Heterotic Non-linear Sigma Models

  • Ilarion V. Melnikov
Part of the Lecture Notes in Physics book series (LNP, volume 951)


In this chapter we discuss the basic features of non-linear sigma models with (0,2) supersymmetry. This is a large universe, and to circumscribe our explorations we mainly stick to the theories relevant to compactifications of the heterotic string. To elucidate the geometric structures it turns out easiest to start with (0,1) supersymmetry. The reader may find it useful to skim through the geometry appendix before diving into the details of this chapter.


  1. 1.
    Adam, I.: On the marginal deformations of general (0,2) non-linear sigma-models. Proc. Symp. Pure Math. 90, 171–179 (2015). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 5.
    Adams, A., Ernebjerg, M., Lapan, J.M.: Linear models for flux vacua.
  3. 7.
    Affleck, I., Dine, M., Seiberg, N.: Dynamical supersymmetry breaking in supersymmetric QCD. Nucl. Phys. B241, 493–534 (1984)ADSCrossRefGoogle Scholar
  4. 8.
    Aldazabal, G., Ibanez, L.E.: A note on 4D heterotic string vacua, FI-terms and the swampland. Phys. Lett. B782, 375–379 (2018).; ADSzbMATHCrossRefGoogle Scholar
  5. 9.
    Alexandrov, S., Louis, J., Pioline, B., Valandro, R.: \(\mathcal N=2\) heterotic-type II duality and bundle moduli. J. High Energy Phys. 08, 092 (2014).;
  6. 10.
    Alvarez-Gaume, L., Ginsparg, P.H.: The structure of gauge and gravitational anomalies. Ann. Phys. 161, 423 (1985)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 11.
    Alvarez-Gaume, L., Witten, E.: Gravitational anomalies. Nucl. Phys. B234, 269 (1984). ADSMathSciNetCrossRefGoogle Scholar
  8. 12.
    Alvarez-Gaume, L., Freedman, D.Z., Mukhi, S.: The background field method and the ultraviolet structure of the supersymmetric nonlinear sigma model. Ann. Phys. 134, 85 (1981). ADSCrossRefGoogle Scholar
  9. 13.
    Anderson, L.B.: Heterotic and M-theory compactifications for string phenomenology. PhD thesis, Oxford University (2008).;
  10. 15.
    Anderson, L.B., Gray, J., Lukas, A., Palti, E.: Heterotic line bundle standard models. J. High Energy Phys. 06, 113 (2012).;
  11. 17.
    Anderson, L.B., Gray, J., Sharpe, E.: Algebroids, heterotic moduli spaces and the Strominger system.
  12. 18.
    Anderson, L.B., Gray, J., Lukas, A., Ovrut, B.: The Atiyah class and complex structure stabilization in heterotic Calabi-Yau compactifications.
  13. 19.
    Angelantonj, C., Israel, D., Sarkis, M.: Threshold corrections in heterotic flux compactifications. J. High Energy Phys. 08, 032 (2017).;
  14. 21.
    Angella, D., Ugarte, L.: On small deformations of balanced manifolds. Differ. Geom. Appl. 54(part B), 464–474 (2017)Google Scholar
  15. 24.
    Argyres, P.C.: An introduction to global supersymmetry. DIY (2000)Google Scholar
  16. 25.
    Argyres, P.C., Plesser, M.R., Seiberg, N.: The moduli space of vacua of N=2 SUSY QCD and duality in N=1 SUSY QCD. Nucl. Phys. B471, 159–194 (1996). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. 26.
    Argyres, P.C., Plesser, M.R., Shapere, A.D.: N=2 moduli spaces and N=1 dualities for SO(n(c)) and USp(2n(c)) superQCD. Nucl. Phys. B483, 172–186 (1997).;
  18. 28.
    Ashmore, A., De La Ossa, X., Minasian, R., Strickland-Constable, C., Svanes, E.E.: Finite deformations from a heterotic superpotential: holomorphic Chern–Simons and an L algebra.
  19. 29.
    Aspinwall, P.S.: A McKay-like correspondence for (0,2)-deformations.
  20. 30.
    Aspinwall, P.S.: D-branes on Calabi-Yau manifolds.
  21. 31.
