Exceptional Calabi-Yau spaces: the geometry of \(\mathcal{N} = 2\) backgrounds with flux. (English) Zbl 1371.83176
Summary: In this paper we define the analogue of Calabi-Yau geometry for generic \(D = 4,\; \mathcal{N} = 2\) flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in \(E_{7(7)} \times \mathbb{R}^+\) generalised geometry. Such “exceptional Calabi-Yau” geometries are determined by two generalised objects that parametrise hyper- and vector-multiplet degrees of freedom and generalise conventional complex, symplectic and hyper-Kähler geometries. The integrability conditions for both hyper- and vector-multiplet structures are given by the vanishing of moment maps for the “generalised diffeomorphism group” of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to \(D = 5\; \text{and}\; D = 6\) flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in \(\operatorname{O}(d,d) \times \mathbb{R}^+\) generalised geometry.
MSC:
| 83E50 | Supergravity |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |
| 32J81 | Applications of compact analytic spaces to the sciences |
| 57S05 | Topological properties of groups of homeomorphisms or diffeomorphisms |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
References:
| [1] | P.Candelas, G.Horowitz, A.Strominger, and E.Witten, “Vacuum configurations for superstrings”, Nucl. Phys.B258, 46-74 (1985). |
| [2] | A.Strominger, “Yukawa couplings in superstring compactification”, Phys. Rev. Lett.55, 2547 (1985). |
| [3] | A.Strominger and E.Witten, “New manifolds for superstring compactification”, Commun. Math. Phys.101, 341 (1985). |
| [4] | P.Candelas and X.de la Ossa, “Moduli space of Calabi-Yau manifolds”, Nucl. Phys.B355, 455-481 (1991). · Zbl 0732.53056 |
| [5] | P.Candelas and D.Raine, “Spontaneous compactification and supersymmetry in \(d = 11\) supergravity”, Nucl. Phys.B248, 415 (1984). |
| [6] | P.Candelas, “Compactification and supersymmetry of chiral \(N = 2, D = 10\) supergravity”, Nucl. Phys.B256, 385 (1985). |
| [7] | J.Maldacena and C.Nunez, “Supergravity description of field theories on curved manifolds and a no go theorem”, Int. J. Mod. Phys.A16, 822-855 (2001), arXiv:hep‐th/0007018. · Zbl 0984.83052 |
| [8] | J.Gauntlett, D.Martelli, S.Pakis, and D.Waldram, “G structures and wrapped NS5‐branes”, Commun. Math. Phys.247, 421-445 (2004), arXiv:hep‐th/0205050. · Zbl 1061.81058 |
| [9] | J.Gauntlett, D.Martelli, and D.Waldram, “Superstrings with intrinsic torsion”, Phys. Rev.D69, 086002 (2004), arXiv:hep‐th/0302158. |
| [10] | S.Ivanov and G.Papadopoulos, “A no‐go theorem for string warped compactifications”, Phys. Lett.B497, 309-316 (2001), arXiv:hep‐th/0008232. · Zbl 0971.83512 |
| [11] | C.Hull, “Generalised geometry for M‐theory”, JHEP07, 079 (2007), arXiv:hep‐th/0701203. |
| [12] | P.Pacheco and D.Waldram, “M‐theory, exceptional generalised geometry and superpotentials”, JHEP09, 123 (2008), arXiv:0804.1362 [hep‐th]. · Zbl 1245.83070 |
| [13] | A.Coimbra, C.Strickland‐Constable, and D.Waldram, “E \({}_{d ( d )} \times \mathbb{R}^+\) generalised geometry, connections and M theory”, JHEP02, 054 (2014), arXiv:1112.3989 [hep‐th]. · Zbl 1333.83220 |
