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Heterotic Non-linear Sigma Models

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An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry

Part of the book series: Lecture Notes in Physics ((LNP,volume 951))

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Abstract

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.

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Notes

  1. 1.

    We will follow a standard abuse of terminology: for us, unless otherwise noted, a compact manifold will mean a compact manifold without boundary.

  2. 2.

    We will always stick to closed Riemann surfaces, but those interested in D-branes and open strings will want to add boundaries and boundary conditions for the fields.

  3. 3.

    Our predilection for a flat worldsheet has caused us to leave out the dilaton coupling that can be added to the action on a curved worldsheet; qualitatively it is of the form ΔS =∫ΣDil(ϕ)R Σ, where Dil(ϕ) is the dilaton field and R Σ is the Ricci form on Σ.

  4. 4.

    More precisely, the β function is an obstruction to scale invariance; it is possible to find non-linear sigma models that are scale invariant but not conformally invariant[244], but the examples given are all non-compact. This is consistent with the theorem that a two-dimensional unitary and compact scale-invariant theory is automatically conformal [326]. We will not discuss non-compact non-linear sigma models and so will conflate conformal invariance with scale invariance.

  5. 5.

    More precisely, X is a submanifold that represents the homology cycle \(H_3(M,{\mathbb {Z}})\) obtained by gluing \(\widehat {\pi }(T)\) and \(\widehat {\pi }'(T)\) along the common boundary. It is always possible to represent such a cycle by a smooth submanifold [92].

  6. 6.

    More details may be found in the geometry appendix.

  7. 7.

    The canonical bundle is trivial on the plane or cylinder (or torus), but it is generally non-trivial on a Riemann surface Σ; for instance, on \(\Sigma = {\mathbb {P}}^1\), we have \(K_\Sigma = {\mathcal {O}}(-2)\), so that \(\overline {K}^{1/2}_\Sigma = \overline {{\mathcal {O}}(-1)}\). On the plane the notation is simply a reminder that ψ is a right-moving (aka anti-holomorphic) fermion, but we keep the more general notation since we will want to define twisted versions of this non-linear sigma model on a compact Riemann surface.

  8. 8.

    If this is confusing, the reader can take a look ahead to where we explicitly make a similar manipulation for the right-moving fermions.

  9. 9.

    For reasons that will become clear when we discuss sigma model anomalies, E is an oriented vector bundle; without that assumption the general structure group would have been \(\operatorname {O{}}(r)\).

  10. 10.

    The reader should be aware of some notational differences in the literature: for us it is \({\mathcal {S}}^{-}\) that shows up in the action, and, as we will see shortly \({\mathcal {S}}^{+}\) that shows up in the Green–Schwarz anomaly. Sometimes conventions have it just the other way around.

  11. 11.

    There is a relation between local Lorentz anomalies and diffeomorphism anomalies—there are local counter-terms that can be added to shift the Lorentz anomaly into a diffeomorphism anomaly. The former is simpler to work with, so we will stick to that. A relevant discussion may be found in the timeless [10, 11].

  12. 12.

    Either a full contraction is impossible, or it is zero due to symmetry properties under a ↔ b.

  13. 13.

    This fundamental result is due to Hopf and should be understood in the larger context of representability of cycles as described by Thom: for \(H_k(M,{\mathbb {Z}})\) with \(\dim M = n\), if k ≤ 6 or if n − 2 ≤ k ≤ n, then every cycle can be represented by a submanifold. A readable summary and references may be found in [92].

  14. 14.

    We are again slightly abusing notation here and conflating the topological invariants \(c_i \in H^{2i}(M,{\mathbb {Z}})\), with the specific representatives in terms of the curvature forms. This usually does not cause trouble, but the reader should be aware of this abuse.

  15. 15.

    We restored units of α′ in restating the quantization condition.

  16. 16.

    We work this out explicitly in an abelian example in Sect. B.2.4.

  17. 17.