    Aspinwall, P.S.: K3 surfaces and string duality.
  22. 32.
    Aspinwall, P.S.: The Moduli space of N=2 superconformal field theories.
  23. 33.
    Aspinwall, P.S., Gaines, B.: Rational curves and (0,2)-deformations. J. Geom. Phys. 88, 1–15 (2014).; ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. 34.
    Aspinwall, P.S., Morrison, D.R.: Chiral rings do not suffice: N=(2,2) theories with nonzero fundamental group. Phys. Lett. B334, 79–86 (1994).; ADSMathSciNetCrossRefGoogle Scholar
  25. 37.
    Aspinwall, P.S., Plesser, M.R.: Elusive worldsheet instantons in heterotic string compactifications.
  26. 40.
    Aspinwall, P.S., Bridgeland, T., Craw, A., Douglas, M.R., Kapustin, A., Moore, G.W., Gross, M., Segal, G., Szendroi, B., Wilson, P.M.H.: Dirichlet Branes and Mirror Symmetry. Clay Mathematics Monographs, vol. 4. AMS, Providence (2009).
  27. 41.
    Aspinwall, P.S., Melnikov, I.V., Plesser, M.R.: (0,2) elephants. J. High Energy Phys. 1201, 060 (2012).
  28. 43.
    Atiyah, M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85(1), 181–207 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 44.
    Atiyah, M., Hitchin, N.J., Singer, I.: Selfduality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. A362, 425–461 (1978)ADSzbMATHCrossRefGoogle Scholar
  30. 46.
    Banks, T., Dixon, L.J.: Constraints on string vacua with space-time supersymmetry. Nucl. Phys. B307, 93–108 (1988)ADSCrossRefGoogle Scholar
  31. 47.
    Banks, T., Seiberg, N.: Nonperturbative infinities. Nucl. Phys. B273, 157 (1986)ADSMathSciNetCrossRefGoogle Scholar
  32. 48.
    Banks, T., Dixon, L.J., Friedan, D., Martinec, E.J.: Phenomenology and conformal field theory or can string theory predict the weak mixing angle? Nucl. Phys. B299, 613–626 (1988)ADSMathSciNetCrossRefGoogle Scholar
  33. 49.
    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces, vol. 4, 2nd edn. Springer, Berlin (2004)zbMATHCrossRefGoogle Scholar
  34. 50.
    Basu, A., Sethi, S.: World-sheet stability of (0,2) linear sigma models. Phys. Rev. D68, 025003 (2003). ADSMathSciNetGoogle Scholar
  35. 57.
    Beasley, C., Witten, E.: Residues and world-sheet instantons. J. High Energy Phys. 10, 065 (2003). ADSMathSciNetCrossRefGoogle Scholar
  36. 59.
    Becker, K., Dasgupta, K.: Heterotic strings with torsion. J. High Energy Phys. 11, 006 (2002).; MathSciNetCrossRefGoogle Scholar
  37. 62.
    Becker, K., Becker, M., Fu, J.-X., Tseng, L.-S., Yau, S.-T.: Anomaly cancellation and smooth non-Kaehler solutions in heterotic string theory. Nucl. Phys. B751, 108–128 (2006). ADSzbMATHCrossRefGoogle Scholar
  38. 64.
    Behtash, A., Dunne, G.V., Schaefer, T., Sulejmanpasic, T., Unsal, M.: Complexified path integrals, exact saddles and supersymmetry. Phys. Rev. Lett. 116(1), 011601 (2016).;
  39. 73.
    Berglund, P., Candelas, P., de la Ossa, X., Derrick, E., Distler, J., et al.: On the instanton contributions to the masses and couplings of E(6) singlets. Nucl. Phys. B454, 127–163 (1995). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. 75.
    Bergshoeff, E., de Roo, M.: The Quartic effective action of the heterotic string and supersymmetry. Nucl. Phys. B328, 439 (1989). ADSMathSciNetCrossRefGoogle Scholar
  41. 78.
    Bershadsky, M., Intriligator, K.A., Kachru, S., Morrison, D.R., Sadov, V., et al.: Geometric singularities and enhanced gauge symmetries. Nucl. Phys. B481, 215–252 (1996). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 82.
