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Purely coclosed G\(_{\mathbf {2}}\)-structures on 2-step nilpotent Lie groups

Abstract

We consider left-invariant (purely) coclosed G\(_2\)-structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G\(_2\)-structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G\(_2\)-structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G\(_2\)-structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.

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References

  1. Bagaglini, L., Fernández, M., Fino, A.: Coclosed G\(_2\)-structures inducing nilsolitons. Forum Math. 30(1), 109–128 (2018)

    MathSciNet  Article  Google Scholar 

  2. Bryant, R.L.: Some remarks on G\(_2\)-structures. In: Proceedings of Gökova Geometry-Topology Conference 2005, pp. 75–109. Gökova Geometry/Topology Conference (GGT), Gökova (2006)

  3. Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G\(_2\) structures. In: Differential geometry. Valencia, 2001, pp. 115–133. World Sci. Publ, River Edge, NJ (2002)

  4. Clarke, A., Garcia-Fernandez, M., Tipler, C.: \(T\)-dual solutions and infinitesimal moduli of the G\(_2\)-Strominger system. arXiv:2005.09977

  5. Crowley, D., Nordström, J.: New invariants of G\(_2\)-structures. Geom. Topol. 19(5), 2949–2992 (2015)

    MathSciNet  Article  Google Scholar 

  6. Di Scala, A.: Invariant metrics on the Iwasawa manifold. Q. J. Math. 64(2), 555–569 (2013)

    MathSciNet  Article  Google Scholar 

  7. Eberlein, P.: Geometry of 2-step nilpotent groups with a left invariant metric. Ann. Sci. École Norm. Sup. 27(5), 611–660 (1994)

    MathSciNet  Article  Google Scholar 

  8. Fernández, M., Gray, A.: Riemannian manifolds with structure group G\(_{2}\). Ann. Mat. Pura Appl. 132, 19–45 (1982)

    MathSciNet  Article  Google Scholar 

  9. Fernández, M., Ivanov, S., Ugarte, L., Vassilev, D.: Quaternionic Heisenberg group and heterotic string solutions with non-constant dilaton in dimensions 7 and 5. Commun. Math. Phys. 339(1), 199–219 (2015)

    MathSciNet  Article  Google Scholar 

  10. Fernández, M., Ivanov, S., Ugarte, L., Villacampa, R.: Compact supersymmetric solutions of the heterotic equations of motion in dimensions 7 and 8. Adv. Theor. Math. Phys. 15(2), 245–284 (2011)

    MathSciNet  Article  Google Scholar 

  11. Fernández-Culma, E.A.: Classification of Nilsoliton metrics in dimension seven. J. Geom. Phys. 86, 164–179 (2014)

    MathSciNet  Article  Google Scholar 

  12. Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6(2), 303–335 (2002)

    MathSciNet  Article  Google Scholar 

  13. Friedrich, T., Ivanov, S.: Killing spinor equations in dimension 7 and geometry of integrable G\(_2\)-manifolds. J. Geom. Phys. 48(1), 1–11 (2003)

    MathSciNet  Article  Google Scholar 

  14. Gong, M.-P.: Classification of Nilpotent Lie Algebras of Dimension 7 (Over Algebraically Closed Fields and \(\mathbb{R}\)). PhD thesis, University of Waterloo (Canada) (1998)

  15. Gray, A.: Vector cross products on manifolds. Trans. Am. Math. Soc. 141, 465–504 (1969)

    MathSciNet  Article  Google Scholar 

  16. Harvey, R., Lawson Jr., H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    MathSciNet  Article  Google Scholar 

  17. Ivanov, S.: Heterotic supersymmetry, anomaly cancellation and equations of motion. Phys. Lett. B 685(2–3), 190–196 (2010)

    MathSciNet  Article  Google Scholar 

  18. Lauret, J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319(4), 715–733 (2001)

    MathSciNet  Article  Google Scholar 

  19. Lauret, J.: Finding Einstein solvmanifolds by a variational method. Math. Z. 241(1), 83–99 (2002)

    MathSciNet  Article  Google Scholar 

  20. Malčev, A.I.: On a class of homogeneous spaces. Am. Math. Soc. Transl. 39, 1951 (1951)

    MathSciNet  Google Scholar 

  21. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)

    MathSciNet  Article  Google Scholar 

  22. Reggiani, S., Vittone, F.: The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups. arXiv:2011.02854

  23. Strominger, A.: Superstrings with Torsion. Nucl. Phys. B 274–253 (1986)

  24. Will, C.: Rank-one Einstein solvmanifolds of dimension 7. Differ. Geom. Appl. 19(3), 307–318 (2003)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

A.R. was supported by GNSAGA of INdAM and by the project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics”. Part of this work was done during a visit of A.R. to the Laboratoire de Mathématiques d’Orsay of the Université Paris-Saclay. He is grateful to the LMO for the hospitality. This work was prepared while V.d.B was working at the Laboratoire de Mathématiques d’Orsay.

