The Anomaly Flow and the Fu-Yau Equation

citation_journal_title=Journal of Geometry and Physics; citation_title=Heterotic non-Kähler geometries via polystable bundles on Calabi-Yau threefolds; citation_author=B Andreas, M Garcia-Fernandez; citation_volume=62; citation_issue=2; citation_publication_date=2012; citation_pages=183-188; citation_doi=10.1016/j.geomphys.2011.10.013; citation_id=CR1 citation_journal_title=Communications in Mathematical Physics; citation_title=Solutions of the Strominger system via stable bundles on Calabi-Yau threefolds; citation_author=B Andreas, M Garcia-Fernandez; citation_volume=315; citation_publication_date=2012; citation_pages=153-168; citation_doi=10.1007/s00220-012-1509-9; citation_id=CR2 citation_journal_title=Nuclear Physics B; citation_title=Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory; citation_author=K Becker, M Becker, JX Fu, LS Tseng, ST Yau; citation_volume=751; citation_issue=1; citation_publication_date=2006; citation_pages=108-128; citation_doi=10.1016/j.nuclphysb.2006.05.034; citation_id=CR3 citation_journal_title=Annals of Mathematics; citation_title=A class of compact complex manifolds which are not algebraic; citation_author=E Calabi, B Eckmann; citation_volume=58; citation_publication_date=1953; citation_pages=494-500; citation_doi=10.2307/1969750; citation_id=CR4 Carlevaro, L., Israel, D.: Heterotic resolved conifolds with torsion, from supergravity to CFT, Journal of High Energy Physics, (1), 1-57 (2010) citation_journal_title=Nucl. Phys. B; citation_title=Vacuum configurations for superstrings; citation_author=P Candelas, G Horowitz, A Strominger, E Witten; citation_volume=258; citation_publication_date=1985; citation_pages=46-74; citation_doi=10.1016/0550-3213(85)90602-9; citation_id=CR6 citation_journal_title=Invent. Math.; citation_title=Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds; citation_author=HD Cao; citation_volume=81; citation_issue=2; citation_publication_date=1985; citation_pages=359-372; citation_doi=10.1007/BF01389058; citation_id=CR7 citation_journal_title=Comm. Pure. Appl. Math.; citation_title=A variational theory of the Hessian equation; citation_author=KS Chou, XJ Wang; citation_volume=54; citation_publication_date=2001; citation_pages=1029-1064; citation_doi=10.1002/cpa.1016; citation_id=CR8 Chu, J.: The parabolic Monge-Ampère equation on compact almost Hermitian manifolds, preprint, arXiv:1607.02608 citation_journal_title=Proc. London Math. Soc. (3); citation_title=Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles; citation_author=SK Donaldson; citation_volume=50; citation_issue=1; citation_publication_date=1985; citation_pages=1-26; citation_doi=10.1112/plms/s3-50.1.1; citation_id=CR10 Dasgupta, K., Rajesh, G., Sethi, S.S.: M-theory, orientifolds and G-flux, Journal of High Energy Physics, no. 08 (1999) citation_journal_title=Advances in Mathematics; citation_title=A construction of non-Kähler Calabi-Yau manifolds and new solutions to the Strominger system; citation_author=T Fei; citation_volume=302; citation_publication_date=2016; citation_pages=529-550; citation_doi=10.1016/j.aim.2016.07.023; citation_id=CR12 citation_journal_title=Comm. Anal. Geom.; citation_title=Some Torsional Local Models of Heterotic Strings; citation_author=T Fei; citation_volume=25; citation_issue=5; citation_publication_date=2017; citation_pages=941-968; citation_doi=10.4310/CAG.2017.v25.n5.a3; citation_id=CR13 citation_journal_title=Comm. Math. Phys.; citation_title=Invariant solutions to the Strominger system on complex Lie groups and their quotients; citation_author=T Fei, ST Yau; citation_volume=338; citation_issue=3; citation_publication_date=2015; citation_pages=1-13; citation_doi=10.1007/s00220-015-2374-0; citation_id=CR14 citation_journal_title=Journal of High Energy Physics; citation_title=Non-Kähler heterotic string solutions with non-zero fluxes and non-constant dilaton; citation_author=M