Abstract
We express covariance of the Batalin–Vilkovisky formalism in classical mechanics by means of the Maurer–Cartan equation in a curved Lie superalgebra, defined using the formal variational calculus and Sullivan’s Thom–Whitney construction. We use this framework to construct a Batalin–Vilkovisky canonical transformation identifying the Batalin–Vilkovisky formulation of the spinning particle with an AKSZ field theory.
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Alexandrov, M., Schwarz, A., Zaboronsky, O., Kontsevich, M.: The geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A 12(7), 1405–1429 (1997)
Axelrod, S., Singer, I.M.: Chern–Simons perturbation theory. In: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, vol. 1, 2, New York, 1991. World Sci. Publ., River Edge, NJ, pp. 3–45 (1992)
Bousfield, A.K., Gugenheim, V.K.A.M.: On PL de Rham theory and rational homotopy type. Mem. Amer. Math. Soc. 8(179), 1 (1976)
Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212(3), 591–611 (2000)
Cattaneo, A.S., Felder, G.: On the AKSZ formulation of the Poisson sigma model. Lett. Math. Phys. 56(2), 163–179 (2001). (EuroConférence Moshé Flato 2000, Part II (Dijon))
Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Classical BV theories on manifolds with boundary. Commun. Math. Phys. 332(2), 535–603 (2014)
Cattaneo, A.S., Schiavina, M.: On time. Lett. Math. Phys. 107(2), 375–408 (2017)
Cattaneo, Alberto S., Schiavina, M., Selliah, I.: BV-equivalence between triadic gravity and BF theory in three dimensions. arXiv:1707.07764
Getzler, E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111(3), 535–560 (2002)
Getzler, E.: The Batalin–Vilkovisky cohomology of the spinning particle. J. High Energy Phys. 2016(6), 1–17 (2016)
Getzler, E.: The spinning particle with curved target. Commun. Math. Phys. 352(1), 185–199 (2017)
Getzler, E., Pohorence, S.: Covariance of the classical Brink–Schwarz superparticle (to appear)
Kruskal, M.D., Miura, R.M., Gardner, C.S., Zablusky, N.J.: Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws. J. Math. Phys. 11, 952–960 (1970)
Olver, P.J.: Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107. Springer, New York (1986)
Sullivan, D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47, 269–331 (1977)
Tao, T.: Hilbert’s Fifth Problem and Related Topics, Graduate Studies in Mathematics, vol. 153. American Mathematical Society, Providence (2014)
Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)
Acknowledgements
I am grateful to Chris Hull for introducing me to the first-order formalism of the spinning particle, and to Si Li, Pavel Mnëv and Sean Pohorence for further insights. This research is partially supported by EPSRC Programme Grant EP/K034456/1 “New Geometric Structures from String Theory”, a Fellowship of the Simons Foundation, and Collaboration Grants #243025 and #524522 of the Simons Foundation. Parts of this paper were written while the author was visiting the Yau Mathematical Sciences Center at Tsinghua University and the Department of Mathematics of Columbia University, as a guest of Si Li and Mohammed Abouzaid, respectively.
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Getzler, E. Covariance in the Batalin–Vilkovisky formalism and the Maurer–Cartan equation for curved Lie algebras. Lett Math Phys 109, 187–224 (2019). https://doi.org/10.1007/s11005-018-1106-8
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DOI: https://doi.org/10.1007/s11005-018-1106-8
Keywords
- Spinning particle
- Batalin–Vilkovisky field theory
- Variational calculus
- Supergravity
- Thom–Whitney normalization