Abstract
Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (−)-connection on 6-dimensional non-Kähler nilmanifold are presented. In the conformally compact case the dilaton is determined by the real slices of the elliptic Weierstrass function. The dilaton of non-compact complete solutions is given by the fundamental solution of the Laplacian on R 4. All solutions satisfy the heterotic equations of motion up to the first order of α ′.
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P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].
J.P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004) 086002 [hep-th/0302158] [INSPIRE].
J. Li, S.-T. Yau, The existence of supersymmetric string theory with torsion, J. Diff. Geom. 70 (2005) 143.
J.-X. Fu and S.-T. Yau, Existence of supersymmetric Hermitian metrics with torsion on non-Kähler manifolds, hep-th/0509028 [INSPIRE].
G. Lopes Cardoso et al., Non-Kähler string backgrounds and their five torsion classes, Nucl. Phys. B 652 (2003) 5 [hep-th/0211118] [INSPIRE].
K. Dasgupta, H. Firouzjahi and R. Gwyn, On the warped heterotic axion, JHEP 06 (2008) 056 [arXiv:0803.3828] [INSPIRE].
C.M. Hull, Anomalies, ambiguities and superstrings, Phys. Lett. B 167 (1986) 51 [INSPIRE].
E.A. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].
J.-X. Fu and S.-T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampere equation, J. Diff. Geom. 78 (2009) 369 [hep-th/0604063] [INSPIRE].
E. Goldstein and S. Prokushkin, Geometric model for complex nonKähler manifolds with SU(3) structure, Commun. Math. Phys. 251 (2004) 65 [hep-th/0212307] [INSPIRE].
S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31 (1978) 339.
K. Uhlenbeck, S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986) S257.
S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1 [INSPIRE].
K. Becker and S. Sethi, Torsional heterotic geometries, Nucl. Phys. B 820 (2009) 1 [arXiv:0903.3769] [INSPIRE].
K. Becker, C. Bertinato, Y.-C. Chung and G. Guo, Supersymmetry breaking, heterotic strings and fluxes, Nucl. Phys. B 823 (2009) 428 [arXiv:0904.2932] [INSPIRE].
C.G. Callan Jr., J.A. Harvey and A. Strominger, Worldbrane actions for string solitons, Nucl. Phys. B 367 (1991) 60 [INSPIRE].
C.G. Callan Jr., J.A. Harvey and A. Strominger, World sheet approach to heterotic instantons and solitons, Nucl. Phys. B 359 (1991) 611 [INSPIRE].
M.J. Duff and J.X. Lu, Elementary five-brane solutions of D = 10 supergravity, Nucl. Phys. B 354 (1991) 141 [INSPIRE].
A. Strominger, Heterotic solitons, Nucl. Phys. B 343 (1990) 167 [Erratum ibid. B 353 (1991) 565] [INSPIRE].
J.-X. Fu, L.-S. Tseng and S.-T. Yau, Local heterotic torsional models, Commun. Math. Phys. 289 (2009) 1151 [arXiv:0806.2392] [INSPIRE].
M. Fernández, S. Ivanov, L. Ugarte and R. Villacampa, Non-Kähler heterotic string compactifications with non-zero fluxes and constant dilaton, Commun. Math. Phys. 288 (2009) 677 [arXiv:0804.1648] [INSPIRE].
T. Kimura and S. Mizoguchi, Chiral generations on intersecting 5-branes in heterotic string theory, JHEP 04 (2010) 028 [arXiv:0912.1334] [INSPIRE].
H. Imazato, S. Mizoguchi and M. Yata, Taub-NUT crystal, Int. J. Mod. Phys. A 26 (2011) 5143 [arXiv:1107.3557] [INSPIRE].
S. Mizoguchi and M. Yata, Family unification via quasi-Nambu-Goldstone fermions in string theory, PTEP 2013 (2013) 053B01 [arXiv:1211.6135] [INSPIRE].
A. Sen, A Note on enhanced gauge symmetries in M and string theory, JHEP 09 (1997) 001 [hep-th/9707123] [INSPIRE].
A. Hanany and B. Kol, On orientifolds, discrete torsion, branes and M-theory, JHEP 06 (2000) 013 [hep-th/0003025] [INSPIRE].
A. Hanany and B. Pioline, (Anti-)instantons and the Atiyah-Hitchin manifold, JHEP 07 (2000) 001 [hep-th/0005160] [INSPIRE].
S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].
C.M. Hull and P.K. Townsend, The two loop β-function for σ models with torsion, Phys. Lett. B 191 (1987) 115 [INSPIRE].
C.M. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357 [INSPIRE].
J. Gillard, G. Papadopoulos and D. Tsimpis, Anomaly, fluxes and (2,0) heterotic string compactifications, JHEP 06 (2003) 035 [hep-th/0304126] [INSPIRE].
C.M. Hull and E. Witten, Supersymmetric σ-models and the heterotic string, Phys. Lett. B 160 (1985) 398 [INSPIRE].
P.S. Howe and G. Papadopoulos, Ultraviolet behavior of two-dimensional supersymmetric nonlinear σ models, Nucl. Phys. B 289 (1987) 264 [INSPIRE].
B. de Wit, D.J. Smit and N.D. Hari Dass, Residual supersymmetry of compactified D = 10 supergravity, Nucl. Phys. B 283 (1987) 165 [INSPIRE].
J.P. Gauntlett, N. Kim, D. Martelli and D. Waldram, Five-branes wrapped on SLAG three cycles and related geometry, JHEP 11 (2001) 018 [hep-th/0110034] [INSPIRE].
