In this thesis I will investigate quantum property of fuzzy sphere.
The matrix model with mass term has a nontrivially classical solution which is known to represent a noncommutative fuzzy
sphere. The fuzzy sphere has a lower energy than that of the trivial solution. In this thesis we investigate the quantum correction of the energy of the fuzzy sphere by using the Gaussian variation technique, in contrast to the other studying in which only the small fluctuation was considered. Our result, which only considers the boson part, shows that the quantum correction does not change the stability of the fuzzy sphere.

ibitem{GSW}M.B. Green, J.H. Schwarz and E. Witten, {it Superstring
Theory} (Cambridge University Press, 1987).
ibitem{POLCHINSKI}J. Polchinski, {it String Theory} (Cambridge
University Press, 1998).
ibitem{sen}A. Sen, ``Recent Developments in Superstring Theory'
Nucl.Phys.Proc.Suppl. {f 94} (2001) 35-48 [hep-lat/0011073]
ibitem{johnson}C.V. Johnson, ``D-Brane Primer', {it Strings, Branes and Gravity}, eds. J.A.~Harvey, S. Kachru and
E. Silverstein (World Scientific, 2001), pp. 129--350 [hep-th/0007170]
ibitem{9510017}J. Polchinski, ``Dirichlet Branes and Ramond-Ramond
Charges', Phys. Rev. Lett. {f 75} (1995) 4724--4727 [hep-th/9510017]
ibitem{9501068}P.~K.~Townsend,``The eleven-dimensional supermembrane revisited,'
Phys. Lett. {f B350}, 184 (1995)[hep-th/9501068].
ibitem{9503124} E.~Witten,``String theory dynamics in various dimensions,'
Nucl. Phys. {f B443}, 85 (1995)[hep-th/9503124]
ibitem{9607201}J.~H.~Schwarz,
``Lectures on superstring and M theory dualities,'
Nucl. Phys. Proc. Suppl. {f 55B}, 1 (1997)[hep-th/9607201]
ibitem{9806199}C. P. Bachas, ``Lectures on D-branes'[hep-th/9806199]
ibitem{0207142}R.J. Szabo,``BUSSTEPP LECTURES ON STRING THEORY:An introduction
to string theory and D-brane dynamics'[hep-th/0207142]
ibitem{ikkt}N. Ishibashi, H. Kawai,I. Kitazawa, and A. Tsuchiya,A large$-$N
reduced model as superstring, Nucl. Phys.{f B492}467-491[hep-th/9612115]
ibitem{bfss} T. Banks, W. Fischler, S.H. Shenker and L. Susskind,
``M-Theory as a Matrix Model: A Conjecture', Phys. Rev. {f D55}
(1997) 5112--5128 [hep-th/9610043].
ibitem{spacetimeuncert}S. Doplicher, K. Fredenhagen and J.E. Roberts,
Spacetime Quantization Induced by Classical Gravity, Phys Lett.{f B331}(1994) 39;
;R.J. Szabo, Quantum Field Theory on Noncommutative Spaces[hep-th/0109162]
ibitem{witten3} E. Witten, Bound States of Strings and p-Branes, Nucl. Phys.{f B460}
(1996) 33[hep-th/9510135].
ibitem{oda}S. Oda and F. Sugino, Gaussian and mean field approximations
for reduced Yang-Mills integrals,JHEP {f 0103}(2001)026,[hep-th/0011175]
ibitem{Nishimura}Jun Nishimura and Fumifiko Sugino, Dynamical Generation
of Four-Dimensional Space-Time in the IIB Matrix Model,JHEP 0205(2002)001
[hep-th/0111102]
ibitem{sugino}F. Sugino, Gaussian and mean field approximations for reduced 4D supersymmetric
Yang-Mills integral,JHEP {f 0107} (2001) 014,[ hep-th/0105284].
ibitem{Kabat}D.Kabat and G. Lifschytz, Approximations for strongly-coupled supersymmetric quantum mechanics,
Nucl. Phys. {f B 571} (2000) 419,[ hep-th/9910001].
ibitem{AABHN}J. Ambjo rn, K.N. Anagnostopoulos, W. Bietenholz,
T. Hotta and J. Nishimura,Large $N$ Dynamics of Dimensionally Reduced 4D SU($N$) Super Yang-Mills
Theory,JHEP {f 0007} (2000) 013,[ hep-th/0003208].

ibitem{1} A.~Y.~Alekseev, A.~Recknagel and V.~Schomerus,
``Non-commutative world-volume geometries: Branes on su(2) and
fuzzy sphere' JHEP {f 9909} (1999) 023, [hep-th/9908040];
``Brane dynamics in background fluxes and non-commutative
geometry,' JHEP {f 0005} (2000) 010, [hep-th/0003187].
ibitem{2} R.~C.~Myers, ``Dielectric-branes,' JHEP {f 9912} (1999) 022,
[hep-th/9910053].

