The goal of this article is to provide an explicit algorithmic construction of formal $F$-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology $\mathbb{H}$ of a smooth projective Calabi-Yau complete intersection variety $X$ defined by homogeneous polynomials $G_1(\underline x), \dots, G_k(\underline x)$. Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra $\mathcal{A}$ obtained from the twisted de Rham complex which computes $\mathbb{H}$. More explicitly, we introduce a notion of \textit{a weak primitive form} associated to a solution of the Maurer-Cartan equation of $\mathcal{A}$ and the Gauss-Manin connection, which is a weakened version of Saito's primitive form (\cite{Saito}). In addition, we provide an explicit algorithm for a weak primitive form based on the Gröbner basis in order to achieve our goal. Our approach through the weak primitive form can be viewed as a unifying link (based on Witten's gauged linear sigma model, \cite{W93}) between the Barannikov-Kontsevich's approach to Frobenius manifolds via dGBV algebras (non-linear topological sigma model, \cite{BK}) and Saito's approach to Frobenius manifolds via primitive forms and higher residue pairings (Landau-Ginzburg model, \cite{ST}).
Extending the notion of Frobenius-splitting, we prove that every finite height Calabi-Yau variety defined over an algebraically closed field of positive characteristic can be lifted to the ring of Witt vectors of length two.
We generalize the toric residue mirror conjecture of Batyrev and Materov to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties.
We prove results that, for a certain class of non-compact Calabi-Yau threefolds, relate the Frobenius action on their $p$-adic cohomology to the Frobenius action on the $p$-adic cohomology of the corresponding curves. In the appendix, we describe our interpretation of the Griffiths-Dwork method.
It is one of the most important problems in mirror symmetry to obtain functorially Frobenius manifolds from smooth compact Calabi-Yau $A_\infty$-categories. This paper gives an approach to this problem based on the theory of primitive forms. Under an assumption on the formality of a certain homotopy algebra, a formal primitive form for a smooth compact Calabi-Yau dg algebra can be constructed, which enable us to have a formal Frobenius manifold.
We describe a variation of Dworks unit-root method to determine the degree four Frobenius polynomial for members of a 1-modulus Calabi-Yau family over $\mathbb{P}^1$ in terms of the holomorphic period near a point of maximal unipotent monodromy. The method is illustrated on a couple of examples.
Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with $r$ Laurent polynomials $f_1,...,f_r$.We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of $f_1,...,f_r$. Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.
We explain the homological relation between the Frobenius structure on the deformation space of Calabi-Yau manifold and the gauge theory of Kodaira-Spencer gravity. We show that the genus zero generating function of descendant invariants on Calabi-Yau manifolds from Barannikov's semi-infinite variation of Hodge structures is equivalent to the Kodaira-Spencer gauge theory at tree level.
Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips its local moduli space with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Fr…
Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This proves a variant of the Frobenius structure conjecture by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is supposed to give the hypothetical mirror family. Although the statement of our theorem involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constan…
We define the notion of mixed Frobenius structure which is a generalization of the structure of a Frobenius manifold. We construct a mixed Frobenius structure on the cohomology of weak Fano toric surfaces and that of the three dimensional projective space using local Gromov-Witten invariants. This is an analogue of the Frobenius manifold associated to the quantum cohomology in the local Calabi-Yau setting.
We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a…
We carry out the explicit computations that are used to write down the integrable hierarchy associated with the quintic Calabi-Yau threefold. We also do the calculations for the geometric structures emerging in the Gromov-Witten theory of the quintic, such as the Frobenius manifold structure and the special Kähler structure.
The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting with the Dolbeault complex of a Calabi-Yau manifold and conjecturally producing the B--side of the Mirror Conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of ``non-linear cohomology''. The comparison of thesestructures sheds some light on the gene…
We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariants
The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. A particular case is Dubrovin's theorem on semisimple Frobenius manifolds. Another special case starts with variations of Hodge structures. This case is used to compare two constructions of Frobenius manifolds, the one in singularity theory and the Barannikov-Kontsevich construction. For homogeneous polynomials which give Calabi-Yau hypersurfaces certain Frobenius submanifolds in both constructions are isomorphic.