| [14] | A.Coimbra, C.Strickland‐Constable, and D.Waldram, “Supergravity as generalised geometry II: E \({}_{d ( d )} \times \mathbb{R}^+\) and M theory”, JHEP03, 019 (2014), arXiv:1212.1586 [hep‐th]. |
| [15] | A.Coimbra, C.Strickland‐Constable, and D.Waldram, “Supersymmetric backgrounds and generalised special holonomy”, arXiv:1411.5721 [hep‐th]. |
| [16] | J.Gauntlett, J.Gutowski, C.Hull, S.Pakis, and H.Reall, “All supersymmetric solutions of minimal supergravity in five dimensions”, Class. Quant. Grav.20, 4587-4634 (2003), arXiv:hep‐th/0209114. · Zbl 1045.83001 |
| [17] | J.Gauntlett and S.Pakis, “The geometry of \(D = 11\) Killing spinors”, JHEP04, 039 (2003), arXiv:hep‐th/0212008. |
| [18] | N.Hitchin, “Generalized Calabi-Yau manifolds”, Quart. J. Math.54, 281-308 (2003), arXiv:math/0209099 [math‐dg]. · Zbl 1076.32019 |
| [19] | M.Gualtieri, “Generalized complex geometry”, arXiv:math/0401221 [math.DG]. |
| [20] | M.Grana, R.Minasian, M.Petrini, and A.Tomasiello, “Supersymmetric backgrounds from generalized Calabi-Yau manifolds”, JHEP08, 046 (2004), arXiv:hep‐th/0406137. |
| [21] | M.Grana, R.Minasian, M.Petrini, and A.Tomasiello, “Generalized structures of \(N = 1\) vacua”, JHEP11, 020 (2005), arXiv:hep‐th/0505212. |
| [22] | A.Tomasiello, “Reformulating supersymmetry with a generalized dolbeault operator”, JHEP02, 010 (2008), arXiv:0704.2613 [hep‐th]. |
| [23] | G.Cavalcanti and M.Gualtieri, “Generalized complex geometry and T‐duality”, arXiv:1106.1747 [math.DG]. |
| [24] | L.‐S.Tseng and S.‐T.Yau, “Generalized cohomologies and supersymmetry”, Commun. Math. Phys.326, 875-885 (2014), arXiv:1111.6968 [hep‐th]. · Zbl 1288.81125 |
| [25] | A.Kapustin, “Topological strings on noncommutative manifolds”, Int. J. Geom. Meth. Mod. Phys.1, 49-81 (2004), arXiv:hep‐th/0310057. · Zbl 1065.81108 |
| [26] | A.Kapustin and Y.Li, “Topological sigma‐models with H‐flux and twisted generalized complex manifolds”, Adv. Theor. Math. Phys.11(2), 269-290 (2007), arXiv:hep‐th/0407249. · Zbl 1192.81310 |
| [27] | V.Pestun and E.Witten, “The Hitchin functionals and the topological B‐model at one loop”, Lett. Math. Phys.74, 21-51 (2005), arXiv:hep‐th/0503083. · Zbl 1101.81098 |
| [28] | M.Grana, R.Minasian, M.Petrini, and A.Tomasiello, “A scan for new \(\mathcal{N} = 1\) vacua on twisted tori”, JHEP05, 031 (2007), arXiv:hep‐th/0609124. |
| [29] | D.Lüst and D.Tsimpis, “Classes of AdS_4 type IIA/IIB compactifications with SU(3)× SU(3) structure”, JHEP04, 111 (2009), arXiv:0901.4474 [hep‐th]. |
| [30] | D.Andriot, “New supersymmetric flux vacua with intermediate SU(2) structure”, JHEP08, 096 (2008), arXiv:0804.1769 [hep‐th]. |
| [31] | R.Minasian, M.Petrini, and A.Zaffaroni, “Gravity duals to deformed SYM theories and generalized complex geometry”, JHEP12, 055 (2006), arXiv:hep‐th/0606257. · Zbl 1226.83087 |
| [32] | A.Butti, D.Forcella, L.Martucci, R.Minasian, M.Petrini, and A.Zaffaroni, “On the geometry and the moduli space of beta‐deformed quiver gauge theories”, JHEP07, 053 (2008), arXiv:0712.1215 [hep‐th]. |
| [33] | M.Gabella, J.Gauntlett, E.Palti, J.Sparks, and D.Waldram, “AdS_5 solutions of type IIB supergravity and generalized complex geometry”, Commun. Math. Phys.299, 365-408 (2010), arXiv:0906.4109 [hep‐th]. · Zbl 1197.83108 |