    There is just one more exception in the case of Minkowski space heterotic string vacua— \({\mathbb {R}}^{1,2}\) with \({\mathcal {N}} = 1\) supersymmetry. All other Minkowski compactifications do require some amount of extended supersymmetry for the internal theory. More discussion of the exceptional \({\mathbb {R}}^{1,2}\) and \({\mathbb {R}}^{1,1}\) cases can be found in [303, 348]; exotic \({\mathbb {R}}^{1,1}\) supersymmetries are discussed in [166].

  18. 18.

    The presence of these operators of weight zero should not bother the reader, because, just as with a free compact boson, a ϕ by itself is not a well-defined local operator; those are constructed by pulling back tensors valued in the bundle E → M and contracting the indices with ∂ϕ, \(\bar {\partial }\phi \), ψ, and λ to construct gauge- and diffeomorphism-invariant operators.

  19. 19.

    Since we are now working with complex M, unless otherwise noted, T M will denote \(T_M^{1,0}\), the holomorphic tangent bundle.

  20. 20.

    This is discussed in the Sect. B.2.1.

  21. 21.

    Incidentally, these were some of the first constructions of complex non-Kähler manifolds [336].

  22. 22.

    Basic tensor identities such as \(g_{a\overline {a}} g_{b\overline {b}} {\epsilon }^{abc} = {\epsilon }_{\overline {a}\overline {b}\overline {c}} g^{\overline {c} c} \det g\) are useful in showing these relations.

  23. 23.

    G-structures and related notions of pure spinors play an important role in supersymmetric solutions of supergravity, both for type II and heterotic theories. We will not delve too far into these spacetime notions and refer the reader to some basic references. A very readable description of G-structures in the supergravity context is given in [190]; the connection to pure spinors and generalized geometry is discussed in many references; we refer to [203] for an introduction and references.

  24. 24.

    It is a familiar string–theoretic fact that the zero mode of the dilaton profile is related to the string coupling via e φ. We will be able to interpret the geometry as a perturbative string theory vacuum when the dilaton profile remains bounded on M and its zero mode is a modulus.

  25. 25.

    This can be contrasted with the result of Gauduchon that every compact complex manifold admits a hermitian metric with \(\partial \bar {\partial } \omega ^{n-1} = 0\).

  26. 26.

    A modern and thorough discussion of these statements is given in [127, 128].

  27. 27.

    As Exercise 4.2 indicates, the restriction to simply connected M is reasonable. When M is a compact Calabi–Yau manifold with exactly \(\operatorname {SU}(3)\) holonomy, then it cannot have continuous isometries and π 1(M) is a finite group. Therefore M has a cover \(\widetilde {M}\) that is simply connected [255]. Our statements then apply to the finite cover, and in principle computations for M can be obtained by taking an orbifold of \(\widetilde {M}\). As shown in, for instance, [34, 100], the orbifold can have important ramifications for the resulting SCFT.

  28. 28.

    This is a simple instance of a calibrated cycle: in this case the holomorphic curve σ( Σ) is said to be calibrated by the Kähler form. A more general discussion of calibrated geometry is given in [256].

  29. 29.

    Here ζ(3) is the Riemann zeta function, and χ(M) = 2(h 1, 1 − h 1, 2) is the Euler characteristic of M.

  30. 30.

    There is a similar ambiguity in the right-moving charge; however, if we flip both charges, we merely describe the moduli space in terms of the complex conjugate coordinates.

  31. 31.

    The Gepner models were classified in [288].

  32. 32.

    The reader may wish to review the spectral flow discussion in Chap. 2.

  33. 33.

    In pondering modular invariance it is useful to think of the internal (2,2) SCFT as “replacing” an \({\mathbb {R}}^6\) or T 6 in the usual construction of the ten-dimensional heterotic string. Spectral flow allows us to define NS/R sectors for the internal theory and show that they contribute in the same way to transformations of the worldsheet one-loop partition function as in the free theory. This is explained in detail in [194, 370].

  34. 34.

