# Correlation functions with fusion-channel multiplicity in \( {\mathcal{W}}_3 \) Toda field theory

## Abstract

Current studies of \( {\mathcal{W}}_N \) Toda field theory focus on correlation functions such that the \( {\mathcal{W}}_N \) highest-weight representations in the fusion channels are multiplicity-free. In this work, we study \( {\mathcal{W}}_3 \) Toda 4-point functions with multiplicity in the fusion channel. The conformal blocks of these 4-point functions involve matrix elements of a fully-degenerate primary field with a highest-weight in the *adjoint* representation of \( \mathfrak{s}{\mathfrak{l}}_3 \), and a fully-degenerate primary field with a highest-weight in the fundamental representation of \( \mathfrak{s}{\mathfrak{l}}_3 \). We show that, when the fusion rules do not involve multiplicities, the matrix elements of the fully-degenerate adjoint field, between two arbitrary descendant states, can be computed explicitly, on equal footing with the matrix elements of the semi-degenerate fundamental field. Using null-state conditions, we obtain a fourth-order Fuchsian differential equation for the conformal blocks. Using Okubo theory, we show that, due to the presence of multiplicities, this differential equation belongs to a class of Fuchsian equations that is different from those that have appeared so far in \( {\mathcal{W}}_N \) theories. We solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions. This computation shows in detail how the ambiguities that are caused by the presence of multiplicities are fixed by requiring monodromy-invariance.

## Keywords

Conformal and W Symmetry Conformal Field Models in String Theory Integrable Field Theories Supersymmetric gauge theory## Notes

