Lens Generalisation of <b>τ</b>-Functions for the Elliptic Discrete Painlevé Equation

Kels, Andrew P.; Yamazaki, Masahito

doi:10.1093/imrn/rnz063

Cited by 2 publications

(2 citation statements)

References 27 publications

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“…There is a Lens generalization of the discrete Painlevé equation [31] whose identification in the Sakai's classification is not clear so far. It may be related to a quantization where q is a root of unity.…”

Section: The Quantum Curve For Ementioning

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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Section: The Quantum Curve For Ementioning

confidence: 99%

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

Moriyama

Yamada

2021

SIGMA

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“…Related to this point, it is worth pointing out that the "lens-elliptic" extension of the elliptic case has been studied in the literature[12], by replacing the theta function by the "lens-theta" functions introduced in[13,14].…”

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confidence: 99%

Toroidal and elliptic quiver BPS algebras and beyond

Galakhov

Yamazaki

2022

J. High Energ. Phys.

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The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.

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Lens Generalisation of τ-Functions for the Elliptic Discrete Painlevé Equation

Cited by 2 publications

References 27 publications

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

Toroidal and elliptic quiver BPS algebras and beyond

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