doi: 10.3934/jgm.2021020
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Symplectic $ {\mathbb Z}_2^n $-manifolds

1. 

Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg

2. 

Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

* Corresponding author: Andrew James Bruce

Received  March 2021 Revised  July 2021 Early access August 2021

Fund Project: The Second author is supported by the Polish National Science Centre grant HARMONIA under the contract number 2016/22/M/ST1/00542

Roughly speaking, $ {\mathbb Z}_2^n $-manifolds are 'manifolds' equipped with $ {\mathbb Z}_2^n $-graded commutative coordinates with the sign rule being determined by the scalar product of their $ {\mathbb Z}_2^n $-degrees. We examine the notion of a symplectic $ {\mathbb Z}_2^n $-manifold, i.e., a $ {\mathbb Z}_2^n $-manifold equipped with a symplectic two-form that may carry non-zero $ {\mathbb Z}_2^n $-degree. We show that the basic notions and results of symplectic geometry generalise to the 'higher graded' setting, including a generalisation of Darboux's theorem.

Citation: Andrew James Bruce, Janusz Grabowski. Symplectic $ {\mathbb Z}_2^n $-manifolds. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021020
References:
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N. Aizawa, K. Amakawa and S. Doi, $\mathcal{N}$-extension of double-graded supersymmetric and superconformal quantum mechanics, J. Phys. A, 53 (2020), 065205. doi: 10.1088/1751-8121/ab661c.  Google Scholar

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N. Aizawa, K. Amakawa and S. Doi, $ {\mathbb Z}_2^n$-graded extensions of supersymmetric quantum mechanics via Clifford algebras, J. Math. Phys., 61 (2020), 052105. doi: 10.1063/1.5144325.  Google Scholar

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N. Aizawa, Z. Kuznetsova, H. Tanak and F. Toppan, $ {\mathbb Z}_2 \times {\mathbb Z}_2$-graded Lie symmetries of the Lévy-Leblond equations, PTEP. Prog. Theor. Exp. Phys., (2016), 123A01. doi: 10.1093/ptep/ptw176.  Google Scholar

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N. Aizawa, Z. Kuznetsova and F. Toppan, $ {\mathbb Z}_2 \times {\mathbb Z}_2$-graded mechanics: The classical theory, Eur. Phys. J. C., 80 (2020). doi: 10.1140/epjc/s10052-020-8242-x.  Google Scholar

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N. Aizawa, Z. Kuznetsova and F. Toppan, $ {\mathbb Z}_2 \times {\mathbb Z}_2$-graded mechanics: The quantization, Nucl. Phys. B, 967 (2021), 115426. doi: 10.1016/j.nuclphysb.2021.115426.  Google Scholar

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M. Asorey and P. M. Lavrov, Fedosov and Riemannian supermanifolds, J. Math. Phys., 50 (2009), 013530. doi: 10.1063/1.3054867.  Google Scholar

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A. J. Bruce, ${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded supersymmetry: 2-d sigma models, J. Phys. A: Math. Theor., 53 (2020), 455201. doi: 10.1088/1751-8121/abb47f.  Google Scholar

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A. J. Bruce and E. Ibarguengoytia, The graded differential geometry of mixed symmetry tensors, Arch. Math. (Brno), 55 (2019), 123-137.  doi: 10.5817/AM2019-2-123.  Google Scholar

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A. J. Bruce, E. Ibarguengoytia and N. Poncin, The Schwarz–Voronov embedding of $ {\mathbb Z}_2^n$-manifolds, SIGMA Symmetry Integrability Geom. Methods Appl., 16 (2020), 47pp. doi: 10.3842/SIGMA.2020.002.  Google Scholar

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A. J. Bruce and N. Poncin, Functional analytic issues in ${\mathbb Z}_2^n$-geometry, Rev. Un. Mat. Argentina, 60 (2019), 611-636.  doi: 10.33044/revuma.v60n2a21.  Google Scholar

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A. J. Bruce and N. Poncin, Products in the category of $\mathbb Z^n_2$-manifolds, J. Nonlinear Math. Phys., 26 (2019), 420-453.  doi: 10.1080/14029251.2019.1613051.  Google Scholar

