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Quillen bundle and geometric prequantization of non-abelian vortices on a Riemann surface

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Abstract.

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from L 2 metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen’s determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrized by Ψ0, a section of a certain bundle. The equivalence of these prequantum bundles are discussed.

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References

  1. Andersen J E, Deformation quantization and geometric quantization of abelian moduli spaces, Commun. Math. Phys. 255 (2005) 727–745

    Article  MATH  Google Scholar 

  2. Andersen J E, Toeplitz operators and Hitchin’s connection, In: The many facets of geometry: A tribute to Nigel Hitchin (eds) J P Bourguignon, O Garcia-Prada and S Salamon (Oxford University Press) (2008)

  3. Baptista J M, Non-abelian vortices on a compact Riemann surface, Commun. Math. Phys. 291 (2009) 799–812

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertram A, Daskalopoulos G and Wentworth R, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Am. Math. Soc. 9(2) (1996) 529–571

    Article  MathSciNet  MATH  Google Scholar 

  5. Bismut J M and Freed D S, The analysis of elliptic families I. Metrics and connections on determinant bundles, Commun. Math. Phys. 106 (1986) 159–176

    Article  MathSciNet  MATH  Google Scholar 

  6. Biswas I and Raghavendra N, The determinant bundle on the moduli space of stable triples over a curve, Proc. Indian Acad. Sci. (Math. Sci.) 112(3) (2002) 367–382

    Article  MathSciNet  MATH  Google Scholar 

  7. Biswas I and Schumacher G, Coupled vortex equations and moduli: Deformation theoretic approach and Kähler geometry, Math. Ann. 343(4) (2009) 825–851

    Article  MathSciNet  MATH  Google Scholar 

  8. Bradlow S, Special metrics and stability for holomorphic bundles with global sections, J. Differ. Geom. 33(1) (1991) 169–213

    MathSciNet  MATH  Google Scholar 

  9. Bradlow S, Daskalopoulos G, Garcia-Prada O and Wentworth R, Stable augmented bundles over Riemann surfaces, Vector Bundles in Algebraic Geometry, LMS Lecture Notes Series, 208 (Cambridge University Press) (1995)

  10. Dey R, Geometric prequantization of the moduli space of the vortex equations on a Riemann surface, J. Math. Phys. 47(10) (2006) 103501–103508, math-phy/0605025

    Article  MathSciNet  Google Scholar 

  11. Dey R, Erratum: Geometric prequantization of the moduli space of the vortex equations on a Riemann surface, J. Math. Phys. 50 (2009) 119901

    Article  MathSciNet  Google Scholar 

  12. Dey R, Geometric prequantization of various moduli spaces (in preparation)

  13. Quillen D, Determinants of Cauchy-Riemann operators over a Riemann surface, Functional Analysis and its Application 19 (1985) 31–34

    Article  MathSciNet  MATH  Google Scholar 

  14. Woodhouse N M J, Geometric quantization (New York: The Clarendon Press, Oxford University Press) (1992)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Harish Chandra Research Institute, Jhusi, Allahabad, 211 019, India

    RUKMINI DEY

  2. S.N. Bose National Centre for Basic Sciences, Sector-III, Block-JD, Salt Lake, Kolkata, 700 098, India

    SAMIR K PAUL

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  1. RUKMINI DEY
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  2. SAMIR K PAUL
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Correspondence to RUKMINI DEY.

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DEY, R., PAUL, S.K. Quillen bundle and geometric prequantization of non-abelian vortices on a Riemann surface. Proc Math Sci 121, 27–35 (2011). https://doi.org/10.1007/s12044-011-0011-1

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  • DOI: https://doi.org/10.1007/s12044-011-0011-1

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