Abstract
It is shown that there is a critical value of the Chern-Simons coupling parameter so that, below the value, there exists self-dual doubly periodic vortex solutions, and, above the value, the vortices are absent. Solutions of such a nature indicate the existence of dyon condensates carrying quantized electric and magnetic charges.
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[Ab] Abrikosov, A.A.: On the magnetic properties of superconductors of the second group, Sov. Phys. JETP5, 1174–1182 (1957)
[A1] Aubin, T.: Nonlinear Analysis on Manifolds: Monge-Ampére Equations. Springer: Berlin Heidelberg, New York, 1982
[A2] Aubin, T.: Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de courburne scalaire. J. Funct. Anal.32, 148–174 (1979)
[Av] Aviles, P.: Conformal complete metrics with prescribed non-negative Gaussian curvature inR 2. Invent. Math.83, 519–544 (1986)
[CH] Chakravarty, S., Hosotani, Y.: Anyon model on a cylinder Phys. Rev. D44, 441–451 (1991)
[CY] Chang, S.Y.A., Yang, P.C.: Prescribing Gaussian curvature onS 2. Acta Math.159, 215–259 (1987)
[CHMcY] Chen, X., Hastings, S., McLeod J.B., Yang, Y.: A nonlinear elliptic equation arising from gauge field theory and cosmology. Proc. R. Soc. London, Series A,446, 453–478 (1994)
[FM1] Fröhlich, J., Marchetti, P.: Bosonization, topological solitons and fractional charges in two-dimensional quantum field theory. Commun. Math. Phys.116, 127–173 (1988)
[FM2] Fröhlich, J., Marchetti, P.: Quantum field theories of vortices and anyons. Commun. Math. Phys.121, 177–223 (1989)
[HKP] Hong, J., Kim, Y., Pac, P.Y.: Multivortex solutions of the Abelian Chern-Simons theory. Phys. Rev. Lett.64, 2230–2233 (1990)
[H] Hosotani, Y.: Gauge invariance in Chern-Simons theory on a torus. Phys. Rev. Lett.62, 2785–2788 (1989)
[IL] Iengo, R., Lechner, K.: Quantum mechanics of anyons on a torus. Nucl. Phys.B 346, 551–575 (1990)
[JLW] Jackiw, R., Lee, K., Weinberg, E.J.: Self-dual Chern-Simons solitions. Phys. Rev.D 42, 3488–3499 (1990)
[JW] Jackiw, R., Weinberg, E.J.: Self-dual Chern-Simons vortices. Phys. Rev. Lett.64, 2234–2237 (1990)
[JT] Jaffe, A., Taubes, C.H.: Vortices and Monopoles Boston: Birkhäuser, 1980
[KW1] Kazdan, J.L., Warner, F.W.: Integrability conditions for Δu=k−Ke 2u with applications to Riemannian geometry. Bull. Am. Math. Soc.77, 819–823 (1971)
[KW2] Kazdan, J.L., Warner, F.W.: Curvature functions for compact 2-manifolds. Ann. Math.99, 14–47 (1974)
[KW3] Kazdan, J.L., Warner, F.W.: Curvature functions for open 2-manifold. Ann. Math.99, 203–219 (1974)
[Mc] McOwen, R.C.: Conformal metrics inR 2 with prescribed Gaussian curvature and positive total curvature. Indiana U. Math. J.34, 97–104 (1985)
[Ni] Ni, W.-M.: On the elliptic equation Δu+K(x)e 2u=0 and conformal metrics with prescribed Gaussian curvature. Invent. Math.66, 343–352 (1982)
[O] Olesen, P.: Solition condensation in some self-dual Chern-Simons theories. Phys. Lett.B 265, 361–365 (1991); Erratum,267, 541 (1991)
[P] Polychronakos, A.: Abelian Chern-Simons theories in 2+1 dimensions. Ann. Phys.203, 231–254 (1990)
[RSS] Randjbar, S., Salam, A., Strathdee, J.A.: Anyons and Chern-Simons theory on compact spaces of finite genus. Phys. Lett.B 240, 121–126 (1990)
['t H] Hooft, G. 't.: A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys.B 153, 141–160 (1979)
[SY1] Spruck, J., Yang, Y.: The existence of non-topological solitons in the self-dual Chern-Simons Theory. Commun. Math. Phys.149, 361–376 (1992)
[SY2] Spruck, J., Yang, Y.: Topological solutions in the self-dual Chern-Simons theory: Existence and approximation. Ann. l'I. H. P.-Anal. non Linéaire, to appear
[T1] Taubes, C.H.: ArbitraryN-vortex solutions to the first order Ginzburg-Landau equations. Commun. Math. Phys.72, 277–292 (1980)
[T2] Taubes, C.H.: On the equivalence of the first and second order equations for gauge theories. Commun. Math. Phys.75, 207–227 (1980)
[WY] Wang, S., Yang, Y.: Abrikosov's vortices in the critical coupling. SIAM J. Math. Anal.23, 1125–1140 (1992)
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Caffarelli, L.A., Yang, Y. Vortex condensation in the Chern-Simons Higgs model: An existence theorem. Commun.Math. Phys. 168, 321–336 (1995). https://doi.org/10.1007/BF02101552
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DOI: https://doi.org/10.1007/BF02101552