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T. Covolo, S. Kwok and N. Poncin, The Frobenius theorem for $ {\mathbb Z}_2^n$-supermanifolds, arXiv: 1608.00961, [math.DG]. Google Scholar

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T. CovoloV. Ovsienko and N. Poncin, Higher trace and Berezinian of matrices over a Clifford algebra, J. Geom. Phys., 62 (2012), 2294-2319.  doi: 10.1016/j.geomphys.2012.07.004.  Google Scholar

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A. S. Galaev, Irreducible holonomy algebras of Riemannian supermanifolds, Ann. Global Anal. Geom., 42 (2012), 1-27.  doi: 10.1007/s10455-011-9299-4.  Google Scholar

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H. M. Khudaverdian, Semidensities on odd symplectic supermanifolds, Comm. Math. Phys., 247 (2004), 353-390.  doi: 10.1007/s00220-004-1083-x.  Google Scholar

[30]

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S. L. Lyakhovich and A. A. Sharapov, Characteristic classes of gauge systems, Nuclear Phys. B, 703 (2004), 419-453.  doi: 10.1016/j.nuclphysb.2004.10.001.  Google Scholar

[33]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.  Google Scholar

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A. Mostafazadeh, Parageneralization of Peierls bracket quantization, Internat. J. Modern Phys. A, 11 (1996), 2941-2955.   Google Scholar

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A. Mostafazadeh, Parabose–parafermi supersymmetry, Internat. J. Modern Phys. A, 11 (1996), 2957-2975.  doi: 10.1142/S0217751X96001449.  Google Scholar

[37]

N. Poncin, Towards integration on colored supermanifolds, Geometry of Jets and Fields, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 110 (2016), 201–217. https://www.impan.pl/en/publishing-house/banach-center-publications/all/110  Google Scholar

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V. Rittenberg and D. Wyler, Generalized superalgebras, Nuclear Phys. B, 139 (1978), 189-202.  doi: 10.1016/0550-3213(78)90186-4.  Google Scholar

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M. Rothstein, The structure of supersymplectic supermanifolds, Differential Geometric Methods in Theoretical Physics (Rapallo, 1990), Lecture Notes in Phys., Springer, Berlin, 375 (1991), 331–343 doi: 10.1007/3-540-53763-5_70.  Google Scholar

[40]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroid, Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Amer. Math. Soc., Providence, RI, 315 (2002), 169-185, doi: 10.1090/conm/315/05479.  Google Scholar

[41]

M. Scheunert, Generalized Lie algebras, J. Math. Phys., 20 (1979), 712-720.  doi: 10.1063/1.524113.  Google Scholar

[42]

A. Schwarz, Geometry of Batalin-Vilkovisky quantization, Comm. Math. Phys., 155 (1993), 249–260,, doi: 10.1007/BF02097392.  Google Scholar

[43]

A. Schwarz, Superanalogs of symplectic and contact geometry and their applications to quantum field theory, Topics in Statistical and Theoretical Physics, Amer. Math. Soc. Transl. Ser. 2, Adv. Math. Sci., Amer. Math. Soc., Providence, RI, 177 (1996), 203-218.  doi: 10.1090/trans2/177/11.  Google Scholar

[44]

V. N. Shander, Analogues of the Frobenius and Darboux theorems for supermanifolds, C. R. Acad. Bulgare Sci., 36 (1983), 309-312.   Google Scholar

[45]

V. N. Tolstoy, Once more on parastatistics, Phys. Part. Nucl. Lett., 11 (2014), 933-937.  doi: 10.1134/S1547477114070449.  Google Scholar

[46]

V. N. Tolstoy, Super-de Sitter and alternative super-Poincaré symmetries, Lie Theory and Its Applications in PhysicsSpringer Proc. Math. Stat., Springer, Tokyo, 111 (2014), 357-367.  doi: 10.1007/978-4-431-55285-7_26.  Google Scholar

[47]