    Bertolini, M., Romo, M.: Aspects of (2,2) and (0,2) hybrid models.
  43. 84.
    Bertolini, M., Melnikov, I.V., Plesser, M.R.: Hybrid conformal field theories.
  44. 91.
    Bogomolov, F.A.: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243(5), 1101–1104 (1978)MathSciNetGoogle Scholar
  45. 92.
    Bohr, C., Hanke, B., Kotschick, D.: Cycles, submanifolds, and structures on normal bundles. Manuscr. Math. 108(4), 483–494 (2002). MathSciNetzbMATHCrossRefGoogle Scholar
  46. 94.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)zbMATHCrossRefGoogle Scholar
  47. 98.
    Braun, V., Kreuzer, M., Ovrut, B.A., Scheidegger, E.: Worldsheet instantons and torsion curves. Part A: direct computation. J. High Energy Phys. 10, 022 (2007). ADSzbMATHCrossRefGoogle Scholar
  48. 100.
    Braun, V., Kreuzer, M., Ovrut, B.A., Scheidegger, E.: Worldsheet instantons and torsion curves.
  49. 103.
    Buchbinder, E., Lukas, A., Ovrut, B., Ruehle, F.: Heterotic instanton superpotentials from complete intersection Calabi-Yau manifolds. J. High Energy Phys. 10, 032 (2017).;
  50. 104.
    Buchbinder, E.I., Lin, L., Ovrut, B.A.: Non-vanishing heterotic superpotentials on elliptic fibrations.
  51. 105.
    Buchdahl, N.P.: Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann. 280(4), 625–648 (1988). MathSciNetzbMATHCrossRefGoogle Scholar
  52. 106.
    Buividovich, P.V., Dunne, G.V., Valgushev, S.N.: Complex path integrals and saddles in two-dimensional gauge theory. Phys. Rev. Lett. 116(13), 132001 (2016).;
  53. 107.
    Callan, J., Curtis, G., Martinec, E., Perry, M., Friedan, D.: Strings in background fields. Nucl. Phys. B262, 593 (1985). ADSMathSciNetCrossRefGoogle Scholar
  54. 108.
    Candelas, P., de la Ossa, X.: Moduli space of Calabi-Yau manifolds. Nucl. Phys. B355, 455–481 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. 109.
    Candelas, P., Horowitz, G.T., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys. B258, 46–74 (1985)ADSMathSciNetCrossRefGoogle Scholar
  56. 110.
    Candelas, P., De La Ossa, X.C., Green, P.S., Parkes, L.: A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B359, 21–74 (1991). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  57. 117.
    Coleman, S.R.: There are no Goldstone bosons in two-dimensions. Commun. Math. Phys. 31, 259–264 (1973). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. 120.
    Cox, D.A., Katz, S.: Mirror Symmetry and Algebraic Geometry, 469pp. AMS, Providence (2000)Google Scholar
  59. 124.
    Dai, X.-z., Freed, D. S.: Eta invariants and determinant lines. J. Math. Phys. 35, 5155–5194 (1994). [Erratum: J. Math. Phys. 42, 2343 (2001)].;
  60. 125.
    Dasgupta, K., Rajesh, G., Sethi, S.: M theory, orientifolds and G-flux. J. High Energy Phys. 08, 023 (1999). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  61. 127.
    de la Ossa, X., Svanes, E.E.: Holomorphic Bundles and the Moduli Space of N=1 Supersymmetric Heterotic Compactifications. J. High Energy Phys. 10, 123 (2014).;
  62. 128.
    de la Ossa, X., Svanes, E.E.: Connections, field redefinitions and heterotic supergravity. J. High Energy Phys. 12, 008 (2014).;
  63. 135.
    Dine, M., Lee, C.: Remarks on (0,2) models and intermediate scale scenarios in string theory. Phys. Lett. B203, 371–377 (1988)ADSCrossRefGoogle Scholar
  64. 137.
    Dine, M., Seiberg, N.: Are (0,2) models string miracles? Nucl. Phys. B306, 137 (1988)ADSMathSciNetCrossRefGoogle Scholar
  65. 138.
    Dine, M., Seiberg, N.: Microscopic knowledge from macroscopic physics in string theory. Nucl. Phys. B301, 357 (1988)ADSMathSciNetCrossRefGoogle Scholar
  66. 139.