Kontsevich suggested that the Landau-Ginzburg model presents a good formalism for homological mirror symmetry. In this paper we propose to investigate the LG theory from the viewpoint of Koopman-von Neumann's construction. New advances are thus provided, namely regarding a conjecture of Kontsevich-Soibelman (on a version of the Strominger-Yau-Zaslow mirror problem). We show that there exists a Monge-Ampère domain Y, generated by a space of probability densities parametrising mirror dual Calabi-Yau manifolds. This provides torus fibrations over Y. The mirror pairs are obtained via the Berglund-…
Building on our earlier work on toric residues and reduction, we give a proof for the mixed toric residue conejecture of Batyrev and Materov. We simplify and streamline our technique of tropical degenerations, which allows one to interpolate between two localization principles: one appearing in the intersection theory toric quotients and the other in the calculus of toric residues. This quickly leads to the proof of the conjecture, which gives a closed form for the summation of a generating series whose coefficients are certain variants of the "numbers" of rational curves on toric complete int…
In this paper we are concerned with the monodromy of Picard-Fuch differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgrou…
We study the space of stability conditions $\Stab(X)$ on the non-compact Calabi-Yau threefold $X$ which is the total space of the canonical bundle of $\PP^2$. We give a combinatorial description of an open subset of $\Stab(X)$ and state a conjecture relating $\Stab(X)$ to the Frobenius manifold obtained from the quantum cohomology of $\PP^2$. We give some evidence from mirror symmetry for this conjecture.
We provide a rigorous perturbative quantization of the B-twisted topological sigma model via a first order quantum field theory on derived mapping space in the formal neighborhood of constant maps. We prove that the first Chern class of the target manifold is the obstruction to the quantization via Batalin-Vilkovisky formalism. When the first Chern class vanishes, i.e. on Calabi-Yau manifolds, the factorization algebra of observables gives rise to the expected topological correlation functions in the B-model. We explain a twisting procedure to generalize to the Landau-Ginzburg case, and show t…
We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau 3-folds. We define relative invariants for the local theory which give rise to a 1+1-dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in case either the target genus or the degree equals 1.
Doran and Morgan have introduced certain rational basis for the monodromy group of the Picard-Fuchs operator of a hypergeometric family of Calabi-Yau threefolds. In this paper we compute numerically the transition matrix between a generalization of the Doran-Morgan basis and the Frobenius basis at a half-conifold point of a one-parameter family of double octic Calabi-Yau threefolds. We identify the entries of this matrix as rational functions in the special values $L(f,1)$ and $L(f,2)$ of the corresponding modular form $f$ and one constant. We also present related results concerning the rank o…
We argue how boundary B-type Landau-Ginzburg models based on matrix factorizations can be used to compute exact superpotentials for intersecting D-brane configurations on compact Calabi-Yau spaces. In this paper, we consider the dependence of open-string, boundary changing correlators on bulk moduli. This determines, via mirror symmetry, non-trivial disk instanton corrections in the A-model. As crucial ingredient we propose a differential equation that involves matrix analogs of Saito's higher residue pairings. As example, we compute from this for the elliptic curve certain quantum products m_…
Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field $k$. Let $X$ be a variety over $k$, $Y$ be an open of $X$, $Y'$ be an open of $Y$ dense in $X$. We extend Kedlaya's full faithfulness as follows (we do not suppose $Y$ to be smooth): the canonical functor $F\text{-}\mathrm{Isoc} ^† (Y,X/K) \to F\text{-}\mathrm{Isoc} ^† (Y,Y/K) $ is fully faithfull. Suppose now $Y$ smooth. We construct the category of partially overcoherent isocrystals over $(Y,X)$ denoted by $\mathrm{Isoc} ^{††} (Y,X/K) $ whose objects are some particular arithmetic $\D$-modules. Fur…
For a nondegenerate homogeneous polynomial $f\in\mathbb{C}[z_0, \dots, z_{n+1}]$ with degree $n+2$, we can obtain a $tt^*$ structure from the Landau-Ginzburg model $(\C^{n+2}, f)$ and a (new) $tt^*$ structure on the Calabi-Yau hypersurface defined by the zero locus of $f$ in $\C P^{n+1}$. We can prove that the big residue map considered by Steenbrink gives an isomorphism between the two $tt^*$ structures. We also build the correspondence for non-Calabi-Yau cases, and it turns out that only partial structure can be preserved. As an application, we show that the $tt^*$ geometry structure of Land…
