The Landau–Ginzburg/Calabi–Yau correspondence and the mirror quintic

Priddis, Nathan

doi:10.2969/aspm/08310357

Cited by 6 publications

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“…Proof. The proof has been given in several places, including [7], [6], [20], so we will not give it in detail here. It consists of checking that over any geometric point (C, p 1 , .…”

Section: Cohft Of Fjrw Theory and Gromov-witten Theorymentioning

confidence: 99%

“…In this section, we reformulate FJRW theory in order to obtain the genus zero potential F ( Ẽ7 ,G max ) 0 from a basic CohFT. This method is also used in [7], [6], [20], so we will be brief. The details can be found in these other articles.…”

Section: Extended Fjrw Correlatorsmentioning

confidence: 99%

“…Proof. This proof is given in [7], and [20], so we only give an outline. Comparing A 3 and A 3 , we see that if m ki ∈…”

Section: Proposition 61 ([7]mentioning

confidence: 99%

“…Second, we give a Givental-type formula, expressing the partition function of FJRW theory of (x 4 + y 4 + z 2 , G max ) via the partition function of the so-called "untwisted theory". This step is connected to the work of [7,20,8]. The partition function of the "untwisted theory" differs from the partition function of the product of the Gromov-Witten theories of a point just by a linear change of the variables.…”

Section: Introductionmentioning

confidence: 99%

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Givental-Type Reconstruction at a Nonsemisimple Point

Basalaev¹,

Priddis²

2018

Michigan Math. J.

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In this paper we consider the orbifold curve, which is a quotient of an elliptic curve E by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold curve via the product of the Gromov-Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Giventals action giving the CY/LG correspondence between the Gromov-Witten theory of the orbifold curve E /Z 4 and FJRW theory of the pair defined by the polynomial x 4 + y 4 + z 2 and the maximal group of diagonal symmetries. The methods we have developed can easily be applied to other finite quotients of an elliptic curve. Using Givental's action we also recover this FJRW theory via the product of the Gromov-Witten theories of a point. Combined with the CY/LG action we get a result in pure Gromov-Witten theory with the help of modern mirror symmetry conjectures. 4,4,2 39References 40 ALEXEY BASALAEV AND NATHAN PRIDDIS pairing η. A cohomological field theory (CohFT for brevity) Λ g,n on (V, η) is a system of linear maps Λ g,n

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“…Proof. The proof has been given in several places, including [7], [6], [20], so we will not give it in detail here. It consists of checking that over any geometric point (C, p 1 , .…”

Section: Cohft Of Fjrw Theory and Gromov-witten Theorymentioning

confidence: 99%

Section: Extended Fjrw Correlatorsmentioning

confidence: 99%

“…Proof. This proof is given in [7], and [20], so we only give an outline. Comparing A 3 and A 3 , we see that if m ki ∈…”

Section: Proposition 61 ([7]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Givental-Type Reconstruction at a Nonsemisimple Point

Basalaev¹,

Priddis²

2018

Michigan Math. J.

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“…correspondence have been formalised in a few ways, but the one of most importance here is that produced by [16] between Gromov-Witten theory on the Calabi-Yau side, and Fan-Jarvis-Ruan-Witten (FJRW) theory on the Landau-Ginzburg side, both defined for a hypersurface of weighted projective space. Such a relationship has been established already for a number of examples, including the quintic [11] mirror quintic [25], general Calabi-Yau hypersurfaces [6], and in a more general form for the classic Calabi-Yau three-fold complete intersections [14], and hypersurfaces of Fano and general type [1]. There is a corresponding mirror symmetry for FJRW theory [22], known as Bergland-Hübsch-Krawitz (BHK) mirror symmetry, forming a square of dualities.…”

Section: Introductionmentioning

confidence: 82%

The Gromov-Witten Theory of Borcea-Voisin Orbifolds and Its Analytic Continuations

Schaug

2015

Preprint

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In the early 1990s, Borcea-Voisin orbifolds were some of the earliest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry. However, their quantum theory has been poorly investigated. We study this in the context of the gauged linear sigma model, which in their case encompasses Gromov-Witten theory and its three companions (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, we calculate all four genus zero theories explicitly. Furthermore, we relate the I-functions of these theories by analytic continuation and symplectic transformation. In particular, the relation between the Gromov-Witten and FJRW theories can be viewed as an example of the Landau-Ginzburg/Calabi-Yau correspondence for complete intersections of toric varieties.

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$6$-dimensional FJRW theories of the simple–elliptic singularities

Basalaev

2022

Asian Journal of Mathematics

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We give explicitly in the closed formulae the genus zero primary potentials of the three 6-dimensional FJRW theories of the simple-elliptic singularity Ẽ7 with the non-maximal symmetry groups. For each of these FJRW theories we establish the CY/LG correspondence to the Gromov-Witten theory of the elliptic orbifold [E/(Z/2Z)] -the orbifold quotient of the elliptic curve by the hyperelliptic involution. Namely, we give explicitly the Givental's group elements, whose actions on the partition function of the Gromov-Witten theory of [E/(Z/2Z)] give up to a linear change of the variables the partition functions of the FJRW theories mentioned. We keep track of the linear changes of the variables needed. We show that using only the axioms of Fan-Jarvis-Ruan, the genus zero potential can only be reconstructed up to a scaling.

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The Landau–Ginzburg/Calabi–Yau correspondence and the mirror quintic

Cited by 6 publications

References 0 publications

Givental-Type Reconstruction at a Nonsemisimple Point

Givental-Type Reconstruction at a Nonsemisimple Point

The Gromov-Witten Theory of Borcea-Voisin Orbifolds and Its Analytic Continuations

$6$-dimensional FJRW theories of the simple–elliptic singularities

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