Letters in Mathematical Physics

, Volume 107, Issue 7, pp 1195–1214 | Cite as

Odd Laplacians: geometrical meaning of potential and modular class



A second-order self-adjoint operator \(\Delta =S\partial ^2+U\) is uniquely defined by its principal symbol S and potential U if it acts on half-densities. We analyse the potential U as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an affine connection compensates the action of coordinate transformations on first derivatives in the first-order operator, a covariant derivative, \(\nabla =\partial +\Gamma \). Usually a potential U is derived from other geometrical constructions such as a volume form, an affine connection, or a Riemannian structure, etc. The story is different if \(\Delta \) is an odd operator on a supermanifold. In this case, the second-order potential becomes a primary object. For example, in the case of an odd symplectic supermanifold, the compensating field of the canonical odd Laplacian depends only on this symplectic structure and can be expressed by the formula obtained by K. Bering. We also study modular classes of odd Poisson manifolds via \(\Delta \)-operators, and consider an example of a non-trivial modular class which is related with the Nijenhuis bracket.


Operators on half-densities Odd Poisson manifold Modular class Odd symplectic manifold Canonical odd Laplacian Compensating field 

Mathematics Subject Classification

53D17 58A50 81R99 



We are happy to acknowledge A. Voronov and B. Kruglikov for encouraging us to begin the work on this article. It is a pleasure to also thank K. Mackenzie and T. Voronov for very useful discussions. One of us (H.Kh.) learnt much about the peculiarities of Poisson geometry through discussions with A. Bolsinov. Many thanks to him.


  1. 1.
    Batalin, I.A., Bering, K.: Odd scalar curvature in field-antifield formalism. J. Math. Phys. 49(1–22), 033515 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Batalin, I.A., Bering, K.: Odd scalar curvature in anti-Poisson geometry. Phys. Lett. B 663(1–2), 132–135 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and quantization. Phys. Lett. 102B, 27–31 (1981)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Berezin, F.A.: Introduction in superanalysis. In: Kirillov, A.A. (ed.) Expanded Translation from the Russian: Introduction to Analysis with Anticommuting Variables. Moscow State University Press, Moscow (1983). Translation edited by Leites, D.A. and Reidel, D., Dordrecht (1987)Google Scholar
  5. 5.
    Bering, K.: A note on semidensities in antisymplectic geometry. J. Math. Phys. 47(1–9), 123513 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bering, K.: Semidensities, second-class constraints, and conversion in anti-Poisson geometry. J. Math. Phys. 49(4), 043516, 31 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hitchin, N.J., Segal, G.B., Ward, R.S.: Integrable Systems. The Clarendon Press, New York (1999)MATHGoogle Scholar
  8. 8.
    Khudaverdian, O.M.: Geometry of superspace with even and odd brackets. J. Math. Phys. 32, 1934–1937, (1991). Preprint of the Geneva University, UGVA–DPT 1989/05–613Google Scholar
  9. 9.
    Khudaverdian, H.M.: Laplacians in odd symplectic geometry. In: Voronov, Th. (ed.) Quantization, Poisson Brackets and Beyond. Contemp. Math., Vol. 315, pp. 199–212, Amer. Math. Soc., Providence, RI (2002)Google Scholar
  10. 10.
    Khudaverdian, H.M.: Semidensities on odd symplectic supermanifold. Commun. Math. Phys. 247, 353–390 (2004). Preprint 2000, arXiv:math/0012256 ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khudaverdian, H.M., Voronov, Th: On odd Laplace operators. Lett. Math. Phys. 62, 127–142 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Khudaverdian, H.M., Voronov, Th: On odd Laplace operators II. Am. Math. Soc. Transl. (2) 212, 179 (2004)MathSciNetMATHGoogle Scholar
  13. 13.
    Khudaverdian, H.M., Voronov, Th: Geometry of differential operators of second order, the algebra of densities, and groupoids. J. Geom. Phys. 64, 31 (2013). See also preprint of Max-Planck-Institut for Math., 73, (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khudaverdian, H.M., Voronov, Th.: Geometric constructions on algebra of densities. In: Buchstaber, V.M., Dubrovin, B.A., Krichever, I.M. (eds.) Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014. AMS Translations, Ser. 2, vol. 234, pp. 221–243, Providence, RI (2014). math-arXiv:1310.0784Google Scholar
  15. 15.
    Kosmann-Schwarzbach, Y., Monterde, J.: Divergence operators and odd Poisson brackets. Ann. Inst. Fourier (Grenoble) 52(2), 419–456 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ovsienko, V., Tabachnikov, S.: Projective Differential Geometry Old and New. From Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
  17. 17.
    Schwarz, A.S.: Geometry of Batalin–Vilkovisky formalism. Commun. Math. Phys. 155, 249–260 (1993)ADSCrossRefMATHGoogle Scholar
  18. 18.
    Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom. 18(3), 523–557 (1983)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23(3–4), 379–394 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Hovhannes M. Khudaverdian
    • 1
  • Matthew T. Peddie
    • 1
  1. 1.School of MathematicsUniversity of ManchesterManchesterUK

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