Landau-Ginzburg mirror symmetry for orbifolded Frobenius algebras

Francis, Amanda; Jarvis, Tyler J.; Johnson, D.; Suggs, R.

doi:10.1090/pspum/085/1389

Cited by 14 publications

(24 citation statements)

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“…The Landau-Ginzburg mirror symmetry conjecture states that the A-model of a pair (W, G) should be isomorphic to the B-model of the dual pair (W T , G T ), and is denoted as A[W, G] ∼ = B[W T , G T ]. This conjecture has been proven in many cases [6,11], although the proof of the full conjecture remains open. To better understand mirror symmetry, it has been fruitful to focus on studying isomorphisms between Landau-Ginzburg models of the same type: either from A to A, or from B to B.…”

Section: Introductionmentioning

confidence: 94%

“…Property (*) in [6] is a generalization of the well behaved condition for a polynomial/group pair (W, G) given in Definition 1.1. We note that for the following polynomials, any possible choice of group (that fixes the polynomial and is contained in SL(n, C)) will form a well-behaved pair: fermats, loops in any number of variables, and any admissible polynomial in two variables.…”

Section: Isomorphism Extension Theoremmentioning

confidence: 99%

“…We note that Definition 1.1 is similar to Property (*) of [6] (see also Definition 3.1). We also require that the particular isomorphism between Milnor rings be equivariant.…”

Section: Introductionmentioning

confidence: 99%

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An isomorphism extension theorem for Landau-Ginzburg B-models

Cordner

2018

Communications in Algebra

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Landau-Ginzburg mirror symmetry studies isomorphisms between A-and B-models, which are graded Frobenius algebras that are constructed using a weighted homogeneous polynomial W and a related group of symmetries G of W . It is known that given two polynomials W1, W2 with the same weights and same group G, the corresponding A-models built with (W1,G) and (W2,G) are isomorphic. Though the same result cannot hold in full generality for B-models, which correspond to orbifolded Milnor rings, we provide a partial analogue. In particular, we exhibit conditions where isomorphisms between unorbifolded B-models (or Milnor rings) can extend to isomorphisms between their corresponding orbifolded B-models (or orbifolded Milnor rings).

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Section: Introductionmentioning

confidence: 94%

Section: Isomorphism Extension Theoremmentioning

confidence: 99%

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An isomorphism extension theorem for Landau-Ginzburg B-models

Cordner

2018

Communications in Algebra

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“…In this article, we will exploit the work of Francis-Jarvis-Johnson-Suggs in [19], who use Krawitz' definition of the B-model Frobenius algebra. This is the only work we know of that really deals with the Frobenius algebra structure for LG mirror symmetry.…”

Section: Landau-ginzburg Algebra Isomorphismmentioning

confidence: 99%

“…See Section 2 for a definition of both models. These are known to be isomorphic as vector spaces (see [33]), and in some cases as Frobenius algebras (see [19]), though there are still some question regarding the proper definitions of the Frobenius algebra structure. We will return to this question later.The LG/CY correspondence predicts a deep relationship between CY orbifolds and LG models in certain cases.…”

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confidence: 99%

Borcea–Voisin mirror symmetry for Landau–Ginzburg models

Francis¹,

Priddis²,

Schaug³

2019

Illinois J. Math.

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FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau-Ginzburg/Calabi-Yau correspondence, several birational morphisms of Calabi-Yau orbifolds should correspond to isomorphisms in FJRW theory. In this paper, we exhibit some of these isomorphisms that are related to Borcea-Voisin mirror symmetry. In particular, we develop a modified version of BHK mirror symmetry for certain LG models. Using these isomorphisms, we prove several interesting consequences in the corresponding geometries.2 AMANDA FRANCIS, NATHAN PRIDDIS, AND ANDREW SCHAUG BV mirror symmetry relies on mirror symmetry for K3 surfaces with involution, as described in the work of Nikulin (see [38]). That is, to each K3 surface S with involution, there is a family of K3 surfaces that are mirror dual to S.With this in mind, the mirror symmetry posited by Borcea and Voisin is described as follows. Let S ′ be a K3 surface with involution σ S ′ mirror dual to S, and consider the crepant resolution Y ′ of the quotientThe pair Y and Y ′ are said to be a Borcea-Voisin mirror pair. (Note that here we may treat elliptic curves as self-mirror.)In a much different way we can define BHK mirror symmetry for certain LG models. An LG model is a pair (W, G) of a quasihomogeneous polynomial W and group of symmetries G. Under certain conditions (see Section 2.2), we can associate to the pair (W, G) another LG model (W T , G T ), called the BHK mirror. It has been predicted and partially verified by mathematicians that the A-model construction for (W, T ) will be equivalent in a certain way (even in non-CY cases) to the B-model construction for (W T , G T ).Although mirror symmetry makes predictions for equivalence of the A-model and the B-model on many levels, in this paper we focus on the state space of these models. In other words, we consider the A-model state space A W,G , constructed by Fan, Jarvis, and Ruan in [17]-and the B-model state space B W T ,G T , given by Saito and Givental in [40,41,42,43,21,22]. See Section 2 for a definition of both models. These are known to be isomorphic as vector spaces (see [33]), and in some cases as Frobenius algebras (see [19]), though there are still some question regarding the proper definitions of the Frobenius algebra structure. We will return to this question later.The LG/CY correspondence predicts a deep relationship between CY orbifolds and LG models in certain cases. On the one side of the LG/CY corresondence, we have the LG model defined by the pair (W, G), and on the CY side, we have the orbifold quotient X W,G := [X W / G] in a quotient of weighted projective space. In this notation, which we use throughout this work, X W is a hypersurface defined by W and the group G is simply the quotient of G by the exponential grading operator. There are certain conditions on W and G, that make X W,G a Calabi-Yau orbifold which we describe in Section 2). On the simplest level, the LG/CY correspo...

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