We study the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant for the Seifert fibered homology spheres with -exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight and . By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large- limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in , is related to the number of integral lattice points inside the -dimensional tetrahedron.
REFERENCES
1.
E.
Witten
, Commun. Math. Phys.
121
, 351
(1989
).2.
M. F.
Atiyah
, The Geometry and Physics of Knots
(Cambridge University Press
, Cambridge, UK
, 1990
).3.
D. S.
Freed
and R. E.
Gompf
, Commun. Math. Phys.
141
, 79
(1991
).4.
N. Yu.
Reshetikhin
and V. G.
Turaev
, Invent. Math.
103
, 547
(1991
).5.
R.
Kirby
and P.
Melvin
, Invent. Math.
105
, 473
(1991
).6.
7.
R.
Lawrence
, J. Math. Phys.
36
, 6106
(1995
).8.
9.
L.
Rozansky
, Proceedings of the Conference on Quantum Topology
, edited by D. N.
Yetter
(World Scientific
, Singapore
, 1994
), pp. 307
–354
.10.
L.
Rozansky
, J. Math. Phys.
35
, 5219
(1994
).11.
L.
Rozansky
, Commun. Math. Phys.
171
, 279
(1995
).12.
13.
L.
Rozansky
, Commun. Math. Phys.
175
, 297
(1996
).14.
15.
16.
17.
K.
Hikami
, Commun. Math. Phys.
(in press).18.
19.
20.
21.
22.
23.
24.
S.
Ramanujan
, The Lost Notebook and Other Unpublished Papers
(Narosa
, New Delhi
, 1987
).25.
26.
27.
P.
Orlik
, Seifert Manifolds
, Lecture Notes Mathematics
Vol. 291
(Springer
, Berlin
, 1972
).28.
N.
Saveliev
, Invariants for Homology 3-Spheres
, Encyclopedia of Mathematical Sciences
Vol. 140
(Springer
, Berlin
, 2002
).29.
H.
Seifert
and W.
Threlfall
, Seifert and Threlfall: A Textbook of Topology
, Pure Applied Mathematics
Vol. 89
(Academic
, New York, 1980
).30.
L. C.
Jeffrey
, Commun. Math. Phys.
147
, 563
(1992
).31.
H.
Rademacher
, Topics in Analytic Number Theory
, Grund. Math. Wiss.
Vol. 169
(Springer
, New York
, 1973
).32.
33.
G. E.
Andrews
, Analytic Number Theory
, edited by M. I.
Knopp
, Lecture Notes in Math
Vol. 899
(Springer
, New York, 1981
), pp. 10
–48
.34.
35.
36.
S.
Lang
, Introduction to Modular Forms
, Grund. Math. Wiss.
Vol. 222
(Springer
, Berlin
, 1976
).37.
38.
K.
Ono
, The Web of Modularity
, CBMS Regional Conference Series in Mathematics
(American Mathematical Society
, Providence
, 2004
), No. 102.39.
N. J. A.
Sloane
, http://www.research.att.com/~njas/sequences/index.html.40.
R. P.
Stanley
, Enumerative Combinatorics I
, no. 49 in Cambridge Studies in Advanced Mathematics (Cambridge University Press
, Cambridge
, 1997
), No. 49.41.
T.
Arakawa
, T.
Ibukiyama
, and M.
Kaneko
, Bernoulli Number and Zeta Function
(Makino Shoten
, Tokyo, Japan
, 2001
) (in Japanese).42.
43.
44.
S.
Fukuhara
, Y.
Matsumoto
, and K.
Sakamoto
, Math. Ann.
287
, 275
(1990
).45.
46.
47.
48.
49.
M.
Beck
and S.
Robins
, Computing the Continuous Discretely. Integer-Point Enumeration in Polyhedra
(Springer
, Berlin
, 2005
).50.
51.
52.
53.
54.
55.
M.
Mariño
, Commun. Math. Phys.
253
, 25
(2005
).56.
57.
M.
Mariño
, Rev. Mod. Phys.
77
, 675
(2005
).58.
H.
Rademacher
and E.
Grosswald
, Dedekind Sums
, Carus Mathematical Monographs
(Mathematical Association of America
, Washington, D.C.
, 1972
), No. 16.59.
K.
Chandrasekharan
, Elliptic Functions
, Grund. Math. Wiss.
Vol. 281
(Springer
, Berlin
, 1985
).60.
© 2006 American Institute of Physics.
2006
American Institute of Physics
You do not currently have access to this content.