    It may be useful to review Exercise 2.9 on twisted fermions to see a convenient definition of fermion numbers and charges in the R sectors of the free theories.

  35. 35.

    We will suppress the omni-present momentum-dependent e ik⋅X factor, and will not write explicit polarization tensors for the gravitino or gauginos.

  36. 36.

    The singularity need not be a singular SCFT: the correlation functions of the SCFT can remain perfectly finite as we approach the origin from either branch, as long as we keep track of the full SCFT spectrum, which will vary smoothly. If we artificially restrict attention to just the marginal operators, there will be a discontinuity at the origin. On the other hand, there are also singularities that cannot be understood in terms of the SCFT degrees of freedom, and in this case, the SCFT will be singular.

  37. 37.

    We refer to [28] and references therein for results obtained in the large radius limit. D-branes in type II theories also offer an example with some similar features, where the general open-close string deformation theory is encoded in a superpotential, which can be directly studied in the framework of topological field theory [40].

  38. 38.

    This was first discussed in the context of large radius compactifications in [384], but the idea applies to general deformations of (0,2) SCFTs.

  39. 39.

    A classic reference for this analysis is [144]; although it uses the non-linear sigma model for the internal SCFT, the worldsheet analysis is essentially identical. The reader may find it instructive to take a look and compare the abstract A/2 and B/2 structures to the non-linear sigma model realizations. We will have more to say on this in the next section.

  40. 40.

    This is a convention for the 8 c and 8 s representations.

  41. 41.

    This discussion closely follows [302].

  42. 42.

    This is not true for a general Hermitian metric, nor for the \({\mathcal {R}}_+\) curvature of the connection with torsion, and it is another indication that the equations themselves must be corrected in the α′ expansion.

  43. 43.

    The decomposition is discussed in Sect. B.3.4.

  44. 44.

    We depart from the notation Γ for the fermi multiplets to avoid confusion with the Levi-Civita connection. The chiral fermi multiplets ΛA will have the components λ A and L A that parallel the γ A and G A familiar from Sect. 1.8.

  45. 45.

    Some examples of these were investigated in unpublished work by McOrist and the author. For instance, one concrete non-trivial scenario involves a rank r = 4 compactification of the \(\operatorname {E}_8\times \operatorname {E}_8\) string on a K3 manifold. The structure group is enlarged as \(\operatorname {SU}(4)\to \operatorname {SO}(7)\), and the spacetime Higgs mechanism is \(\operatorname {\mathfrak {so}}(10) \to \operatorname {\mathfrak {so}}(9)\). A discussion of an r = 3 case on a Calabi–Yau 3-fold, where the terms give another parametrization of the \(\operatorname {\mathfrak {e}}_6\to \operatorname {\mathfrak {so}}(10)\) Higgs branch described above, can be found in [1]. Another study of these couplings, in the context of an orbifold compactification, is given in [317].

  46. 46.

    There is another equivalence relation possible as well— we can shift \({\mathcal {O}} \to {\mathcal {O}}+ \partial X'\), where ∂X′ is a well-defined chiral operator. The resulting “deformation” is a total derivative. Such operators are intimately tied to the right-moving chiral currents and should be considered in discussing theories with (0,4) or larger superconformal invariance, but they are not present in the \(\operatorname {SU}(3)\) structure non-linear sigma model of interest here [299].

  47. 47.

    Results highlighting the differences in this deformation theory and the more familiar Calabi–Yau case are presented in [251].

  48. 48.

    We confront here the confusion between the six-dimensional and two-dimensional supersymmetry labels. We will try to stick to (1,0) as the supersymmetry with eight chiral supercharges in six dimensions.

  49. 49.

    When R i is real or complex, N i is the number of hypermultiplets; when R i is pseudoreal, then representation is actually 2N i \({\textstyle \frac {1}{2}}\)-hypermultiplets, each in representation R i.

  50. 50.

    There is a nice presentation of these ideas in the Appendix of [327].

  51. 51.

