) intertwines the pairings χ and [·, ·) (see (1)), it suffices √ −1 −π −1(r−1)σ 1 p ι⋆ on cohomology preserves the pairing [·, ·). to show that the map (r!) ǫr e ⋆ r n−1 For α, β ∈ ∧ H (P ), we have h √ √ (r!)−1 ǫr e−π −1(r−1)σ1 p⋆ ι⋆ (α), (r!)−1 ǫr e−π −1(r−1)σ1 p⋆ ι⋆ (β) Z ǫ 2 √ √ 1 r π −1c1 (G) π −1 deg /2 √ (e p⋆ ι⋆ α) ∪ p⋆ ι⋆ (β) e = r! (2π −1)dim G G r Z √ √ (−1)(2) 1 ⋆ π −1c1 (P) π −1 deg /2 √ e α) ∪ p⋆ ι⋆ (β) (p ι e = ⋆ (r!)2 (2π −1)dim P G r Z √ √ (−1)(2) 1 π −1c1 (P) π −1 deg /2 √ (e = e α) ∪ ι⋆ p⋆ p⋆ ι⋆ (β). (r!)2 (2π −1)dim P P r By the formulas in Lemma 9.4 and Lemma 9.5, it follows easily that ι⋆ p⋆ p⋆ ι⋆ β = r!(−1)(2) β for the antisymmetric element β. Therefore this gives 1/r! of the pairing [·, ·) on P and the conclusion follows. Corollary 9.8 ([62]). For µ1 > µ2 > · · · > µr and ν1 > ν2 > · · · > νr χ(Eµ , Eν ) = det (χ(OPn−1 (li ), OPn−1 (ki ))16i,j6r ) where li = µi + r − i and ki = νi + r − i. Remark 9.9. The map (r!)−1 p⋆ ι⋆ : ∧r H (Pn−1 ) → H (G) sending xµ1 1 +r−1 ∧ xµ2 2 +r−2 ∧ · · · ∧ xµr r to σµ is called the Satake identification in [26]. 9.4. Quantum cohomology central charge. Here we restate the abelian/non-abelian correspondence of the J-functions in terms of quantum cohomology central charges. Recall from Remark 5.2 that the central charge Z G (E) of a vector bundle E → G is given by h √ √ bG Ch(E) , (33) Z G (E)(t) = (2π −1)dim G JG (eπ −1 t), Γ where [·, ·) is the pairing defined in (2). This is a function on the universal cover of the punctured t-plane C× . A branch of this function is determined when we specify arg t ∈ R. When t ∈ R>0 , we regard arg t = 0 unless otherwise specified. For α ∈ K 0 (P), we define h √ √ √ bP Ch(α) Z P (α) := (2π −1)dim P JP (eπ −1 t1 , . . . , eπ −1 tr ), Γ GAMMA CONJECTURE VIA MIRROR SYMMETRY 31 where [·, ·) is the pairing on H (P). When α = α1 ∧ α2 ∧ · · · ∧ αr ∈ ∧r K 0 (Pn−1 ), we have n−1 Z P (α) = det Z P (αi )(tj ) 16i,j6r where Z Pn−1 is the quantum cohomology central charge for Pn−1 . Proposition 9.10. Let Z G and Z P denote the quantum cohomology central charges of G and P respectively. Then we have the equality:   Y 1  (θi − θj ) · Z P (α) , Z G (p⋆ (R ⊗ ι⋆ α)) = √ r ( ) 2 (2π −1n) i<j where α ∈ ∧r K 0 (Pn−1 ) and ξ := eπ √ t1 =···=tr =ξt −1(r−1)/n . Proof. By Lemma 9.5, the abelian/non-abelian correspondence (Theorem 9.1) can be written in the form: √ 1 JG (t) = e−π −1(r−1)σ1 p⋆ ι⋆ Je(t) r! with   Y r e = n−(2) (θi − θj )JP (t1 , . . . , tr ) . J(t) i<j t1 =···=tr =ξt By Proposition 9.6 (3), we have b G Ch(p⋆ (R ⊗ ι⋆ α)) = ǫr e−π Γ √ −1(r−1)σ1 √ b P Ch(α) . p⋆ ι⋆ Γ By the above equations and the fact that ǫr!r e−π −1(r−1)σ1 p⋆ ι⋆ preserves the pairing [·, ·) (Proposition 9.7), we obtain h √ r e Γ bP Ch(α) Z G (p⋆ (R ⊗ ι⋆ α)) = (2π −1)dim G+(2) r! J(t), where the pairing [·, ·) on the right-hand side denotes the pairing (32) on ∧r H (Pn−1 ) (which r r is 1/r! of the pairing [·, ·) on H (P)). Using dim G + 2 = dim P − 2 , we obtain the formula in the proposition. 