R. Trostel, Color analysis, variational self-adjointness, and color Poisson (super)algebras, J. Math. Phys., 25 (1984), 3183-3189.  doi: 10.1063/1.526088.  Google Scholar

[48]

D. V. Volkov, On the quantization of half-integer spin fields, Soviet Physics. JETP, 9 (1959), 1107-1111.   Google Scholar

[49]

T. T. Voronov, Q-manifolds and Mackenzie theory, Comm. Math. Phys., 315 (2012), 279-310.  doi: 10.1007/s00220-012-1568-y.  Google Scholar

[50]

W. Yang and S. Jing, A new kind of graded Lie algebra and parastatistical supersymmetry, Sci. China Ser. A, 44 (2001), 1167-1173.  doi: 10.1007/BF02877435.  Google Scholar

show all references

References:
[1]

N. Aizawa, K. Amakawa and S. Doi, $\mathcal{N}$-extension of double-graded supersymmetric and superconformal quantum mechanics, J. Phys. A, 53 (2020), 065205. doi: 10.1088/1751-8121/ab661c.  Google Scholar

[2]

N. Aizawa, K. Amakawa and S. Doi, $ {\mathbb Z}_2^n$-graded extensions of supersymmetric quantum mechanics via Clifford algebras, J. Math. Phys., 61 (2020), 052105. doi: 10.1063/1.5144325.  Google Scholar

[3]

N. Aizawa, Z. Kuznetsova, H. Tanak and F. Toppan, $ {\mathbb Z}_2 \times {\mathbb Z}_2$-graded Lie symmetries of the Lévy-Leblond equations, PTEP. Prog. Theor. Exp. Phys., (2016), 123A01. doi: 10.1093/ptep/ptw176.  Google Scholar

[4]

N. Aizawa, Z. Kuznetsova and F. Toppan, $ {\mathbb Z}_2 \times {\mathbb Z}_2$-graded mechanics: The classical theory, Eur. Phys. J. C., 80 (2020). doi: 10.1140/epjc/s10052-020-8242-x.  Google Scholar

[5]

N. Aizawa, Z. Kuznetsova and F. Toppan, $ {\mathbb Z}_2 \times {\mathbb Z}_2$-graded mechanics: The quantization, Nucl. Phys. B, 967 (2021), 115426. doi: 10.1016/j.nuclphysb.2021.115426.  Google Scholar

[6]

M. Asorey and P. M. Lavrov, Fedosov and Riemannian supermanifolds, J. Math. Phys., 50 (2009), 013530. doi: 10.1063/1.3054867.  Google Scholar

[7]

P. J. M. Bongaarts and H. G. J. Pijls, Almost commutative algebra and differential calculus on the quantum hyperplane, J. Math. Phys., 35 (1994), 959-970.  doi: 10.1063/1.530888.  Google Scholar

[8]

A. J. Bruce, On a $ {\mathbb Z}_2^n$-graded version of supersymmetry, Symmetry, 11 (2019), 116. doi: 10.3390/sym11010116.  Google Scholar

[9]

A. J. Bruce, ${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded supersymmetry: 2-d sigma models, J. Phys. A: Math. Theor., 53 (2020), 455201. doi: 10.1088/1751-8121/abb47f.  Google Scholar

[10]

A. J. Bruce and S. Duplij, Double-graded supersymmetric quantum mechanics, J. Math. Phys., 61 (2020), 063503. doi: 10.1063/1.5118302.  Google Scholar

[11]

A. J. BruceK. Grabowska and G. Moreno, On a geometric framework for Lagrangian supermechanics, J. Geom. Mech., 9 (2017), 411-437.  doi: 10.3934/jgm.2017016.  Google Scholar

[12]

A. J. Bruce and J. Grabowski, Riemannian Structures on $ {\mathbb Z}_2^n$-manifolds, Mathematics, 8 (2020), 1469. doi: 10.3390/math8091469.  Google Scholar

[13]

A. J. Bruce and J. Grabowski, Odd connections on supermanifolds: Existence and relation with affine connections, J. Phys. A: Math. Theor., 53 (2020), 455203. doi: 10.1088/1751-8121/abb9f0.  Google Scholar