    Dine, M., Seiberg, N., Wen, X.G., Witten, E.: Nonperturbative effects on the string world sheet. Nucl. Phys. B278, 769 (1986)ADSMathSciNetCrossRefGoogle Scholar
  67. 140.
    Dine, M., Seiberg, N., Wen, X.G., Witten, E.: Nonperturbative effects on the string world sheet. 2. Nucl. Phys. B289, 319 (1987)Google Scholar
  68. 141.
    Dine, M., Seiberg, N., Witten, E.: Fayet-Iliopoulos terms in string theory. Nucl. Phys. B289, 589 (1987)ADSMathSciNetCrossRefGoogle Scholar
  69. 142.
    Distler, J.: Resurrecting (2,0) compactifications. Phys. Lett. B188, 431–436 (1987)ADSMathSciNetCrossRefGoogle Scholar
  70. 144.
    Distler, J., Greene, B.R.: Aspects of (2,0) string compactifications. Nucl. Phys. B304, 1 (1988)ADSMathSciNetCrossRefGoogle Scholar
  71. 147.
    Distler, J., Sharpe, E.: Heterotic compactifications with principal bundles for general groups and general levels. Adv. Theor. Math. Phys. 14, 335–398 (2010). MathSciNetzbMATHCrossRefGoogle Scholar
  72. 148.
    Dixon, L.J.: Some world sheet properties of superstring compactifications, on orbifolds and otherwise. Lectures given at the 1987 ICTP Summer Workshop in High Energy Phsyics and Cosmology, Trieste, 29 June–7 Aug 1987Google Scholar
  73. 150.
    Dixon, L.J., Kaplunovsky, V., Louis, J.: On effective field theories describing (2,2) vacua of the heterotic string. Nucl. Phys. B329, 27–82 (1990)ADSMathSciNetCrossRefGoogle Scholar
  74. 155.
    Donaldson, S.K., Kronheimer, P.B.: The Geometry Of Four-Manifolds. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1990)zbMATHGoogle Scholar
  75. 161.
    Eguchi, T., Gilkey, P.B., Hanson, A.J.: Gravitation, gauge theories and differential geometry. Phys. Rep. 66, 213 (1980). ADSMathSciNetCrossRefGoogle Scholar
  76. 164.
    Fei, T., Huang, Z., Picard, S.: A construction of infinitely many solutions to the Strominger system.
  77. 165.
    Ferrara, S., Lust, D., Theisen, S.: World sheet versus spectrum symmetries in heterotic and type II superstrings. Nucl. Phys. B325, 501 (1989)ADSMathSciNetCrossRefGoogle Scholar
  78. 166.
    Florakis, I., Garcia-Etxebarria, I., Lust, D., Regalado, D.: 2d orbifolds with exotic supersymmetry.
  79. 168.
    Freed, D.: Determinants, torsion, and strings. Commun. Math. Phys. 107, 483–513 (1986). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  80. 169.
    Freed, D.S.: Special Kaehler manifolds. Commun. Math. Phys. 203, 31–52 (1999). ADSzbMATHCrossRefGoogle Scholar
  81. 170.
    Freed, D., Harvey, J.A.: Instantons and the spectrum of Bloch electrons in a magnetic field. Phys. Rev. B41, 11328 (1990). ADSMathSciNetCrossRefGoogle Scholar
  82. 172.
    Friedan, D.H.: Nonlinear models in two + epsilon dimensions. Ann. Phys. 163, 318 (1985). Ph.D. Thesis.
  83. 175.
    Friedan, D., Martinec, E.J., Shenker, S.H.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B271, 93 (1986)ADSMathSciNetCrossRefGoogle Scholar
  84. 176.
    Fu, J.-X., Yau, S.-T.: The theory of superstring with flux on non-Kaehler manifolds and the complex Monge-Ampere equation. J. Differ. Geom. 78, 369–428 (2009). zbMATHCrossRefGoogle Scholar
  85. 177.
    Fu, J., Yau, S.-T.: A note on small deformations of balanced manifolds. C. R. Math. Acad. Sci. Paris 349(13–14), 793–796 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 188.
