Mock theta functions of order 2 and their shadow computations
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Zwegers showed that a mock theta function can be completed to form essentially a real analytic modular form of weight $1/2$ by adding a period integral of a certain weight $3/2$ unary theta series. This theta series is related to the holomorphic modular form called the shadow of the mock theta function. In this paper, we discuss the computation of shadows of the second order mock theta functions and show that they share the same shadow with a mock theta function which appears in the Mathieu moonshine phenomenon.
Abstract
Zwegers showed that a mock theta function can be completed to form essentially a real analytic modular form of weight $1/2$ by adding a period integral of a certain weight $3/2$ unary theta series. This theta series is related to the holomorphic modular form called the shadow of the mock theta function. In this paper, we discuss the computation of shadows of the second order mock theta functions and show that they share the same shadow with a mock theta function which appears in the Mathieu moonshine phenomenon.
참고문헌 (23)
- G. E. Andrews, "Mordell integrals and Ramanujan's "lost" notebook, Analytic number theory (Philadelphia, PA., 1980)", Lecture Notes in Math., 899, pp.10~18 (1981), Springer
- G. E. Andrews, "Modularity of the concave composition generating function", Algebra Number Theory, 7(9), pp.2103~2139 (2013)
- M. C. N. Cheng, "K3 surfaces, N = 4 Dysons, and the Mathieu group M24", arxiv:1005.5415, ~ (2010)
- M. C. N. Cheng, "Umbral moonshine", Commun. Number Theory Phys, 8(2), pp.101~242 (2014)
- , ~
- J. F. R. Duncan, "Proof of the umbral moonshine conjecture", Res. Math. Sci, 2, pp.47~ (2015)
- T. Eguchi, "Notes on the K3 surface and the Mathieu group M24", Exp. Math, 20(1), pp.91~96 (2011)
- T. Eguchi, "Superconformal algebra and string compacti cation on manifolds with SU(N) Holonomy", Nuclear Phys. B, 315(1), pp.193~221 (1989)
- T. Eguchi, "On the unitary representation of N = 2 and N = 4 superconformal algebras", Phys. Lett. B, 210(1-2), pp.125~132 (1988)
- A. Folsom, "Quantum mock modular forms arising from eta-theta fnctions", Res. Number Theory, 2, pp.41~ (2016)
- B. Gordon, "A survey of classical mock theta functions, Partitions, q-Series, and Modular Forms", pp.95~144 (2012), Springer
- K. Hikami, "Transformation formula of the second order mock theta functions", Lett. Math. Phys, 75(1), pp.93~98 (2006)
- S.-Y. Kang, "Generalizations of Ramanujan's reciprocity theorem and their applications", J. Lond. Math. Soc, 75(2), pp.18~34 (2007)
- S.-Y. Kang, "Mock Jacobi forms in basic hypergeometric series", Compos. Math, 145(3), pp.553~565 (2009)
- R. J. McIntosh, "Second order mock theta functions", Canad. Math. Bull, 50(2), pp.284~290 (2007)
- K. Ono, "Current developments in mathematics, 2008", pp.347~454 (2009), Int. Press
- S. Ramanujan, "The Lost Notebook and Other Unpublished Papers" (1988), Narosa
- G. N. Watson, "The final problem: An account of the mock theta functions", J. London Math. Soc, 11, pp.55~80 (1936)
- D. Zagier, "Ramanujan's mock theta functions and their applications", Seminaire Bourbaki 60eme annee, (986) (2007)
- S. P. Zwegers, "Mock Theta Functions and Real Analytic Modular Forms, q-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000)", Con-temp. Math., 291, pp.269~277 (2001), Amer. Math. Soc.[DOI]
- S. P. Zwegers, "Mock Theta Functions", pp.2002
- "Multivariable Appell Functions" (2010)[URL]
- S. P. Zwegers, "Geometry and Physics" (2011)