    Taking a look at the characteristic classes discussion above, it is easy to see that for a holomorphic bundle over any complex surface \(\operatorname {ch}({\mathcal {E}}\otimes {\mathcal {E}}^\ast ) = r^2 +r(c_1({\mathcal {E}})^2-2 c_2({\mathcal {E}})).\) To obtain the counting above it is also important to remember that the worldsheet states count \({\textstyle \frac {1}{2}}\)-hypermultiplets.

  52. 52.

    We denote the intersection product \(H^2(M,{\mathbb {Z}}) \times H^2(M,{\mathbb {Z}}) \to H^4(M,{\mathbb {Z}})\) by ⋅; the product is symmetric and has signature (3,  19).

References

  1. Adam, I.: On the marginal deformations of general (0,2) non-linear sigma-models. Proc. Symp. Pure Math. 90, 171–179 (2015). http://arxiv.org/abs/1710.07431

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, A., Ernebjerg, M., Lapan, J.M.: Linear models for flux vacua. http://arxiv.org/abs/hep-th/0611084

  3. Affleck, I., Dine, M., Seiberg, N.: Dynamical supersymmetry breaking in supersymmetric QCD. Nucl. Phys. B241, 493–534 (1984)

    Article  ADS  Google Scholar 

  4. Aldazabal, G., Ibanez, L.E.: A note on 4D heterotic string vacua, FI-terms and the swampland. Phys. Lett. B782, 375–379 (2018). http://dx.doi.org/10.1016/j.physletb.2018.05.061; http://arxiv.org/abs/1804.07322

    Article  ADS  MATH  Google Scholar 

  5. 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). http://dx.doi.org/10.1007/JHEP08(2014)092; http://arxiv.org/abs/1405.4792

  6. Alvarez-Gaume, L., Ginsparg, P.H.: The structure of gauge and gravitational anomalies. Ann. Phys. 161, 423 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Alvarez-Gaume, L., Witten, E.: Gravitational anomalies. Nucl. Phys. B234, 269 (1984). http://dx.doi.org/10.1016/0550-3213(84)90066-X

    Article  ADS  MathSciNet  Google Scholar 

  8. 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). http://dx.doi.org/10.1016/0003-4916(81)90006-3

    Article  ADS  Google Scholar 

  9. Anderson, L.B.: Heterotic and M-theory compactifications for string phenomenology. PhD thesis, Oxford University (2008). http://arxiv.org/abs/0808.3621; https://inspirehep.net/record/793857/files/arXiv:0808.3621.pdf

  10. Anderson, L.B., Gray, J., Lukas, A., Palti, E.: Heterotic line bundle standard models. J. High Energy Phys. 06, 113 (2012). http://dx.doi.org/10.1007/JHEP06(2012)113; http://arxiv.org/abs/1202.1757

  11. Anderson, L.B., Gray, J., Sharpe, E.: Algebroids, heterotic moduli spaces and the Strominger system. http://arxiv.org/abs/1402.1532

  12. Anderson, L.B., Gray, J., Lukas, A., Ovrut, B.: The Atiyah class and complex structure stabilization in heterotic Calabi-Yau compactifications. http://arxiv.org/abs/1107.5076

  13. Angelantonj, C., Israel, D., Sarkis, M.: Threshold corrections in heterotic flux compactifications. J. High Energy Phys. 08, 032 (2017). http://dx.doi.org/10.1007/JHEP08(2017)032; http://arxiv.org/abs/1611.09442

  14. Angella, D., Ugarte, L.: On small deformations of balanced manifolds. Differ. Geom. Appl. 54(part B), 464–474 (2017)

    Google Scholar 

  15. Argyres, P.C.: An introduction to global supersymmetry. DIY (2000)

    Google Scholar 

  16. 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). http://arxiv.org/abs/hep-th/9603042

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. 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). http://dx.doi.org/10.1016/S0550-3213(96)00583-4; http://arxiv.org/abs/hep-th/9608129

