Abstract
By an odd structure we mean an algebraic structure in the category of graded vector spaces whose structure operations have odd degrees. Particularly important are odd modular operads which appear as Feynman transforms of modular operads and, as such, describe some structures of string field theory. We will explain how odd structures are affected by the choice of the monoidal structure of the underlying category. We will then present two ‘natural’ and ‘canonical’ constructions of an odd modular endomorphism operad leading to different results, only one being correct. This contradicts the generally accepted belief that the systematic use of the Koszul sign rule leads to correct signs.
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Doubek, M., Jurčo, B., Muenster, K.: Modular operads and the quantum openclosed homotopy algebra. J. High Energ. Phys. 2015, 158 (2015)
Getzler E., Kapranov M.M.: Modular operads. Compos. Math. 110(1), 65–126 (1998)
Kaufmann, R.M., Ward, B.C., Zuniga, J.J.: The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators and the master equations. J. Math. Phys. 56, 103504 (2015)(40 pages)
Markl M.: Loop homotopy algebras in closed string field theory. Commun. Math. Phys. 221(2), 367–384 (2001)
Markl M., Shnider S., Stasheff J.D.: Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence (2002)
Muenster C., Sachs I.: On homotopy algebras and quantum string field theory. Miscolc. Math. Notes 14(2), 433–443 (2013)
Münster K., Sachs I.: Quantum open-closed homotopy algebra and string field theory. Commun. Math. Phys. 321(3), 769–801 (2013)
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The author was supported by the Eduard Čech Institute P201/12/G028 and RVO: 67985840.
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Markl, M. Odd Structures Are Odd. Adv. Appl. Clifford Algebras 27, 1567–1580 (2017). https://doi.org/10.1007/s00006-016-0720-8
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DOI: https://doi.org/10.1007/s00006-016-0720-8