9.5. Integral representation of quantum cohomology central charges. In this section we give integral representations of quantum cohomology central charges in terms of the Hori– Vafa mirrors. Let f : (C× )n−1 → C be the mirror Laurent polynomial of Pn−1 (see §5): 1 f (y) = y1 + · · · + yn−1 + . y1 y2 · · · yn−1 The critical values of f are given as vk := ne−2π √ −1k/n , k ∈ Z/nZ. Let φ ∈ R be an admissible phase for {v0 , v1 , . . . , vn−1 } (see §4.2). Let Γk (φ) ⊂ (C× )n−1 denote the Lefschetz thimble for f associated with the critical value vk and the vanishing √ path vk + R>0 e −1φ (see Figure 2). The Lefschetz thimble Γk (φ) is homeomorphic to Rn−1 √ and fibers over the straight half-line f (Γk ) = vk + R>0 e −1φ ; the fiber is a vanishing cycle the “opposite” Lefschetz thimble associated to homeomorphic to S n−2 . We write Γ∨ k (φ) for √ −1φ the critical value vk and the path vk − R>0 e . We choose orientation of Lefschetz thimbles (n−1)(n−2)/2 . (φ)) = (−1) such that ♯(Γk (φ) ∩ Γ∨ k 32 S. GALKIN AND H. IRITANI • • • • • ✶ admissible direction φ • Figure 2. Vanishing paths in the admissible direction φ Proposition 9.11. Let φ ∈ R be an admissible phase for {v0 , v1 , . . . , vn−1 }. There exist K-classes Fk (φ), Gk (φ) ∈ K 0 (Pn−1 ) for k ∈ Z/nZ such that Z dy n−1 e−tf (y) , Z P (Fk (φ))(t) = y Γ (φ) Z k (34) √ dy n−1 etf (y) , Z P (Gk (φ))(e−π −1 t) = y Γ∨ k (φ) Vn−1 dyi hold when | arg t + φ| < π2 , where dy i=1 yi . Moreover, we have: y = (1) χ(Fk (φ), Gl (φ)) = δkl ; π π (2) when |k 2π n + φ| < 2 + n , we have Fk (φ) = OPn−1 (k). Proof. This follows from the result in [42, 47]. (See §6 and [26, §5] for a closely related discussion). By [42, Theorems 4.11, 4.14], we have an isomorphism K 0 (Pn−1 ) ∼ = Hn−1 ((C× )n−1 , {y : Re(tf (y)) > M }), α 7→ Γ(α, arg t) depending on arg t ∈ R, such that Γ(α, arg t) is Gauss–Manin flat with respect to the variation of arg t and that Z dy Pn−1 e−tf (y) . (α)(t) = Z y Γ(α,arg t) Here M is a sufficiently big positive number (it is sufficient that M > 2n|t|). Moreover, this isomorphism intertwines the Euler pairing with the intersection pairing: χ(α, β) = (−1) (n−1)(n−2) 2 ♯ (Γ(α, arg t + π) ∩ Γ(β, arg t)) . Therefore equation (34) and part (1) of the proposition hold for Fk , Gk such that Γ(Fk (φ), −φ) = Γk (φ), Γ(Gk (φ), −φ − π) = Γ∨ k (φ). (We only need to check (34) at arg t = −φ; it then follows by analytic continuation for other t ∈ (−φ − π2 , −φ + π2 ).) Recall from (20) that we have Z dy Pn−1 e−tf (y) (OPn−1 )(t) = Z y Γ0 (0) when arg t = 0. Therefore, when arg t = 2πk/n, we have by setting t′ = e−2π ZP n−1 n−1 √ (OPn−1 )(t′ ) Z ′ dy dy e−t f (y) e−tf (y) = y y Γ0 (0) Γk (−2πk/n) (OPn−1 (k))(t) = Z P Z = −1k/n t, GAMMA CONJECTURE VIA MIRROR SYMMETRY 33 where in the first line we used (18) and the √ definition of the central charge, and in the −1k/n y of variables and used the fact that 2π second i √ line we performed the change yi → e −2π −1k/n e Γ0 (0) = Γk (−2πk/n). From this it follows