Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras

Chekhov, Leonid; Mazzocco, Marta; Rubtsov, Vladimir

doi:10.1016/j.aim.2020.107442

Cited by 12 publications

(10 citation statements)

References 49 publications

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“…We further note that Chekhov et al [4] conjectured explicit affine Del Pezzo surfaces of degree three as the monodromy manifolds of the q-Painlevé equations higher up in Sakai's classification scheme [34] than qP VI .…”

Section: 2mentioning

confidence: 89%

On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem

Joshi¹,

Roffelsen²

2022

Preprint

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We study the sixth q-difference Painlevé equation (qP VI ) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of qP VI with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explicitly in terms of certain orthogonal polynomials yielding special function solutions of qP VI .

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Section: 2mentioning

confidence: 89%

On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem

Joshi¹,

Roffelsen²

2022

Preprint

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“…Eliminating θ4 in (5), we obtain the same equations as in [8, Corollary 4.6], obtained there using a totally different approach based on the fact that the cubic equation obtained from (2) by eliminating ϑ 4 is the defining equation of the wild character variety arising as the monodromy manifold of the Painlevé IV equation.…”

mentioning

confidence: 89%

“…This gives a non-commutative algebra, generated by quantum theta functions. There is a totally different approach to construct deformation quantizations using the realization of the mirror as the monodromy manifold of the Painlevé IV equation [5,8]. We show that these two approaches agree.…”

mentioning

confidence: 89%

The Quantum Mirror to the Quartic del Pezzo Surface

Argüz¹

2021

Preprint

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A log Calabi-Yau surface (X, D) is given by a smooth projective surface X, together with an anti-canonical cycle of rational curves D ⊂ X. The homogeneous coordinate ring of the mirror to such a surface -or to the complement X \ D -is constructed using wall structures and is generated by theta functions [6,7]. In [1], we provide a recipe to concretely compute these theta functions from a combinatorially constructed wall structure in R 2 , called the heart of the canonical wall structure. In this paper, we first apply this recipe to obtain the mirror to the quartic del Pezzo surface, denoted by dP 4 , together with an anti-canonical cycle of 4 rational curves. We afterwards describe the deformation quantization of this coordinate ring, following [4]. This gives a non-commutative algebra, generated by quantum theta functions. There is a totally different approach to construct deformation quantizations using the realization of the mirror as the monodromy manifold of the Painlevé IV equation [5,8]. We show that these two approaches agree.

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“…Symbols A n , D n , E n represent the types of the symmetry (affine in the Painlevé equations) and correspond to the (non-affine) flavor symmetry of the gauge theory. 9 There are two standard ways to realize the above geometry, namely (i) nine-point blow-up of P 2 or (ii) eight-point blow-up of P 1 × P 1 . In the most generic case, these points determine an elliptic curve and we have the elliptic Painlevé equation.…”

Section: B Standard Realizations In Commutative Casementioning

confidence: 99%

“…Recently, this subject attracts various interests due to its relation to conformal field theories, gauge theories and topological strings. Despite some interesting pioneering works [5,6,7,9,19,37], there remain many problems to be studied especially on the quantization of the discrete Painlevé equations. One of the main problems is to establish the quantization compatible with the geometric formulation in [29,49].…”

Section: Introductionmentioning

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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Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras

Cited by 12 publications

References 49 publications

On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem

On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem

The Quantum Mirror to the Quartic del Pezzo Surface

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

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