Complex hyperkähler structures defined by Donaldson-Thomas invariants.(English)Zbl 1475.14108
The notion of a Joyce structure, introduced byT. Bridgeland [“Geometry from Donaldson-Thomas invariants”, Preprint,arXiv:1912.06504], is a combination of geometric structures which are expected to exist on the space of stability conditions on a CY3 triangulated category, encoded by the Donaldson-Thomas invariants. The name originates in the work ofD. Joyce [Geom. Topol. 11, 667–725 (2007;Zbl 1141.14023)]. The original definition in loc.cit.is rather complicated and given in terms of non-geometric viewpoint. The paper under this review gives a more geometric reformulation of a slightly restricted class, called a strong Joyce structure on a complex manifold \(M\) (Definition 3.3). It consists of a period structure on \(M\), introduced in §3.1, and a complex hyperkähler structure on the holomorphic tangent bundle \(X=T M\) satisfying certain compatibility conditions. Associated to it is a locally-defined holomorphic function \(W\) on \(X\) satisfying a system of differential equations (see (1) in §1 and (13), (14) in §3.2), which is nothing but the Joyce function appearing in the original definition. Moreover, the paper discusses a relation between a quantum deformation of the above-mentioned differential equations and the Moyal quantization of Plebański’s heavenly equation. It also includes a brief but valuable review of the original definition of a Joyce structure.
Reviewer: Shintaro Yanagida (Nagoya)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |
| 14J42 | Holomorphic symplectic varieties, hyper-Kähler varieties |
Citations:
Zbl 1141.14023Cite
References:
| [1] | Balser, W.; Jurkat, WB; Lutz, DA, Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl., 71, 1, 48-94 (1979) ·Zbl 0415.34008 ·doi:10.1016/0022-247X(79)90217-8 |
| [2] | Boyer, C.; Plebański, J., An infinite hierarchy of conservation laws and nonlinear superposition principles for self-dual Einstein spaces, J. Math. Phys., 26, 229 (1985) ·Zbl 0555.53044 ·doi:10.1063/1.526652 |
| [3] | Bridgeland, T., Stability conditions on triangulated categories, Ann. Math., 166, 31-345 (2007) ·Zbl 1137.18008 ·doi:10.4007/annals.2007.166.317 |
| [4] | Bridgeland, T.; Toledano Laredo, V., Stability conditions and Stokes factors, Invent. Math., 187, 1, 61-98 (2012) ·Zbl 1239.14008 ·doi:10.1007/s00222-011-0329-4 |
| [5] | Bridgeland, T., Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. Math., 216, 69-124 (2019) ·Zbl 1423.14309 ·doi:10.1007/s00222-018-0843-8 |
| [6] | Bridgeland, T.: Geometry from Donaldson-Thomas invariants, preprint arXiv:1912.06504 ·Zbl 1451.14159 |
| [7] | Bridgeland, T., Barbieri, A., Stoppa, J.: A quantized Riemann-Hilbert problem in Donaldson-Thomas theory, preprint arXiv:1905.00748 (to appear IMRN) |
| [8] | Bridgeland, T.: On the monodromy of the deformed cubic oscillator, with an appendix by D. Masoero, preprint arXiv:2006.10648 |
| [9] | Bridgeland, T.: A complexification of the Hitchin system, in preparation |
| [10] | Dunajski, M.: Null Kähler geometry and isomonodromic deformations, preprint arXiv:2010.11216 |
| [11] | Dunajski, M.; Tod, KP, Einstein-Weyl structures from hyper-Kähler Metrics with conformal Killing vectors, Diff. Geom. Appl., 14, 39-55 (2001) ·Zbl 0997.53036 ·doi:10.1016/S0926-2245(00)00037-1 |
