Abstract
We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an SU(3) × Spin(6 + n) structure within O(6, 6 + n) × ℝ+ generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines a natural Hitchin functional for heterotic geometries. Intriguingly, this formulation suggests new connections to geometric invariant theory and an extended notion of stability. Finally we show that the analysis of infinitesimal deformations of these geometric structures naturally reproduces the known cohomologies that count the massless moduli of supersymmetric heterotic backgrounds.
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References
C. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357.
A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].
A. Adams, M. Ernebjerg and J.M. Lapan, Linear models for flux vacua, Adv. Theor. Math. Phys. 12 (2008) 817 [hep-th/0611084] [INSPIRE].
M. Kreuzer, J. McOrist, I.V. Melnikov and M. Plesser, (0, 2) deformations of linear σ-models, JHEP 07 (2011) 044 [arXiv:1001.2104] [INSPIRE].
J. McOrist, The revival of (0, 2) linear σ-models, Int. J. Mod. Phys. A 26 (2011) 1 [arXiv:1010.4667] [INSPIRE].
M. Beccaria, M. Kreuzer and A. Puhm, Counting charged massless states in the (0, 2) heterotic CFT/geometry connection, JHEP 01 (2011) 077 [arXiv:1010.4564] [INSPIRE].
J. McOrist and I.V. Melnikov, Old issues and linear σ-models, Adv. Theor. Math. Phys. 16 (2012) 251 [arXiv:1103.1322] [INSPIRE].
I.V. Melnikov and E. Sharpe, On marginal deformations of (0, 2) non-linear σ-models, Phys. Lett. B 705 (2011) 529 [arXiv:1110.1886] [INSPIRE].
M. Blaszczyk, S. Groot Nibbelink and F. Ruehle, Green-Schwarz mechanism in heterotic (2, 0) gauged linear σ-models: torsion and NS5 branes, JHEP 08 (2011) 083 [arXiv:1107.0320] [INSPIRE].
C. Quigley and S. Sethi, Linear σ-models with torsion, JHEP 11 (2011) 034 [arXiv:1107.0714] [INSPIRE].
S. Groot Nibbelink and L. Horstmeyer, Super Weyl invariance: BPS equations from heterotic worldsheets, JHEP 07 (2012) 054 [arXiv:1203.6827] [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 nonKähler complex manifolds. 2, Nucl. Phys. B 678 (2004) 19 [hep-th/0310058] [INSPIRE].
K. Becker and L.-S. Tseng, Heterotic flux compactifications and their moduli, Nucl. Phys. B 741 (2006) 162 [hep-th/0509131] [INSPIRE].
A. Chatzistavrakidis, O. Lechtenfeld and A.D. Popov, Nearly Kähler heterotic compactifications with fermion condensates, JHEP 04 (2012) 114 [arXiv:1202.1278] [INSPIRE].
D. Lüst, Compactification of ten-dimensional superstring theories over Ricci flat coset spaces, Nucl. Phys. B 276 (1986) 220 [INSPIRE].
E. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439.
T. Friedrich and S. Ivanov, Parallel spinors and connections with skew symmetric torsion in string theory, Asian J. Math. 6 (2002) 303 [math/0102142] [INSPIRE].
K. Becker and K. Dasgupta, Heterotic strings with torsion, JHEP 11 (2002) 006 [hep-th/0209077] [INSPIRE].
J.P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004) 086002 [hep-th/0302158] [INSPIRE].
G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, Heterotic string theory on nonKähler manifolds with H flux and gaugino condensate, Fortsch. Phys. 52 (2004) 483 [hep-th/0310021] [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, 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].
M. Garcia-Fernandez, Torsion-free generalized connections and Heterotic Supergravity, Commun. Math. Phys. 332 (2014) 89 [arXiv:1304.4294] [INSPIRE].
J. Li and S.T. Yau, The existence of supersymmetric string theory with torsion, J. Diff. Geom. 70 (2005) 143.
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 (2008) 369 [hep-th/0604063] [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].
K. Becker and S. Sethi, Torsional heterotic geometries, Nucl. Phys. B 820 (2009) 1 [arXiv:0903.3769] [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].
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
V. Braun, P. Candelas, R. Davies and R. Donagi, The MSSM spectrum from (0, 2)-deformations of the heterotic standard embedding, JHEP 05 (2012) 127 [arXiv:1112.1097] [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A standard model from the E8 × E8 heterotic superstring, JHEP 06 (2005) 039 [hep-th/0502155] [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, The exact MSSM spectrum from string theory, JHEP 05 (2006) 043 [hep-th/0512177] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing the complex structure in heterotic Calabi-Yau vacua, JHEP 02 (2011) 088 [arXiv:1010.0255] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Two hundred heterotic standard models on smooth Calabi-Yau threefolds, Phys. Rev. D 84 (2011) 106005 [arXiv:1106.4804] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic line bundle standard models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing all geometric moduli in heterotic Calabi-Yau vacua, Phys. Rev. D 83 (2011) 106011 [arXiv:1102.0011] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, The Atiyah class and complex structure stabilization in heterotic Calabi-Yau compactifications, JHEP 10 (2011) 032 [arXiv:1107.5076] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stability walls in heterotic theories, JHEP 09 (2009) 026 [arXiv:0905.1748] [INSPIRE].
