F-theory and \(\mathrm{AdS}_{3}/CFT_{2}\). (English) Zbl 1381.81110

Couzens, Christopher; Lawrie, Craig; Martelli, Dario; Schäfer-Nameki, Sakura; Wong, Jin-Mann

J. High Energy Phys. 2017, No. 8, Paper No. 43, 72 p. (2017).

Summary: We construct supersymmetric \(\mathrm{AdS}_{3}\) solutions in F-theory, that is Type IIB supergravity with varying axio-dilaton, which are holographically dual to 2d \(\;\mathcal{N}=\left(0,4\right) \) superconformal field theories with small superconformal algebra. In F-theory these arise from D3-branes wrapped on curves in the base of an elliptically fibered Calabi-Yau threefold \(Y_3\) and correspond to self-dual strings in the 6d \(\;\mathcal{N}=\left(1,0\right) \) theory obtained from F-theory on \(Y_3\). The non-trivial fibration over the wrapped curves implies a varying coupling of the \( \mathcal{N}=4 \) Super-Yang-Mills theory on the D3-branes. We compute the holographic central charges and show that these agree with the field theory and with the anomalies of self-dual strings in 6d. We complement our analysis with a discussion of the dual M-theory solutions and a comparison of the central charges.

MSC:

81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
83E50	Supergravity
14J32	Calabi-Yau manifolds (algebro-geometric aspects)
32Q25	Calabi-Yau theory (complex-analytic aspects)

Keywords:

AdS-CFT correspondence; F-theory

References:

[1]	C. Vafa, Evidence for F-theory, Nucl. Phys.B 469 (1996) 403 [hep-th/9602022] [INSPIRE]. · Zbl 1003.81531
[2]	D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys.B 473 (1996) 74 [hep-th/9602114] [INSPIRE]. · Zbl 0925.14005
[3]	D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys.B 476 (1996) 437 [hep-th/9603161] [INSPIRE]. · Zbl 0925.14005
[4]	M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4D SYM to 2D σ-models, Nucl. Phys.B 448 (1995) 166 [hep-th/9501096] [INSPIRE]. · Zbl 1009.58502
[5]	B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-strings, Commun. Math. Phys.334 (2015) 779 [arXiv:1305.6322] [INSPIRE]. · Zbl 1393.81031
[6]	B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of minimal 6d SCFTs, Fortsch. Phys.63 (2015) 294 [arXiv:1412.3152] [INSPIRE]. · Zbl 1338.81324
[7]	M. Del Zotto and G. Lockhart, On exceptional instanton strings, arXiv:1609.00310 [INSPIRE]. · Zbl 1382.83109
[8]	B. Haghighat, S. Murthy, C. Vafa and S. Vandoren, F-theory, spinning black holes and multi-string branches, JHEP01 (2016) 009 [arXiv:1509.00455] [INSPIRE]. · Zbl 1388.83817
[9]	C. Lawrie, S. Schäfer-Nameki and T. Weigand, Chiral 2d theories from N = 4 SYM with varying coupling, JHEP04 (2017) 111 [arXiv:1612.05640] [INSPIRE]. · Zbl 1378.81141
[10]	D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys.10 (2012) 1072 [arXiv:1201.1943] [INSPIRE]. · Zbl 1255.81210
[11]	L. Martucci, Topological duality twist and brane instantons in F-theory, JHEP06 (2014) 180 [arXiv:1403.2530] [INSPIRE]. · Zbl 1333.81349
[12]	B. Assel and S. Schäfer-Nameki, Six-dimensional origin of N = 4 SYM with duality defects, JHEP12 (2016) 058 [arXiv:1610.03663] [INSPIRE]. · Zbl 1390.81560
[13]	F. Bonetti and T.W. Grimm, Six-dimensional (1,0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP05 (2012) 019 [arXiv:1112.1082] [INSPIRE]. · Zbl 1348.81353
[14]	A. Grassi, On minimal models of elliptic threefolds, Math. Ann.290 (1991) 287. · Zbl 0719.14006
[15]	A. Grassi, The singularities of the parameter surface of a minimal elliptic threefold, Internat. J. Math.4 (1993) 203. · Zbl 0802.14020
[16]	K. Kodaira, On compact analytic surfaces II, Ann. Math.77 (1963) 563. · Zbl 0118.15802
[17]	A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux (in French), Inst. Hautes Études Sci. Publ. Math.21 (1964) 128. · Zbl 0132.41403
[18]	E. Ó. Colgáin, J.-B. Wu and H. Yavartanoo, Supersymmetric AdS3× S2M-theory geometries with fluxes, JHEP08 (2010) 114 [arXiv:1005.4527] [INSPIRE]. · Zbl 1290.81109
[19]	Ö. Kelekci, Y. Lozano, J. Montero, E. Ó. Colgáin and M. Park, Large superconformal near-horizons from M-theory, Phys. Rev.D 93 (2016) 086010 [arXiv:1602.02802] [INSPIRE].