  18. 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. http://arxiv.org/abs/1806.08367

  19. Aspinwall, P.S.: A McKay-like correspondence for (0,2)-deformations. http://arxiv.org/abs/1110.2524

  20. Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. http://arxiv.org/abs/hep-th/0403166

  21. Aspinwall, P.S.: K3 surfaces and string duality. http://arxiv.org/abs/hep-th/9611137

  22. Aspinwall, P.S.: The Moduli space of N=2 superconformal field theories. http://arxiv.org/abs/hep-th/9412115

  23. Aspinwall, P.S., Gaines, B.: Rational curves and (0,2)-deformations. J. Geom. Phys. 88, 1–15 (2014). http://dx.doi.org/10.1016/j.geomphys.2014.09.012; http://arxiv.org/abs/1404.7802

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. 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). http://dx.doi.org/10.1016/0370-2693(94)90594-0; http://arxiv.org/abs/hep-th/9406032

    Article  ADS  MathSciNet  Google Scholar 

  25. Aspinwall, P.S., Plesser, M.R.: Elusive worldsheet instantons in heterotic string compactifications. http://arxiv.org/abs/1106.2998

  26. 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). http://people.maths.ox.ac.uk/cmi/library/monographs/cmim04c.pdf

  27. Aspinwall, P.S., Melnikov, I.V., Plesser, M.R.: (0,2) elephants. J. High Energy Phys. 1201, 060 (2012). http://arxiv.org/abs/1008.2156

  28. Atiyah, M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85(1), 181–207 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  29. Atiyah, M., Hitchin, N.J., Singer, I.: Selfduality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. A362, 425–461 (1978)

    Article  ADS  MATH  Google Scholar 

  30. Banks, T., Dixon, L.J.: Constraints on string vacua with space-time supersymmetry. Nucl. Phys. B307, 93–108 (1988)

    Article  ADS  Google Scholar 

  31. Banks, T., Seiberg, N.: Nonperturbative infinities. Nucl. Phys. B273, 157 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  32. 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)

    Article  ADS  MathSciNet  Google Scholar 

  33. Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces, vol. 4, 2nd edn. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  34. Basu, A., Sethi, S.: World-sheet stability of (0,2) linear sigma models. Phys. Rev. D68, 025003 (2003). http://arxiv.org/abs/hep-th/0303066

    ADS  MathSciNet  Google Scholar 

  35. Beasley, C., Witten, E.: Residues and world-sheet instantons. J. High Energy Phys. 10, 065 (2003). http://arxiv.org/abs/hep-th/0304115

    Article  ADS  MathSciNet  Google Scholar 

  36. Becker, K., Dasgupta, K.: Heterotic strings with torsion. J. High Energy Phys. 11, 006 (2002). http://dx.doi.org/10.1088/1126-6708/2002/11/006; http://arxiv.org/abs/hep-th/0209077

    Article  MathSciNet  Google Scholar 

  37. 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). http://arxiv.org/abs/hep-th/0604137

    Article  ADS  MATH  Google Scholar 

  38. 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). http://dx.doi.org/10.1103/PhysRevLett.116.011601; http://arxiv.org/abs/1510.00978

  39. 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). http://arxiv.org/abs/hep-th/9505164

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Bergshoeff, E., de Roo, M.: The Quartic effective action of the heterotic string and supersymmetry. Nucl. Phys. B328, 439 (1989). http://dx.doi.org/10.1016/0550-3213(89)90336-2

    Article  ADS  MathSciNet  Google Scholar 

  41. 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). http://arxiv.org/abs/hep-th/9605200

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Bertolini, M., Romo, M.: Aspects of (2,2) and (0,2) hybrid models. http://arxiv.org/abs/1801.04100