X. de la Ossa and E.E. Svanes, Holomorphic bundles and the moduli space of N = 1 supersymmetric heterotic compactifications, JHEP 10 (2014) 123 [arXiv:1402.1725] [INSPIRE].
L.B. Anderson, J. Gray and E. Sharpe, Algebroids, heterotic moduli spaces and the Strominger system, JHEP 07 (2014) 037 [arXiv:1402.1532] [INSPIRE].
X. de la Ossa, E. Hardy and E.E. Svanes, The heterotic superpotential and moduli, JHEP 01 (2016) 049 [arXiv:1509.08724] [INSPIRE].
M. Garcia-Fernandez, R. Rubio and C. Tipler, Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry, Math. Ann. 369 (2017) 2 [arXiv:1503.07562] [INSPIRE].
M. Becker, L.-S. Tseng and S.-T. Yau, Moduli space of torsional manifolds, Nucl. Phys. B 786 (2007) 119 [hep-th/0612290] [INSPIRE].
M. Cyrier and J.M. Lapan, Towards the massless spectrum of non-Kähler heterotic compactifications, Adv. Theor. Math. Phys. 10 (2006) 853 [hep-th/0605131] [INSPIRE].
J. McOrist and R. Sisca, Small gauge transformations and universal geometry in heterotic theories, arXiv:1904.07578 [INSPIRE].
A. Ashmore, X. De La Ossa, R. Minasian, C. Strickland-Constable and E.E. Svanes, Finite deformations from a heterotic superpotential: holomorphic Chern-Simons and an L∞ algebra, JHEP 10 (2018) 179 [arXiv:1806.08367] [INSPIRE].
M. Garcia-Fernandez, R. Rubio and C. Tipler, Holomorphic string algebroids, Trans. Am. Math. Soc. 373 (2020) 7347 [arXiv:1807.10329] [INSPIRE].
M. Garcia-Fernandez, R. Rubio, C. Shahbazi and C. Tipler, Canonical metrics on holomorphic Courant algebroids, arXiv:1803.01873 [INSPIRE].
I. Benmachiche, J. Louis and D. Martinez-Pedrera, The effective action of the heterotic string compactified on manifolds with SU(3) structure, Class. Quant. Grav. 25 (2008) 135006 [arXiv:0802.0410] [INSPIRE].
S. Gurrieri, A. Lukas and A. Micu, Heterotic on half-flat, Phys. Rev. D 70 (2004) 126009 [hep-th/0408121] [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].
A. Ashmore, C. Strickland-Constable, D. Tennyson and D. Waldram, Generalising G2 geometry: involutivity, moment maps and moduli, arXiv:1910.04795 [INSPIRE].
A. Ashmore and D. Waldram, Exceptional Calabi-Yau spaces: the geometry of \( \mathcal{N} \) = 2 backgrounds with flux, Fortsch. Phys. 65 (2017) 1600109 [arXiv:1510.00022] [INSPIRE].
A. Coimbra and C. Strickland-Constable, Supersymmetric backgrounds, the Killing superalgebra, and generalised special holonomy, JHEP 11 (2016) 063 [arXiv:1606.09304] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry II: Ed(d) × ℝ+ and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Ed(d) × ℝ+ generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supersymmetric backgrounds and generalised special holonomy, Class. Quant. Grav. 33 (2016) 125026 [arXiv:1411.5721] [INSPIRE].
M. Graña, J. Louis, A. Sim and D. Waldram, E7(7) formulation of N = 2 backgrounds, JHEP 07 (2009) 104 [arXiv:0904.2333] [INSPIRE].
P. Pires Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].
C. Hull and U. Lindström, The generalised complex geometry of (p, q) Hermitian geometries, Commun. Math. Phys. 375 (2019) 479 [arXiv:1810.06489] [INSPIRE].
A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014) 160 [arXiv:1407.7542] [INSPIRE].
C. Hull and U. Lindström, Strong Kähler with torsion as generalised geometry, PoS CORFU2018 (2019) 121 [arXiv:1904.03412] [INSPIRE].
M. Garcia-Fernandez, R. Rubio and C. Tipler, Gauge theory for string algebroids, arXiv:2004.11399 [INSPIRE].
C.M. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357 [INSPIRE].
S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010) 190 [arXiv:0908.2927] [INSPIRE].
D. Martelli and J. Sparks, Non-Kähler heterotic rotations, Adv. Theor. Math. Phys. 15 (2011) 131 [arXiv:1010.4031] [INSPIRE].
S. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1.
