September  2021, 13(3): 285-311. doi: 10.3934/jgm.2021020

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 Published  September 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, 2021, 13 (3) : 285-311. doi: 10.3934/jgm.2021020
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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[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.

[31]

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

[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.

[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.

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

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

[36]

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

[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

[38]

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

[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.

[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.

[41]

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

[42]

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

[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.

[44]

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

[45]

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

[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.

[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.

[48]

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

[49]

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

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[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.

[31]

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

[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.

[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.

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

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

[36]

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

[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

[38]

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

[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.

[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.

[41]

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

[42]

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

[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.

[44]

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

[45]

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

[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.

[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.

[48]

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

[49]

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

[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.

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