2000 character limit reached
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.
- I. Agricola. The Srní lectures on non-integrable geometries with torsion. Arch. Math. (Brno) 42, 5–84, 2006.
- I. Agricola, T. Friedrich. A note on flat metric connections with antisymmetric torsion. Differential Geom. Appl. 28, 480–487, 2010.
- D. V. Alekseevsky, B. N. Kimelfeld. Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funktional. Anal. i Priložen 9, 5–11, 1975.
- (0,2)02(0,2)( 0 , 2 ) Mirror Symmetry on homogeneous Hopf surfaces. arXiv:2012.01851. To appear in International Mathematics Research Notices.
- L. Bagaglini, A. Fino. The Laplacian coflow on almost-abelian Lie groups. Ann. Mat. Pura Appl. 197, 1855–1873, 2018.
- L. Bedulli, L. Vezzoni. The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57, 1125–1146, 2007.
- M. Berger. Les variètès riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Scuola Norm. Sup. Pisa 15, 179–246, 1961.
- J.-M. Bismut. A local index theorem for non-Kähler manifolds,. Math. Ann. 284, 681–699, 1989.
- 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.
- 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.
- 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.
- 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.
- 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.
- T. Friedrich. G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-manifolds with parallel characteristic torsion. Differential Geom. Appl. 25, 632–648, 2007.
- T. Friedrich, S. Ivanov. Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6, 303–335, 2002.
- 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.
- M. Garcia-Fernández. Ricci flow, Killing spinors, and T-duality in generalized geometry. Adv. Math. 350, 1059–1108, 2019.
- Canonical metrics on holomorphic Courant algebroids. Proc. London Math. Soc. 125, 700–758, 2022.
- M. Garcia-Fernández, J. Streets. Generalized Ricci Flow. AMS University Lecture Series 76, 2021.
- P. Gauduchon. Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B 11, 257–288, 1997.
- G-Structures and Wrapped NS5-Branes. Commun. Math. Phys. 247, 421–445, 2004.
- 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.
- S. Ivanov. Heterotic supersymmetry, anomaly cancellation and equations of motion. Phys. Lett. B 685, 190–196, 2010.
- S. Ivanov, G. Papadopoulos. Vanishing theorems and string backgrounds. Classical Quantum Gravity 18, 1089–1110, 2001.
- 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.
- 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.
- 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.
- 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.
- J. Milnor. Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329, 1976.
- S. Picard, C. Suan. Flows of G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-Structures associated to Calabi-Yau Manifolds. arXiv:2209.03411.
- 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.
- 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.
- J. Streets. Generalized geometry, T-duality, and renormalization group flow. J. Geom. Phys. 114, 506–522, 2017.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
