Witten’s D 4 integrable hierarchies conjecture

Fan, Huijun; Francis, Amanda; Jarvis, Tyler J.; Merrell, Evan; Ruan, Yongbin

doi:10.1007/s11401-016-0944-x

Cited by 23 publications

(27 citation statements)

References 13 publications

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“…For all (w, G) except the D 4 singularity with the group G = J , the Reconstruction Theorem implies that both theories are determined by the Frobenius algebra structure on the state space along with a certain four-point correlator which can computed using the Concavity property. In the remaining case (D 4 , J ) a similar statement is true as follows from [16,Thm. 4.5].…”

Section: Calculations For Genus Zero and Three Pointssupporting

confidence: 61%

Matrix factorizations and cohomological field theories

Polishchuk

Vaintrob

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

110

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We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W . This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W -structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov-Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the DeligneMumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools.

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Section: Calculations For Genus Zero and Three Pointssupporting

confidence: 61%

Matrix factorizations and cohomological field theories

Polishchuk

Vaintrob

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

110

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“…After the Γreduction theorem, the only remaining part of the proof is to compare the Γaction on genus zero part of FJRW-theory with its counterpart in integrable hierarchies. We can use the mirror theorems of [FJR2,FFJ] to compare the action of FJRW theory with the action of the mirror Frobenius manifold structures. In this subsection, we match the Γ-action on the Frobenius manifold structure of the ADE singularities with that of the semiclassical limit of the Drinfeld-Sokolov hierarchy.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…The above theorem provides an alternative proof of the ADE Witten conjecture by directly connecting the FJRW invariants to the Drinfeld-Sokolov hierarchies. Proof: According to [FJR1,FJR2,FFJ], the generating function of the genus zero FJRW invariants of an ADE singularity coincides with the genus zero free energy of the semisimple Frobenius manifold of the mirror singularity. Furthermore, the all genus generating function satisfies the Virasoro constraints.…”

Section: In An Extended Version Of [Dz3]mentioning

confidence: 99%

BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

2014

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2 Cohomological field theory with a finite symmetryThe first appearance of integrable hierarchies in Gromov-Witten theory is the KdV-hierarchy. Its geometric counterpart is the intersection theory on the moduli space of stable Riemann surfaces M g,k . The latter can be treated as the Gromov-Witten theory of the zero dimensional manifold or FJRWtheory of A 1 = x 2 singularity. Let's review it in more detail.6 Zhang, Y.: On the CP 1 topological sigma model and the Toda lattice hierarchy.

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“…Notice that this CohFT was proved to be isomorphic to the quantum singularity theory of Fan-Jarvis-Ruan-Witten [FJR07,FJR13] for the simple singularity W = x 3 + xy 2 , with respect to the non-maximal diagonal symmetry group J = Z/3Z [FFJMR10].…”

Section: Classical Double Ramification Hierarchy For the D 4 Dubrovin-saito Cohftmentioning

confidence: 99%

“…The case of D 4 is in many ways the most subtle among the ADE singularities. For instance the general method of proof for the mirror symmetry result of [FJR13] did not work in the D 4 case because of the specific form of the CohFT and its phase space, and indeed the proof for the mirror theorem was completed in [FFJMR10]. In essence the complication (specifically the appearance of the so called broad sectors of the phase space) originates from the peculiar symmetry of the D 4 singularity.…”

Section: Introductionmentioning

confidence: 99%

Quantum D4 Drinfeld–Sokolov hierarchy and quantum singularity theory

Villeneuve

Rossi

2019

Journal of Geometry and Physics

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In this paper we compute explicitly the double ramification hierarchy and its quantization for the D 4 Dubrovin-Saito cohomological field theory obtained applying the Givental-Teleman reconstruction theorem to the D 4 Coxeter group Frobenius manifold, or equivalently the D 4 Fan-Jarvis-Ruan-Witten cohomological field theory (with respect to the non-maximal diagonal symmetry group J = Z/3Z). We then prove its equivalence to the corresponding Dubrovin-Zhang hierarchy, which was known to coincide with the D 4 Drinfeld-Sokolov hierarchy. Our techniques provide hence an explicit quantization of the D 4 Drinfeld-Sokolov hierarchy. Moreover, since the DR hierarchy is well defined for partial CohFTs too, our approach immediately computes the DR hierarchies associated to the invariant sectors of the D 4 CohFT with respect to folding of the Dynkin diagram, the B 3 and G 2 Drinfeld-Sokolov hierarchies. Contents 15 2.5. B 3 and G 2 double ramification hierarchies 17 3. Quantum double ramification hierarchy for the D 4 Dubrovin-Saito CohFT 18 3.1. Quantum double ramification hierarchy 18 3.2. Quantum systems of DR type and the quantum D 4 hierarchy 19 References 20 1. Drinfeld-Sokolov D 4 hierarchy A. du Crest de Villeneuve: LAREMA,

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Witten’s D 4 integrable hierarchies conjecture

Cited by 23 publications

References 13 publications

Matrix factorizations and cohomological field theories

Matrix factorizations and cohomological field theories

BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

Quantum D4 Drinfeld–Sokolov hierarchy and quantum singularity theory

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