K. Uhlenbeck and S.T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986) 257.
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford University Press, Oxford U.K. (2003) [math/0401221] [INSPIRE].
P. Ševera, Letters to Alan Weinstein about Courant algebroids, arXiv:1707.00265 [INSPIRE].
O. Hohm and S.K. Kwak, Double field theory formulation of heterotic strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].
J. McOrist, On the effective field theory of heterotic vacua, Lett. Math. Phys. 108 (2018) 1031 [arXiv:1606.05221] [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. 608 (2001) 477] [hep-th/9906070] [INSPIRE].
S. Gukov, Solitons, superpotentials and calibrations, Nucl. Phys. B 574 (2000) 169 [hep-th/9911011] [INSPIRE].
A. Borel, Kählerian coset spaces of semisimple Lie groups, Proc. Natl. Acad. Sci. 40 (1954) 1147.
S. Gurrieri, J. Louis, A. Micu and D. Waldram, Mirror symmetry in generalized Calabi-Yau compactifications, Nucl. Phys. B 654 (2003) 61 [hep-th/0211102] [INSPIRE].
S. Gurrieri, A. Lukas and A. Micu, Heterotic string compactifications on half-flat manifolds. II., JHEP 12 (2007) 081 [arXiv:0709.1932] [INSPIRE].
P. Candelas, X. de la Ossa and J. McOrist, A metric for heterotic moduli, Commun. Math. Phys. 356 (2017) 567 [arXiv:1605.05256] [INSPIRE].
A. Strominger, Yukawa couplings in superstring compactification, Phys. Rev. Lett. 55 (1985) 2547 [INSPIRE].
L.J. Dixon, V. Kaplunovsky and J. Louis, On effective field theories describing (2, 2) vacua of the heterotic string, Nucl. Phys. B 329 (1990) 27 [INSPIRE].
P. Candelas and X. de la Ossa, Moduli space of Calabi-Yau manifolds, Nucl. Phys. B 355 (1991) 455 [INSPIRE].
J.P. Derendinger, L.E. Ibáñez and H.P. Nilles, On the low-energy limit of superstring theories, Nucl. Phys. B 267 (1986) 365 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
R. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, math/0512411.
M.F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957) 181.
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 (1989) 703.
A. Fujiki, Moduli space of polarized algebraic manifolds and Kähler metrics, Sugaku Exp. 5 (1992) 173.
S. Donaldson, Remarks on gauge theory, complex geometry and 4-manifold topology, World Scienfitic Series on 20th Century Mathematics volume 5, World Scientific, Singapore (1997).
X.-X. Chen, S. Donaldson and S. Sun, Kähler-Einstein metrics and stability, arXiv:1210.7494.
S. Trautwein, A survey of the GIT picture for the Yang-Mills equation over Riemann surfaces, arXiv:1511.08122 [INSPIRE].
N.P. Buchdahl, Hermitian-Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann. 280 (1988) 625.
S.T. Yau and J. Li, Hermitian-Yang-Mills connections on non-Kähler manifolds, World Scientific, London U.K. (1987).
M. Garcia-Fernandez, Lectures on the Strominger system, Travaux Math. Special Issue XXIV (2016) 7.
M. Gualtieri, M. Matviichuk and G. Scott, Deformation of Dirac structures via L∞ algebras, Int. Math. Res. Not. 2020 (2020) 4295 [arXiv:1702.08837] [INSPIRE].
K. Dasgupta, G. Rajesh and S. Sethi, M theory, orientifolds and G-flux, JHEP 08 (1999) 023 [hep-th/9908088] [INSPIRE].
I.V. Melnikov, R. Minasian and S. Sethi, Heterotic fluxes and supersymmetry, JHEP 06 (2014) 174 [arXiv:1403.4298] [INSPIRE].
A. Ashmore, M. Petrini and D. Waldram, The exceptional generalised geometry of supersymmetric AdS flux backgrounds, JHEP 12 (2016) 146 [arXiv:1602.02158] [INSPIRE].
A. Ashmore, M. Gabella, M. Graña, M. Petrini and D. Waldram, Exactly marginal deformations from exceptional generalised geometry, JHEP 01 (2017) 124 [arXiv:1605.05730] [INSPIRE].
A. Ashmore, Marginal deformations of 3d \( \mathcal{N} \) = 2 CFTs from AdS4 backgrounds in generalised geometry, JHEP 12 (2018) 060 [arXiv:1809.03503] [INSPIRE].
P. Candelas, X. De La Ossa, J. McOrist and R. Sisca, The universal geometry of heterotic vacua, JHEP 02 (2019) 038 [arXiv:1810.00879] [INSPIRE].
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Ashmore, A., Strickland-Constable, C., Tennyson, D. et al. Heterotic backgrounds via generalised geometry: moment maps and moduli. J. High Energ. Phys. 2020, 71 (2020). https://doi.org/10.1007/JHEP11(2020)071
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DOI: https://doi.org/10.1007/JHEP11(2020)071