Abstract
We report on some recent work on deformation of spaces, notably deformation of spheres, describing two classes of examples.
Preview
Unable to display preview. Download preview PDF.
References
P. Aschieri, F. Bonechi, On the Noncommutative Geometry of Twisted Spheres, Lett. Math. Phys. 59 (2002) 133–156.
M.F. Atiyah, Global theory of elliptic operators, in: ‘Proc. Intn’l Conf. on functional analysis and related topics’, Tokyo 1969, Univ. of Tokyo Press 1970, pp. 21–30.
M.F. Atiyah, Geometry of Yang-Mills Fields, Accad. Naz. Dei Lincei, Scuola Norm. Sup. Pisa, 1979.
F. Bonechi, N. Ciccoli, M. Tarlini, Quantum even spheres Σ 2n q from Poisson double suspension, Commun. Math. Phys., 243 (2003) 449–459.
J. Brodzki, An Introduction to K-theory and Cyclic Cohomology, Polish Scientific Publishers 1998.
P.S. Chakraborty, A. Pal, Equivariant Spectral triple on the Quantum SU(2)-group, math.KT/0201004.
P.S. Chakraborty, A. Pal, Spectral triples and associated Connes-de Rham complex for the quantum SU(2) and the quantum sphere, math.QA/0210049.
A. Connes, C* algèbres et géométrie différentielle, C.R. Acad. Sci. Paris, Ser. A-B, 290 (1980) 599–604.
A. Connes, Noncommutative differential geometry, Inst. Hautes Etudes Sci. Publ. Math., 62 (1985) 257–360.
A. Connes, Noncommutative geometry, Academic Press 1994.
A. Connes, Noncommutative geometry and reality, J. Math. Physics, 36 (1995) 6194–6231.
A. Connes, Gravity coupled with matter and foundation of noncommutative geometry, Commun. Math. Phys., 182 (1996) 155–176.
A. Connes, A short survey of noncommutative geometry, J. Math. Physics, 41 (2000) 3832–3866.
A. Connes, Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq(2), math.QA/0209142,
A. Connes, M. Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Commun. Math. Phys. 230 (2002) 539–579.
A. Connes, M. Dubois-Violette, Moduli space and structure of noncommutative 3-spheres, math.QA/0308275.
A. Connes, G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, Commun. Math. Phys. 221 (2001) 141–159.
A. Connes, H. Moscovici, The local index formula in noncommutative geometry, GAFA 5 (1995) 174–243.
A. Connes, M. Rieffel, Yang-Mills for noncommutative two tori, In: Operator algebras and mathematical physics (Iowa City, Iowa, 1985). Contemp. Math. Oper. Algebra Math. Phys., 62, Amer. Math. Soc., Providence, 1987, 237–266.
A. Connes, D. Sullivan, N. Teleman, Quasi-conformal mappings, operators on Hilbert spaces and a local formula for characteristic classes, Topology 33 (1994) 663–681.
J. Cuntz, On the homotopy groups for the space of endomorphisms of a C*-algebra (with applications to topological Markov chains), in Operator algebras and group representations, Pitman 1984, pp. 124–137
L. Dabrowski, The Garden of Quantum Spheres, Banach Center Publications, 61 (2003) 37–48.
L. Dabrowski, G. Landi, Instanton algebras and quantum 4-spheres, Differ. Geom. Appl. 16 (2002) 277–284.
L. Dabrowski, G. Landi, T. Masuda, Instantons on the quantum 4-spheres S 4 q , Commun. Math. Phys. 221 (2001) 161–168.
L. Dabrowski, A. Sitarz, Dirac Operator on the Standard Podleś Quantum Sphere, Banach Center Publications, 61 (2003) 49–58.
K.R. Davidson, C*-algebras by example, American Mathematical Society 1996.
J. Dixmier, Existence de traces non normales, C.R. Acad. Sci. Paris, Ser. A-B, 262 (1966) 1107–1108.
M. Dubois-Violette, Equations de Yang et Mills, modèles σ à deux dimensions et généralisation, in ‘Mathématique et Physique’, Progress in Mathematics, vol. 37, Birkhäuser 1983, pp. 43–64.
M. Dubois-Violette, Y. Georgelin, Gauge Theory in Terms of Projector Valued Fields, Phys. Lett. 82B (1979) 251–254.
L.D. Faddeev, N.Y. Reshetikhin, L.A. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math. Jour. 1 (1990) 193.
G. Fiore, The Euclidean Hopf algebra U q (eN) and its fundamental Hilbert space representations, J. Math. Phys. 36 (1995) 4363.
G. Fiore, The q-Euclidean algebra U q (eN) and the corresponding q-Euclidean lattice, Int. J. Mod. Phys. A11 (1996) 863.
I. Frenkel, M. Jardim Quantum Instantons with Classical Moduli Spaces, Commun. Math. Phys., 237 (2003) 471–505.
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicæ, 82 (1985) 307–347.
J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, Boston, 2001.
