Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

Moriyama, Sanefumi; Yamada, Yasuhiko

doi:10.48550/arxiv.2104.06661

Cited by 3 publications

(3 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• Another set of generalised problems which it would be interesting to investigate are these connected to q-deformed Painlevé equations and five-dimensional gauge theories [85,123,[128][129][130][131][132][133]. In this case the relevant quantum spectral problems are the ones associated to relativistic quantum integrable systems.…”

Section: Some Further Comments and Generalisationsmentioning

confidence: 99%

Quantum Spectral Problems and Isomonodromic Deformations

Bershtein

Gavrylenko

Grassi

2022

Commun. Math. Phys.

View full text Add to dashboard Cite

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $$2\times 2$$ 2 × 2 linear systems (Riemann–Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé $$\mathrm {III}_3$$ III 3 equation. Then we extend the analysis to the 2-particle elliptic Calogero–Moser operator, which corresponds to a linear system on the torus. By using the Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions ($$\epsilon _1+\epsilon _2=0$$ ϵ 1 + ϵ 2 = 0 ). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations ($$\epsilon _2=0$$ ϵ 2 = 0 ). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the $$\epsilon _2\rightarrow 0$$ ϵ 2 → 0 limit of the blowup relations from the regularized action functional and CFT arguments.

show abstract

Section: Some Further Comments and Generalisationsmentioning

confidence: 99%

Quantum Spectral Problems and Isomonodromic Deformations

Bershtein

Gavrylenko

Grassi

2022

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…Especially, the brane configurations associated to the del Pezzo geometries of genus one enjoy large symmetries of exceptional Weyl groups. These group-theoretical structures have played central roles in previous studies of the matrix models [8,9,[33][34][35][36][37][38]. For those with interpretations of the del Pezzo geometries enjoying symmetries of Weyl groups, the above questions on duality cascades were answered by discovering the structure of affine Weyl groups [9].…”

Section: Introductionmentioning

confidence: 99%

Duality Cascades and Parallelotopes

Furukawa¹,

Moriyama²,

Sasaki³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Duality cascades are a series of duality transformations in field theories, which can be realized as the Hanany-Witten transitions in brane configurations on a circle. From the physical requirement that duality cascades always end and the final destination depends only on the initial brane configuration, we propose that the fundamental domain of supersymmetric brane configurations in duality cascades can tile the whole parameter space of relative ranks by translations, hence is a parallelotope. We provide our arguments for the proposal.

show abstract

“…For this reason, it is interesting to work out the affine Weyl groups from the quantum curves explicitly. The E 8 quantum curve of the highest rank was constructed in [35] and the complete quantum affine Weyl group including tau variables was worked out recently in [39].…”

Section: Introductionmentioning

confidence: 99%

Duality Cascades and Affine Weyl Groups

Furukawa,

Matsumura,

Moriyama

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Brane configurations in a circle allow subsequent applications of the Hanany-Witten transitions, which are known as duality cascades. By studying the process of duality cascades corresponding to quantum curves with symmetries of Weyl groups, we find a hidden structure of affine Weyl groups. Namely, the fundamental domain of duality cascades consisting of all the final destinations is characterized by the affine Weyl chamber and the duality cascades are realized as translations of the affine Weyl group, where the overall rank in the brane configuration associates to the grading operator of the affine algebra. The structure of the affine Weyl group guarantees the finiteness of the process and the uniqueness of the endpoint of the duality cascades. In addition to the original duality cascades, we can generalize to the cases with Fayet-Iliopoulos parameters. There we can utilize the Weyl group to analyze the fundamental domain similarly and find that the fundamental domain continues to be the affine Weyl chamber. We further interpret the Weyl group we impose as a "half" of the Hanany-Witten transition.

show abstract

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

Cited by 3 publications

References 46 publications

Quantum Spectral Problems and Isomonodromic Deformations

Quantum Spectral Problems and Isomonodromic Deformations

Duality Cascades and Parallelotopes

Duality Cascades and Affine Weyl Groups

Contact Info

Product

Resources

About