[20]	J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP12 (1997) 002 [hep-th/9711053] [INSPIRE]. · Zbl 0951.83034
[21]	C. Vafa, Black holes and Calabi-Yau threefolds, Adv. Theor. Math. Phys.2 (1998) 207 [hep-th/9711067] [INSPIRE]. · Zbl 0956.83029
[22]	R. Minasian, G.W. Moore and D. Tsimpis, Calabi-Yau black holes and (0, 4) σ-models, Commun. Math. Phys.209 (2000) 325 [hep-th/9904217] [INSPIRE]. · Zbl 0960.83022
[23]	I. Bena, D.-E. Diaconescu and B. Florea, Black string entropy and Fourier-Mukai transform, JHEP04 (2007) 045 [hep-th/0610068] [INSPIRE].
[24]	M. Ademollo et al., Supersymmetric strings and color confinement, Phys. Lett.B 62 (1976) 105 [INSPIRE].
[25]	J.A. Harvey, R. Minasian and G.W. Moore, Non-Abelian tensor multiplet anomalies, JHEP09 (1998) 004 [hep-th/9808060] [INSPIRE]. · Zbl 0953.81093
[26]	D. Martelli and J. Sparks, G structures, fluxes and calibrations in M-theory, Phys. Rev.D 68 (2003) 085014 [hep-th/0306225] [INSPIRE].
[27]	J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS5solutions of M-theory, Class. Quant. Grav.21 (2004) 4335 [hep-th/0402153] [INSPIRE]. · Zbl 1059.83038
[28]	J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS5solutions of type IIB supergravity, Class. Quant. Grav.23 (2006) 4693 [hep-th/0510125] [INSPIRE]. · Zbl 1096.83069
[29]	M. Gabella, D. Martelli, A. Passias and J. Sparks, N = 2 supersymmetric AdS4solutions of M-theory, Commun. Math. Phys.325 (2014) 487 [arXiv:1207.3082] [INSPIRE]. · Zbl 1282.83010
[30]	F. Apruzzi, M. Fazzi, D. Rosa and A. Tomasiello, All AdS7solutions of type-II supergravity, JHEP04 (2014) 064 [arXiv:1309.2949] [INSPIRE].
[31]	F. Apruzzi, M. Fazzi, A. Passias, D. Rosa and A. Tomasiello, AdS6solutions of type-II supergravity, JHEP11 (2014) 099 [Erratum ibid.05 (2015) 012] [arXiv:1406.0852] [INSPIRE]. · Zbl 1061.81058
[32]	F. Apruzzi, M. Fazzi, A. Passias and A. Tomasiello, Supersymmetric AdS5solutions of massive IIA supergravity, JHEP06 (2015) 195 [arXiv:1502.06620] [INSPIRE]. · Zbl 1388.83723
[33]	C. Couzens, C. Lawrie, D. Martelli and S. Schäfer-Nameki, in progress. · Zbl 1372.83076
[34]	H.J. Boonstra, B. Peeters and K. Skenderis, Brane intersections, anti-de Sitter space-times and dual superconformal theories, Nucl. Phys.B 533 (1998) 127 [hep-th/9803231] [INSPIRE]. · Zbl 0956.81060
[35]	D. Tong, The holographic dual of AdS3× S3× S3× S1, JHEP04 (2014) 193 [arXiv:1402.5135] [INSPIRE].