    Garcia-Fernandez, M.: Lectures on the Strominger system.
  87. 190.
    Gauntlett, J.P., Martelli, D., Waldram, D.: Superstrings with intrinsic torsion. Phys. Rev. D69, 086002 (2004). ADSMathSciNetGoogle Scholar
  88. 192.
    Gepner, D.: Exactly solvable string compactifications on manifolds of SU(N) holonomy. Phys. Lett. B199, 380–388 (1987)ADSMathSciNetCrossRefGoogle Scholar
  89. 193.
    Gepner, D.: Space-time supersymmetry in compactified string theory and superconformal models. Nucl. Phys. B296, 757 (1988)ADSMathSciNetCrossRefGoogle Scholar
  90. 194.
    Gepner, D.: Lectures on N=2 string theory. Lectures at Spring School on Superstrings, Trieste, 3–14 Apr 1989Google Scholar
  91. 199.
    Goldstein, E., Prokushkin, S.: Geometric model for complex non-Kaehler manifolds with SU(3) structure. Commun. Math. Phys. 251, 65–78 (2004). ADSzbMATHCrossRefGoogle Scholar
  92. 202.
    Gomis, J., Komargodski, Z., Ooguri, H., Seiberg, N., Wang, Y.: Shortening anomalies in supersymmetric theories. J. High Energy Phys. 01, 067 (2017).;
  93. 203.
    Grana, M., Minasian, R., Petrini, M., Waldram, D.: T-duality, generalized geometry and non-geometric backgrounds. J. High Energy Phys. 04, 075 (2009).; MathSciNetCrossRefGoogle Scholar
  94. 204.
    Green, M.B., Seiberg, N.: Contact interactions in superstring theory. Nucl. Phys. B299, 559 (1988). ADSMathSciNetCrossRefGoogle Scholar
  95. 205.
    Green, M.B., Schwarz, J.H., West, P.C.: Anomaly free chiral theories in six-dimensions. Nucl. Phys. B254, 327–348 (1985). ADSMathSciNetCrossRefGoogle Scholar
  96. 206.
    Green, M., Schwarz, J., Witten, E.: Superstring Theory, Volume 1. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  97. 207.
    Green, M., Schwarz, J., Witten, E.: Superstring Theory, Volume 2. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  98. 209.
    Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys. B338, 15–37 (1990)ADSMathSciNetCrossRefGoogle Scholar
  99. 210.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)zbMATHGoogle Scholar
  100. 211.
    Grisaru, M.T., van de Ven, A., Zanon, D.: Two-dimensional supersymmetric sigma models on Ricci flat Kahler manifolds are not finite. Nucl. Phys. B277, 388 (1986). ADSCrossRefGoogle Scholar
  101. 212.
    Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  102. 213.
    Gross, D.J., Harvey, J.A., Martinec, E.J., Rohm, R.: Heterotic string theory. 1. The free heterotic string. Nucl. Phys. B256, 253 (1985). ADSMathSciNetCrossRefGoogle Scholar
  103. 220.
    Harlow, D., Maltz, J., Witten, E.: Analytic continuation of Liouville theory. J. High Energy Phys. 12, 071 (2011).;
  104. 223.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  105. 224.
    Hatcher, A.: Vector Bundles and K-Theory. Online, 2.1 edn. (2009)Google Scholar
  106. 228.
    Hitchin, N.J.: Lectures on special Lagrangian submanifolds. In: Proceedings, Winter School on Mirror Symmetry and Vector Bundles, Cambridge, MA, 4–15 Jan 1999, pp. 151–182. zbMATHGoogle Scholar
  107. 229.
    Hohenberg, P.C.: Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967). ADSCrossRefGoogle Scholar
  108. 231.
    Honecker, G.: Massive U(1)s and heterotic five-branes on K3. Nucl. Phys. B748, 126–148 (2006). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  109. 237.
    Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror Symmetry. Clay Mathematics Monographs, vol. 1. American Mathematical Society, Providence (2003). With a preface by VafaGoogle Scholar
  110. 239.
    Howe, P.S., Papadopoulos, G.: Anomalies in two-dimensional supersymmetric nonlinear sigma models. Class. Quantum Gravity 4, 1749–1766 (1987)ADSzbMATHCrossRefGoogle Scholar
  111. 240.