ibitem{connes}A. Connes, M. R. Douglas and A. Schwarz, Noncommutative Geometry
and Matrix Theory: Compacitification on Tori, JHEP 9802:003(1998),[hep-th/9711162];
B. Morariu and B. Zumino, Super Yang-Mills on Noncommutative Torus,[hep-th/9807198];
C. Hofman and E.Verlinde, U-duality of Born-Infeld on the Noncommutative Two Torus,JHEP 9812,010(1998),[hep-th/9810116]
ibitem{seiberg}N. Seiberg and E.Witten, String Thoery and Noncommutative Geometry, JHEP 9909, 032(1999),[hep-th/9908142]
ibitem{strassler}J. Polchinski and M. Strassler, The String Duality of a Confining Four-Dimensional
Gauge Theory, [hep-th/0003136]
ibitem{iso}S. Iso, Y. Kimura,K. Tanaka, and K. Wakatsuki, Noncommutative Gauge Theoryon Fuzzy Spere from Matrix Model,
nucl. Phys 604(2001)121 [hep-th/0101102]
ibitem{imai}T. Imai, Y. Kitazaya, Y. Takayama, and D. Tomino,
Quantum Correction on Fuzzy sphere, [hep-th/0303120]

ibitem{3} Pei-Ming Ho, ``Fuzzy Sphere from Matrix
Model' JHEP {f 0012} (2000) 015, [hep-th/0010165].
ibitem{4}C.~Bachas, M.~Douglas and C.~Schweigert, ``Flux stabilization of
D-branes,' JHEP {f 0005} (2000) 048, [hep-th/0003037].
ibitem{5} K. Hashimoto and K. Krasnov, ``D-brane Solutions in Non-Commutative
Gauge Theory on Fuzzy Sphere', Phys. Rev. {f D64}(2001) 046007,
[hep-th/0101145]; S. Iso, Y. Kimura, K. Tanaka, and K. Wakatsuki,
``Noncommutative Gauge Theory on Fuzzy Sphere from Matrix Model',
Nucl. Phys. {f 604}(2001) 121, [hep-th/0101102].
ibitem{6} Y. Kimura, ``Noncommutative Gauge Theory on Fuzzy Sphere and fuzzy
torus from Matrix Model', Prog. theor. Phys. {f 106}(2001) 445,
[hep-th/0103192].
ibitem{7} H.~Grosse, C.~Klimcik and P.~Presnajder,
``Towards finite quantum field theory in noncommutative
geometry,' Int. J. Theor. Phys. {f 35}, 231 (1996),
[hep-th/9505175].
ibitem{8} S.~Vaidya, ``Perturbative dynamics on
fuzzy $S^2$ and $RP^2$,' Phys.Lett. B512, 403 (2001),
[hep-th/0102212]
ibitem{9} C. S~Chu, J. Madore and H. Steinacker,
``Scaling Limits of the fuzzy sphere at one loop', JHEP {f
0108}, 038 (2001), [hep-th/0106205].
ibitem{10} B. P. Dolan, D. O'Connor and P. Presnajder, ``Matrix $phi ^4$ models on the fuzzy
sphere and their continuum limits', [hep-th/0109084].
ibitem{11} S.~Minwalla, M.~Van Raamsdonk and N.~Seiberg, ``Noncommutative
perturbative dynamics,' JHEP {f 0002}, 020 (2000), [hep-th/9912072].
ibitem{} M.~R.~Douglas and N.~A.~Nekrasov,
``Noncommutative field theory,' Rev.Mod.Phys. {f 73} (2002) 97,
[hep-th/0106048].

ibitem W. Fischler, E. Gorbatov, A. Kashani-Poor, S. Paban, P.
Pouliot and J. Gomis, Evidence for Winding States in
Noncommutative Quantum Field Theory, JHEP {f 05} (2000) 024,
[hep-th/0002067].
ibitem W. Fischler, E. Gorbatov, A. Kashani-Poor,
R. McNees, S. Paban, P. Pouliot and J. Gomis, The interplay
between $ heta$ and T, JHEP {f 06} (2000) 032,
[hep-th/0003216].
ibitem{whh1} W. H. Huang, ``High-Temperature Effective
Potential of Noncommutative Scalar Field Theory: Reduction of
Degrees of Freedom by Noncommutativity', Phys. Rev. {f D 63}
(2001) 125004, [hep-th/0101040]; W. H. Huang, ``Two-loop effective
potential in noncommutative scalar field theory', Phys.Lett. {f
B496} (2000) 206, [hep-th/0009067].
ibitem{whh2} W. H. Huang,
``Finite-Temperature Casimir Effect on the Radius Stabilization of
Noncommutative Torus',JHEP {f 0011}, 040 (2000),
[hep-th/0011037]; W. H. Huang, ``Casimir Effect on the Radius
Stabilization of the Noncommutative Torus', Phys.Lett. {f B497}
(2001) 317, [hep-th/0010160].
ibitem{whh3} W. H. Huang, ``Quantum Stabilization of Compact Space by Extra Fuzzy Space',
Phys.Lett.{f B537} (2002) 311, [hep-th/0203176].
W. H. Huang, Effective Potential on Fuzzy Sphere, [hep-th/0203051]
ibitem U. C.Watamura and S. Watamura, ``Noncommutative Geometry and Gauge
Theory on Fuzzy Sphere', Commun.Math.Phys. {f 212} (2000) 395,
[hep-th/9801195].