Addendum: On the geometric quantization of twisted Poisson manifolds [J. Math.Phys.48, 083502 (2007)]Fani Petalidou Citation: Journal of Mathematical Physics 49, 033520 (2008); doi: 10.1063/1.2901067 View online: http://dx.doi.org/10.1063/1.2901067 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/49/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Erratum: “Geometric prequantization of the moduli space of the vortex equations on a Riemann surface” [J. Math.Phys.47, 103501 (2006)] J. Math. Phys. 50, 119901 (2009); 10.1063/1.3255495 Abelian gauge theories on compact manifolds and the Gribov ambiguity J. Math. Phys. 49, 052302 (2008); 10.1063/1.2909197 Erratum: Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral ofMotion [J. Math. Phys.48, 012902 (2007)] J. Math. Phys. 49, 019901 (2008); 10.1063/1.2831929 Comment on “Double product integrals and Enriquez quantization of Lie bialgebras I: The quasitriangularidentities” [Hudson and Pulmannová, J. Math. Phys. 45, 2090 (2004)] J. Math. Phys. 45, 2106 (2004); 10.1063/1.1695095 On the geometric quantization of Jacobi manifolds J. Math. Phys. 38, 6185 (1997); 10.1063/1.532207

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Addendum: On the geometric quantization of twistedPoisson manifolds †J. Math. Phys. 48, 083502 „2007…‡

Fani Petalidoua�

Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, Nicosia1678, Cyprus

�Received 16 February 2008; accepted 22 February 2008; published online 27 March 2008�

�DOI: 10.1063/1.2901067�

I would like to make an addendum that resulted from my further study of the quantizationproblem of a twisted Poisson manifold in Theorem 4.1 of my paper “On the geometric quantiza-tion of twisted Poisson manifolds” �J. Math. Phys. 48, 083502 �2007��. We consider a twistedPoisson manifold �M ,� ,��, i.e.,

12 ��,�� = �#��� ,

where � is a bivector field on M and � is a closed 3-form. Theorem 4.1 states the following: Atwisted Poisson manifold �M ,� ,�� is prequantizable if, and only if, there exist a vector field Z onM and a closed 2-form � on M, which represents an integral cohomology class of M, such thatthe following relation holds on M:

� + ��Z = �#��� . �28�

Taking into account that �i� the operator ��, which is defined in Sec. II B of the paper, is thederivative on ��∧TM� and verifies the relation �� ���=−�� �d and �ii� � is a closed 2-form, wecan easily check condition �28� means that � must be a cocycle of the Lichnerowicz-twistedPoisson cohomology H

L-tP* �M� of �M ,� ,��. Indeed,

��� =�28�

����#��� − ��Z� = − �#�d�� − ��2Z = − 0 − 0 = 0 . ���

However, a straightforward calculation of ��� yields the following: for any �1 ,�2 ,�3���T*M�,

�����1,�2,�3� = �#��1� · ���2,�3� − �#��2� · ���1,�3� + �#��3� · ���1,�2� − ����1,�2��,�3�

+ ����1,�3��,�2� − ����2,�3��,�1�

=�6�

�#��1� · ���2,�3� − �#��2� · ���1,�3� + �#��3� · ���1,�2� − ����1,�2�,�3�

+ ����1,�3�,�2� − ����2,�3�,�1� − �����#��1�,�#��2�, · �,�3�

+ �����#��1�,�#��3�, · �,�2� − �����#��2�,�#��3�, · �,�1�

= ��,����1,�2,�3� − 3�#�����1,�2,�3�

= 2�#�����1,�2,�3� − 3�#�����1,�2,�3� = − �#�����1,�2,�3� ,

i.e.,

a�Electronic mail: petalido@ucy.ac.cy.

JOURNAL OF MATHEMATICAL PHYSICS 49, 033520 �2008�

49, 033520-10022-2488/2008/49�3�/033520/2/$23.00 © 2008 American Institute of Physics

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��� = − �#��� . �� � �

Hence, by comparing relations ��� and ����, we conclude that �28� has a solution �Z ,�� only in thecase where �����=0, which means that �M ,� ,�� must be a Poisson manifold.

033520-2 Fani Petalidou J. Math. Phys. 49, 033520 �2008�

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