| [12] | Fairlie, DB; Zachos, C., Infinite-dimensional algebras, sine brackets, and \(SU(\infty )\), Phys. Lett. B, 224, 1-2, 101-107 (1989) ·Zbl 0689.17019 ·doi:10.1016/0370-2693(89)91057-5 |
| [13] | Gaiotto, D.; Moore, G.; Neitzke, A., Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys., 299, 1, 163-224 (2010) ·Zbl 1225.81135 ·doi:10.1007/s00220-010-1071-2 |
| [14] | Gaiotto, D.; Moore, G.; Neitzke, A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math., 234, 239-403 (2013) ·Zbl 1358.81150 ·doi:10.1016/j.aim.2012.09.027 |
| [15] | Gindikin, SG, A construction of hyper-Kähler metrics, Funct. Anal. Appl., 20, 3, 238-240 (1986) ·Zbl 0641.53063 ·doi:10.1007/BF01078477 |
| [16] | Joyce, D.: Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. in Math. 217, 125-204 (2008) ·Zbl 1134.14008 |
| [17] | Joyce, D., Song, Y.: A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217, no. 1020, iv+199 pp (2012) ·Zbl 1259.14054 |
| [18] | Joyce, D., Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds, Geom. Topol., 11, 667-725 (2007) ·Zbl 1141.14023 ·doi:10.2140/gt.2007.11.667 |
| [19] | Kapustin, A.; Kuznetsov, A.; Orlov, D., Noncommutative instantons and twistor transform, Commun. Math. Phys., 221, 385-432 (2001) ·Zbl 0989.81127 ·doi:10.1007/PL00005576 |
| [20] | Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arxiv 0811.2435 ·Zbl 1248.14060 |
| [21] | LeBrun, C.: Spaces of complex null geodesics in complex-Riemannian geometry, Trans. A.M.S. 278(1), 209-231 (1983) ·Zbl 0562.53018 |
| [22] | Mason, LJ; Newman, ET, A connection between the Einstein and Yang-Mills equations, Commun. Math. Phys., 121, 659-668 (1989) ·Zbl 0668.53048 ·doi:10.1007/BF01218161 |
| [23] | Han Park, Q., Self-dual gravity as a large-N limit of the 2D non-linear sigma model, Phys. Lett., 238B, 287-90 (1990) ·Zbl 1332.83038 ·doi:10.1016/0370-2693(90)91737-V |
| [24] | Penrose, R., Non-linear gravitons and curved twistor theory, Gen. Relativ. Gravit., 7, 31-52 (1976) ·Zbl 0354.53025 ·doi:10.1007/BF00762011 |
| [25] | Plebański, JF, Some solutions of complex Einstein equations, J. Math. Phys., 16, 2395-2402 (1975) ·doi:10.1063/1.522505 |
| [26] | Plebański, JF; Przanowski, M.; Rajca, B.; Tosiek, J., The moyal deformation of the second heavenly equation, Acta Phys. Pol. B, 26, 889 (1995) ·Zbl 0966.53529 |
| [27] | Strachan, IAB, The Moyal algebra and integrable deformations of the self-dual Einstein equations, Phys. Lett. B, 283, 1-2, 63-66 (1992) ·doi:10.1016/0370-2693(92)91427-B |
| [28] | Strachan, IAB, A geometry for multidimensional integrable systems, J. Geom. Phys., 21, 3, 255-278 (1997) ·Zbl 0893.58061 ·doi:10.1016/S0393-0440(96)00019-8 |
| [29] | Takasaki, K., An infinite number of hidden variables in hyper-Kähler metrics, J. Math. Phys., 30, 1515-1521 (1989) ·Zbl 0683.53017 ·doi:10.1063/1.528283 |
| [30] | Takasaki, K., Dressing operator approach to Moyal algebraic deformations of selfdual gravity, J. Geom. Phys., 14, 3, 111 (1994) ·Zbl 0803.58059 ·doi:10.1016/0393-0440(94)90003-5 |
| [31] | Ward, R.S.: Integrable and solvable systems, and relations among them, Phil. Trans. Roy. Soc. A, (315), issue 1533, 451-457 (1985) ·Zbl 0579.35078 |
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