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Appendix A.: The classification of 7-dimensional 2-step nilpotent Lie algebras

Appendix A.: The classification of 7-dimensional 2-step nilpotent Lie algebras

The isomorphism classes of 7-dimensional nilpotent Lie algebras were determined in [14]. Here, we recall the classification of those that are 2-step nilpotent.

The notation we use is consistent with [14]: \(\mathfrak {n}_{n,t}\) or \(\mathfrak {n}_{n,t,\bullet }\) means that the Lie algebra has dimension n and derived algebra of dimension t, while different capital letters in the third argument are used to distinguish non-isomorphic Lie algebras whose derived algebras have the same dimension. We also denote by \(\mathfrak h _n\) the Heisenberg Lie algebra of dimension n and by \(\mathfrak {h}_3^{\mathbb {C}}\) the real Lie algebra underlying the complex Heisenberg Lie algebra.

For each Lie algebra \(\mathfrak {n}\), the structure equations are written with respect to a basis \(\{f^1,\ldots ,f^7\}\) of the dual Lie algebra \(\mathfrak {n}^*\).

\(\bullet \):

7-dimensional 2-step nilpotent Lie algebras \(\mathfrak n \) with \(\dim (\mathfrak n ')=1\):

$$\begin{aligned} \mathfrak {h}_3\oplus \mathbb {R}^4= & {} \left( 0,0,0,0,0,0,f^{12}\right) ,\\ \mathfrak {h}_5\oplus \mathbb {R}^2= & {} \left( 0,0,0,0,0,0,f^{12}+f^{34}\right) ,\\ \mathfrak h _7= & {} \left( 0,0,0,0,0,0,f^{12}+f^{34}+f^{56}\right) . \end{aligned}$$

The Heisenberg Lie algebra \(\mathfrak h _7\) is the only indecomposable one in the above list.

\(\bullet \):

7-dimensional 2-step nilpotent Lie algebras \(\mathfrak n \) with \(\dim (\mathfrak n ')=2\):

$$\begin{aligned} \mathfrak {n}_{5,2}\oplus \mathbb {R}^2= & {} \left( 0,0,0,0,f^{12},f^{13},0\right) ,\\ \mathfrak {h}_3\oplus \mathfrak {h}_3\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{12},f^{34},0\right) ,\\ \mathfrak {h}_3^{\mathbb {C}}\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{13}-f^{24},f^{14}+f^{23},0\right) ,\\ \mathfrak {n}_{6,2}\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{12},f^{14}+f^{23},0\right) ,\\ \mathfrak {n}_{7,2,A}= & {} \left( 0,0,0,0,0,f^{12},f^{14}+f^{35}\right) ,\\ \mathfrak {n}_{7,2,B}= & {} \left( 0,0,0,0,0,f^{12}+f^{34},f^{15}+f^{23}\right) . \end{aligned}$$

The only indecomposable Lie algebras in the above list are \(\mathfrak {n}_{7,2,A}\) and \(\mathfrak {n}_{7,2,B}\).

\(\bullet \):

7-dimensional 2-step nilpotent Lie algebras \(\mathfrak n \) with \(\dim (\mathfrak n ')=3\):

$$\begin{aligned} \mathfrak {n}_{6,3}\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{12},f^{13},f^{23}\right) ,\\ \mathfrak {n}_{7,3,A}= & {} \left( 0,0,0,0,f^{12},f^{23},f^{24}\right) ,\\ \mathfrak {n}_{7,3,B}= & {} \left( 0,0,0,0,f^{12},f^{23},f^{34}\right) ,\\ \mathfrak {n}_{7,3,B_1}= & {} \left( 0,0,0,0,f^{12}-f^{34},f^{13}+f^{24},f^{14}\right) \\ \mathfrak {n}_{7,3,C}= & {} \left( 0,0,0,0,f^{12}+f^{34},f^{23},f^{24}\right) ,\\ \mathfrak {n}_{7,3,D}= & {} \left( 0,0,0,0,f^{12}+f^{34},f^{13},f^{24}\right) ,\\ \mathfrak {n}_{7,3,D_1}= & {} \left( 0,0,0,0,f^{12}-f^{34},f^{13}+f^{24},f^{14}-f^{23}\right) . \end{aligned}$$

The only decomposable Lie algebra in the above list is \(\mathfrak {n}_{6,3}\oplus \mathbb {R}\).

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del Barco, V., Moroianu, A. & Raffero, A. Purely coclosed G\(_{\mathbf {2}}\)-structures on 2-step nilpotent Lie groups. Rev Mat Complut 35, 323–359 (2022). https://doi.org/10.1007/s13163-021-00392-0

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Keywords

  • Purely coclosed G\(_2\)-structure
  • 2-Step nilpotent Lie algebra
  • Metric Lie algebra
  • G\(_2\)-Strominger system

Mathematics Subject Classification

  • 53C15
  • 22E25
  • 53C30