    Howe, P.S., Papadopoulos, G.: Further remarks on the geometry of two-dimensional nonlinear sigma models. Class. Quantum Gravity 5, 1647–1661 (1988)ADSzbMATHCrossRefGoogle Scholar
  112. 243.
    Hull, C.: Compactifications of the Heterotic Superstring. Phys. Lett. B178, 357 (1986). ADSMathSciNetCrossRefGoogle Scholar
  113. 244.
    Hull, C.M., Townsend, P.K.: Finiteness and conformal invariance in nonlinear σ models. Nucl. Phys. B274, 349–362 (1986). ADSMathSciNetCrossRefGoogle Scholar
  114. 245.
    Hull, C.M., Townsend, P.K.: World sheet supersymmetry and anomaly cancellation in the heterotic string. Phys. Lett. B178, 187 (1986)ADSMathSciNetCrossRefGoogle Scholar
  115. 246.
    Hull, C.M., Witten, E.: Supersymmetric sigma models and the heterotic string. Phys. Lett. B160, 398–402 (1985)ADSMathSciNetCrossRefGoogle Scholar
  116. 249.
    Israel, D., Sarkis, M.: New supersymmetric index of heterotic compactifications with torsion. J. High Energy Phys. 12, 069 (2015).;
  117. 250.
    Israel, D., Sarkis, M.: Dressed elliptic genus of heterotic compactifications with torsion and general bundles. J. High Energy Phys. 08, 176 (2016).;
  118. 251.
    Ivanov, S., Ugarte, L.: On the Strominger system and holomorphic deformations.
  119. 252.
    Jardine, I.T., Quigley, C.: Conformal invariance of (0, 2) sigma models on Calabi-Yau manifolds. J. High Energy Phys. 03, 090 (2018).;
  120. 254.
    Jockers, H., Kumar, V., Lapan, J.M., Morrison, D.R., Romo, M.: Two-sphere partition functions and Gromov-Witten invariants.
  121. 255.
    Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  122. 256.
    Joyce, D.D.: Riemannian Holonomy Groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics, vol. 12. Oxford University Press, Oxford (2007)Google Scholar
  123. 258.
    Kachru, S., Vafa, C.: Exact results for N=2 compactifications of heterotic strings. Nucl. Phys. B450, 69–89 (1995).; ADSMathSciNetzbMATHCrossRefGoogle Scholar
  124. 261.
    Kapustin, A.: Chiral de Rham complex and the half-twisted sigma-model.
  125. 267.
    Ketov, S.: Quantum Non-linear Sigma Models. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
  126. 271.
    Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics. Springer, Berlin (2005)zbMATHCrossRefGoogle Scholar
  127. 278.
    Kumar, V., Taylor, W.: Freedom and constraints in the K3 landscape. J. High Energy Phys. 0905, 066 (2009).; ADSMathSciNetCrossRefGoogle Scholar
  128. 281.
    Lawson, H.B., Jr., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)Google Scholar
  129. 283.
    Lerche, W., Vafa, C., Warner, N.P.: Chiral rings in N=2 superconformal theories. Nucl. Phys. B324, 427 (1989)ADSMathSciNetCrossRefGoogle Scholar
  130. 284.
    Li, J.: Hermitian-Yang-Mills connections and beyond. In: Surveys in Differential Geometry 2014. Regularity and Evolution of Nonlinear Equations. Surveys in Differential Geometry, vol. 19, pp. 139–149. International Press, Somerville (2015). MathSciNetzbMATHCrossRefGoogle Scholar
  131. 285.
    Li, J., Yau, S.-T.: Hermitian-Yang-Mills connection on non-Kähler manifolds. In: Mathematical Aspects of String Theory (San Diego, California, 1986). Advanced Series in Mathematical Physics, vol. 1, pp. 560–573. World Scientific Publishing, Singapore (1987)Google Scholar
  132. 288.
    Lutken, C., Ross, G.G.: Taxonomy of heterotic superconformal field theories. Phys. Lett. B213, 152 (1988). ADSMathSciNetCrossRefGoogle Scholar
  133. 293.