| [34] | B.de Wit and H.Nicolai, “\( d = 11\) supergravity with local SU(8) invariance”, Nucl. Phys.B274, 363 (1986). |
| [35] | P.West, “Hidden superconformal symmetry in M theory”, JHEP08, 007 (2000), arXiv:hep‐th/0005270. · Zbl 0989.81101 |
| [36] | P.West, “E_11 and M theory”, Class. Quant. Grav.18, 4443-4460 (2001), arXiv:hep‐th/0104081. · Zbl 0992.83079 |
| [37] | P.West, “E_11, SL(32) and central charges”, Phys. Lett.B575, 333-342 (2003), arXiv:hep‐th/0307098. · Zbl 1031.22010 |
| [38] | K.Koepsell, H.Nicolai, and H.Samtleben, “An exceptional geometry for \(D = 11\) supergravity?”, Class. Quant. Grav.17, 3689-3702 (2000), arXiv:hep‐th/0006034. · Zbl 0971.83078 |
| [39] | T.Damour, M.Henneaux, and H.Nicolai, “E_10 and a ”small tension expansion’ of M theory”, Phys. Rev. Lett.89, 221601 (2002), arXiv:hep‐th/0207267. · Zbl 1267.83103 |
| [40] | T.Damour, M.Henneaux, and H.Nicolai, “Cosmological billiards”, Class. Quant. Grav.20, R145-R200 (2003), arXiv:hep‐th/0212256. · Zbl 1138.83306 |
| [41] | C.Hillmann, “Generalized E_(7(7)) coset dynamics and \(D = 11\) supergravity”, JHEP03, 135 (2009), arXiv:0901.1581 [hep‐th]. |
| [42] | D.Berman and M.Perry, “Generalized geometry and M theory”, JHEP06, 074 (2011), arXiv:1008.1763 [hep‐th]. · Zbl 1298.81244 |
| [43] | D.Berman, H.Godazgar, and M.Perry, “SO(5,5) duality in M‐theory and generalized geometry”, Phys. Lett.B700, 65-67 (2011), arXiv:1103.5733 [hep‐th]. |
| [44] | D.Berman, H.Godazgar, M.Perry, and P.West, “Duality invariant actions and generalised geometry”, JHEP02, 108 (2012), arXiv:1111.0459 [hep‐th]. · Zbl 1309.81201 |
| [45] | G.Aldazabal, M.Grana, D.Marqués, and J. A.Rosabal, “Extended geometry and gauged maximal supergravity”, JHEP06, 046 (2013), arXiv:1302.5419 [hep‐th]. · Zbl 1342.83439 |
| [46] | O.Hohm and H.Samtleben, “Exceptional form of \(D = 11\) supergravity”, Phys. Rev. Lett.111, 231601 (2013), arXiv:1308.1673 [hep‐th]. |
| [47] | D.Berman, M.Cederwall, and M.Perry, “Global aspects of double geometry”, JHEP09, 066 (2014), arXiv:1401.1311 [hep‐th]. · Zbl 1333.81286 |
| [48] | M.Cederwall, “The geometry behind double geometry”, JHEP09, 070 (2014), arXiv:1402.2513 [hep‐th]. · Zbl 1333.81287 |
| [49] | C.Hull, “Finite gauge transformations and geometry in double field theory”, JHEP04, 109 (2015), arXiv:1406.7794 [hep‐th]. · Zbl 1388.81723 |
| [50] | K.Lee, C.Strickland‐Constable, and D.Waldram, “New gaugings and non‐geometry”, arXiv:1506.03457 [hep‐th]. |
| [51] | M.Grana, J.Louis, A.Sim, and D.Waldram, “E_7(7) formulation of \(\mathcal{N} = 2\) backgrounds”, JHEP07, 104 (2009), arXiv:0904.2333 [hep‐th]. |
| [52] | M.Grana and F.Orsi, “\( \mathcal{N} = 1\) vacua in exceptional generalized geometry”, JHEP08, 109 (2011), arXiv:1105.4855 [hep‐th]. · Zbl 1298.81288 |
| [53] | M.Grana and F.Orsi, “\( \mathcal{N} = 2\) vacua in generalized geometry”, JHEP11, 052 (2012), arXiv:1207.3004 [hep‐th]. · Zbl 1397.81245 |
| [54] | M.Grana and H.Triendl, “Generalized \(N = 1\) and \(N = 2\) structures in M‐theory and type II orientifolds”, JHEP03, 145 (2013), arXiv:1211.3867 [hep‐th]. · Zbl 1342.81429 |