Fernandez, S Ivanov, L Ugarte, D Vassilev; citation_volume=6; citation_publication_date=2014; citation_pages=1-23; citation_id=CR15 citation_journal_title=Comm. Math. Phys.; citation_title=Non-Kähler heterotic string compactifications with non-zero fluxes and constant dilaton; citation_author=M Fernandez, S Ivanov, L Ugarte, R Villacampa; citation_volume=288; citation_publication_date=2009; citation_pages=677-697; citation_doi=10.1007/s00220-008-0714-z; citation_id=CR16 citation_journal_title=Communications in Mathematical Physics; citation_title=Local heterotic torsional models; citation_author=JX Fu, LS Tseng, ST Yau; citation_volume=289; citation_publication_date=2009; citation_pages=1151-1169; citation_doi=10.1007/s00220-009-0789-1; citation_id=CR17 citation_journal_title=J. Differential Geom; citation_title=The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation; citation_author=JX Fu, ST Yau; citation_volume=78; citation_issue=3; citation_publication_date=2008; citation_pages=369-428; citation_doi=10.4310/jdg/1207834550; citation_id=CR18 citation_journal_title=Comm. Anal. Geom.; citation_title=A Monge-Ampère type equation motivated by string theory; citation_author=JX Fu, ST Yau; citation_volume=15; citation_issue=1; citation_publication_date=2007; citation_pages=29-76; citation_doi=10.4310/CAG.2007.v15.n1.a2; citation_id=CR19 citation_journal_title=Comm. Anal. Geom.; citation_title=Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds; citation_author=M Gill; citation_volume=19; citation_issue=2; citation_publication_date=2011; citation_pages=277-303; citation_doi=10.4310/CAG.2011.v19.n2.a2; citation_id=CR20 citation_journal_title=Comm. Math. Phys; citation_title=Geometric model for complex non-Kähler manifolds with SU(3) structure; citation_author=E Goldstein, S Prokushkin; citation_volume=251; citation_publication_date=2004; citation_pages=65-78; citation_doi=10.1007/s00220-004-1167-7; citation_id=CR21 citation_journal_title=Advances in Mathematics; citation_title=Geometry of compact complex homogeneous spaces with vanishing first Chern class; citation_author=G Grantcharov; citation_volume=226; citation_publication_date=2011; citation_pages=3136-3159; citation_doi=10.1016/j.aim.2010.10.005; citation_id=CR22 Green, M., Schwarz, J.: Anomaly cancellations in supersymmetric $$D = 10$$ gauge theory and superstring theory, Physics Letters B. 149, 117-122 citation_journal_title=Anal. PDE.; citation_title=On estimates for fully nonlinear parabolic equations on Riemannian manifolds; citation_author=B Guan, S Shi, Z Sui; citation_volume=8; citation_issue=5; citation_publication_date=2015; citation_pages=1145-1164; citation_doi=10.2140/apde.2015.8.1145; citation_id=CR24 citation_journal_title=Phys. Lett; citation_title=Compactifications of the heterotic superstring; citation_author=C Hull; citation_volume=178 B; citation_publication_date=1986; citation_pages=357-364; citation_doi=10.1016/0370-2693(86)91393-6; citation_id=CR25 citation_title=Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics; citation_publication_date=1996; citation_id=CR26; citation_author=NV Krylov; citation_publisher=American Mathematical Society citation_journal_title=Izv. Akad. Nauk; citation_title=Certain properties of solutions of parabolic equations with measurable coefficients; citation_author=NV Krylov, MV Safonov; citation_volume=44; citation_publication_date=1980; citation_pages=161-175; citation_id=CR27 citation_title=Second Order Parabolic Differential Equations; citation_publication_date=1996; citation_id=CR28; citation_author=GM Lieberman; citation_publisher=World Scientific citation_journal_title=J. Differential Geom.; citation_title=The existence of supersymmetric string theory with torsion; citation_author=J Li, ST Yau; citation_volume=70; citation_issue=1; citation_publication_date=2005; citation_pages=143-181; citation_doi=10.4310/jdg/1143572017; citation_id=CR29 