The goal of this paper is to compute the zeta function determinant for the massive Laplacian on Riemann caps (or spherical suspensions). These manifolds are defined as compact and boundaryless D-dimensional manifolds deformed by a singular Riemannian structure. The deformed spheres, considered previously in the literature, belong to this class. After presenting the geometry and discussing the spectrum of the Laplacian, we illustrate a method to compute its zeta regularized determinant. The special case of the deformed sphere is recovered as a limit of our general formulas.
REFERENCES
1.
R.
Critchley
and J. S.
Dowker
, Phys. Rev. D
13
, 3224
(1976
).2.
S. W.
Hawking
, Commun. Math. Phys.
55
, 133
(1977
).3.
D. B.
Ray
and I. M.
Singer
, Adv. Math.
7
, 145
(1971
).4.
L.
Parker
and D. J.
Toms
, Quantum Field Theory in Curved Spacetime
(Cambridge University Press
, Cambridge, England
, 2009
).5.
J. S.
Dowker
, Commun. Math. Phys.
162
, 633
(1994
).6.
P.
Chang
and J. S.
Dowker
, Nucl. Phys. B
395
, 407
(1993
).7.
M.
Spreafico
, Rocky Mt. J. Math.
33
, 1499
(2003
).8.
M.
Bordag
, B.
Geyer
, K.
Kirsten
, and E.
Elizalde
, Commun. Math. Phys.
179
, 215
(1996
).9.
R.
Camporesi
, Phys. Rep.
196
, 1
(1990
).10.
M.
Bordag
, K.
Kirsten
, and J. S.
Dowker
, Commun. Math. Phys.
182
, 371
(1996
).11.
R.
Forman
, Invent. Math.
88
, 447
(1987
).12.
M.
Spreafico
and S.
Zerbini
, Commun. Math. Phys.
273
, 677
(2007
).13.
A. O.
Barvinsky
, A. Yu.
Kamenshchik
, and I. P.
Karmazin
, Ann. Phys.
219
, 201
(1992
).14.
A.
Flachi
, A.
Knapman
, W.
Naylor
, and M.
Sasaki
, Phys. Rev. D
70
, 124011
(2004
).15.
A.
Flachi
and M.
Sasaki
, in Proceedings of the Fourteenth Workshop on General Relativity and Gravitation in Japan
(Kyoto University
, Kyoto, Japan
, November 29–December 3, 2004
), pp. 239
–243
.16.
F. W. J.
Olver
, Asymptotics and Special Functions
, edited by A. K.
Peters
(AKP Classics
, Natick
, 1997
).17.
18.
N. R.
Khusnutdinov
, J. Math. Phys.
44
, 2320
(2003
).19.
R. C.
Thorne
, Philos. Trans. R. Soc. London
249
, 597
(1957
).20.
P. B.
Gilkey
, Invariance Theory the Heat Equation and the Atiyah-Singer Index Theorem
(CRC
, Boca Raton
, 1995
).21.
J.
Brüning
, Math. Ann.
268
173
(1984
).22.
J.
Brüning
and R.T.
Seeley
, J. Funct. Anal.
73
, 369
(1987
).23.
K.
Kirsten
, Spectral Functions in Mathematics and Physics
, (CRC
, Boca Raton
, 2001
).24.
G.
Cognola
, E.
Elizalde
, and S.
Zerbini
, J. Math. Phys.
47
, 083516
(2006
).25.
G.
Cognola
and S.
Zerbini
, Phys. Rev. D
69
, 024004
(2004
).26.
K.
Kirsten
, P.
Loya
, and J.
Park
, J. Math. Phys.
47
, 043506
(2006
).27.
E.
Elizalde
, Ten Physical Applications of the Spectral Zeta Function
(Springer-Verlag
, Berlin
, 1995
).28.
S.
Leseduarte
and A.
Romeo
, J. Phys. A
26
, 2483
(1994
).29.
I. S.
Gradshtein
and I. M.
Ryzhik
, Table of Integrals, Series and Products
, edited by A.
Jeffrey
and D.
Zwillinger
(Academic
, Oxford
, 2007
).30.
G.
Fucci
and K.
Kirsten
, J. Phys. A: Math. Theor.
43
, 365204
(2010
).31.
G.
Esposito
, A. Yu.
Kamenshchik
, and G.
Pollifrone
, Euclidean Quantum Gravity on Manifolds with Boundaries
(Kluwer Academic
, The Netherlands
, 1997
).32.
D. V.
Vassilevich
, Phys. Rep.
388
, 279
(2003
).33.
A. O.
Barvinsky
, A.
Yu
. Kamenshchik, I. P.
Karmazin
, and I. V.
Mishakov
, Class. Quantum Grav.
9
, L27
(1992
).34.
35.
J. S.
Dowker
and K.
Kirsten
, Anal. Appl.
3
, 45
(2005
).36.
G.
Fucci
, J. Math. Phys.
50
, 102301
(2009
).37.
J.
Miller
and V. S.
Adamick
, J. Comput. Appl. Math.
100
, 201
(1998
).38.
C.
De Rahm
, J. High Energy Phys.
60
(1
), 1
(2008
).39.
C. P.
Burgess
, D.
Hoover
, C.
De Rahm
, and G.
Tasinato
, J. High Energy Phys.
124
(3
), 1
(2009
).40.
T.
Kobayashi
, Phys. Rev. D
78
, 084018
–1
(2008
)41.
M.
Minamitsuji
and D.
Langlois
, Phys. Rev. D
76
, 084031
–1
(2007
).42.
E.
Elizalde
, L.
Vanzo
, and S.
Zerbini
, Commun. Math. Phys.
194
, 613
(1998
).43.
E.
Elizalde
, A.
Filippi
, L.
Vanzo
, and S.
Zerbini
, Phys. Rev. D
57
, 7430
(1998
).© 2011 American Institute of Physics.
2011
American Institute of Physics
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