| [55] | F.Kirwan, “Momentum maps and reduction in algebraic geometry”, Differ. Geom. Appl.9(1‐2), 135-171 (1998). · Zbl 0955.37031 |
| [56] | R.Thomas, “Notes on GIT and symplectic reduction for bundles and varieties”, arXiv:math/0512411 [math.DG]. |
| [57] | M.Atiyah and R.Bott, “The Yang-Mills equations over Riemann surfaces”, Phil. Trans. R. Soc. A308(1505), 523-615 (1983). · Zbl 0509.14014 |
| [58] | S.Donaldson, “Anti self‐dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles”, Proc. Lond. Math. Soc.50(1), 1-26 (1985). · Zbl 0529.53018 |
| [59] | K.Uhlenbeck and S.‐T.Yau, “On the existence of Hermitian‐Yang-Mills connections in stable vector bundles”, Comm. Pure Appl. Math.39(S1), 257-293 (1986). · Zbl 0615.58045 |
| [60] | K.Uhlenbeck and S.‐T.Yau, “A note on our previous paper: On the existence of Hermitian Yang-Mills connections in stable vector bundles”, Comm. Pure Appl. Math.42(5), 703-707 (1989). · Zbl 0678.58041 |
| [61] | N.Hitchin, “The self‐duality equations on a Riemann surface”, Proc. Lond. Math. Soc.55, 59-131 (1987). · Zbl 0634.53045 |
| [62] | A.Fujiki, “Moduli space of polarized algebraic manifolds and Kähler metrics”, Sugaku Expositions5(2), 173-191 (1992). Sugaku Expositions. · Zbl 0796.32009 |
| [63] | S.Donaldson, Remarks on gauge theory, complex geometry and 4‐manifold topology, vol. 5 of World Sci. Ser. 20th Century Math., ch. 42, pp. 384-403 (World Sci. Publ., River Edge, NJ, 1997). |
| [64] | A.Coimbra and C.Strickland‐Constable, “Generalised structures for \(\mathcal{N} = 1\) AdS backgrounds”, arXiv:1504.02465 [hep‐th]. |
| [65] | M.Grana, J.Louis, and D.Waldram, “Hitchin functionals in \(\mathcal{N} = 2\) supergravity”, JHEP01, 008 (2006), arXiv:hep‐th/0505264. |
| [66] | M.Grana, J.Louis, and D.Waldram, “SU(3)× SU(3) compactification and mirror duals of magnetic fluxes”, JHEP04, 101 (2007), arXiv:hep‐th/0612237. |
| [67] | L.Andrianopoli, M.Bertolini, A.Ceresole, R.D’Auria, S.Ferrara, P.Fré, and T.Magri, “\( N = 2\) supergravity and \(N = 2\) super Yang-Mills theory on general scalar manifolds: symplectic covariance, gaugings and the momentum map”, J. Geom. Phys.23, 111-189 (1997), arXiv:hep‐th/9605032. · Zbl 0899.53073 |
| [68] | P.Koerber and L.Martucci, “From ten to four and back again: How to generalize the geometry”, JHEP08, 059 (2007), arXiv:0707.1038 [hep‐th]. · Zbl 1326.81162 |
| [69] | K.Hristov, H.Looyestijn, and S.Vandoren, “Maximally supersymmetric solutions of \(D = 4 \mathcal{N} = 2\) gauged supergravity”, JHEP11, 115 (2009), arXiv:0909.1743 [hep‐th]. |
| [70] | J.Louis, P.Smyth, and H.Triendl, “Supersymmetric vacua in \(\mathcal{N} = 2\) supergravity”, JHEP08, 039 (2012), arXiv:1204.3893 [hep‐th]. |
| [71] | S.deAlwis, J.Louis, L.McAllister, H.Triendl, and A.Westphal, “Moduli spaces in AdS_4 supergravity”, JHEP05, 102 (2014), arXiv:1312.5659 [hep‐th]. · Zbl 1333.83228 |
| [72] | A.Ashmore, M.Petrini, and D.Waldram, “The exceptional generalised geometry of supersymmetric AdS flux backgrounds”, arXiv:1602.02158 [hep‐th]. |