that Z dy Pn−1 e−tf (y) (OPn−1 (k))(t) = Z y Γk (φ) π when |φ + k 2π n | < 2 + and part (2) follows. π n and arg t = −φ. This implies Fk (φ) = OPn−1 (k) for such φ and k Remark 9.12. The sign (−1)(n−1)(n−2)/2 in the intersection pairing was missing in [42], and this has been corrected in [44, footnote 16]. Remark 9.13. Part (2) of the proposition determines roughly half of Fk (φ)’s. The whole collection {Fk (φ) : k ∈ Z/nZ} is obtained from the exceptional collection {O(k) : −φ − π < k 2π n < −φ + π} by a sequence of mutations (see [26, §5]), and thus is an exceptional collection. b Pn−1 Ch(Fk (φ)) : k ∈ Z/nZ} Because Fk (φ) is mirror to the Lefschetz thimble Γk (φ), the set {Γ n−1 gives the asymptotic basis of P at phase φ (see the proof of Theorem 6.4) and the Gamma conjecture II (§4.3) holds for Pn−1 . The Hori–Vafa mirror g̃ of P = (Pn−1 )r is given by: g̃(y) := f (~y1 ) + f (~y2 ) + · · · + f (~yr ) r X = yi,1 + yi,2 + · · · + yi,n−1 + i=1 1 yi,1 yi,2 · · · yi,n−1 where y = (~y1 , . . . , ~yr ) ∈ (C× )r(n−1) and ~yi = (yi,1 , . . . , yi,n−1 ) ∈ (C× )n−1 . Critical values of g̃ are given by ṽK := vk1 + · · · + vkr for K = (k1 , . . . , kr ) ∈ Zr . The product Γk1 (φ) × · · · × Γkr (φ) of Lefschetz thimbles for√f gives a Lefschetz thimble for g̃ associated to the critical value ṽK and the path ṽK + R>0 e −1φ . The Hori–Vafa mirror of G is obtained from the mirror of P by shifting the phase by (r − 1)π/n and restricting to “anti-symmetrized” Lefschetz thimbles (see [37, 62, 49]). We set g(y) := ξg̃(y) with ξ = eπ √ −1(r−1)/n . Then critical values of g(y) are vK := ξṽK for K = (k1 , . . . , kr ) ∈ Zr . For a tuple K = (k1 , . . . , kr ) ∈ Zr and φ ∈ R, we define the anti-symmetrized Lefschetz thimble for g(y) as: X ΓK (φ) := Γk1 (φ′ ) ∧ · · · ∧ Γkr (φ′ ) = sgn(σ)Γkσ(1) (φ′ ) × · · · × Γkσ(r) (φ′ ) σ∈Sr ∨ ′ ∨ ′ Γ∨ K (φ) := Γk1 (φ ) ∧ · · · ∧ Γkr (φ ) = with φ′ = φ − (r−1)π n . X σ∈Sr ′ ∨ ′ sgn(σ)Γ∨ kσ(1) (φ ) × · · · × Γkσ(r) (φ ) They are elements of the relative homology groups Hr(n−1) ((C× )r(n−1) , {y : ± Re(e− √ −1φ g(y)) > M }) 34 S. GALKIN AND H. IRITANI with M sufficiently large. We also define K-classes FK (φ), GK (φ) ∈ K 0 (G) as: 1 FK (φ) := p⋆ R ⊗ ι⋆ (Fk1 (φ′ ) ∧ · · · ∧ Fkr (φ′ )) r! (35) 1 GK (φ) := p⋆ R ⊗ ι⋆ (Gk1 (φ′ ) ∧ · · · ∧ Gkr (φ′ )) r! with φ′ = φ − (r−1)π n , where Fk (φ) and Gk (φ) are as in Proposition 9.11. Note that {FK (φ)}K or {GK (φ)}K gives an integral basis of K 0 (G) by Proposition 9.6. ′ , we can also write: Remark 9.14. By the change of variables yi,j = ξyi,j g(y) = r X i=1 (−1)r−1 ′ ′ ′ yi,1 + yi,2 + · · · + yi,n−1 + ′ ′ ′ yi,1 yi,2 · · · yi,n−1 ! Theorem 9.15. Let φ ∈ R be such that φ′ = φ − (r−1)π is admissible for {v0 , v1 , . . . , vn−1 }. n For a mutually distinct r-tuple K = (k1 , . . . , kr ) ∈ (Z/nZ)r , we have (r) Z 2 −ξt dy −tg(y) Y 1 G √ e · (f (~yi ) − f (~yj )) Z (FK (φ))(t) = r! 2π −1n ΓK (φ) y i<j Z G (GK (φ))(e−π √ −1 t) = 1 r! for | arg t + φ| < π2 , where ξ := eπ √ ξt √ 2π −1n −1(r−1)/n (r) Z 2 and dy y Γ∨ K (φ) := dy tg(y) Y e · (f (~yi ) − f (~yj )) y i<j Vr i=1 χ(FK (φ), GL (φ)) = δK,L Vn−1 dyi,j j=1 yi,j . Moreover, we have when the tuples K, L are ordered with respect to a fixed choice of a total order of Z/nZ. Proof. Combining Proposition 9.10 and Proposition 9.11, we have that   Y 1  (θi − θj ) · Z P (Fk1 (φ′ ) ∧ · · · ∧ Fkr (φ′ )) Z G (FK (φ))(t) = √ r ( ) r!