Correlation functions with fusion-channel multiplicity in W 3 $$ {\mathcal{W}}_3 $$ Toda field theory

Belavin, A. A.; Estienne, Benoît; Foda, Omar; Santachiara, Raoul

doi:10.1007/jhep06(2016)137

Cited by 14 publications

(28 citation statements)

References 39 publications

(94 reference statements)

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“…In the large distance limit, we have Let us briefly review some basic formulas related to the W 3 algebra. For the conventions and the notations used here, we refer the reader to [37] and references therein.…”

Section: B4 An Excursion In the Complex Planementioning

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: B4 An Excursion In the Complex Planementioning

confidence: 99%

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

Ribault²,

2019

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“…In Virasoro CFT's, the Shapolavov matrix and the 3-point functions are completely determined by the Virasoro algebra (1). Note that this is not true anymore for more general conformal chiral algebras such as the W N algebras [1], [2].…”

Section: The Virasoro Conformal Blocksmentioning

confidence: 99%

“…Using the state-field correspondence, we use Φ ∆ (x) for the primary field of conformal dimension ∆, and L −Y Φ ∆ (x) for the descendant fields. We parametrize the conformal dimension ∆ by the parameter Q, (2), and the charge α,…”

Section: Verma Modulesmentioning

confidence: 99%

Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models

Javerzat

Santachiara

Foda

2018

J. High Energ. Phys.

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We study Virasoro minimal-model 4-point conformal blocks on the sphere and 0-point conformal blocks on the torus (the Virasoro characters), as solutions of Zamolodchikov-type recursion relations. In particular, we study the singularities due to resonances of the dimensions of conformal fields in minimal-model representations, that appear in the intermediate steps of solving the recursion relations, but cancel in the final results. arXiv:1806.02790v3 [hep-th] 15 Aug 2018 1.0.3 The singularitiesThe 4-point conformal block on the sphere is a function of six parameters: the Virasoro central charge, the conformal dimension of the Virasoro representation that flows in the internal channel, and the conformal dimensions of the four external fields. The solution of the elliptic recursion relation displays a rich structure of poles. These poles are physical in the sense that they correspond to the propagation of states for suitable choices of the central charge and conformal dimensions. In a numerical 2D bootstrap based on Virasoro conformal blocks that are computed using the elliptic recursion relation, one must deal with these poles when exploring the space of possible crossing-symmetric CFT solutions. This happens, for example, in studies of percolation in the 2D Ising model [9]. When the central charge is such that one deals with minimal-model conformal blocks, additional poles appear. These additional poles are non-physical and appear due to resonances of conformal dimensions (see equation (11)) at rational values of the central charge. This complication requires a careful study of the pole structure of the elliptic recursion relation in the case of Virasoro minimal models, which is the aim of the present work. The present work.We study the cancellation of the non-physical poles in computations of minimal-model conformal blocks using Zamolodchikov's elliptic recursion relation for the 4-point conformal block on the sphere. But the 4-point conformal block on the sphere is not the only or the simplest conformal block that can be computed using a recursion relation. In 2009, Poghossian [10], and independently Fateev and Litvinov [3] proposed recursion relations to compute Liouville 1-point conformal blocks on the torus. These recursion relations are equivalent [6], and we use the Fateev-Litvinov version to study minimalmodel 1-point functions on the torus, and their 0-point limits (when the vertex operator insertion is the identity) which are Virasoro minimal-model characters, as the simplest examples of solutions of a Zamolodchikov-type elliptic recursion relation. Outline of contentsIn section 2, we recall basic facts related to the Virasoro algebra, its representations, and conformal blocks. In section 3, we consider the 4-point conformal blocks on the sphere as solutions of the recursion relation, study their singularities and their behaviour in the context of the Virasoro generalized minimal models and minimal models. In section 4, we consider the 1-point conformal block on the torus as solutions of the Fateev-Litvi...

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“…This is why a generic four-point function involving a fully degenerate field Φ bω 1 does not obey a differential equation of order three. If one of the other fields is semi-degenerate though, a differential equation can be found, but its order depends on the semi-degenerate field [25,26]. For semi-degenerate operators, similar (if less strict) restrictions exist.…”

Section: Fusion In Wmentioning

confidence: 99%

“…The semi-and fully degenerate representations of higher level can be described explicitly by studying level-N null vectors. The fields Φ κω 2 +bh 1 and Φ κω 2 +bh 3 , which appear in the previous fusions are also semi-degenerate, see for example [25] for an explicit derivation.…”

Section: Now Using the Null-vector Conditionmentioning

confidence: 99%

The imaginary Toda field theory

Dupic¹,

Estienne²,

Ikhlef³

2019

J. Phys. A: Math. Theor.

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We consider the two-dimensional sl n quantum Toda field theory with an imaginary background charge. This conformal field theory has a higher spin symmetry (W n algebra), a central charge c ≤ n − 1 and a continuous spectrum. Using the conformal bootstrap, we compute structure constants involving two arbitrary scalar fields and a semi-degenerate field of Wyllard type. The solution obtained is not the analytic continuation of the usual Toda three-point function. Non-scalar primary fields and their three-point functions are also discussed. Non-scalar primary fields are classified by conjugacy classes of the permutation group S n , and their structure constants are computed explicitly, up to an overall factor. arXiv:1809.05568v2 [math-ph]

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Correlation functions with fusion-channel multiplicity in W 3 $$ {\mathcal{W}}_3 $$ Toda field theory

Cited by 14 publications

References 39 publications

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

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

Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models

The imaginary Toda field theory

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