Abstract
By using the theory of vertex operator algebras, we gave a new proof of the famous Ramanujan’s modulus 5 modular equation from his ‘‘Lost Notebook’’ (p. 139 in [R]). Furthermore, we obtained an infinite list of q-identities for all odd moduli; thus, we generalized the result of Ramanujan.
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Andrews, G.: The Theory of Partitions, Reprint of the 1976 original. Cambridge Mathematical Library. Cambridge: Cambridge University Press, 1998
Andrews, G.: Ramanujan’s “lost” notebook, I, II, III. Adv. Math. 41, 137–172, 173–185, 186–208 (1981)
Bailey, W.: A note on two of Ramanujan’s formulae. Quart. J. Math. Oxford Ser. (2) 3, 29–31 (1952)
Bailey, W.: Further note on two of Ramanujan’s formulae. Quart. J. Math. Oxford Ser. (2) 3, 158–160 (1952)
Berndt, B.: Ramanujan’s notebooks Part III. New York: Springer-Verlag, 1991
Berndt, B., Ono, K.: Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary. Sem. Lotharingien de Combinatoire 42 (1999). In: The Andrews Festschrift, D. Foata, G.-N. Han (eds.), Berlin: Springer-Verlag, 2001, pp. 39–110
Beilinson, A., Feigin, B., Mazur, B.: Algebraic field theory on curves. Preprint, 1991
Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984)
Chan, H.-H.: New proofs of Ramanujan’s partition identities for moduli 5 and 7. J. Number Theory 53, 144–158 (1995)
Chan, H.-H.: On the equivalence of Ramanujan’s partition identities and a connection with the Rogers-Ramanujan continued fraction. J. Math. Analy. and Appl. 198, 111–120 (1996)
Dong, C., Li, H., Mason, G.: Modular–invariance of trace functions in orbifold theory generalized Moonshine. Commun. Math. Phys. 214, 1–56 (2000)
Dong, C., Li, H., Mason, G.: Regularity of rational vertex operator algebras. Adv. Math. 132, 148–166 (1997)
Dong, C., Mason, G., Nagatomo, K.: Quasi-modular forms and trace functions associated to free boson and lattice vertex operator algebras. I. M. R. N. 8, 409–427 (2001)
Feigin, B., Frenkel, E.: Coinvariants of nilpotent subalgebras of the Virasoro algebra and partition identities. In: I. M. Gelfand Seminar, Adv. Soviet Math. 16(1), 139–148 (1993)
Feigin, B., Fuchs, D.: Representations of the Virasoro algebra. In: Representation of Lie groups and related topics. Adv. Stud. Contemp. Math. 7, New York: Gordon and Breach, 1990, pp. 465–554
Feigin, B., Fuchs, D.: Verma modules over the Virasoro algebra. Lecture Notes in Math 1060, 230–245 (1982)
Feigin, B., Fuchs, D.: Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras. J. Geom. Phys. 5, 209–235 (1988)
Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 494, 1993
Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Appl. Math., 134, New York: Academic Press, 1988
Frenkel, I., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)
Hille, E.: Ordinary differential equations in the complex domain. New York: Dover, 1997
Huang, Y.-Z.: Two-dimensional Conformal Geometry and Vertex Operator Algebras. Progress in Math., 148, Boston: Birkhäuser, 1997
Kac, V., Raina, A.K.: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. Advanced Series in Mathematical Physics 2. Riveredge, NJ: World Scientific Publishing, 1987
Kac, V., Wakimoto, M.: Modular invariant representations of infinite dimensional Lie algebras and superalgebras. Proc. Natl. Acad. Sci. USA 85, 4956–4960 (1988)
Kaneko, M., Zagier, D.: Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. In: Computational perspectives on number theory (Chicago, IL, 1995), AMS-IP Studies in Adv. Math., Vol. 7, 1998, pp. 97–126
Lepowsky, J.: Remarks on vertex operator algebras and moonshine. In: Proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, S. Catto, A. Rocha (ed.), Singapore: World Scientific, 1992, pp. 362–370
Lepowsky, J., Wilson, R.L.: Construction of the affine Lie algebra A1(1). Commun. Math. Phys. 62, 43–53 (1978)
Milas, A.: Formal differential operators, vertex operator algebras and zeta–values, II. J. Pure Appl. Alg. 183, 191–244 (2003)
Milas, A.: Virasoro algebra, Dedekind eta-function and Specialized Macdonald’s identities. Transf. Groups 9, 273–288 (2004)
Milne, S.: Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6, 7–149 (2002)
Miyake, T.: Modular Forms. Berlin-Heidelberg-New York: Springer Verlag, 1989
Raghavan, S.: On certain identities due to Ramanujan. Quart. J. Math. Oxford Ser. (2) 37, 221–229 (1986)
Ramanujan, S.: Lost Notebook and Other Unpublished Papers. New Delhi: Narosa, 1988.
Rocha-Caridi, A.: Vacuum vector representations of the Virasoro algebra. In: Vertex operators in mathematics and physics (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ. 3, New York: Springer, 1985, pp. 451–473
Wang, W.: Rationality of Virasoro vertex operator algebras. I.M.R.N. 7, 197–211 (1993)
Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9, 237–307 (1996)
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Communicated by L. Takhtajan
Acknowledgements It was indeed hard to trace all the known proofs of (1.1), (1.2) and (1.3). We apologize if some important references are omitted. We would like to thank Jim Lepowsky for conversations on many related subjects. A few years ago Lepowsky and the author were trying to relate classical Rogers-Ramanujan identities and Zhu’s work [Z]. We also thank Bruce Berndt for pointing us to [BrO] and Steve Milne for bringing [Mi] to our attention.
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Milas, A. Ramanujan’s ‘‘Lost Notebook’’ and the Virasoro Algebra. Commun. Math. Phys. 251, 567–588 (2004). https://doi.org/10.1007/s00220-004-1179-3
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DOI: https://doi.org/10.1007/s00220-004-1179-3