| [73] | E.Bergshoeff, R.Kallosh, T.Ortín, D.Roest, and A.VanProeyen; “New formulations of D = 10 supersymmetry and D8‐O8 domain walls”, Class. Quant. Grav.18, 3359-3382 (2001), arXiv:hep‐th/0103233. · Zbl 1004.83053 |
| [74] | P.Kaste, R.Minasian, and A.Tomasiello, “Supersymmetric M‐theory compactifications with fluxes on seven‐manifolds and G‐structures”, JHEP07, 004 (2003), arXiv:hep‐th/0303127. |
| [75] | K.Behrndt and C.Jeschek, “Fluxes in M‐theory on seven‐manifolds and G structures”, JHEP04, 002 (2003), arXiv:hep‐th/0302047. |
| [76] | G.Dall’Agata and N.Prezas, “\( \mathcal{N} = 1\) geometries for M theory and type IIA strings with fluxes”, Phys. Rev.D69, 066004 (2004), arXiv:hep‐th/0311146. |
| [77] | A.Lukas and P.Saffin, “M‐theory compactification, fluxes and AdS_4”, Phys. Rev.D71, 046005 (2005), arXiv:hep‐th/0403235. |
| [78] | F.Witt, “Generalised G_2‐manifolds”, Commun. Math. Phys.265, 275-303 (2006), arXiv:math/0411642 [math‐dg]. · Zbl 1154.53014 |
| [79] | M.Grana, R.Minasian, M.Petrini, and D.Waldram, “T‐duality, generalized geometry and non‐geometric backgrounds”, JHEP04, 075 (2008), arXiv:0807.4527 [hep‐th]. |
| [80] | R.Gilmore, Lie groups, physics, and geometry (Cambridge University Press, 2008). · Zbl 1157.00009 |
| [81] | K.Lee, C.Strickland‐Constable, and D.Waldram, “Spheres, generalised parallelisability and consistent truncations”, arXiv:1401.3360 [hep‐th]. |
| [82] | D.Baraglia, “Leibniz algebroids, twistings and exceptional generalized geometry”, J. Geom. Phys.62, 903-934 (2012), arXiv:1101.0856 [math.DG]. · Zbl 1253.53081 |
| [83] | J.Wolf, “Complex homogeneous contact manifolds and quaternionic symmetric spaces”, J. Math. Mech.14(6), 1033-1047 (1965). · Zbl 0141.38202 |
| [84] | D.Alekseevsky, “Compact quaternion spaces”, Funct. Anal. Appl.2(2), 106-114 (1968). |
| [85] | D.Alekseevsky and V.Cortés, “Classification of pseudo‐Riemannian symmetric spaces of quaternionic Kähler type”, Amer. Math. Soc. Transl. Ser. 2213(2), 33-62 (2005). · Zbl 1077.53040 |
| [86] | A.Swann, “HyperKähler and quaternionic Kähler geometry”, Math. Ann.289(1), 421-450 (1991). · Zbl 0711.53051 |
| [87] | P.Kobak and A.Swann, “The HyperKähler geometry associated to Wolf spaces”, B. Unione Mat. Ital.4‐B(3), 587-595 (2001), arXiv:math/0001025 [math.DG]. · Zbl 1182.53041 |
| [88] | B.DeWitt, “Quantum theory of gravity I. The canonical theory”, Phys. Rev.160, 1113-1148 (1967). · Zbl 0158.46504 |
| [89] | D.Ebin, “On the space of Riemannian metrics”, Bull. Amer. Math. Soc.74(5), 1001-1003 (1968). · Zbl 0172.22905 |
| [90] | O.Gil‐Medrano and P.Michor, “The Riemannian manifold of all Riemannian metrics”, Quart. J. Math.42, 183-202 (1991), arXiv:math/9201259 [math.DG]. · Zbl 0739.58010 |
| [91] | B.de Wit, B.Kleijn, and S.Vandoren, “Superconformal hypermultiplets”, Nucl. Phys.B568, 475-502 (2000), arXiv:hep‐th/9909228. · Zbl 0951.81042 |
| [92] | B.deWit, M.Roček, and S.Vandoren, “Hypermultiplets, hyper‐Kähler cones and quaternion‐Kähler geometry”, JHEP02, 039 (2001), arXiv:hep‐th/0101161. |
| [93] | B.de Wit, M.Roček, and S.Vandoren, “Gauging isometries on hyperkähler cones and quaternion‐Kähler manifolds”, Phys. Lett.B511, 302-310 (2001), arXiv:hep‐th/0104215. · Zbl 0969.83505 |
