FJRW-Rings and Mirror Symmetry

Krawitz, Marc; Priddis, Nathan; Acosta, Pedro; Bergin, Natalie; Rathnakumara, Himal

doi:10.1007/s00220-009-0929-7

Cited by 70 publications

(177 citation statements)

References 2 publications

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“…In Sect. 1, we reformulate Kreuzer's and Krawitz's explicit description [25,27] of the state space of invertible polynomials using a natural bookkeeping device of decorated graphs, and we describe the moduli space of W -spin curves. In Sect.…”

Section: 3mentioning

confidence: 99%

“…By [25,Lemma 1.7], the set of all the elements e(C γ ), with a diagonal automorphism γ and an admissible and balanced decoration C γ , forms a basis of the state space,…”

Section: Fermatmentioning

confidence: 99%

“…These polynomials were completely classified by Kreuzer and Skarke [28] as Thom-Sebastiani sums 2 of terms of type Fermat, chain or loop (a) x a+1 ; (b) x with l ≥ 2. Clearly, if only Fermat terms occur, the weights w j divide the degree d. On the other hand, the Gorenstein condition may fail when chain terms appear, consider for instance the chain polynomial W = x Under the Calabi-Yau assumption, these polynomials allow an explicit and elementary construction of the (conjectural) mirror, by Berglund-Hübsch [2] and by Krawitz [25]:…”

Section: Introductionmentioning

confidence: 99%

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A Landau–Ginzburg mirror theorem without concavity

Guéré¹

2016

Duke Math. J.

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Abstract. We provide a mirror symmetry theorem in a range of cases where the state-of-the-art techniques relying on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials named after Fan, Jarvis, Ruan, and Witten's quantum singularity theory and viewed as the counterpart of a non-convex Gromov-Witten potential via the physical LG/CY correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus zero. In the non-concave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory. In the last two decades, mirror symmetry has been a central statement in theoretical physics and a fundamental driving force for several developments in mathematics. For instance it can be phrased mathematically as a prediction on GromovWitten invariants, namely the intersection numbers attached to curves traced on a Calabi-Yau variety. In this form, it has been proven in a vast range of concrete cases: the most famous example provides a full computation of the genus-zero invariants enumerating rational curves on the quintic threefold [21,29]. Even in this case, it is often pointed out how we still lack a complete computation in higher genus (only the genus-one case was completely proven by Zinger [35]).But even in genus zero the problem of computing Gromov-Witten invariants of projective varieties is far from being completely solved; indeed, most known techniques focus on computing Gromov-Witten invariants attached to cohomology classes which lie in the so-called ambient part of cohomology: the restriction to classes from the ambient projective space. For the quintic threefold, working with ambient cohomology classes turns out to determine the entire theory; however, in general, this scheme covers only a tiny portion of quantum cohomology. Remarkably, even the ambient cohomology classes may pose problem as soon as we work with orbifolds.It is interesting to notice that these gaps in Gromov-Witten computation all arise from the same phenomena: as we argue below, certain positivity or negativity conditions named convexity and concavity are not always satisfied, making the virtual cycle 1 challenging to compute. In genus one, this difficulty was overcome by Zinger after a great deal of hard work, but we still lack a comprehensive approach for higher genus. Guided by the frame of ideas of mirror symmetry and the Landau-Ginzburg/Calabi-Yau correspondence, we switch to the quantum theory of singularities introduced by Fan, Jarvis, and Ruan [17,18] based on ideas of Witten [34] (FJRW theory). Single non-concave quantum invariants were inferred from concave ones using tautological relation (e.g. using WDVV equation as in [16,17,26]), but so far no systematic approach tackling directly the virtual cycle has been taken. Polishchuk and Vaintrob recently opened the way to an algebraic computation: their construction [32] of a virtual cycle is given by applyin...

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Section: 3mentioning

confidence: 99%

“…By [25,Lemma 1.7], the set of all the elements e(C γ ), with a diagonal automorphism γ and an admissible and balanced decoration C γ , forms a basis of the state space,…”

Section: Fermatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Landau–Ginzburg mirror theorem without concavity

Guéré¹

2016

Duke Math. J.

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“…One can check that G T is a group and that the definition is independent of the presentation of the elements of G. Additionally, the transpose group has the following properties, which are verified in [Kra10]:…”

Section: Review Of the Constructionsmentioning

confidence: 90%

“…The product was first written down explicitly in [Kra10], following ideas from [Kau06]. To define the product on B W,G , first note that for all g there is a surjective homomorphism of Milnor rings Q e → Q g given by setting to zero all variables not fixed by g. Thus, Q g may be thought of as a cyclic Q e -module with generator ⌈1 ; g⌋.…”

Section: This Gives An Isomorphismmentioning

confidence: 99%

Landau-Ginzburg mirror symmetry for orbifolded Frobenius algebras

Francis¹,

Jarvis²,

Johnson³

et al. 2012

Proceedings of Symposia in Pure Mathematics

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Abstract. We prove the Landau-Ginzburg Mirror Symmetry Conjecture at the level of (orbifolded) Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups. Specifically, we show that for a quasi-homogeneous polynomial W and an admissible group G within the class, the Frobenius algebra arising in the FJRW theory [FJR07b]

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