Abstract
We show that the conformal characters of various rational models ofW-algebras can be already uniquely determined if one merely knows the central charge and the conformal dimensions. As a side result we develop several tools for studying representations of SL(2,ℤ) on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, may be useful for the analysis of many others.
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Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory. Nucl. Phys.B 241, 333–380 (1984)
Witten, E.: Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys.121, 351–399 (1989)
Cardy, J.L.: Operator Content of Two-Dimensional Conformally Invariant Theories. Nucl. Phys.B 270, 186–204 (1986)
Bouwknegt, P., Schoutens, K.:W-Symmetry in Conformal Field Theory. Phys. Rep.223, 183–276 (1993)
Nahm, W., Recknagel, A., Terhoeven, M.: Dilogarithm Identities in Conformal Field Theory. Mod. Phys. Lett.A 8, 1835–1847 (1993)
Kellendonk, J., Rösgen, M., Varnhagen, R.: Path Spaces andW-Fusion in Minimal Models. Int. J. Mod. Phys.A 9, 1009–1023 (1994)
Eholzer, E., Skoruppa, N.-P.: Conformal Characters and Theta Series. Preprint MSRI 012-95 BONN-TH-94-24, Lett. Math. Phys. (to appear)
Frenkel, I.B., Huang, Y., Lepowsky, J.: On Axiomatic Approaches to Vertex Operator Algebras and Modules. Mem. Am. Math. Soc.104, Number 494, Providence, Rhode Island: Am. Math. Soc., 1993
Frenkel, I.B., Zhu, Y.: Vertex Operator Algebras Associated to Representations of Affine and Virasoro Algebras. Duke Math. J.66(1), 123–168 (1992)
Nahm, W.: Chiral Algebras of Two-Dimensional Chiral Field Theories and Their Normal Ordered Products Proc. of the Trieste Conference on “Recent Developments in Conformal Field Theories”, Singapore: World Scientific, 1989
Féher, L., O'Raifeartaigh, L., Tsutsui, I.: The Vacuum Preserving Lie Algebra of a ClassicalW-algebra. Phys. Lett.B 316, 275–281 (1993)
Zhu, Y.: Vertex Operator Algebras, Elliptic Functions, and Modular Forms. Ph.D. thesis, Yale University, 1990
Wang, W.: Rationality of Virasoro Vertex Operator Algebras. Int. Research Notices in Duke Math. J.7, 197–211 (1993)
Eholzer, W., Flohr, M., Honecker, A., Hübel, R., Nahm, W., Varnhagen R.: Representations ofW-Algebras with Two Generators and New Rational Models. Nucl. Phys.B 383, 249–288 (1992)
Anderson, G., Moore, G.: Rationality in Conformal Field Theory. Commun. Math. Phys.117, 441–450 (1988)
Rocha-Caridi, A.: Vacuum Vector Representations of the Virasoro Algebra. In: Vertex Operators in Mathematics and Physics. Mandelstam, S., Singer, I.M. (eds.), MSRI Publications Nr.3, Berlin, Heidelberg, New York: Springer 1984
Cappelli, A., Itzykson, C., Zuber, J.B.: The A-D-E Classification of Minimal andA (1)1 Conformal Invariant Theories. Commun. Math. Phys.113, 1–26 (1987)
Frenkel, E., Kac, V., Wakimono, M.: Characters and Fusion Rules forW-Algebras via Quantized Drinfeld-Sokolov Reduction. Commun. Math. Phys.147, 295–328 (1992)
Kac, V.: Infinite Dimensional Lie Algebras and Groups. Singapore: World Scientific, 1989
Eholzer, W., Honecker, A., Hübel, R.: How Complete is the Classification ofW-Symmetries. Phys. Lett.B 308, 42–50 (1993)
Eholzer, W.: Fusion Algebras Induced by Representations of the Modular Group. Int. J. Mod. Phys.A 8, 3495–3507 (1993)
Eholzer, W.: Exzeptionelle und SupersymmetrischeW-Algebren in Konformer Quantenfeldtheorie. Diplomarbeit BONN-IR-92-10
Skoruppa, N.-P.: Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts. Bonner Mathematische Schriften159 (1985)
Wohlfahrt, K.: An extension of F. Klein's level concept. Illinois J. Math.8, 529–535 (1964)
Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Sholten, Tokyo, Japan: Princeton Press, 1971
Nobs, A.: Die irreduziblen Darstellungen der GruppenSL 2(Z p ) insbesondereSL 2(Z 2) I, Comment. Math. Helvetici51, 465–489 (1976); Nobs, A., Wolfart, J.: Die irreduziblen Darstellungen der GruppenSL 2(Z p ) insbesondereSL 2(Z 2) II, Comment. Math. Helvetici51, 491–526 (1976)
Dornhoff, L.: Group Representation Theory. New York: Marcel Dekker, 1971
Batut, C., Bernardi, D., Cohen, H., Olivier, M.: PARI-GP (1989), Université Bordeaux 1, Bordeaux
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Communicated by R.H. Dijkgraaf
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Eholzer, W., Skoruppa, NP. Modular invariance and uniqueness of conformal characters. Commun.Math. Phys. 174, 117–136 (1995). https://doi.org/10.1007/BF02099466
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DOI: https://doi.org/10.1007/BF02099466