Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

Belavin, A. A.; Haraoka, Yoshishige; Santachiara, Raoul

doi:10.1007/s00220-018-3274-x

Cited by 8 publications

(6 citation statements)

References 51 publications

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“…We will determine our block by solving a third-order BPZ differential equation. Rather than a direct derivation from the singular vector, we will use knowledge of the fusion rules for deriving this equation, in the spirit of [26]. The fusion rules determine the characteristic exponents of our Fuchsian differential equation at each one of the three singularities z = 0, 1, ∞.…”

Section: (42)mentioning

confidence: 99%

On four-point connectivities in the critical 2d Potts model

Ribault²,

2019

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We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states Q ∈ (0, 4) that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the 2 to 4 significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases Q = 0, 3, 4. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane {ℜc < 13}, where c is the central charge of the Virasoro symmetry algebra.

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Section: (42)mentioning

confidence: 99%

On four-point connectivities in the critical 2d Potts model

Ribault²,

2019

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“…The properties of semi-degenerate W Nconformal blocks describing correlation functions of the Toda CFT and their gauge theory counterparts have been studied, for instance, in [33,16]. An interesting direction which is in a sense close to ours is the construction of integral representation of the 4-point conformal blocks of W 3 -algebra involving one semi-degenerate field of higher level (whose matrix elements cannot be reduced to primary ones only) and one fundamental degenerate field [4]. This construction is based on the middle convolution from the Katz theory of rigid systems.…”

Section: Introductionmentioning

confidence: 99%

Higher-rank isomonodromic deformations and W-algebras

2019

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We construct the general solution of a class of Fuchsian systems of rank N as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of W N -algebra with central charge c = N − 1. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the W N -algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for c = N − 1.

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“…We could not determine the constant α. Analogously to the determination of β, see (3.8), this requires to find the connection of a Fuchsian system. However, differently from the case N = 3, for N ≥ 4 this is a non-rigid rank 3 Fuchsian system and finding the connection of this system is a very hard problem, see the discussion in [29]. We let α undetermined and we use it as a fit parameter in the comparison with Monte-Carlo results.…”

Section: N (X) Gmentioning

confidence: 99%

Spin interfaces and crossing probabilities of spin clusters in parafermionic models

Fukusumi,

Picco,

Santachiara

2020

Preprint

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We consider fractal curves in two-dimensional Z N spin lattice models. These are N states spin models that undergo a continuous ferromagnetic-paramagnetic phase transition described by the Z N parafermionic field theory. The main motivation here is to investigate the correspondence between Schramm-Loewner evolutions (SLE) and conformal field theories with extended conformal algebras (ECFT). By using Monte-Carlo simulation, we compute the fractal dimension of different spin interfaces for the N = 3 and N = 4 spin models that correspond respectively to the Q = 3 Potts model and to the Ashkin-Teller model at the Fateev-Zamolodchikov point. These numerical measures, that improve and complete the ones presented in the previous works [1,2], are shown to be consistent with SLE/ECFT predictions. We consider then the crossing probability of spin clusters in a rectangular domain. Using a multiple SLE approach, we provide crossing probability formulas for Z N parafarmionic theories. The parafermionic conformal blocks that enter the crossing probability formula are computed by solving a Knhiznik-Zamolodchikov system of rank 3. In the Q = 3 Potts model case (N = 3), where the parafermionic blocks coincide with the Virasoro ones, we rederive the crossing formula found in [3] that is in good agreement with our measures. For N ≥ 4 where the crossing probability satisfies a third order differential equation instead of a second order one, our formulas are new. The theoretical predictions are compared to Monte-Carlo measures taken at N = 4 and a fair agreement is found.

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Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

Cited by 8 publications

References 51 publications

On four-point connectivities in the critical 2d Potts model

On four-point connectivities in the critical 2d Potts model

Higher-rank isomonodromic deformations and W-algebras

Spin interfaces and crossing probabilities of spin clusters in parafermionic models

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