[36]	N. Kim, AdS3solutions of IIB supergravity from D3-branes, JHEP01 (2006) 094 [hep-th/0511029] [INSPIRE].
[37]	J.P. Gauntlett, O.A.P. Mac Conamhna, T. Mateos and D. Waldram, New supersymmetric AdS3solutions, Phys. Rev.D 74 (2006) 106007 [hep-th/0608055] [INSPIRE].
[38]	J.P. Gauntlett, N. Kim and D. Waldram, Supersymmetric AdS3, AdS2and bubble solutions, JHEP04 (2007) 005 [hep-th/0612253] [INSPIRE].
[39]	J.P. Gauntlett and O.A.P. Mac Conamhna, AdS spacetimes from wrapped D3-branes, Class. Quant. Grav.24 (2007) 6267 [arXiv:0707.3105] [INSPIRE]. · Zbl 1133.83012
[40]	A. Donos, J.P. Gauntlett and N. Kim, AdS solutions through transgression, JHEP09 (2008) 021 [arXiv:0807.4375] [INSPIRE]. · Zbl 1245.83060
[41]	F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP06 (2013) 005 [arXiv:1302.4451] [INSPIRE]. · Zbl 1390.83325
[42]	N. Bobev, K. Pilch and O. Vasilakis, (0, 2) SCFTs from the Leigh-Strassler fixed point, JHEP06 (2014) 094 [arXiv:1403.7131] [INSPIRE]. · Zbl 1333.81158
[43]	F. Benini, N. Bobev and P.M. Crichigno, Two-dimensional SCFTs from D3-branes, JHEP07 (2016) 020 [arXiv:1511.09462] [INSPIRE]. · Zbl 1390.83088
[44]	F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett.110 (2013) 061601 [arXiv:1211.4030] [INSPIRE]. · Zbl 1390.83325
[45]	O. Aharony, A. Fayyazuddin and J.M. Maldacena, The large-N limit of N = 2, N = 1 field theories from three-branes in F-theory, JHEP07 (1998) 013 [hep-th/9806159] [INSPIRE]. · Zbl 0958.81084
[46]	C.-H. Ahn, K. Oh and R. Tatar, The large-N limit of N = 1 field theories from F-theory, Mod. Phys. Lett.A 14 (1999) 369 [hep-th/9808143] [INSPIRE].
[47]	O. Aharony and Y. Tachikawa, A holographic computation of the central charges of D = 4, N = 2 SCFTs, JHEP01 (2008) 037 [arXiv:0711.4532] [INSPIRE].
[48]	J. Erdmenger, N. Evans, I. Kirsch and E. Threlfall, Mesons in gauge/gravity duals — a review, Eur. Phys. J.A 35 (2008) 81 [arXiv:0711.4467] [INSPIRE].
[49]	E.I. Buchbinder, J. Gomis and F. Passerini, Holographic gauge theories in background fields and surface operators, JHEP12 (2007) 101 [arXiv:0710.5170] [INSPIRE]. · Zbl 1246.81356
[50]	J.A. Harvey and A.B. Royston, Gauge/gravity duality with a chiral N = (0, 8) string defect, JHEP08 (2008) 006 [arXiv:0804.2854] [INSPIRE].
[51]	J.A. Harvey and A.B. Royston, Localized modes at a D-brane-O-plane intersection and heterotic Alice atrings, JHEP04 (2008) 018 [arXiv:0709.1482] [INSPIRE]. · Zbl 1246.81340
[52]	E. D’Hoker, M. Gutperle, A. Karch and C.F. Uhlemann, Warped AdS6× S2in type IIB supergravity I: local solutions, JHEP08 (2016) 046 [arXiv:1606.01254] [INSPIRE].
