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Conformally and Projective Covariant Differential Operators

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Abstract

Motivated by the structure of conformal anomalies in two-dimensional gravity and its generalizations, the projective and conformal covariance properties of linear, bilinear and trilinear differential operators are investigated in some detail and the triviality of the covariant trilinear operators is demonstrated.

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References

  1. Knecht, M., Lazzarini, S., and Thuillier, F.: Shifting the Weyl anomaly to the chirally split diffeomorphism anomaly in two dimension, Phys Lett. B 251 (1990), 279–283.

    Google Scholar 

  2. Polyakov, A.M.: Gauge transformations and diffeomorphisms, Internat. J. Math. Phys. A 5 (1990), 833–842.

    Google Scholar 

  3. Polyakov, A. M.: Quantum Gravity in two dimensions, Modern Phys. Lett. A 2 (1987), 893–898.

    Google Scholar 

  4. Belavin, A., Polyakov, A., and Zamolodchikov, A. B.: Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333–380.

    Google Scholar 

  5. Zamolodochikov. B.: Infinite additional symmetries in two-dimensional conformal quantum field theory, Theoret. Math. Phys. 65 (1985), 1205–1213.

    Google Scholar 

  6. Ooguri, H., Schoutens, K., Sevrin, A., and van Nieuwenhuizen, P.: The induced action of W3 gravity, Comm. Math. Phys. 145 (1992), 515–539.

    Google Scholar 

  7. Bilal, A., Fock,V. V., and Kogan, I. I.: On the origin of W-algebras, Nuclear Phys. B 359 (1991), 635–672.

    Google Scholar 

  8. Grimm, R., Lazzarini, S., and Garajeu, D.: W-gauge structures and their anomalies: an algebraic approach, J. Math. Phys. 36(12) (1995), 7043–7072.

    Google Scholar 

  9. Wess, J. and Zumino, B.: Consequences of anomalous ward identities, Phys. Lett. B 37 (1971), 95–97.

    Google Scholar 

  10. Becchi, C., Rouet, A., and Stora, R.: Renormalization of gauge theories, Ann. Phys. 98 (1976), 287–321.

    Google Scholar 

  11. Bol, G.: Invarianten Linearer Differentialgleichungen, Abh. Math. Sem. Univ. Hamburger Univ. 16 (1949), 1–28.

    Google Scholar 

  12. Gustafsson, B. and Peetre, J.: Notes on projective structures on complex manifolds, Nagoya Math. J. 116 (1989), 63–88.

    Google Scholar 

  13. Gieres, F.: Conformally covariant operators on Riemann surfaces (with applications to conformal and integrable models), Internat. J. Modern Phys. A 8 (1993), 1–58.

    Google Scholar 

  14. Drinfel'd V. G. and Sokolov, V. V.: Lie algebras and equations of Korteweg–de Vries type, J. Soviet Math. 30 (1985), 1975–2036.

    Google Scholar 

  15. Scherer, W.: Covariant differential operators for densities and the KdVequation as a flow on diffeomorphism groups, Lett. Math. Phys. 17 (1989), 45–49.

    Google Scholar 

  16. Henderson, R. J. and Rajeev S. G.: Quantum gravity on a circle and the diffeomorphism invariance of the Schrödinger equation, Classical Quantum Gravity 11 (1994), 1631–1651.

    Google Scholar 

  17. Gordan P.: Invariantentheorie, Chelsea Publ. Co., New York, 1987.

    Google Scholar 

  18. Grozman, P. Y.: Classification of bilinear invariants of operators on tensor fields, Funct. Anal. Appl. 14 (1981), 127–128.

    Google Scholar 

  19. Feigin, B. L. and Fuks, D. B.: Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra, Funct Anal. Appl. 16(2) (1982), 47–63.

    Google Scholar 

  20. Ovsienko, O. D. and Ovsienko, V.Y.: Lie derivatives of order n on the line. Tensor meaning of the Gelfand–Dikii bracket, Adv. Soviet Math. 2 (1991), 221–231.

    Google Scholar 

  21. Bauer, M., Di Francesco, P., Itzykson, C. and Zuber, J.-B.: Covariant differential equations and singular vectors in Virasoro representations, Nuclear Phys. B 362 (1991), 515–562.

    Google Scholar 

  22. Garajeu, D.: Conditions de courbure zéro covariantes conformes et structures de jauge W, PhD thesis, Centre de Physique Théorique, Marseille, September 1997.

    Google Scholar 

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Garajeu, D. Conformally and Projective Covariant Differential Operators. Letters in Mathematical Physics 47, 293–306 (1999). https://doi.org/10.1023/A:1007505930234

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