Unitons and their moduli
Author:
Christopher Kumar Anand
Journal:
Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 7-16
MSC (1991):
Primary 58E20, 58D27, 58G37
DOI:
https://doi.org/10.1090/S1079-6762-96-00002-9
MathSciNet review:
1405964
Full-text PDF Free Access
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Abstract: We sketch the proof that unitons (harmonic spheres in $\operatorname {U}(N)$) correspond to holomorphic ‘uniton bundles’, and that these admit monad representations analogous to Donaldson’s representation of instanton bundles. We also give a closed-form expression for the unitons involving only matrix operations, a finite-gap result (two-unitons have energy $\ge 4$), computations of fundamental groups of energy $\le 4$ components, new methods of proving discreteness of the energy spectrum and of Wood’s Rationality Conjecture, a discussion of the maps into complex Grassmannians and some open problems.
- C. K. Anand, Uniton bundles, McGill University Ph.D. Thesis 1994.
- ---, Uniton bundles, Comm. Anal. Geom. 3 (1995), 371–419.
- ---, A closed form for unitons, in preparation.
- M. F. Atiyah, N. J. Hitchin, V. G. Drinfel′d, and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. MR 598562, DOI https://doi.org/10.1016/0375-9601%2878%2990141-X
- C. P. Boyer, J. C. Hurtubise, B. M. Mann, and R. J. Milgram, The topology of instanton moduli spaces. I. The Atiyah-Jones conjecture, Ann. of Math. (2) 137 (1993), no. 3, 561–609. MR 1217348, DOI https://doi.org/10.2307/2946532
- Francis E. Burstall and John H. Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Mathematics, vol. 1424, Springer-Verlag, Berlin, 1990. With applications to harmonic maps of Riemann surfaces. MR 1059054
- S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), no. 4, 453–460. MR 763753
- Mikio Furuta, Martin A. Guest, Motoko Kotani, and Yoshihiro Ohnita, On the fundamental group of the space of harmonic $2$-spheres in the $n$-sphere, Math. Z. 215 (1994), no. 4, 503–518. MR 1269487, DOI https://doi.org/10.1007/BF02571727
- N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83 (1982), no. 4, 579–602. MR 649818
- G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 689–713. MR 169877, DOI https://doi.org/10.1112/plms/s3-14.4.689
- Jacques Hurtubise, Instantons and jumping lines, Comm. Math. Phys. 105 (1986), no. 1, 107–122. MR 847130
- M. K. Murray, Nonabelian magnetic monopoles, Comm. Math. Phys. 96 (1984), no. 4, 539–565. MR 775045
- Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
- K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), no. 3, 207–221. MR 408535
- J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI https://doi.org/10.2307/1971131
- Graeme Segal, Loop groups and harmonic maps, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 153–164. MR 1055875, DOI https://doi.org/10.1017/CBO9780511662614.015
- Karen Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), no. 1, 1–50. MR 1001271
- Giorgio Valli, On the energy spectrum of harmonic $2$-spheres in unitary groups, Topology 27 (1988), no. 2, 129–136. MR 948176, DOI https://doi.org/10.1016/0040-9383%2888%2990032-8
- Giorgio Valli, Interpolation theory, loop groups and instantons, J. Reine Angew. Math. 446 (1994), 137–163. MR 1256151, DOI https://doi.org/10.1515/crll.1994.446.137
- R. S. Ward, Classical solutions of the chiral model, unitons, and holomorphic vector bundles, Comm. Math. Phys. 128 (1990), no. 2, 319–332. MR 1043523
- John C. Wood, Explicit construction and parametrization of harmonic two-spheres in the unitary group, Proc. London Math. Soc. (3) 58 (1989), no. 3, 608–624. MR 988105, DOI https://doi.org/10.1112/plms/s3-58.3.608
- C. K. Anand, Uniton bundles, McGill University Ph.D. Thesis 1994.
- ---, Uniton bundles, Comm. Anal. Geom. 3 (1995), 371–419.
- ---, A closed form for unitons, in preparation.
- M. F. Atiyah, V. G. Drinfeld, N. J. Hitchin, and Yu. I. Manin, Construction of instantons, Phys. Lett. 65A (1978), 185–187.
- C. P. Boyer, J. C. Hurtubise, B. M. Mann, and R. J. Milgram, The topology of instanton moduli spaces. I: The Atiyah-Jones conjecture, Annals of Math.(2) 137 (1993), 561–609.
- F. E. Burstall and J. H. Rawnsley, Twistor theory for Riemannian symmetric spaces, LNM 1424, Springer-Verlag, 1990.
- S. K. Donaldson, Instantons and geometric invariant theory, Commun. Math. Phys. 93 (1984), 453–460.
- M. Furuta, M. A. Guest, M. Kotani, and Y. Ohnita, On the fundamental group of the space of harmonic $2$-spheres in the $n$-sphere, Math. Z. 215 (1994), 503–518.
- N. J. Hitchin, Monopoles and geodesics, Commun. Math. Phys. 83 (1982), 579–602.
- G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. Lond. Math. Soc. (3) 14 (1964), 689-713.
- J. C. Hurtubise, Instantons and jumping lines, Commun. Math. Phys. 105 (1986), 107–122.
- M. K. Murray, Non-Abelian magnetic monopoles, Commun. Math. Phys. 96 (1984), 539–565.
- C. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces, Birkhauser, Boston, 1980.
- K. Pohlmeyer, Integrable Hamiltonian systems and interactions through constraints, Comm. Math. Phys. 46 (1976), 207–221.
- J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Annals of Math. 113 (1981), 1–24.
- G. Segal, Loop groups and harmonic maps, Advances in Homotopy Theory (Cortona 1988), London Math. Soc. Lecture Note 139, Cambridge Univ. Press, 1989, pp. 153–164.
- K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geometry 30 (1989), 1–50.
- G. Valli, On the energy spectrum of harmonic 2-spheres in unitary groups, Topology 27 (1988), 129–136.
- ---, Interpolation theory, loop groups and instantons, J. Reine Angew. Math. 446 (1994), 137–163.
- R. S. Ward, Classical solutions of the chiral model, unitons, and holomorphic vector bundles, Commun. Math. Phys. 123 (1990), 319–332.
- J. C. Wood, Explicit construction and parametrisation of harmonic two-spheres in the unitary group, Proc. London Math. Soc. (3) 58 (1989), 608–624.
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Additional Information
Christopher Kumar Anand
Affiliation:
Mathematics Research Centre, University of Warwick, Coventry CV4 7AL, UK
Email:
anand@maths.warwick.ac.uk
Keywords:
Uniton,
harmonic map,
chiral field,
sigma model
Received by editor(s):
September 19, 1995
Additional Notes:
Research supported by NSERC and FCAR scholarships.
Communicated by:
Eugenio Calabi
Article copyright:
© Copyright 1996
American Mathematical Society