### **Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

- [1]V.A. Fateev and S.L. Lukyanov,
*The Models of Two-Dimensional Conformal Quantum Field Theory with Z(n) Symmetry*,*Int. J. Mod. Phys.***A 3**(1988) 507 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [2]P. Bouwknegt and K. Schoutens,
*W symmetry*,*Adv. Ser. Math. Phys.***22**(1995) 1.MathSciNetMATHGoogle Scholar - [3]A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov,
*Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory*,*Nucl. Phys.***B 241**(1984) 333 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [4]V.A. Fateev and A.V. Litvinov,
*Correlation functions in conformal Toda field theory I*,*JHEP***11**(2007) 002 [arXiv:0709.3806] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [5]V.A. Fateev and A.V. Litvinov,
*Correlation functions in conformal Toda field theory II*,*JHEP***01**(2009) 033 [arXiv:0810.3020] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [6]S. Kanno, Y. Matsuo and S. Shiba,
*Analysis of correlation functions in Toda theory and AGT-W relation for*SU(3)*quiver*,*Phys. Rev.***D 82**(2010) 066009 [arXiv:1007.0601] [INSPIRE].ADSMathSciNetGoogle Scholar - [7]P. Furlan and V.B. Petkova,
*On some 3-point functions in the W*_{4}*CFT and related braiding matrix*,*JHEP***12**(2015) 079 [arXiv:1504.07556] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [8]A. Mironov and A. Morozov,
*On AGT relation in the case of U(3)*,*Nucl. Phys.***B 825**(2010)1 [arXiv:0908.2569] [INSPIRE]. - [9]G. Bonelli, A. Tanzini and J. Zhao,
*Vertices, Vortices and Interacting Surface Operators*,*JHEP***06**(2012) 178 [arXiv:1102.0184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [10]J. Gomis and B. Le Floch,
*M2-brane surface operators and gauge theory dualities in Toda*,*JHEP***04**(2016) 183 [arXiv:1407.1852] [INSPIRE].ADSCrossRefGoogle Scholar - [11]O. Alekseev and F. Novaes,
*Wilson loop invariants from W*_{N}*conformal blocks*,*Nucl. Phys.***B 901**(2015) 461 [arXiv:1505.06221] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [12]M. Kohno,
*Global analysis in linear differential equations*, Mathematics and its Applications, volume 471, Springer Netherlands (1999).Google Scholar - [13]V.A. Fateev,
*Normalization factors, reflection amplitudes and integrable systems*, hep-th/0103014 [INSPIRE]. - [14]V.S. Dotsenko,
*Série de cours sur la théorie conforme*, https://cel.archives-ouvertes.fr/cel-00092929 (2006). - [15]V.A. Fateev and A.V. Litvinov,
*Coulomb integrals in Liouville theory and Liouville gravity*,*JETP Lett.***84**(2007) 531 [INSPIRE].ADSCrossRefGoogle Scholar - [16]P. Bowcock and G.M.T. Watts,
*Null vectors of the W(3) algebra*,*Phys. Lett.***B 297**(1992) 282 [hep-th/9209105] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [17]Z. Bajnok, L. Palla and G. Takács,
*A*_{2}*Toda theory in reduced WZNW framework and the representations of the W algebra*,*Nucl. Phys.***B 385**(1992) 329 [hep-th/9206075] [INSPIRE].ADSCrossRefGoogle Scholar - [18]P. Bowcock and G.M.T. Watts,
*Null vectors, three point and four point functions in conformal field theory*,*Theor. Math. Phys.***98**(1994) 350 [hep-th/9309146] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar - [19]G.M.T. Watts,
*Fusion in the W(3) algebra*,*Commun. Math. Phys.***171**(1995) 87 [hep-th/9403163] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [20]A. Alex, M. Kalus, A. Huckleberry and J. von Delft,
*A Numerical algorithm for the explicit calculation of*SU(*N*)*and*SL(*N, ℂ*)*Clebsch-Gordan coefficients*,*J. Math. Phys.***52**(2011) 023507 [arXiv:1009.0437] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [21]B. Estienne, V. Pasquier, R. Santachiara and D. Serban,
*Conformal blocks in Virasoro and W theories: Duality and the Calogero-Sutherland model*,*Nucl. Phys.***B 860**(2012) 377 [arXiv:1110.1101] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [22]V. Belavin, O. Foda and R. Santachiara,
*AGT, N-Burge partitions and*\( {\mathcal{W}}_N \)*minimal models*,*JHEP***10**(2015) 073 [arXiv:1507.03540] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [23]M. Yoshida,
*Fuchsian differential equations*, Aspects of Mathematics, volume E 11, Vieweg+Teubner Verlag (1987).Google Scholar - [24]K. Okubo,
*On the group of Fuchsian equations*, in*Seminar Reports of Tokyo Metropolitan University*(1987).Google Scholar - [25]K. Okubo,
*Connection problems for systems of linear differential equations*, in*Japan-United States Seminar on Ordinary Differential and Functional Equations*, Springer (1971) pp. 238-248.Google Scholar - [26]N.M. Katz,
*Rigid local systems. (AM-139)*, Princeton University Press, Princeton U.S.A. (1996).Google Scholar - [27]E. Ince,
*Ordinary differential equations*, Courier Corporation, North Chelmsford U.S.A. (1956).Google Scholar - [28]Y. Haraoka,
*Canonical forms of differential equations free from accessory parameters*,*SIAM J. Math. Anal.***25**(1994) 1203.MathSciNetCrossRefMATHGoogle Scholar - [29]B. Estienne and R. Santachiara,
*Relating Jack wavefunctions to W A*_{k−1}*theories*,*J. Phys.***A 42**(2009) 445209 [arXiv:0906.1969] [INSPIRE]. - [30]L.F. Alday, D. Gaiotto and Y. Tachikawa,
*Liouville Correlation Functions from Four-dimensional Gauge Theories*,*Lett. Math. Phys.***91**(2010) 167 [arXiv:0906.3219] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [31]N. Wyllard,
*A(N-1) conformal Toda field theory correlation functions from conformal N*= 2 SU(*N*)*quiver gauge theories*,*JHEP***11**(2009) 002 [arXiv:0907.2189] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [32]V. Mitev and E. Pomoni,
*Toda 3-Point Functions From Topological Strings*,*JHEP***06**(2015) 049 [arXiv:1409.6313] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [33]M. Isachenkov, V. Mitev and E. Pomoni,
*Toda 3-Point Functions From Topological Strings II*, arXiv:1412.3395 [INSPIRE]. - [34]B. Belavin, B. Estienne, O. Foda and R. Santachiara,
*W*_{3}*semi-degenerate fields in higher representations: differential equations and fusion rules*in preparation.Google Scholar - [35]K.B. Alkalaev and V.A. Belavin,
*Conformal blocks of W*_{N}*minimal models and AGT correspondence*,*JHEP***07**(2014) 024 [arXiv:1404.7094] [INSPIRE].ADSCrossRefGoogle Scholar - [36]M. Bershtein and O. Foda,
*AGT, Burge pairs and minimal models*,*JHEP***06**(2014) 177 [arXiv:1404.7075] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [37]V.S. Dotsenko and V.A. Fateev,
*Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge C*≤ 1,*Nucl. Phys.***B 251**(1985) 691 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [38]V. Dotsenko, M. Picco and P. Pujol,
*Renormalization group calculation of correlation functions for the*2 −*D random bond Ising and Potts models*,*Nucl. Phys.***B 455**(1995) 701 [hep-th/9501017] [INSPIRE].ADSCrossRefGoogle Scholar