| [94] | S.Ferrara and M.Gunaydin, “Orbits of exceptional groups, duality and BPS states in string theory”, Int. J. Mod. Phys.A13, 2075-2088 (1998), arXiv:hep‐th/9708025 [hep‐th]. · Zbl 0936.81032 |
| [95] | B.Craps, F.Roose, W.Troost, and A.VanProeyen, “What is special Kähler geometry?”, Nucl. Phys.B503, 565-613 (1997), arXiv:hep‐th/9703082. · Zbl 0951.81088 |
| [96] | D.Freed, “Special Kähler manifolds”, Commun. Math. Phys.203, 31-52 (1999), arXiv:hep‐th/9712042. · Zbl 0940.53040 |
| [97] | S.Cecotti, S.Ferrara, and L.Girardello, “Geometry of type II superstrings and the moduli of superconformal field theories”, Int. J. Mod. Phys.A4, 2475 (1989). · Zbl 0681.58044 |
| [98] | S.Cecotti, “Homogeneous Kähler manifolds and T algebras in \(N = 2\) supergravity and superstrings”, Commun. Math. Phys.124, 23-55 (1989). · Zbl 0685.53040 |
| [99] | N.Hitchin, “The geometry of three‐forms in six dimensions”, J. Diff. Geom.55, 547-576 (2000). · Zbl 1036.53042 |
| [100] | M.Sato and T.Kimura, “A classification of irreducible prehomogeneous vector spaces and their relative invariants”, Nagoya Math. J.65, 1-155 (1977). · Zbl 0321.14030 |
| [101] | S.Chiossi and S.Salamon, “The intrinsic torsion of SU(3) and G_2 structures”, in Differential geometry, Valencia 2001 (World Sci. Publishing, 2002). arXiv:math/0202282 [math.DG]. · Zbl 1024.53018 |
| [102] | K.Behrndt, M.Cvetič, and T.Liu, “Classification of supersymmetric flux vacua in M‐theory”, Nucl. Phys.B749, 25-68 (2006), arXiv:hep‐th/0512032. · Zbl 1214.81189 |
| [103] | M.Grana and J.Polchinski, “Supersymmetric three‐form flux perturbations on AdS_5”, Phys. Rev.D63, 026001 (2001), arXiv:hep‐th/0009211. |
| [104] | S.Gubser, “Supersymmetry and F theory realization of the deformed conifold with three form flux”, arXiv:hep‐th/0010010. |
| [105] | M.Grana and J.Polchinski, “Gauge‐gravity duals with a holomorphic dilaton”, Phys. Rev.D65, 126005 (2002), arXiv:hep‐th/0106014. |
| [106] | M.Bertolini, V.Campos, G.Ferretti, P.Fré, P.Salomonson, and M.Trigiante, “Supersymmetric 3‐branes on smooth ALE manifolds with flux”, Nucl. Phys.B617, 3-42 (2001), arXiv:hep‐th/0106186. · Zbl 0974.83049 |
| [107] | A.Coimbra, R.Minasian, H.Triendl, and D.Waldram, “Generalised geometry for string corrections”, JHEP11, 160 (2014), arXiv:1407.7542 [hep‐th]. · Zbl 1333.83179 |
| [108] | B.deWit, H.Samtleben, and M.Trigiante, “Magnetic charges in local field theory”, JHEP09, 016 (2005), arXiv:hep‐th/0507289. |
| [109] | B.deWit and M.VanZalk; “Electric and magnetic charges in \(N = 2\) conformal supergravity theories”, JHEP10, 050 (2011), arXiv:1107.3305 [hep‐th]. · Zbl 1303.81113 |
| [110] | A.Dancer and A.Swann, “The structure of quaternionic Kähler quotients”, in Geometry and Physics, vol. 184 of Lecture Notes in Pure and Appl. Math., pp. 313-320, (Dekker, New York, 1997). · Zbl 0869.53042 |
| [111] | R.Sjamaar and E.Lerman, “Stratified symplectic spaces and reduction”, Ann. of Math.134(2), 365-422 (1991). · Zbl 0759.58019 |
| [112] | P.Koerber and D.Tsimpis, “Supersymmetric sources, integrability and generalized‐structure compactifications”, JHEP08, 082 (2007), arXiv:0706.1244 [hep‐th]. · Zbl 1326.81207 |