E. Hawkins, G. Landi, Fredholm Modules for Quantum Euclidean Spheres, J. Geom. Phys. 49 (2004) 272–293.
T. Hadfield, Fredholm Modules over Certain Group C*-algebras, math.OA/0101184.
T. Hadfield, K-homology of the Rotation Algebra A_θ, math.OA/0112235.
T. Hadfield, The Noncommutative Geometry of the Discrete Heisenberg Group, math.OA/0201093.
J.H. Hong, W. Szymański, Quantum spheres and projective spaces as graph algebras, Commun. Math. Phys. 232 (2002) 1, 157–188.
G. Kasparov, Topological invariants of elliptic operators, I. K-homology, Math. URSS Izv. 9 (1975) 751–792.
U. Krähmer, Dirac Operators on Quantum Flag Manifolds, math.QA/0305071.
G. Landi, An Introduction to Noncommutative Spaces and Their Geometries, Springer, 1997; Online corrected edition, Springer Server, September 2002.
A preliminary version is available as hep-th/9701078.
G. Landi, Deconstructing Monopoles and Instantons, Rev. Math. Phys. 12 (2000) 1367–1390.
G. Landi, talk at the Mini-workshop on Noncommutative Geometry Between Mathematics and Physics, Ancona, February 23–24, 2001.
G. Landi, J. Madore, Twisted Configurations over Quantum Euclidean Spheres, J. Geom. Phys. 45 (2003) 151–163.
B. Lawson, M.L. Michelsohn, Spin Geometry, Princeton University Press, 1989.
J.-L. Loday, Cyclic Homology, Springer, Berlin, 1992.
D. McDuff, D. Salamon, Introduction to Symplectic Topology, Oxford University Press, Oxford 1995.
T. Masuda, Y. Nakagami, J. Watanabe, Noncommutative differential geometry on the quantum SU(2). I: An algebraic viewpoint, K-Theory 4 (1990) 157.
T. Masuda, Y. Nakagami, J. Watanabe, Noncommutative differential geometry on the quantum two sphere of P. Podleś, I: An algebraic viewpoint, K-Theory 5 (1991) 151.
T. Natsume, C.L. Olsen, A new family of noncommutative 2-spheres, J. Func. Anal., 202 (2003) 363–391. T. Natsume, this proceedings.
N. Nekrasov, A. Schwarz, Instantons in noncommutative IR4 and (2,0) superconformal six dimensional theory, Commun. Math. Phys. 198 (1998) 689–703.
P. Podleś, Quantum spheres, Lett. Math. Phys. 14 (1987) 193.
I. Raeburn, W. Szymański, Cuntz-Krieger algebras of infinite graphs and matrices, University of Newcastle Preprint, December 1999.
M.A. Rieffel, C*-algebras associated with irrational rotations, Pac. J. Math. 93 (1981) 415–429.
M.A. Rieffel, Deformation Quantization for Actions of IRd, Memoirs of the Amer. Math. Soc. 506, Providence, RI, 1993.
M.A. Rieffel, K-groups of C*-algebras deformed by actions of IRd, J. Funct. Anal. 116 (1993) no. 1, 199–214.
S. Sakai, C*-Algebras and W*-Algebras, Springer 1998.
K. Schmüdgen, E. Wagner, Dirac operator and a twisted cyclic cocycle on the standard Podleś quantum sphere, math.QA/0305051.
B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, 1979.
A. Sitarz, Twists and spectral triples for isospectral deformations, Lett. Math. Phys. 58 (2001) 69–79.
A. Sitarz, Dynamical Noncommutative Spheres, Commun. Math. Phys., 241 (2003) 161–175.
L.L. Vaksman, Y.S. Soibelman, The algebra of functions on quantum SU(n+1) group and odd-dimensional quantum spheres, Leningrad Math. Jour. 2 (1991) 1023.
J.C. Várilly, Quantum symmetry groups of noncommutative spheres, Commun. Math. Phys. 221 (2001) 511–523.
S. Waldmann, Morita Equivalence, Picard Groupoids and Noncommutative Field Theories, math.QA/0304011 and this proceedings.
M. Welk, Differential Calculus on Quantum Spheres, Czechoslovak J. Phys. 50 (2000) no. 11, 1379–1384.
M. Wodzicki, Noncommutative residue, Part I. Fundamentals, In K-theory, arithmetic and geometry. Lecture Notes Math., 1289, Springer, 1987, 320–399.
S.L. Woronowicz, Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987) 117.
Author information
Authors and Affiliations
Editor information
Rights and permissions
About this chapter
Cite this chapter
Landi, G. Noncommutative Spheres and Instantons. In: Carow-Watamura, U., Maeda, Y., Watamura, S. (eds) Quantum Field Theory and Noncommutative Geometry. Lecture Notes in Physics, vol 662. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11342786_1
Download citation
DOI: https://doi.org/10.1007/11342786_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23900-0
Online ISBN: 978-3-540-31526-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)