Landau–Ginzburg/Calabi–Yau correspondence for the complete intersections X3,3 and X2,2,2,2

Clader, Emily

doi:10.1016/j.aim.2016.11.010

Cited by 33 publications

(65 citation statements)

References 16 publications

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“…Mathematically, the LG/CY correspondence relates the Gromov-Witten theory of Z to the quantum singularity theory of W , and it was proved in varying levels of generality by Chiodo-Ruan [8], Chiodo-Iritani-Ruan [6], and Lee-Priddis-Shoemaker [23]. It has been extended to the non-Calabi-Yau setting by Acosta [1] and Acosta-Shoemaker [2], and to very special complete intersections by the first author [11].…”

Section: Introductionmentioning

confidence: 99%

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Sigma Models and Phase Transitions for Complete Intersections

Clader

Ross

2017

International Mathematics Research Notices

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We study a one-parameter family of gauged linear sigma models (GLSMs) naturally associated to a complete intersection in weighted projective space. In the positive phase of the family we recover Gromov-Witten theory of the complete intersection, while in the negative phase we obtain a Landau-Ginzburg-type theory. Focusing on the negative phase, we develop foundational properties which allow us to state and prove a genus-zero comparison theorem that generalizes the multiple log-canonical correspondence and should be viewed as analogous to quantum Serre duality in the positive phase. Using this comparison result, along with the crepant transformation conjecture and quantum Serre duality, we prove a genus-zero correspondence between the GLSMs which arise at the two phases, thereby generalizing the Landau-Ginzburg/Calabi-Yau correspondence to complete intersections.The right-hand vertical arrow is quantum Serre duality (QSD) and the quantum Lefschetz hyperplane principle, developed by Coates-Givental [14] and Tseng [26]. It is an identification of a T-equivariant extension of the Gromov-Witten theory of Z to the T-equivariant Gromov-Witten theory of X + . The top of the diagram is the crepant transformation conjecture (CTC), proved in this setting by Coates-Iritani-Jiang [15].The equivalence in the left-hand vertical arrow is new and is the technical heart of our paper. It generalizes the multiple log-canonical (MLK) correspondence of [23] and serves as the analogue in the negative phase of quantum Serre duality:

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Section: Introductionmentioning

confidence: 99%

“…Localization graphs. Virtual localization expresses the genuszero GLSM correlators(11) φ 1 ψ a 1 · · · φ n ψ an X,W,T 0,n,β as a sum of contributions from the T-fixed loci on QM W 0,n (X, β). The fixed loci of QM W 0,n (X, β) are indexed by decorated trees Γ.…”

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Sigma Models and Phase Transitions for Complete Intersections

Clader

Ross

2017

International Mathematics Research Notices

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“…In this expression, G max W1 acts trivially on the ambient classes. Comparing expressions (13) and (14), we see that we only need to compare the narrow sectors and the ambient classes. The degree shift for the broad sectors and for the primitive classes in the Chen-Ruan cohomology agree because the two age shifts agree (see [10,Lemma 22]).…”

Section: The Isomorphism Comparing Expressionsmentioning

confidence: 99%

“…It is expected that the correspondence will hold for these complete intersections as well, though it has not been proven. There is some work of Chiodo-Nagel (see [12]) and Clader (see [13]) in this direction.In particular, the mirror symmetry we have described for CY threefolds is related by the LG/CY correspondence to mirror symmetry for LG models. Using this relationship, we will extend BHK mirror symmetry to certain LG models.…”

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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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“…. , W n ) developed in this paper, have already been used to provide a quantum correspondence between Calabi-Yau complete intersections inside projective spaces and hybrid Landau-Ginzburg models, see Clader [11]. Recently, Fan, Jarvis and Ruan provided a generalisation of their theory based on the state space H(W 1 , .…”

Section: Introductionmentioning

confidence: 97%

The hybrid Landau–Ginzburg models of Calabi–Yau complete intersections

Chiodo¹,

Nagel²

2018

Proceedings of Symposia in Pure Mathematics

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We observe that the state space of Landau-Ginzburg isolated singularities is simply a special case of Chen-Ruan orbifold cohomology relative to the generic fibre of the potential. This leads to the definition of the cohomology of hybrid Landau-Ginzburg models and its identification via an explicit isomorphism to the cohomology of Calabi-Yau complete intersections inside weighted projective spaces. The combinatorial method used in the case of hypersurfaces proven by the first named author in collaboration with Ruan is streamlined and generalised after an orbifold version of the Thom isomorphism and of the Tate twist.

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Landau–Ginzburg/Calabi–Yau correspondence for the complete intersections X3,3 and X2,2,2,2

Cited by 33 publications

References 16 publications

Sigma Models and Phase Transitions for Complete Intersections

Sigma Models and Phase Transitions for Complete Intersections

Borcea–Voisin mirror symmetry for Landau–Ginzburg models

The hybrid Landau–Ginzburg models of Calabi–Yau complete intersections

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