[14]

A. J. Bruce and E. Ibarguengoytia, The graded differential geometry of mixed symmetry tensors, Arch. Math. (Brno), 55 (2019), 123-137.  doi: 10.5817/AM2019-2-123.  Google Scholar

[15]

A. J. Bruce, E. Ibarguengoytia and N. Poncin, The Schwarz–Voronov embedding of $ {\mathbb Z}_2^n$-manifolds, SIGMA Symmetry Integrability Geom. Methods Appl., 16 (2020), 47pp. doi: 10.3842/SIGMA.2020.002.  Google Scholar

[16]

A. J. Bruce and N. Poncin, Functional analytic issues in ${\mathbb Z}_2^n$-geometry, Rev. Un. Mat. Argentina, 60 (2019), 611-636.  doi: 10.33044/revuma.v60n2a21.  Google Scholar

[17]

A. J. Bruce and N. Poncin, Products in the category of $\mathbb Z^n_2$-manifolds, J. Nonlinear Math. Phys., 26 (2019), 420-453.  doi: 10.1080/14029251.2019.1613051.  Google Scholar

[18]

T. Covolo, J. Grabowski and N. Poncin, The category of $\mathbb{Z}_2^n$-supermanifolds, J. Math. Phys., 57 (2016), 073503. doi: 10.1063/1.4955416.  Google Scholar

[19]

T. CovoloJ. Grabowski and N. Poncin, Splitting theorem for $\mathbb{Z}_2^n$-supermanifolds, J. Geom. Phys., 110 (2016), 393-401.  doi: 10.1016/j.geomphys.2016.09.006.  Google Scholar

[20]

T. Covolo, S. Kwok and N. Poncin, Differential calculus on $ {\mathbb Z}_2^n$-supermanifolds, arXiv: 1608.00949, [math.DG]. Google Scholar

[21]

T. Covolo, S. Kwok and N. Poncin, The Frobenius theorem for $ {\mathbb Z}_2^n$-supermanifolds, arXiv: 1608.00961, [math.DG]. Google Scholar

[22]

T. CovoloV. Ovsienko and N. Poncin, Higher trace and Berezinian of matrices over a Clifford algebra, J. Geom. Phys., 62 (2012), 2294-2319.  doi: 10.1016/j.geomphys.2012.07.004.  Google Scholar

[23]

A. S. Galaev, Irreducible holonomy algebras of Riemannian supermanifolds, Ann. Global Anal. Geom., 42 (2012), 1-27.  doi: 10.1007/s10455-011-9299-4.  Google Scholar

[24]

S. Garnier and T. Wurzbacher, The geodesic flow on a Riemannian supermanifold, J. Geom. Phys., 62 (2012), 1489-1508.  doi: 10.1016/j.geomphys.2012.02.002.  Google Scholar

[25]

O. Goertsches, Riemannian supergeometry, Math. Z., 260 (2008), 557-593.  doi: 10.1007/s00209-007-0288-z.  Google Scholar

[26]

H. S. Green, A generalized method of field quantization, Phys. Rev. (2), 90 (1953), 270-273.  doi: 10.1103/PhysRev.90.270.  Google Scholar

[27]

O. W. Greenberg and A. M. L. Messiah, Selection rules for parafields and the absence of para particles in nature, Phys. Rev. (2), 138 (1965). doi: 10.1103/PhysRev.138.B1155.  Google Scholar

[28]

A. J. Kálnay, Parastatistics and Dirac Brackets, Int. J. Theor. Phys., 6 (1972), 415-424.  doi: 10.1007/BF00712262.  Google Scholar

[29]

H. M. Khudaverdian, Semidensities on odd symplectic supermanifolds, Comm. Math. Phys., 247 (2004), 353-390.  doi: 10.1007/s00220-004-1083-x.  Google Scholar

[30]

B. Kostant, Graded manifolds, graded Lie theory, and prequantization, In Differential Geometrical Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Math., Springer, Berlin, 570 (1977), 177–306. doi: 10.1007/BFb0087788.  Google Scholar