[53]	E. D’Hoker, M. Gutperle and C.F. Uhlemann, Warped AdS6× S2in type IIB supergravity II: global solutions and five-brane webs, JHEP05 (2017) 131 [arXiv:1703.08186] [INSPIRE]. · Zbl 1380.83282
[54]	D. Bak, M. Gutperle and S. Hirano, A dilatonic deformation of AdS5and its field theory dual, JHEP05 (2003) 072 [hep-th/0304129] [INSPIRE].
[55]	E. D’Hoker, J. Estes and M. Gutperle, Exact half-BPS type IIB interface solutions I. Local solution and supersymmetric Janus, JHEP06 (2007) 021 [arXiv:0705.0022] [INSPIRE].
[56]	A.B. Clark, D.Z. Freedman, A. Karch and M. Schnabl, Dual of the Janus solution: an interface conformal field theory, Phys. Rev.D 71 (2005) 066003 [hep-th/0407073] [INSPIRE]. · Zbl 1096.83069
[57]	C. Bachas, E. D’Hoker, J. Estes and D. Krym, M-theory solutions invariant under D(2, 1; γ) ⊕ D(2, 1; γ), Fortsch. Phys.62 (2014) 207 [arXiv:1312.5477] [INSPIRE]. · Zbl 1338.81310
[58]	D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the θ-angle in N = 4 super Yang-Mills theory, JHEP06 (2010) 097 [arXiv:0804.2907] [INSPIRE]. · Zbl 1290.81065
[59]	F. Denef, Les Houches lectures on constructing string vacua, in String theory and the real world: from particle physics to astrophysics. Proceedings, Summer School in Theoretical Physics, 87thSession, Les Houches France, 2-27 July 2007, pg. 483 [arXiv:0803.1194] [INSPIRE]. · Zbl 0934.81061
[60]	T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav.27 (2010) 214004 [arXiv:1009.3497] [INSPIRE]. · Zbl 1204.83007
[61]	J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yau’s, JHEP11 (2011) 098 [arXiv:1108.1794] [INSPIRE]. · Zbl 1306.81258
[62]	B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau, Stringy cosmic strings and noncompact Calabi-Yau manifolds, Nucl. Phys.B 337 (1990) 1 [INSPIRE]. · Zbl 0744.53045
[63]	M. Gross and P.M.H. Wilson, Large complex structure limits of K3 surfaces, J. Diff. Geom.55 (2000) 475. · Zbl 1027.32021
[64]	P.S. Aspinwall et al., Dirichlet branes and mirror symmetry, in Clay Mathematics Monographs4, American Mathematical Society, Providence RI U.S.A., (2009) [INSPIRE]. · Zbl 1188.14026
[65]	R. Wazir, Arithmetic on elliptic threefolds, Compos. Math.140 (2004) 567. · Zbl 1060.11039
[66]	K. Kodaira, On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. Math.60 (1954) 28. · Zbl 0057.14102
[67]	K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys.B 497 (1997) 56 [hep-th/9702198] [INSPIRE]. · Zbl 0934.81061
[68]	T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP03 (2012) 027 [arXiv:1111.1232] [INSPIRE]. · Zbl 1309.81218
[69]	H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box graphs and singular fibers, JHEP05 (2014) 048 [arXiv:1402.2653] [INSPIRE]. · Zbl 1333.81369
[70]	A. Schwimmer and N. Seiberg, Comments on the N = 2, N = 3, N = 4 superconformal algebras in two-dimensions, Phys. Lett.B 184 (1987) 191 [INSPIRE].
[71]	J.H. Schwarz, Covariant field equations of chiral N = 2 D = 10 supergravity, Nucl. Phys.B 226 (1983) 269 [INSPIRE].
[72]	P.S. Howe and P.C. West, The complete N = 2, D = 10 supergravity, Nucl. Phys.B 238 (1984) 181 [INSPIRE].