(2π −1n) 2 i<j  Y 1  = (θi − θj ) · √ r r!(2π −1n)(2) i<j = 1 r! −ξt √ 2π −1n (r) Z 2 ΓK (φ) t1 =···=tr =ξt  dy e−(t1 f (~y1 )+···+tr f (~yr ))  y ΓK (φ) Z dy −tg(y) Y (f (~yi ) − f (~yj )), e y t1 =···=tr =ξt i<j √ where in the last line, we used Lemma 9.16 below. The formula for Z G (GK (φ))(e−π −1 t) follows similarly. The orthogonality χ(FK (φ), GL (φ)) = δK,L follows easily from Propositions 9.7 and 9.11. Lemma 9.16. We have   Pr Pr Y Y  (θi − θj ) e i=1 αi ti = e i=1 αi ti (αi ti − αj tj ) i<j where θi = ti ∂t∂ i . i<j GAMMA CONJECTURE VIA MIRROR SYMMETRY 35 Pr Proof. The left-hand side can be written in the Q form ϕ(α1 t1 , . . . , αr tr )e i=1 αi ti for some polynomial ϕ and the highest order term of ϕ is i<j (αi ti − αj tj ). On the other hand, ϕ Q is anti-symmetric in the arguments, and thus should be divisible by i<j (αi ti − αj tj ). This implies the lemma. Remark 9.17. The collection {FK (φ)}K , where K ranges over distinct r elements of Z/nZ, b G Ch(FK (φ))}K of G at phase φ. This follows either by studying yields the asymptotic basis {Γ mirror oscillatory integrals in more details or by combining [26, Proposition 6.5.1], Remark 9.13 and Proposition 9.6. It follows from the deformation argument in [26, §6] that this is mutation equivalent to Kapranov’s exceptional collection {Eµ : n − r > µ1 > · · · > µr > 0} [46]. 9.6. Eguchi–Hori–Xiong mirror and quantum period. There is another mirror description for Grassmannians due to Eguchi–Hori–Xiong [22] (see also [6]). This is a Laurent polynomial mirror of dimension dim G = r(n − r). We use this description to show that the limit sup in (10) can be replaced with the limit for Grassmannians. We introduce r(n − r) independent variables Xi,j with 1 6 i 6 r, 1 6 j 6 n − r. Define the Laurent polynomial W (X) in these variables by W (X) = r n−r−1 X X Xi,j+1 i=1 j=1 Xi,j + n−r X r−1 X Xi+1,j j=1 i=1 Xi,j + 1 X1,n−r + 1 . Xr,1 Batyrev–Ciocan-Fontanine–Kim–van-Straten [6] showed that the Newton polytope of W equals the fan polytope of a toric degeneration of G = Gr(r, n) and conjectured that W (X) is a weak Landau–Ginzburg model of G. Recently, Marsh and Rietsch [54] proved this conjecture by constructing a compactification of the Eguchi–Hori–Xiong mirror and showing an isomorphism between the quantum connection and the Gauss–Manin connection. Theorem 9.18 ([54]). The quantum period (9) GG (t) = h[pt], JG (t)i of the Grassmannian G is given by the constant term series of the Eguchi–Hori–Xiong mirror W (X), i.e. GG (t) = P ∞ 1 i i i=0 i! Const(W )t . P kn with It is easy to check that the constant term series of W (X) is of the form ∞ k=0 ak t ak 6= 0 for all k ∈ Z>0 (see [6, §5.2] for the explicit form). Therefore, by applying Lemma 3.13, we obtain the following. (It seems difficult to deduce this from Hori–Vafa mirrors). P kn be the quantum period of G. Proposition 9.19. Let GG (t) = h[pt], JG (t)i = ∞ k=0 Gkn t p Then Gkn > 0 for all k ∈ Z>0 and limk→∞ kn (kn)!Gkn exists. 