    McOrist, J., Melnikov, I.V.: Old issues and linear sigma models. Adv. Theor. Math. Phys. 16, 251–288 (2012). MathSciNetzbMATHCrossRefGoogle Scholar
  134. 295.
    Melnikov, I.V., Minasian, R.: Heterotic sigma models with N=2 space-time supersymmetry. J. High Energy Phys. 1109, 065 (2011).;
  135. 299.
    Melnikov, I.V., Sharpe, E.: On marginal deformations of (0,2) non-linear sigma models. Phys. Lett. B705, 529–534 (2011).;
  136. 300.
    Melnikov, I.V., Quigley, C., Sethi, S., Stern, M.: Target spaces from chiral gauge theories. J. High Energy Phys. 1302, 111 (2013).;
  137. 301.
    Melnikov, I.V., Minasian, R., Theisen, S.: Heterotic flux backgrounds and their IIA duals. J. High Energy Phys. 07, 023 (2014).;
  138. 302.
    Melnikov, I.V., Minasian, R., Sethi, S.: Heterotic fluxes and supersymmetry. J. High Energy Phys. 06, 174 (2014).;
  139. 303.
    Melnikov, I.V., Minasian, R., Sethi, S.: Spacetime supersymmetry in low-dimensional perturbative heterotic compactifications.
  140. 304.
    Mermin, N.D., Wagner, H.: Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966). ADSCrossRefGoogle Scholar
  141. 305.
    Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  142. 307.
    Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton (1974)Google Scholar
  143. 308.
    Moore, G.W., Nelson, P.C.: The etiology of sigma model anomalies. Commun. Math. Phys. 100, 83 (1985). ADSzbMATHCrossRefGoogle Scholar
  144. 315.
    Nekrasov, N.A.: Lectures on curved beta-gamma systems, pure spinors, and anomalies.
  145. 316.
    Nemeschansky, D., Sen, A.: Conformal invariance of supersymmetric sigma models on Calabi-Yau manifolds. Phys. Lett. B178, 365 (1986). ADSCrossRefGoogle Scholar
  146. 317.
    Nibbelink, S.G.: Heterotic orbifold resolutions as (2,0) gauged linear sigma models. Fortschr. Phys. 59, 454–493 (2011).; ADSMathSciNetzbMATHCrossRefGoogle Scholar
  147. 318.
    Nibbelink, S.G., Horstmeyer, L.: Super Weyl invariance: BPS equations from heterotic worldsheets.
  148. 326.
    Polchinski, J.: Scale and conformal invariance in quantum field theory. Nucl. Phys. B303, 226 (1988). ADSMathSciNetCrossRefGoogle Scholar
  149. 327.
    Polchinski, J.: String Theory, Volume 2. Cambridge University Press, Cambridge (1998)zbMATHCrossRefGoogle Scholar
  150. 328.
    Polchinski, J.: String Theory. Volume 1: An Introduction to the Bosonic String. Cambridge University Press, Cambridge (2007)Google Scholar
  151. 334.
    Rohm, R., Witten, E.: The antisymmetric tensor field in superstring theory. Ann. Phys. 170, 454 (1986). ADSMathSciNetCrossRefGoogle Scholar
  152. 335.
    Sagnotti, A.: A note on the Green-Schwarz mechanism in open string theories. Phys. Lett. B294, 196–203 (1992).; ADSCrossRefGoogle Scholar
  153. 336.
    Salamon, S.M.: Hermitian geometry. In: Invitations to Geometry and Topology. Oxford Graduate Texts in Mathematics, vol. 7, pp. 233–291. Oxford University Press, Oxford (2002)Google Scholar
  154. 342.
    Seiberg, N., Tachikawa, Y., Yonekura, K.: Anomalies of duality groups and extended conformal manifolds.
  155. 343.
    Sen, A.: (2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory. Nucl. Phys. B278, 289 (1986)ADSMathSciNetCrossRefGoogle Scholar
  156. 344.
    Sen, A.: Supersymmetry restoration in superstring perturbation theory. J. High Energy Phys. 12, 075 (2015).;
  157. 348.
    Shatashvili, S.L., Vafa, C.: Superstrings and manifold of exceptional holonomy. Sel. Math. 1, 347 (1995).; MathSciNetzbMATHCrossRefGoogle Scholar
  158. 350.