| [113] | B.deWit and A.VanProeyen; “Special geometry, cubic polynomials and homogeneous quaternionic spaces”, Commun. Math. Phys.149, 307-334 (1992), arXiv:hep‐th/9112027. · Zbl 0824.53043 |
| [114] | B.de Wit and A.VanProeyen, “Special geometry and symplectic transformations”, Nucl. Phys. Proc. Suppl.45BC, 196-206 (1996), arXiv:hep‐th/9510186. · Zbl 0991.53506 |
| [115] | M.Günaydin, G.Sierra, and P.Townsend, “The geometry of N = 2 Maxwell‐Einstein supergravity and Jordan algebras”, Nucl. Phys.B242, 244-268 (1984). |
| [116] | E.Bergshoeff, S.Cucu, T.deWit, J.Gheerardyn, S.Vandoren, and A.VanProeyen; “N = 2 supergravity in five dimensions revisited”, Class. Quant. Grav.21, 3015-3042 (2004), arXiv:hep‐th/0403045. · Zbl 1061.83061 |
| [117] | D.Alekseevsky and V.Cortés; “Geometric construction of the r‐map: from affine special real to special Kähler manifolds”, Commun. Math. Phys.291, 579-590 (2009), arXiv:0811.1658 [math.DG]. · Zbl 1205.53024 |
| [118] | K.Behrndt and S.Gukov, “Domain walls and superpotentials from M theory on Calabi-Yau three‐folds”, Nucl. Phys.B580, 225-242 (2000), arXiv:hep‐th/0001082. · Zbl 1071.81559 |
| [119] | J.Gauntlett, D.Martelli, J.Sparks, and D.Waldram, “Supersymmetric AdS_5 solutions of M‐theory”, Class. Quant. Grav.21, 4335-4366 (2004), arXiv:hep‐th/0402153. · Zbl 1059.83038 |
| [120] | L.Romans, “Self‐duality for interacting fields: covariant field equations for six‐dimensional chiral supergravities”, Nucl. Phys.B276, 71 (1986). |
| [121] | M.Gunaydin, H.Samtleben, and E.Sezgin, “On the magical supergravities in six dimensions”, Nucl. Phys.B848, 62-89 (2011), arXiv:1012.1818 [hep‐th]. · Zbl 1215.83054 |
| [122] | C.Callan, J.Harvey, and A.Strominger, “Supersymmetric string solitons”, in String theory and quantum gravity (World Sci. Publ., River Edge, NJ, 1992). arXiv:hep‐th/9112030. |
| [123] | A.Strominger, “Superstrings with torsion”, Nucl. Phys.B274, 253 (1986). |
| [124] | A.Coimbra, C.Strickland‐Constable, and D.Waldram, “Supergravity as generalised geometry I: Type II theories”, JHEP11, 091 (2011), arXiv:1107.1733 [hep‐th]. · Zbl 1306.81205 |
| [125] | W.Siegel, “Manifest duality in low‐energy superstrings”, arXiv:hep‐th/9308133. |
| [126] | O.Hohm and S. K.Kwak, “Frame‐like geometry of double field theory”, J. Phys.A44, 085404 (2011), arXiv:1011.4101 [hep‐th]. · Zbl 1209.81168 |
| [127] | M.Gualtieri, “Branes on Poisson varieties”, in The many facets of geometry, pp. 368-394, (Oxford Univ. Press, Oxford, 2010). arXiv:0710.2719 [math.DG]. · Zbl 1239.53104 |
| [128] | I.Jeon, K.Lee, and J.‐H.Park, “Differential geometry with a projection: application to double field theory”, JHEP04, 014 (2011), arXiv:1011.1324 [hep‐th]. · Zbl 1250.81085 |
| [129] | I.Jeon, K.Lee, and J.‐H.Park, “Stringy differential geometry, beyond Riemann”, Phys. Rev.D84, 044022 (2011), arXiv:1105.6294 [hep‐th]. |
| [130] | C.Strickland‐Constable, “Subsectors, Dynkin diagrams and new generalised geometries”, arXiv:1310.4196 [hep‐th]. |
| [131] | D.Berman, M.Cederwall, A.Kleinschmidt, and D.Thompson, “The gauge structure of generalised diffeomorphisms”, JHEP01, 064 (2013), arXiv:1208.5884 [hep‐th]. |