9.7. Apéry constants. In this section we prove Gamma conjecture I’ (Conjecture 3.11) for G = Gr(r, n). Theorem 9.20. The Grassmannian Gr(r, n) satisfies Gamma conjecture I’. The rest of the paper is devoted to the proof of Theorem 9.20. Since Gr(r, n) ∼ = Gr(n−r, n), we may assume that r 6 n/2. We fix a sufficiently small phase φ > 0 such that φ′ = φ− (r−1)π n is admissible for {v0 , v1 , . . . , vn−1 }. Let Λ denote the index set of mutually distinct r-tuples K of elements of Z/nZ: Λ = {K = (k1 , . . . , kr ) ∈ (Z/nZ)r : n − 1 > k1 > k2 > · · · > kr > 0}. For K ∈ Λ, we write FK = FK (φ) and GK = GK (φ) for the elements in (35). Lemma 9.21. Set T := max{|vK | : K ∈ Λ}. 36 S. GALKIN AND H. IRITANI (1) We have T = n sin(πr/n) sin(π/n) . (2) If T = |vK | for K ∈ Λ, then K is given by a consecutive r-tuple of elements in Z/nZ √ −2π −1k/n and vK = T e for some k ∈ Z. (3) We have T = vK0 for K0 := (r − 1, r − 2, . . . , 1, 0). Moreover, FK0 = OG . Proof. Parts (1) and (2) follow easily from the definition of vK . To see Part (3), note that (since |φ| Proposition 9.11 (2) gives Fk (φ′ ) = OPn−1 (k) for 0 6 k 6 r − 1 and φ′ = φ − (r−1)π n is small and r 6 n/2). Then Proposition 9.6 (2) implies the conclusion. Let α ∈ H2|α| (G) be a homology class such that c1 (G) ∩ α = 0. We want to show that the limit formula (see (11)) hα, Jkn i bG i lim = hα, Γ n→∞ h[pt], Jkn i P kn holds where we set JG (t) = ec1 (G) log t ∞ k=0 Jkn t . We start with noting that it suffices to bG i = 0. Indeed, we obtain the general case by applying the show this formula when hα, Γ bG i[pt] (which satisfies hα′ , Γ bG i = 0). formula to α′ = α − hα, Γ 0 Let α b ∈ K (G) ⊗ C be a complexified K-class such that b G Ch(b PD(α) = Γ α). Then we have, using the definition of Z G (see (33) and (2)), Z JG (t) ∪ PD(α) hα, JG (t)i = G Z √ √ √ dim G−2|α| = ( −1) JG (t) ∪ e−π −1µ eπ −1c1 (G) PD(α) G h √ √ dim G |α| bG Ch(b JG (t), Γ α) = (−1)|α| Z G (b α)(e−π −1 t). = (−1) (2π −1) Using the dual bases {FK }K∈Λ and {GK }K∈Λ of K 0 (G), we can expand X α b= χ(FK , α b)GK . K∈Λ Thus by Theorem 9.15, we obtain the integral representation for hα, JG (t)i: Z X r dy tg(y) Y (f (~yi ) − f (~yj )) e χ(FK , α b) · t(2) (36) hα, JG (t)i = (−1)|α| cn,r y (φ) Γ∨ K i<j K∈Λ √ r 2 for | arg t + φ| < π/2, where we set cn,r = r!1 (ξ/(2π −1n))( ) . Let CK (λ) denote the “antisymmetrized” vanishing cycle in the fiber g−1 (λ): X −1 sgn(σ)Ckσ(1) (λ) × · · · × Ckσ(r) (λ) CK (λ) = Γ∨ K (φ) ∩ g (λ) = σ∈Sr √ where λ ∈ vK − R>0 e −1φ and Cki (λ) = Γki (φ′ ) ∩ f −1 (ξ −1 λ) is the vanishing cycle for f . We define the period integral PK (λ) as: Vr Vn−1 dyi,j Z Y i=1 j=1 yi,j . (f (~yi ) − f (~yj )) PK (λ) := dg −1 CK (λ)⊂g (λ) i<j g −1 (λ) GAMMA CONJECTURE VIA MIRROR SYMMETRY 37 Then we may rewrite (36) as the Laplace transform of the period: Z X r |α| χ(FK , α b) (37) hα, JG (t)i = (−1) cn,r dλ · t(2) etλ PK (λ). √ vK −e K∈Λ −1φ R >0 Applying the Laplace transformation ϕ(t) 7→ Z ∞ ϕ(t)e−ut dt 0 to both sides of (37), we obtain (38) ∞ X k=0 −kn−1 (kn)! hα, Jkn i u = (−1)|α| c′n,r X K∈Λ χ(FK , α b) Z vK √ −e −1φ R dλ >0 PK (λ) r (u − λ)(2)+1 when Re(u − vK ) > 0 for all K ∈ Λ, where c′n,r = 2r !cn,r . Note that the right-hand side can be analytically continued to a holomorphic function outside the branch cut: √ [ vK − e −1φ R>0 . K∈Λ See Figure 3. Moreover, this can be analytically continued to the universal cover of C \ {vK : K ∈ Λ}. Since the left-hand side is regular at u = ∞, it follows from Lemma 9.21 that the • • • • • • • • • • • • • • • • • • • • u−1 -plane Figure 3. Branch cut in the u-plane (left) and the 6, r = 2). In the right picture, we set φ = 0 for simplicity. • 1/T 0 • •T • • (right) (n = convergence radius of the left-hand side of (38) (as a power series in u−1 ) is bigger than or equal to 1 1 sin(π/n) = . T n sin(πr/n) By Cauchy’s integral formula, it follows that the jump of the function (38) across the cut √ r vK − e −1φ R>0 is proportional to χ(FK , α b) · (∂u )(2) PK (u). From this it follows that: √ (a) if α = [pt], we have χ(F , α b ) = (2π −1)− dim G χ(OG , Opt ) 6= 0 and thus the converK 0 P∞ gence radius of k=0 (kn)! h[pt], Jkn i xkn is exactly 1/T ; bG i = 0, then we have χ(FK , α b G , α) = 0 (by (1)); this implies (b) if hα, Γ b) = [Γ 0 b ) = χ(OG , α that (38) is holomorphic at u = vK0 = T ; since the left-hand side of (38) is a power series in u−n (multiplied by u−1 ), it is holomorphic at any other vK with |vK | = T P∞ and the series k=0 (kn)! hα, Jkn i xkn has a convergence radius R(α) > 1/T . 38 S. GALKIN AND H. IRITANI Here K0 = (r − 1, r − 2, . . . , 1, 0) and recall from Lemma 9.21 that FK0 = OG . From part (a) and Proposition 9.19, it follows that p lim kn (kn)! h[pt], Jkn i = T. k→∞ bG i = 0, From part (b), it follows that, when hα, Γ hα, Jkn i (kn)! hα, Jkn i = lim = 0. k→∞ h[pt], Jkn i k→∞ (kn)! h[pt], Jkn i lim This completes the proof of Theorem 9.20. 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[63] Yiannis Vlassopoulos, Quantum Cohomology and Morse Theory on the Loop Space of Toric Varieties, arXiv:math.AG/0203083. [64] Wolfgang Wasow: Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1965. National Research University Higher School of Economics, Faculty of Mathematics and Laboratory of Algebraic Geometry, Vavilova str. 7, Moscow 117312, Russia E-mail address: sergey.galkin@phystech.edu Department of Mathematics, Graduate School of Science, Kyoto University, KitashirakawaOiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan E-mail address: iritani@math.kyoto-u.ac.jp

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