W(E10) symmetry, M-theory and Painlevé equations

Mizoguchi, Shun’ya; Yamada, Yasuhiko

doi:10.1016/s0370-2693(02)01870-1

Cited by 13 publications

(13 citation statements)

References 38 publications

(25 reference statements)

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“…• The relation between autonomous limit of Painlevé and the SW hamiltonians is known [38,45,58]. The fact of appearance of the Newton polygons from figure 1 in the Painlevé theory was also noticed in [58].…”

Section: Jhep02(2018)077mentioning

confidence: 90%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

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Section: Jhep02(2018)077mentioning

confidence: 90%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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“…We hope the blowup equations we find in this paper can shed some new light on this subject. In particular, the E-string blowup equations are expected to produce the bilinear relations of the elliptic Painlevé equations [73], while the M-string blowup equations are expected to produce those of the isomonodromic system on the onepunctured torus [74].…”

Section: Jhep07(2020)135mentioning

confidence: 99%

Elliptic blowup equations for 6d SCFTs. Part III. E-strings, M-strings and chains

Haghighat

Klemm

et al. 2020

J. High Energ. Phys.

101

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We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E-M and M string chains.

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“…This list is the same as the degeneration scheme of the E-string. The classes of "elliptic", "multiplicative" and "additive" mean the types of the difference equation and correspond to the gauge theories in 6D/5D/4D (see, e.g., [8,40]). The cases in the box admit the continuous flows (of the original Painlevé equation), and the relation between their Hamiltonians and the D = 4, SU(2) Seiberg-Witten curves was observed in [28].…”

Section: B Standard Realizations In Commutative Casementioning

confidence: 99%

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

Moriyama

Yamada

2021

SIGMA

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We study a quantum (non-commutative) representation of the affine Weyl group mainly of type E(1) 8 , where the representation is given by birational actions on two variables x, y with q-commutation relations. Using the tau variables, we also construct quantum "fundamental" polynomials F (x, y) which completely control the Weyl group actions. The geometric properties of the polynomials F (x, y) for the commutative case is lifted distinctively in the quantum case to certain singularity structures as the q-difference operators. This property is further utilized as the characterization of the quantum polynomials F (x, y). As an application, the quantum curve associated with topological strings proposed recently by the first named author is rederived by the Weyl group symmetry. The cases of type D(1)5 , E

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W(E10) symmetry, M-theory and Painlevé equations

Cited by 13 publications

References 38 publications

Cluster integrable systems, q-Painlevé equations and their quantization

Cluster integrable systems, q-Painlevé equations and their quantization

Elliptic blowup equations for 6d SCFTs. Part III. E-strings, M-strings and chains

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

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