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The twisted G$_2$ equation for strong G$_2$-structures with torsion (2306.07128v2)

Published 12 Jun 2023 in math.DG

Abstract: We discuss general properties of strong G$_2$-structures with torsion and we investigate the twisted G$_2$ equation, which represents the G$_2$-analogue of the twisted Calabi-Yau equation for SU$(n)$-structures introduced by Garcia-Fern\'andez - Rubio - Shahbazi - Tipler. In particular, we show that invariant strong G$_2$-structures with torsion do not occur on compact non-flat solvmanifolds. This implies the non-existence of non-trivial solutions to the twisted Calabi-Yau equation on compact solvmanifolds of dimensions $4$ and $6$. More generally, we prove that a compact, connected homogeneous space admitting invariant strong G$_2$-structures with torsion is diffeomorphic either to $S3 \times T4$ or to $S3 \times S3 \times S1$, up to a covering, and that in both cases solutions to the twisted G$_2$ equation exist. Finally, we discuss the behavior of the homogeneous Laplacian coflow for strong G$_2$-structures with torsion on these spaces.

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References (33)
  1. I. Agricola. The Srní lectures on non-integrable geometries with torsion. Arch. Math. (Brno) 42, 5–84, 2006.
  2. I. Agricola, T. Friedrich. A note on flat metric connections with antisymmetric torsion. Differential Geom. Appl. 28, 480–487, 2010.
  3. D. V. Alekseevsky, B. N. Kimelfeld. Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funktional. Anal. i Priložen 9, 5–11, 1975.
  4. (0,2)02(0,2)( 0 , 2 ) Mirror Symmetry on homogeneous Hopf surfaces. arXiv:2012.01851. To appear in International Mathematics Research Notices.
  5. L. Bagaglini, A. Fino. The Laplacian coflow on almost-abelian Lie groups. Ann. Mat. Pura Appl. 197, 1855–1873, 2018.
  6. L. Bedulli, L. Vezzoni. The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57, 1125–1146, 2007.
  7. M. Berger. Les variètès riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Scuola Norm. Sup. Pisa 15, 179–246, 1961.
  8. J.-M. Bismut. A local index theorem for non-Kähler manifolds,. Math. Ann. 284, 681–699, 1989.
  9. R. L. Bryant. Some remarks on G22{}_{2}start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT-structures. In Proceedings of Gökova Geometry-Topology Conference 2005, pages 75–109. Gökova Geometry/Topology Conference (GGT), Gökova, 2006.
  10. S.. G. Chiossi, S. Salamon. The intrinsic torsion of SU⁢(3)SU3\rm SU(3)roman_SU ( 3 ) and G22{}_{2}start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT structures. In Differential geometry, Valencia, 2001, pages 115–133. World Sci. Publ., River Edge, NJ, 2002.
  11. S. G. Chiossi, A. Swann. G22{}_{2}start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT-structures with torsion from half-integrable nilmanifolds. J. Geom. Phys. 54, 262–285, 2005.
  12. A. Clarke, M. Garcia-Fernandez, C. Tipler. T-Dual solutions and infinitesimal moduli of the G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-Strominger system. arXiv:2005.09977. To appear in Advances in Theoretical and Mathematical Physics.
  13. M. Fernández, A. Gray. Riemannian manifolds with structure group G2subscriptG2{\mathrm{G}}_{2}roman_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Ann. Mat. Pura Appl. 32, 19–45, 1982.
  14. T. Friedrich. G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-manifolds with parallel characteristic torsion. Differential Geom. Appl. 25, 632–648, 2007.
  15. T. Friedrich, S. Ivanov. Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6, 303–335, 2002.
  16. T. Friedrich, S. Ivanov. Killing spinor equations in dimension 7 and geometry of integrable G22{}_{2}start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT-manifolds. J. Geom. Phys. 48, 1–11, 2003.
  17. M. Garcia-Fernández. Ricci flow, Killing spinors, and T-duality in generalized geometry. Adv. Math. 350, 1059–1108, 2019.
  18. Canonical metrics on holomorphic Courant algebroids. Proc. London Math. Soc. 125, 700–758, 2022.
  19. M. Garcia-Fernández, J. Streets. Generalized Ricci Flow. AMS University Lecture Series 76, 2021.
  20. P. Gauduchon. Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B 11, 257–288, 1997.
  21. G-Structures and Wrapped NS5-Branes. Commun. Math. Phys. 247, 421–445, 2004.
  22. N. Hitchin. Stable forms and special metrics. Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math., vol. 288, AMS, Providence, RI, 2001, pp. 70–89.
  23. S. Ivanov. Heterotic supersymmetry, anomaly cancellation and equations of motion. Phys. Lett. B 685, 190–196, 2010.
  24. S. Ivanov, G. Papadopoulos. Vanishing theorems and string backgrounds. Classical Quantum Gravity 18, 1089–1110, 2001.
  25. Soliton solutions for the Laplacian co-flow of some G2subscriptG2{\mathrm{G}}_{2}roman_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-structures with symmetry. Differential Geom. Appl. 30, 318–333, 2012.
  26. A. Kennon, J. D. Lotay. Geometric Flows of G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-Structures on 3333-Sasakian 7777-Manifolds. J. Geom. Phys. 187, 104793, 2023.
  27. H. V. Lê and M. Munir. Classification of compact homogeneous spaces with invariant G2subscriptG2\rm G_{2}roman_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-structures. Adv. Geom. 12, 302–328, 2012.
  28. Flows of G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-structures on contact Calabi-Yau 7-manifolds. Ann. Glob. Anal. Geom. 62, 367–389, 2022.
  29. J. Milnor. Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329, 1976.
  30. S. Picard, C. Suan. Flows of G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-Structures associated to Calabi-Yau Manifolds. arXiv:2209.03411.
  31. L. Schäfer, F. L. Schulte-Hengesbach. Nearly pseudo-Kähler and nearly para-Kähler six-manifolds. In Handbook of pseudo-Riemannian geometry and supersymmetry, vol. 16 of IRMA Lect. Math. Theor. Phys., pp. 425–453. Eur. Math. Soc., Zürich, 2010.
  32. F. L. Schulte-Hengesbach. Half-flat structures on Lie groups, PhD Thesis (2010), Hamburg. Available at math.uni-hamburg.de/home/schulte-hengesbach/diss.pdf.
  33. J. Streets. Generalized geometry, T-duality, and renormalization group flow. J. Geom. Phys. 114, 506–522, 2017.
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