[73]	C. Voisin, Hodge theory and complex algebraic geometry I, in Cambridge Studies in Advanced Mathematics76, Cambridge University Press, Cambridge U.K., (2007). · Zbl 1129.14019
[74]	J.P. Gauntlett, D. Martelli, S. Pakis and D. Waldram, G structures and wrapped NS5-branes, Commun. Math. Phys.247 (2004) 421 [hep-th/0205050] [INSPIRE]. · Zbl 1061.81058
[75]	J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE]. · Zbl 0969.81047
[76]	H. Ooguri and C. Vafa, Summing up D instantons, Phys. Rev. Lett.77 (1996) 3296 [hep-th/9608079] [INSPIRE]. · Zbl 0944.81528
[77]	A.W. Peet, TASI lectures on black holes in string theory, in Strings, branes and gravity. Proceedings, Theoretical Advanced Study Institute, TASI’99, Boulder U.S.A., 31 May-25 June 1999, pg. 353 [hep-th/0008241] [INSPIRE]. · Zbl 1004.83002
[78]	J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys.104 (1986) 207 [INSPIRE]. · Zbl 0584.53039
[79]	P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP01 (2006) 022 [hep-th/0508218] [INSPIRE].
[80]	M.B. Green, J.A. Harvey and G.W. Moore, I-brane inflow and anomalous couplings on D-branes, Class. Quant. Grav.14 (1997) 47 [hep-th/9605033] [INSPIRE]. · Zbl 0867.53063
[81]	J. Hansen and P. Kraus, Generating charge from diffeomorphisms, JHEP12 (2006) 009 [hep-th/0606230] [INSPIRE]. · Zbl 1226.83062
[82]	A. Dabholkar, J. Gomes, S. Murthy and A. Sen, Supersymmetric index from black hole entropy, JHEP04 (2011) 034 [arXiv:1009.3226] [INSPIRE]. · Zbl 1250.81105
[83]	S.T. Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A.74 (1977) 1798 [INSPIRE]. · Zbl 0355.32028
[84]	E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys.22 (1997) 1 [hep-th/9609122] [INSPIRE]. · Zbl 0908.53065
[85]	P. Kraus and F. Larsen, Microscopic black hole entropy in theories with higher derivatives, JHEP09 (2005) 034 [hep-th/0506176] [INSPIRE].
[86]	A.A. Tseytlin, R4terms in 11 dimensions and conformal anomaly of (2, 0) theory, Nucl. Phys.B 584 (2000) 233 [hep-th/0005072] [INSPIRE]. · Zbl 0984.81147
[87]	M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string-string duality: a one loop test, Nucl. Phys.B 452 (1995) 261 [hep-th/9506126] [INSPIRE]. · Zbl 0925.81148
[88]	R. Bott and A.S. Cattaneo, Integral invariants of 3-manifolds II, J. Diff. Geom.53 (1999) 1. · Zbl 1036.57500
[89]	A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett.B 379 (1996) 99 [hep-th/9601029] [INSPIRE]. · Zbl 1376.83026
[90]	E. Witten, Five-brane effective action in M-theory, J. Geom. Phys.22 (1997) 103 [hep-th/9610234] [INSPIRE]. · Zbl 0878.58063
[91]	D. Freed, J.A. Harvey, R. Minasian and G.W. Moore, Gravitational anomaly cancellation for M-theory five-branes, Adv. Theor. Math. Phys.2 (1998) 601 [hep-th/9803205] [INSPIRE]. · Zbl 0971.81152
[92]	K.A. Intriligator, Anomaly matching and a Hopf-Wess-Zumino term in 6d, N = (2, 0) field theories, Nucl. Phys.B 581 (2000) 257 [hep-th/0001205] [INSPIRE]. · Zbl 0984.81145
[93]	D.S. Berman and J.A. Harvey, The self-dual string and anomalies in the M 5-brane, JHEP11 (2004) 015 [hep-th/0408198] [INSPIRE].