Let $A$ be a Koszul Artin-Schelter regular algebra with Nakayama automorphism $ξ$. We show that the Yoneda Ext-algebra of the skew polynomial algebra $A[z;ξ]$ is a trivial extension of a Frobenius algebra. Then we prove that $A[z;ξ]$ is Calabi-Yau; and hence each Koszul Artin Schelter regular algebra is a subalgebra of a Koszul Calabi-Yau algebra. A superpotential $\hat{w}$ is also constructed so that the Calabi-Yau algebra $A[z;ξ]$ is isomorphic to the derivation quotient of $\hat{w}$. The Calabi-Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Arti…
We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a toric Calabi-Yau hypersurface, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an important role in recent study of D-branes [25]. Using the var…
Let $C$ be a smooth projective curve over $\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $ϕ:\mathcal{E}\to C$ its minimal regular model. For each $P\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\mathcal{E}_P=ϕ^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\mathcal{E}_P$ over the residue field $κ_P$ of $P$ are of the form $q_P^{1/2}e^{iθ_P},q_{P}e^{-iθ_P}$, where $q_P=q^{°(P)}$ and $0\leθ_P\leπ$. The goal of this note is to determine given an integer $B\ge 1$, $α,β\in[0,π]$ the number of $P\in C$ where the reduction of $E$ is goo…
Categorically, we introduce the Calabi-Yau-$\mathbb{X}$ categories $\mathcal{D}_{\mathbb{X}}$ of a graded marked surface $\mathbf{S}^λ$, as a $q$-deformation of the topological Fukaya category $\mathcal{D}_\infty$ of $\mathbf{S}^λ$. We show that $\mathcal{D}_\infty$ can be identified with the cluster-$\mathbb{X}$ category associated to $\mathcal{D}_{\mathbb{X}}$. Geometrically, we construct and identify the space of $q$-quadratic differentials on the logarithm surface $\operatorname{log}_{\mathbf{c}} \mathbf{S}_\vartriangle^λ$ with the space of induced $q$-stability conditions on $\mathcal{D}_…
Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；University of Sheffield, School of Mathematical and Physical Sciences（谢菲尔德大学数学与物理科学学院）
In this paper we prove that the GW invariants of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are quasi-modular forms. Our method is based on Givental's higher genus reconstruction formalism applied to the settings of Saito's Frobenius structures for simple elliptic singularities. Our results are part of a larger project whose goal is to prove the Landau-Ginzburg/Calabi-Yau correspondence for simple elliptic singularities. The correspondence describes a relation between Gromov-Witten theory (of a certain hypersurface) and Fan-Jarvis-Ruan-Witten theory (of a certain La…
We present a construction of (faithful) group actions via derived equivalences in the general categorical setting of algebraic 2-Calabi--Yau triangulated categories. To each algebraic 2-Calabi--Yau category $\mathscr{C}$ satisfying standard mild assumptions, we associate a groupoid $\mathscr{G}_{ \mathscr{C} }$, named the green groupoid of $\mathscr{C}$, defined in an intrinsic homological way. Its objects are given by a set of representatives $\operatorname{mrig} \mathscr{C}$ of the equivalence classes of basic maximal rigid objects of $\mathscr{C}$, arrows are given by mutation, and relation…
Gauge Origami of the Conifold $\times$ $\mathbb{C}$
Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学，法国国家科学研究中心，IMB UMR 5584）；Institut Universitaire de France (IUF)（法国高等研究院）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
Solutions of Calabi-Yau Differential Operators as Truncated p-adic Series and Efficient Computation of Zeta Functions
University of Sheffield（谢菲尔德大学）；PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical Physics Johannes Gutenberg-Universität Mainz（PRISMA卓越集群与美因茨理论物理研究所 约翰内斯·古腾堡美因茨大学）；Okinawa Institute of Science and Technology（冲绳科学技术大学院大学）；Cornell University（康奈尔大学）
We show that Ambro-Kawamata's non-vanishing conjecture holds true for a quasi-smooth WCI X which is Fano or Calabi-Yau, i.e. we prove that, if H is an ample Cartier divisor on X, then |H| is not empty. If X is smooth, we further show that the general element of |H| is smooth. We then verify Ambro-Kawamata's conjecture for any quasi-smooth weighted hypersurface. We also verify Fujita's freeness conjecture for a Gorenstein quasi-smooth weighted hypersurface. For the proofs, we introduce the arithmetic notion of regular pairs and enlighten some interesting connection with the Frobenius coin probl…
We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau $d$-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polyto…
