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Exact results for supersymmetric \(\sigma\) models. (English) Zbl 0969.81634
Summary: We show that the metric and Berry’s curvature for the ground states of \(N=2\) supersymmetric \(\sigma\)-models can be computed exactly as one varies the Kähler structure. For the case of \(\mathbf C\text{P}^n\) these are related to special solutions of affine Toda equations. This allows us to extract exact results. We find that the ground-state metric is nonsingular as the size of the manifold shrinks to zero, suggesting that \(2\)D quantum field theory makes sense even beyond zero radius. Thus it seems that manifolds with zero size are nonsingular as target spaces for string theory (even when they are not conformal).

MSC:
81T60 Supersymmetric field theories in quantum mechanics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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References:
[1] F. Wilczek, Phys. Rev. Lett. 52 pp 2111– (1984) · doi:10.1103/PhysRevLett.52.2111
[2] E. Witten, Commun. Math. Phys. 118 pp 411– (1988) · Zbl 0674.58047 · doi:10.1007/BF01466725
[3] E. Witten, Nucl. Phys. B340 pp 281– (1990) · doi:10.1016/0550-3213(90)90449-N
[4] A. D’Adda, Nucl. Phys. B222 pp 45– (1983) · doi:10.1016/0550-3213(83)90608-9
[5] L. Alvarez-Gaumé, Commun. Math. Phys. 102 pp 311– (1985) · Zbl 0597.53070 · doi:10.1007/BF01229382
[6] B. M. McCoy, J. Math. Phys. 18 pp 1058– (1977) · Zbl 0353.33008 · doi:10.1063/1.523367
[7] A. R. Its, in: Lecture Notes in Mathematics (1986)
[8] A. V. Kitaev, in: Zap. Nauch Semin. LOMI
[9] T. T. Wu, Phys. Rev. B 13 pp 316– (1976) · doi:10.1103/PhysRevB.13.316
[10] E. Abdalla, Nucl. Phys. B256 pp 145– (1985) · doi:10.1016/0550-3213(85)90389-X
[11] E. Abdalla, Phys. Rev. D 29 pp 1851– (1984) · doi:10.1103/PhysRevD.29.1851
[12] V. Kurak, Phys. Rev. D 36 pp 627– (1987) · doi:10.1103/PhysRevD.36.627
[13] P. Candelas, Phys. Rev. Lett. 62 pp 1956– (1989) · doi:10.1103/PhysRevLett.62.1956
[14] P. Candelax, Nucl. Phys. B359 pp 21– (1991) · Zbl 1098.32506 · doi:10.1016/0550-3213(91)90292-6
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