Second level semi-degenerate fields in 


                                    

W
3



$$ {\mathcal{W}}_3 $$
 Toda theory: matrix element and differential equation

Belavin, A. A.; Cao, Xiangyu; Estienne, Benoît; Santachiara, Raoul

doi:10.1007/jhep03(2017)008

Cited by 7 publications

(18 citation statements)

References 30 publications

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“…and associated to representation with dimension ∆ m,n + (p − m)(p − n). The column k = 2 is obtained by applying the transformations v (l) 1 and v…”

mentioning

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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“…and associated to representation with dimension ∆ m,n + (p − m)(p − n). The column k = 2 is obtained by applying the transformations v (l) 1 and v…”

mentioning

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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“…As we have argued in Section 2.2 and proven in [26], the Fuchsian system satisfied by B (1,s) M (x) is of spectral type (222, 321, 222), which has the index of rigidity ι = (1 − 2) × 6 2 + 2(2 2 + 2 2 + 2 2 ) + (3 2 + 2 2 + 1 2 ) = 2 and hence is rigid. This system is obtained by the following chain The system of spectral type (2 * 11, 2 * 11, 211) coincides with the system (4.29).…”

Section: Chain Of Addition and Middle Convolution Transformationsmentioning

confidence: 81%

“…In this section we recall basic elements of W 3 CFT necessary for our forthcoming discussion (for reviews, see, e.g., [27] and [28]). For more detailed discussion of the W 3 chiral algebra, its representation modules and its conformal blocks, consistent with the notations and normalizations used here, the reader is referred to [2,26].…”

Section: Differential Equations For W 3 Conformal Blocksmentioning

confidence: 99%

“…In section 4 we briefly review the Katz theory on rigid local systems, and then show how to apply it to reproduce known results in CFT. In section 5 we apply the Katz theory to solve the 6-th order ODE found in [26] and we provide new explicit factorizations (1.1) for correlation function with additional parameters. In section 6 we resume the results and discuss some future perspectives.…”

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confidence: 99%

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Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

Belavin

Haraoka

Santachiara

2018

Commun. Math. Phys.

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We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of W 3 Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one nth level semi-degenerate field and a fully-degenerate one in the fundamental sl 3 representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. This approach allows to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of W 3 correlation functions as well shift relations between structure constants are also provided.

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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 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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Second level semi-degenerate fields in W 3 $$ {\mathcal{W}}_3 $$ Toda theory: matrix element and differential equation

Cited by 7 publications

References 30 publications

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

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

Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

The imaginary Toda field theory

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