Wall-crossing in genus zero Landau–Ginzburg theory

Ross, Dustin; Ruan, Yongbin

doi:10.1515/crelle-2015-0005

Cited by 13 publications

(33 citation statements)

References 23 publications

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“…Rather, we recover a theory built from moduli spaces of weighted spin curves, which were introduced by the second author and Ruan in [25]. Nonetheless, the main result of [25] states that the theory built from weighted spin curves is equivalent, in genus zero, to the usual singularity theory, and it is through this equivalence that Theorem 1 generalizes the LG/CY correspondence.…”

Section: Introductionmentioning

confidence: 98%

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: 98%

Sigma Models and Phase Transitions for Complete Intersections

Clader

Ross

2017

International Mathematics Research Notices

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“…The ϕ-fields identically vanish in the genus-0 case for degree reasons. The proof is the same as that of Lemma 1.5 of [29]. Hence we have M 1/r,ϕ 0,γ = M 1/r 0,γ .…”

Section: Theorem 18 Impliesmentioning

confidence: 71%

“…We expect that the LG side and the CY side are more directly related near ǫ = 0. Using wall-crossing Ross-Ruan proved the LG/CY correspondence in genus 0 [29]. In genus 1, Guo-Ross used the wall-crossing formula to compute the FJRW invariants of the quintic 3-fold explicitly [18] and verified the genus-1 LG/CY correspondence [19].…”

Section: Introductionmentioning

confidence: 99%

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Higher-genus wall-crossing in Landau–Ginzburg theory

Zhou¹

2020

Advances in Mathematics

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For a Fermat quasi-homogeneous polynomial, we study the associated weighted Fan-Jarvis-Ruan-Witten theory with narrow insertions. We prove a wall-crossing formula in all genera via localization on a master space, which is constructed by introducing an additional tangent vector to the moduli problem. This is a Landau-Ginzburg theory analogue of the higher-genus quasi-map wall-crossing formula proved by Ciocan-Fontanine and Kim. It generalizes the genus-0 result by Ross-Ruan and the genus-1 result by Guo-Ross.It is easy to see that the isomorphism is compatible with the cosections.Then we consider the τ * T M term of the distinguished triangle (15). By Lemma 13, the fixed part is τ * T F 1/r J . Moreover, since pr 1 isétale, we have T F 1/r J ∼ = pr * 1 T M 1/r g,γ J . This finishes the proof that pr * 1 ([M 1/r,ϕ g,γJ ] vir loc ) = [F ϕ J ] vir loc .2 The C * -action on L is only well-defined up to µ µ µr. Nevertheless we can compose the action with the r-th power map C * → C * so that it is well-defined. 1/r 0,γ . However, we can pushforward everything to a point instead. For any non-negative integer c this gives (28) J ψ c φ a , µ + J (−ψ)|(φ bj ) j ∈J 0 0,2|(n−|J|) = µ B (z) z −c−1 ,

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“…Proof By , the

I

‐function

I^{false[{double-struckC}^{3} / Z_{3} false]}

lies on the Lagrangian cone

L^{false[{double-struckC}^{3} / Z_{3} false]}

encoding the genus 0 Gromov–Witten theory of

[C^{3} / Z^{3}]

. By standard properties of the Lagrangian cone, we obtain the following results, see, for example, :

\begin{matrix} true \\ right S^{false[{double-struckC}^{3} / Z_{3} false]} (Θ false(θ false), z) (ϕ_{0}) & = & left I^{false[{double-struckC}^{3} / Z_{3} false]}, \\ right S^{false[{double-struckC}^{3} / Z_{3} false]} (Θ false(θ false), z) (ϕ_{1}) & = & left \frac{z sans-serifD S^{false[{double-struckC}^{3} / Z_{3} false]} (ϕ_{0})}{C_{1}}, \\ right S^{false[{double-struckC}^{3} / Z_{3} false]} (Θ false(θ false), z) (ϕ_{2}) & = & left z sans-serifD S^{false[{double-struckC}^{3} /} \end{matrix}

…”

Section: Genus 0 Theory For [C3/double-struckz3]mentioning

confidence: 98%

Crepant resolution and the holomorphic anomaly equation for [C3/Z3]

Lho

Pandharipande

2019

Proc. London Math. Soc.

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We study the orbifold Gromov–Witten theory of the quotient [C3/double-struckZ3] in all genera. Our first result is a proof of the holomorphic anomaly equations in the precise form predicted by B‐model physics. Our second result is an exact crepant resolution correspondence relating the Gromov–Witten theories of [C3/double-struckZ3] and local double-struckP2. The proof of the correspondence requires an identity proven in the Appendix by T. Coates and H. Iritani.

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Wall-crossing in genus zero Landau–Ginzburg theory

Cited by 13 publications

References 23 publications

Sigma Models and Phase Transitions for Complete Intersections

Sigma Models and Phase Transitions for Complete Intersections

Higher-genus wall-crossing in Landau–Ginzburg theory

Crepant resolution and the holomorphic anomaly equation for [C3/Z3]

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