[31]

D. A. Leites, Introduction to the theory of supermanifolds, Russ. Math. Surv., 211 (1980), 3-57.  doi: 10.1070/RM1980v035n01ABEH001545.  Google Scholar

[32]

S. L. Lyakhovich and A. A. Sharapov, Characteristic classes of gauge systems, Nuclear Phys. B, 703 (2004), 419-453.  doi: 10.1016/j.nuclphysb.2004.10.001.  Google Scholar

[33]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.  Google Scholar

[34] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511613104.  Google Scholar
[35]

A. Mostafazadeh, Parageneralization of Peierls bracket quantization, Internat. J. Modern Phys. A, 11 (1996), 2941-2955.   Google Scholar

[36]

A. Mostafazadeh, Parabose–parafermi supersymmetry, Internat. J. Modern Phys. A, 11 (1996), 2957-2975.  doi: 10.1142/S0217751X96001449.  Google Scholar

[37]

N. Poncin, Towards integration on colored supermanifolds, Geometry of Jets and Fields, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 110 (2016), 201–217. https://www.impan.pl/en/publishing-house/banach-center-publications/all/110  Google Scholar

[38]

V. Rittenberg and D. Wyler, Generalized superalgebras, Nuclear Phys. B, 139 (1978), 189-202.  doi: 10.1016/0550-3213(78)90186-4.  Google Scholar

[39]

M. Rothstein, The structure of supersymplectic supermanifolds, Differential Geometric Methods in Theoretical Physics (Rapallo, 1990), Lecture Notes in Phys., Springer, Berlin, 375 (1991), 331–343 doi: 10.1007/3-540-53763-5_70.  Google Scholar

[40]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroid, Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Amer. Math. Soc., Providence, RI, 315 (2002), 169-185, doi: 10.1090/conm/315/05479.  Google Scholar

[41]

M. Scheunert, Generalized Lie algebras, J. Math. Phys., 20 (1979), 712-720.  doi: 10.1063/1.524113.  Google Scholar

[42]

A. Schwarz, Geometry of Batalin-Vilkovisky quantization, Comm. Math. Phys., 155 (1993), 249–260,, doi: 10.1007/BF02097392.  Google Scholar

[43]

A. Schwarz, Superanalogs of symplectic and contact geometry and their applications to quantum field theory, Topics in Statistical and Theoretical Physics, Amer. Math. Soc. Transl. Ser. 2, Adv. Math. Sci., Amer. Math. Soc., Providence, RI, 177 (1996), 203-218.  doi: 10.1090/trans2/177/11.  Google Scholar

[44]

V. N. Shander, Analogues of the Frobenius and Darboux theorems for supermanifolds, C. R. Acad. Bulgare Sci., 36 (1983), 309-312.   Google Scholar

[45]

V. N. Tolstoy, Once more on parastatistics, Phys. Part. Nucl. Lett., 11 (2014), 933-937.  doi: 10.1134/S1547477114070449.  Google Scholar

[46]

V. N. Tolstoy, Super-de Sitter and alternative super-Poincaré symmetries, Lie Theory and Its Applications in PhysicsSpringer Proc. Math. Stat., Springer, Tokyo, 111 (2014), 357-367.  doi: 10.1007/978-4-431-55285-7_26.  Google Scholar

[47]

R. Trostel, Color analysis, variational self-adjointness, and color Poisson (super)algebras, J. Math. Phys., 25 (1984), 3183-3189.  doi: 10.1063/1.526088.  Google Scholar

[48]

D. V. Volkov, On the quantization of half-integer spin fields, Soviet Physics. JETP, 9 (1959), 1107-1111.   Google Scholar

[49]

T. T. Voronov, Q-manifolds and Mackenzie theory, Comm. Math. Phys., 315 (2012), 279-310.  doi: 10.1007/s00220-012-1568-y.  Google Scholar

[50]

W. Yang and S. Jing, A new kind of graded Lie algebra and parastatistical supersymmetry, Sci. China Ser. A, 44 (2001), 1167-1173.  doi: 10.1007/BF02877435.  Google Scholar

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