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].
G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, BPS action and superpotential for heterotic string compactifications with fluxes, JHEP 10 (2003) 004 [hep-th/0306088] [INSPIRE].
K. Becker, M. Becker, K. Dasgupta and P.S. Green, Compactifications of heterotic theory on nonKähler complex manifolds. 1, JHEP 04 (2003) 007 [hep-th/0301161] [INSPIRE].
K. Becker, M. Becker, P.S. Green, K. Dasgupta and E. Sharpe, Compactifications of heterotic strings on non-Kähler complex manifolds. 2, Nucl. Phys. B 678 (2004) 19 [hep-th/0310058] [INSPIRE].
K. Becker, M. Becker, K. Dasgupta and S. Prokushkin, Properties of heterotic vacua from superpotentials, Nucl. Phys. B 666 (2003) 144 [hep-th/0304001] [INSPIRE].
K. Becker, M. Becker, J.-X. Fu, L.-S. Tseng and S.-T. Yau, Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory, Nucl. Phys. B 751 (2006) 108 [hep-th/0604137] [INSPIRE].
U. Gran and G. Papadopoulos, Solution of heterotic Killing spinor equations and special geometry, AIP Conf. Proc. 1093 (2009) 144 [arXiv:0811.1539] [INSPIRE].
U. Gran, G. Papadopoulos and D. Roest, Supersymmetric heterotic string backgrounds, Phys. Lett. B 656 (2007) 119 [arXiv:0706.4407] [INSPIRE].
U. Gran, G. Papadopoulos, D. Roest and P. Sloane, Geometry of all supersymmetric type-I backgrounds, JHEP 08 (2007) 074 [hep-th/0703143] [INSPIRE].
U. Gran, P. Lohrmann and G. Papadopoulos, The spinorial geometry of supersymmetric heterotic string backgrounds, JHEP 02 (2006) 063 [hep-th/0510176] [INSPIRE].
G. Papadopoulos, New half supersymmetric solutions of the heterotic string, Class. Quant. Grav. 26 (2009) 135001 [arXiv:0809.1156] [INSPIRE].
B. Andreas and M. Garcia-Fernandez, Heterotic non-Kähler geometries via polystable bundles on Calabi-Yau threefolds, J. Geom. Phys. 62 (2012) 183 [arXiv:1011.6246] [INSPIRE].
B. Andreas and M. Garcia-Fernandez, Solutions of the Strominger system via stable bundles on Calabi-Yau threefolds, Commun. Math. Phys. 315 (2012) 153 [arXiv:1008.1018] [INSPIRE].
I. Biswas and A. Mukherjee, Solutions of Strominger system from unitary representations of cocompact lattices of SL(2, C), Commun. Math. Phys. 322 (2013) 373 [arXiv:1301.0375] [INSPIRE].
A. Sen, (2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory, Nucl. Phys. B 278 (1986) 289 [INSPIRE].
P. Ivanov and S. Ivanov, SU(3) instantons and G 2 , Spin(7) heterotic string solitons, Commun. Math. Phys. 259 (2005) 79 [math/0312094] [INSPIRE].
T. Kimura and P. Yi, Comments on heterotic flux compactifications, JHEP 07 (2006) 030 [hep-th/0605247] [INSPIRE].
D. Martelli and J. Sparks, Non-Kähler heterotic rotations, Adv. Theor. Math. Phys. 15 (2011) 131 [arXiv:1010.4031] [INSPIRE].
K. Dasgupta, G. Rajesh and S. Sethi, M theory, orientifolds and G-flux, JHEP 08 (1999) 023 [hep-th/9908088] [INSPIRE].
S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010) 190 [arXiv:0908.2927] [INSPIRE].
S. Ivanov and G. Papadopoulos, Vanishing theorems and string backgrounds, Class. Quant. Grav. 18 (2001) 1089 [math/0010038] [INSPIRE].
S. Ivanov and G. Papadopoulos, A no go theorem for string warped compactifications, Phys. Lett. B 497 (2001) 309 [hep-th/0008232] [INSPIRE].
S. Chiossi and S. Salamon, The intrinsic torsion of SU(3) and G 2 -structures, World Scientific, Singapore (2002).
L. Ugarte and R. Villacampa, Balanced hermitian geometry on 6-dimensional nilmanifolds, arXiv:1104.5524 [INSPIRE].
L. Ugarte and R. Villacampa, Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, arXiv:0912.5110 [INSPIRE].
M. Fernández, A. Tomassini, L. Ugarte and R. Villacampa, Balanced Hermitian metrics from SU(2)-structures, J. Math. Phys. 50 (2009) 033507 [INSPIRE].
A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions. Volumes I, II and III, Robert E. Krieger Publishing Co. Inc., Melbourne, U.S.A. (1981).
L.V. Ahlfors, Complex analysis: an introduction of the theory of analytic functions of one complex variable, McGraw-Hill Book Co., U.S.A. (1966).
M.F. Atiyah and N. Hitchin, The geometry and dynamics of magnetic monopoles, M.B. Porter Lectures, Princeton University Press, Princeton U.S.A. (1988).
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Fernández, M., Ivanov, S., Ugarte, L. et al. Non-Kaehler heterotic string solutions with non-zero fluxes and non-constant dilaton. J. High Energ. Phys. 2014, 73 (2014). https://doi.org/10.1007/JHEP06(2014)073
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DOI: https://doi.org/10.1007/JHEP06(2014)073