Abstract
The ring (Frobenius algebra) of local observables for topological σ-models on ℙ1 with values in the grassmannianG(s, n) is known to be “the same as” the quotient of the homology ring of the target space by the (inhomogeneous) ideal generated by the so-called quantum correction. While the need for a quantum correction comes from algebraic motivations in field theory, the aim of this paper is to understand its geometric meaning. The simple examples of ℙ1 → ℙn models tell us that the quantum correction comes by restriction on the boundary of the moduli spaces which allows to compute intersections on moduli spaces of lower degrees. We will check this point of view for the case of ℙ1 →G(s,n) models, yielding a proof of the algebraic result from physics in terms of the geometry of the σ-model itself.
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References
[ACGH] Arbarello, E., Cornalba, M., Griffiths, P. Harris, J.: The geometry of algebraic curves. Berlin, Heidelberg, New York: Springer
[D] Dubrovin, B.A.: Commun. Math. Phys.145, 195 (1992)
[FR] Franco, D., Reina, C.: In: XII DGMTP, Keller, J. et al. (eds.), Univ. Nacional Autónoma de México (1994)
[G] Gepner, D.: Commun. Math. Phys.141, 195 (1991)
[GH] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: J. Wiley
[Gr] Grothendieck, A.: Sem Bourbaki221 (1960–61)
[H] Hartshorne, R.: Algebraic Geometry. GTM Berlin, Heidelberg, New York: Springer
[I] Intriligator, K.: HUPT-91/A041
[S] Stromme, S.: In: Space curves, Ghione, F. et al. (eds.), Berlin, Heidelberg, New York: Springer LNM, 1987
[V] Vafa, C.: Mod. Phys. Lett.A6, 337 (1991)
[W] Witten, E.: Commun. Math. Phys.117, 353 (1988), ibid. Witten, E.: Commun. Math. Phys.118, 411 (1988); Nucl. Phys.B340, 281 (1990)
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Communicated by G. Felder
Work partially supported by National Project 40% “Probabilistic and geometrical methods in Mathematical Physics” and by CNR-Gruppo Nazionale di Fisica Matematica.
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Franco, D., Reina, C. The geometry of the quantum correction for topological σ-models. Commun.Math. Phys. 174, 137–148 (1995). https://doi.org/10.1007/BF02099467
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DOI: https://doi.org/10.1007/BF02099467