[94]	H. Shimizu and Y. Tachikawa, Anomaly of strings of 6d N = (1, 0) theories, JHEP11 (2016) 165 [arXiv:1608.05894] [INSPIRE]. · Zbl 1390.81470
[95]	G. Lopes Cardoso, B. de Wit and T. Mohaupt, Macroscopic entropy formulae and nonholomorphic corrections for supersymmetric black holes, Nucl. Phys.B 567 (2000) 87 [hep-th/9906094] [INSPIRE]. · Zbl 0951.81039
[96]	N. Lambert, The M 5-brane on K3 × T2, JHEP02 (2008) 060 [arXiv:0712.3166] [INSPIRE].
[97]	P. Putrov, J. Song and W. Yan, (0, 4) dualities, JHEP03 (2016) 185 [arXiv:1505.07110] [INSPIRE].
[98]	Y. Lozano, N.T. Macpherson, J. Montero and E. Ó. Colgáin, New AdS3× S2T-duals with N = (0,4) supersymmetry, JHEP08(2015) 121[arXiv:1507.02659] [INSPIRE]. · Zbl 1388.81861
[99]	L. Wulff, All symmetric space solutions of eleven-dimensional supergravity, J. Phys.A 50 (2017) 245401 [arXiv:1611.06139] [INSPIRE]. · Zbl 1372.83076
[100]	S. Schäfer-Nameki and T. Weigand, F-theory and 2d (0, 2) theories, JHEP05 (2016) 059 [arXiv:1601.02015] [INSPIRE]. · Zbl 1388.81189
[101]	F. Apruzzi, F. Hassler, J.J. Heckman and I.V. Melnikov, UV completions for non-critical strings, JHEP07 (2016) 045 [arXiv:1602.04221] [INSPIRE]. · Zbl 1390.81296
[102]	A. Fayyazuddin and M. Spalinski, Large-N superconformal gauge theories and supergravity orientifolds, Nucl. Phys.B 535 (1998) 219 [hep-th/9805096] [INSPIRE]. · Zbl 1080.81571
[103]	A. Kehagias, New type IIB vacua and their F-theory interpretation, Phys. Lett.B 435 (1998) 337 [hep-th/9805131] [INSPIRE].
[104]	J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP05 (2014) 028 [Erratum ibid.06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
[105]	H. Lü, C.N. Pope and J. Rahmfeld, A construction of Killing spinors on Sn, J. Math. Phys.40 (1999) 4518 [hep-th/9805151] [INSPIRE]. · Zbl 0979.53053
[106]	M.F. Sohnius, Introducing supersymmetry, Phys. Rept.128 (1985) 39 [INSPIRE].
[107]	V. Balasubramanian, E.G. Gimon, D. Minic and J. Rahmfeld, Four-dimensional conformal supergravity from AdS space, Phys. Rev.D 63 (2001) 104009 [hep-th/0007211] [INSPIRE].
[108]	T. Hubsch, Calabi-Yau manifolds: a bestiary for physicists, World Scientific, Singapore, (1994) [INSPIRE]. · Zbl 0771.53002
[109]	C. Voisin, Hodge theory and complex algebraic geometry II, in Cambridge Studies in Advanced Mathematics77, Cambridge University Press, Cambridge U.K., (2007). · Zbl 1129.14020
[110]	Y. Nakai, A criterion of an ample sheaf on a projective scheme, Amer. J. Math.85 (1963) 14. · Zbl 0112.13102
[111]	B.G. Moishezon, A projectivity criterion of complete algebraic abstract varieties, Izv. Akad. Nauk SSSR Ser. Mat.28 (1964) 179.
[112]	R. Lazarsfeld, Positivity in algebraic geometry I, in A Series of Modern Surveys in Mathematics48, Springer-Verlag, Berlin Germany, (2004). · Zbl 1066.14021
[113]	R. Hartshorne, Algebraic geometry, Springer-Verlag, New York U.S.A. and Heidelberg Germany, (1977). · Zbl 0367.14001