Abstract
We study the twist-field representations of W-algebras and generalize construction of the corresponding vertex operators to D- and B-series. It is shown, how the computation of characters of these representations leads to nontrivial identities involving lattice theta-functions. We also propose a way to calculate their exact conformal blocks, expressing them for D-series in terms of geometric data of the corresponding Prym variety for covering curve with involution.
Article PDF
Similar content being viewed by others
References
A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Conformal Quantum Field Theory Models in Two-Dimensions Having Z(3) Symmetry, Nucl. Phys. B 280 (1987) 644 [INSPIRE].
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].
A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, hep-th/0302191 [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
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].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].
M. Henneaux and S.-J. Rey, Nonlinear W ∞ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, American Mathematical Society, (2004).
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].
I.B. Frenkel and V.G. Kac, Basic Representations of Affine Lie Algebras and Dual Resonance Models, Invent. Math. 62 (1980) 23.
P.G. Gavrylenko and A.V. Marshakov, Free fermions, W-algebras and isomonodromic deformations, Theor. Math. Phys. 187 (2016) 649 [arXiv:1605.04554] [INSPIRE].
O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP 10 (2012) 038 [Erratum ibid. 10 (2012) 183] [arXiv:1207.0787] [INSPIRE].
N. Iorgov, O. Lisovyy and J. Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Commun. Math. Phys. 336 (2015) 671 [arXiv:1401.6104] [INSPIRE].
P. Gavrylenko, Isomonodromic τ -functions and W N conformal blocks, JHEP 09 (2015) 167 [arXiv:1505.00259] [INSPIRE].
P. Gavrylenko and A. Marshakov, Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations, JHEP 02 (2016) 181 [arXiv:1507.08794] [INSPIRE].
A.B. Zamolodchikov, Conformal Scalar Field on the Hyperelliptic Curve and Critical Ashkin-teller Multipoint Correlation Functions, Nucl. Phys. B 285 (1987) 481 [INSPIRE].
A.B. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, JETP 90 (1986) 1808.
S. Apikyan and Al. Zamolodchikov, Conformal blocks, related to conformally invariant Ramond states of a free scalar field, JETP 92 (1987) 34.
V. Kac, Infinite dimensional Lie algebras, Cambridge University Press, (1990).
B. Bakalov and V. Kac, Twisted Modules over Lattice Vertex Algebras, in Proc. V Internat. Workshop “Lie Theory and Its Applications in Physics” (Varna, June 2003), eds. H.-D. Doebner and V.K. Dobrev, World Scientific, Singapore, (2004) [math/0402315].
I. Macdonald, Affine root systems and Dedekind’s η-function, Invent. Math. 15 (1972) 91.
V. Kac, Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula, Adv. Math. 30 (1978) 85.
B. Feigin and E. Frenkel, Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, Int. J. Mod. Phys. A 7S1A (1992) 197 [INSPIRE].
H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, U.S.A. (1939).
E. Frenkel, V. Kac, A. Radul and W.-Q. Wang, W 1+∞ and W (gl N ) with central charge N, Commun. Math. Phys. 170 (1995) 337 [hep-th/9405121] [INSPIRE].
V.G. Kac and J.W. van de Leur, The n-component KP hierarchy and representation theory, J. Math. Phys. 44 (2003) 3245 [hep-th/9308137] [INSPIRE].
O. Alvarez, P. Windey and M.L. Mangano, Vertex operator construction of the So(2n+1) Kac-Moody algebra and its spinor representation,, Nucl. Phys. B 277 (1986) 317 [INSPIRE].
R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, The Operator Algebra of Orbifold Models, Commun. Math. Phys. 123 (1989) 485 [INSPIRE].
R. Dijkgraaf, V. Pasquier and P. Roche, Quasi-quantum groups related to orbifold models, Nucl. Phys. Proc. Suppl. B 18 (1990) 60.
V.G. Knizhnik, Analytic Fields on Riemann Surfaces. II, Commun. Math. Phys. 112 (1987) 567 [INSPIRE].
V.G. Knizhnik, Multiloop amplitudes in the theory of quantum strings and complex geometry, Sov. Phys. Usp. 32 (1989) 945.
A. Marshakov, A. Mironov, A. Morozov and M. Olshanetsky, c = r G theories of W G gravity: The set of observables as a model of simply laced G, Nucl. Phys. B 404 (1993) 427 [hep-th/9203044] [INSPIRE].
T. Arakawa, Quantized Reductions and Irreducible Representations of W-Algebras, math/0403477.
T. Arakawa, Representation Theory of W-Algebras, Invent. Math. 169 2 (2007) 219 [math/0506056].
E. Frenkel, V. Kac and M. Wakimoto, Characters and fusion rules for W algebras via quantized Drinfeld-Sokolov reductions, Commun. Math. Phys. 147 (1992) 295 [INSPIRE].
V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
V.G. Kac and D.H. Peterson, 112 constructions of the basic representation of the loop group of E 8, in Proc. of the conf. “Anomalies, Geometry, Topology”, Argonne, March 1985, World Scientific, (1985), pp. 276-298.
J. Lepowsky and R.L. Wilson, Construction of the affine Lie algebra \( \mathfrak{s}\mathfrak{l}(2) \), Commun. Math. Phys. 62 (1978) 43 [INSPIRE].
V.G. Kac, D.A. Kazhdan, J. Lepowsky and R.L. Wilson, Realization of the basic representations of the Euclidean Lie algebras, Adv. Math. 42 (1981) 83.
S. Lukyanov, Additional Symmetries and Exactly Solvable Models in Two Dimensional Conformal Field Theory, Ph.D. Thesis (in Russian), (1989).
G.M.T. Watts, WB algebra representation theory, Nucl. Phys. B 339 (1990) 177 [INSPIRE].
J. Fay, Theta-functions on Riemann surfaces, Lect. Notes Math. 352, Springer, N.Y., U.S.A., (1973).
A. Kokotov and D. Korotkin, Tau-function on Hurwitz spaces, Math. Phys. Anal. Geom. 7 (2004) 47 [math-ph/0202034].
P. Di Francesco, H. Saleur and J.B. Zuber, Critical Ising Correlation Functions in the Plane and on the Torus, Nucl. Phys. B 290 (1987) 527 [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 Conformal Field Theories on Riemann Surfaces, Commun. Math. Phys. 115 (1988) 649 [INSPIRE].
D. Bernard, ℤ2 -twisted fields and bosonization on Riemann surfaces, Nucl. Phys. B 302 (1988) 251 [INSPIRE].
A.A. Belavin, M.A. Bershtein and G.M. Tarnopolsky, Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity, JHEP 03 (2013) 019 [arXiv:1211.2788] [INSPIRE].
J. Fuchs, B. Schellekens and C. Schweigert, From Dynkin diagram symmetries to fixed point structures, Commun. Math. Phys. 180 (1996) 39 [hep-th/9506135] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1705.00957
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bershtein, M., Gavrylenko, P. & Marshakov, A. Twist-field representations of W-algebras, exact conformal blocks and character identities. J. High Energ. Phys. 2018, 108 (2018). https://doi.org/10.1007/JHEP08(2018)108
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)108