| [132] | O.Hohm and H.Samtleben, “Exceptional field theory. III. E_8(8)”, Phys. Rev.D90, 066002 (2014), arXiv:1406.3348 [hep‐th]. |
| [133] | J.Rosabal, “On the exceptional generalised Lie derivative for \(d \geq 7\)”, arXiv:1410.8148 [hep‐th]. |
| [134] | M.Cederwall and J.Rosabal, “E_8 geometry”, JHEP07, 007 (2015), arXiv:1504.04843 [hep‐th]. |
| [135] | N.Hitchin, “Stable forms and special metrics”, arXiv:math/0107101 [math.DG]. |
| [136] | N.Hitchin, “Brackets, forms and invariant functionals”, arXiv:math/0508618 [math.DG]. |
| [137] | N.Hitchin, “The geometry of three‐forms in six and seven dimensions”, arXiv:math/0010054 [math.DG]. |
| [138] | R.Rubio, “B_n‐generalized geometry and \(G_2^2\)‐structures”, J. Geom. Phys.73, 150-156 (2013), arXiv:1301.3330 [math.DG]. · Zbl 1284.53077 |
| [139] | R.Rubio, Generalized geometry of type \(B_n\). PhD thesis, University of Oxford, 2014. |
| [140] | M.Manetti, “Lectures on deformations of complex manifolds (deformations from differential graded viewpoint)”, Rend. Mat. Appl. (7)24(1), 1-183 (2004). · Zbl 1066.58010 |
| [141] | M.Gualtieri, “Generalized geometry and the Hodge decomposition”, arXiv:math/0409093 [math.DG]. |
| [142] | G.Cavalcanti, “Formality in generalized Kähler geometry”, Topol. Appl.154(6), 1119-1125 (2007), arXiv:math/0603596 [math.DG]. · Zbl 1123.53014 |
| [143] | A.Ashmore, M.Gabella, M.Grana, M.Petrini, and D.Waldram, “Exactly marginal deformations from exceptional generalised geometry”, arXiv:1605.05730 [hep‐th]. |
| [144] | H.Bursztyn, G.Cavalcanti, and M.Gualtieri, “Reduction of Courant algebroids and generalized complex structures”, Adv. Math.211(2), 726-765 (2006), arXiv:math/0509640. · Zbl 1115.53056 |
| [145] | R.Dijkgraaf, S.Gukov, A.Neitzke, and C.Vafa, “Topological M‐theory as unification of form theories of gravity”, Adv. Theor. Math. Phys.9(4), 603-665 (2005), arXiv:hep‐th/0411073. · Zbl 1129.81066 |
| [146] | N.Nekrasov, “À la recherche de la M‐théorie perdue Z theory: chasing M/F theory”, arXiv:hep‐th/0412021. |
| [147] | A.Fayyazuddin, T.Husain, and I.Pappa, “Geometry of wrapped M5‐branes in Calabi-Yau 2‐folds”, Phys. Rev.D73, 126004 (2006), arXiv:hep‐th/0509018. |
| [148] | A.Fayyazuddin and D.Smith, “Localized intersections of M5‐branes and four‐dimensional superconformal field theories”, JHEP04, 030 (1999), arXiv:hep‐th/9902210. · Zbl 0956.83056 |
| [149] | K.Sakamoto, “On the topology of quaternion Kähler manifolds”, Tohoku Math. J. (2)26(3), 389-405 (1974). · Zbl 0297.53015 |
| [150] | M.Konishi, “On manifolds with Sasakian 3‐structure over quaternion Kaehler manifolds”, Kodai Math. Sem. Rep.26(2‐3), 194-200 (1975). · Zbl 0308.53035 |
| [151] | D.Grandini, “Quaternionic Kähler reductions of Wolf spaces”, Ann. Global Anal. Geom.32(3), 225-252 (2007), arXiv:math/0608365 [math.DG]. · Zbl 1133.53038 |
| [152] | N.Hitchin, “The moduli space of complex Lagrangian submanifolds”, Asian J. Math3, 77-91 (1999), arXiv:math/9901069 [math.DG]. · Zbl 0958.53057 |
| [153] | N.Hitchin, A.Karlhede, U.Lindström, and M.Roček, “Hyperkähler metrics and supersymmetry”, Commun. Math. Phys.108, 535 (1987). · Zbl 0612.53043 |
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