    Silverstein, E., Witten, E.: Criteria for conformal invariance of (0,2) models. Nucl. Phys. B444, 161–190 (1995). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  159. 352.
    Strominger, A.: Superstrings with torsion. Nucl. Phys. B274, 253 (1986)ADSMathSciNetCrossRefGoogle Scholar
  160. 353.
    Strominger, A.: Special geometry. Commun. Math. Phys. 133, 163–180 (1990)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  161. 356.
    Tachikawa, Y.: N=2 Supersymmetric Dynamics for Pedestrians, vol. 890. Springer, Berlin (2014).;
  162. 357.
    Tan, M.-C.: Two-dimensional twisted sigma models and the theory of chiral differential operators. Adv. Theor. Math. Phys. 10, 759–851 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  163. 358.
    Tan, M.-C., Yagi, J.: Chiral algebras of (0,2) sigma models: beyond perturbation theory. Lett. Math. Phys. 84, 257–273 (2008). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  164. 359.
    Taubes, C.H.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differ. Geom. 17(1), 139–170 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  165. 360.
    Taylor, W.: TASI lectures on supergravity and string vacua in various dimensions.
  166. 364.
    Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. In: Mathematical Aspects of String Theory (San Diego, California, 1986). Advanced Series in Mathematical Physics, vol. 1, pp. 629–646. World Scientific Publishing, Singapore (1987)Google Scholar
  167. 365.
    Todorov, A.: Weil-Petersson volumes of the moduli spaces of CY manifolds. Commun. Anal. Geom. 15(2), 407–434 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  168. 367.
    Tseytlin, A.A.: σ model Weyl invariance conditions and string equations of motion. Nucl. Phys. B294, 383–411 (1987).
  169. 369.
    Uhlenbeck, K., Yau, S.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39, S257–S293 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  170. 370.
    Vafa, C.: String vacua and orbifoldized L-G models. Mod. Phys. Lett. A4, 1169 (1989)ADSMathSciNetCrossRefGoogle Scholar
  171. 375.
    Weinberg, S.: The Quantum Theory of Fields, vol. 2. Cambridge University Press, Cambridge (1996)zbMATHCrossRefGoogle Scholar
  172. 376.
    West, P.: Introduction to Supersymmetry and Supergravity. World Scientific, Singapore (1990)zbMATHCrossRefGoogle Scholar
  173. 377.
    Wilson, P.M.H.: Erratum: “The Kähler cone on Calabi-Yau threefolds” [Invent. Math. 107 (1992), no. 3, 561–583; MR1150602 (93a:14037)]. Invent. Math. 114(1), 231–233 (1993)Google Scholar
  174. 380.
    Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  175. 381.
    Witten, E.: Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  176. 382.
    Witten, E.: Global gravitational anomalies. Commun. Math. Phys. 100, 197 (1985). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  177. 383.
    Witten, E.: Global anomalies in string theory. In: Bardeen, W.A. (ed.) Argonne Symposium on Geometry, Anomalies and Topology. Argonne, Lemont (1985)Google Scholar
  178. 384.
    Witten, E.: New issues in manifolds of SU(3) holonomy. Nucl. Phys. B268, 79 (1986)ADSMathSciNetCrossRefGoogle Scholar
  179. 386.
    Witten, E.: Topological sigma models. Commun. Math. Phys. 118, 411 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  180. 390.
    Witten, E.: Mirror manifolds and topological field theory.
  181. 393.
    Witten, E.: Two-dimensional models with (0,2) supersymmetry: perturbative aspects.
  182. 394.
    Yagi, J.: Chiral algebras of (0,2) models.
  183. 395.
    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978). zbMATHCrossRefGoogle Scholar
  184. 396.
    Yau, S.-T.: A survey of Calabi-Yau manifolds. In: Surveys in Differential Geometry. Vol. XIII. Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry. Surveys in Differential Geometry, vol. 13, pp. 277–318. Internatinal Press, Somerville (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  185. 399.
    Zumino, B.: Supersymmetry and Kahler manifolds. Phys. Lett. 87B, 203 (1979). ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ilarion V. Melnikov
    • 1
  1. 1.Department of Physics and AstronomyJames Madison UniversityHarrisonburgUSA

Personalised recommendations