citation_journal_title=Nuclear Physics B; citation_title=Invariant solutions to the Strominger system and the heterotic equations of motion on solvmanifolds; citation_author=A Otal, L Ugarte, R Villacampa; citation_volume=920; citation_publication_date=2017; citation_pages=442-474; citation_doi=10.1016/j.nuclphysb.2017.04.021; citation_id=CR30 citation_journal_title=Math Z.; citation_title=Geometric flows and Strominger systems; citation_author=DH Phong, S Picard, XW Zhang; citation_volume=288; citation_publication_date=2018; citation_pages=101-113; citation_doi=10.1007/s00209-017-1879-y; citation_id=CR31 Phong, D.H., Picard, S., Zhang, X.W.: Anomaly Flows, to appear in Comm. Anal. Geom. ( arXiv:1610.02739 ) Phong, D.H., Picard, S., Zhang, X.W.: On estimates for the Fu-Yau generalization of a Strominger system, to appear J. für die reine und angewandte mathematic (Crelles Journal), also arXiv:1507.08193 citation_journal_title=Invent. Math.; citation_title=The Fu-Yau equation with negative slope parameter; citation_author=DH Phong, S Picard, XW Zhang; citation_volume=209; citation_issue=2; citation_publication_date=2017; citation_pages=541-576; citation_doi=10.1007/s00222-016-0715-z; citation_id=CR34 citation_journal_title=Analysis and PDE; citation_title=A second order estimate for general complex Hessian equations; citation_author=DH Phong, S Picard, XW Zhang; citation_volume=9; citation_issue=7; citation_publication_date=2016; citation_pages=1693-1709; citation_doi=10.2140/apde.2016.9.1693; citation_id=CR35 citation_journal_title=Nuclear Phys. B; citation_title=Superstrings with torsion; citation_author=A Strominger; citation_volume=274; citation_issue=2; citation_publication_date=1986; citation_pages=253-284; citation_doi=10.1016/0550-3213(86)90286-5; citation_id=CR36 citation_journal_title=New York J. Math.; citation_title=Local Calabi and curvature estimates for the Chern-Ricci flow; citation_author=M Sherman, B Weinkove; citation_volume=19; citation_publication_date=2013; citation_pages=565-582; citation_id=CR37 citation_journal_title=J. Eur. Math. Soc. (JEMS); citation_title=Hermitian curvature flow; citation_author=J Streets, G Tian; citation_volume=13; citation_issue=3; citation_publication_date=2011; citation_pages=601-634; citation_doi=10.4171/JEMS/262; citation_id=CR38 citation_journal_title=Calc. Var. Partial Differential Equations; citation_title=Parabolic complex Monge-Ampère type equations on closed Hermitian manifolds; citation_author=W Sun; citation_volume=54; citation_issue=4; citation_publication_date=2015; citation_pages=3715-3733; citation_doi=10.1007/s00526-015-0919-x; citation_id=CR39 citation_journal_title=J. Differential Geom.; citation_title=On the evolution of a Hermitian metric by its Chern-Ricci form; citation_author=V Tosatti, B Weinkove; citation_volume=99; citation_publication_date=2015; citation_pages=125-163; citation_doi=10.4310/jdg/1418345539; citation_id=CR40 citation_journal_title=Compos. Math.; citation_title=The Chern-Ricci flow on complex surfaces; citation_author=V Tosatti, B Weinkove; citation_volume=149; citation_issue=12; citation_publication_date=2013; citation_pages=2101-2138; citation_doi=10.1112/S0010437X13007471; citation_id=CR41 citation_journal_title=Asian Jour. Math.; citation_title=Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry; citation_author=L Ugarte, R Villacampa; citation_volume=18; citation_issue=2; citation_publication_date=2014; citation_pages=229-246; citation_doi=10.4310/AJM.2014.v18.n2.a3; citation_id=CR42 Uhlenbeck, K., Yau, S.T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257-S293. Frontiers of the mathematical sciences: 1985 (New York, 1985) citation_journal_title=Comm. Pure Appl. Math.; citation_title=On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I; citation_author=ST Yau; citation_volume=31; citation_publication_date=1978; citation_pages=339-411; citation_doi=10.1002/cpa.3160310304; citation_id=CR44