  43. Bertolini, M., Melnikov, I.V., Plesser, M.R.: Hybrid conformal field theories. http://arxiv.org/abs/1307.7063

  44. Bogomolov, F.A.: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243(5), 1101–1104 (1978)

    MathSciNet  Google Scholar 

  45. Bohr, C., Hanke, B., Kotschick, D.: Cycles, submanifolds, and structures on normal bundles. Manuscr. Math. 108(4), 483–494 (2002). https://doi.org/10.1007/s002290200279

    Article  MathSciNet  MATH  Google Scholar 

  46. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)

    Book  MATH  Google Scholar 

  47. 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). http://arxiv.org/abs/hep-th/0703182

    Article  ADS  MATH  Google Scholar 

  48. Braun, V., Kreuzer, M., Ovrut, B.A., Scheidegger, E.: Worldsheet instantons and torsion curves. http://arxiv.org/abs/0801.4154

  49. Buchbinder, E., Lukas, A., Ovrut, B., Ruehle, F.: Heterotic instanton superpotentials from complete intersection Calabi-Yau manifolds. J. High Energy Phys. 10, 032 (2017). http://dx.doi.org/10.1007/JHEP10(2017)032; http://arxiv.org/abs/1707.07214

  50. Buchbinder, E.I., Lin, L., Ovrut, B.A.: Non-vanishing heterotic superpotentials on elliptic fibrations. http://arxiv.org/abs/1806.04669

  51. Buchdahl, N.P.: Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann. 280(4), 625–648 (1988). http://dx.doi.org/10.1007/BF01450081

    Article  MathSciNet  MATH  Google Scholar 

  52. 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). http://dx.doi.org/10.1103/PhysRevLett.116.132001; http://arxiv.org/abs/1512.09021

  53. Callan, J., Curtis, G., Martinec, E., Perry, M., Friedan, D.: Strings in background fields. Nucl. Phys. B262, 593 (1985). http://dx.doi.org/10.1016/0550-3213(85)90506-1

    Article  ADS  MathSciNet  Google Scholar 

  54. Candelas, P., de la Ossa, X.: Moduli space of Calabi-Yau manifolds. Nucl. Phys. B355, 455–481 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Candelas, P., Horowitz, G.T., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys. B258, 46–74 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  56. 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). http://dx.doi.org/10.1016/0550-3213(91)90292-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Coleman, S.R.: There are no Goldstone bosons in two-dimensions. Commun. Math. Phys. 31, 259–264 (1973). http://dx.doi.org/10.1007/BF01646487

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. Cox, D.A., Katz, S.: Mirror Symmetry and Algebraic Geometry, 469pp. AMS, Providence (2000)

    Google Scholar 

  59. 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)]. http://dx.doi.org/10.1063/1.530747; http://arxiv.org/abs/hep-th/9405012

  60. Dasgupta, K., Rajesh, G., Sethi, S.: M theory, orientifolds and G-flux. J. High Energy Phys. 08, 023 (1999). http://arxiv.org/abs/hep-th/9908088

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. 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). http://dx.doi.org/10.1007/JHEP10(2014)123; http://arxiv.org/abs/1402.1725

  62. de la Ossa, X., Svanes, E.E.: Connections, field redefinitions and heterotic supergravity. J. High Energy Phys. 12, 008 (2014). http://dx.doi.org/10.1007/JHEP12(2014)008; http://arxiv.org/abs/1409.3347

  63. Dine, M., Lee, C.: Remarks on (0,2) models and intermediate scale scenarios in string theory. Phys. Lett. B203, 371–377 (1988)

    Article  ADS  Google Scholar 

  64. Dine, M., Seiberg, N.: Are (0,2) models string miracles? Nucl. Phys. B306, 137 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  65. Dine, M., Seiberg, N.: Microscopic knowledge from macroscopic physics in string theory. Nucl. Phys. B301, 357 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  66. Dine, M., Seiberg, N., Wen, X.G., Witten, E.: Nonperturbative effects on the string world sheet. Nucl. Phys. B278, 769 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  67. 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. Dine, M., Seiberg, N., Witten, E.: Fayet-Iliopoulos terms in string theory. Nucl. Phys. B289, 589 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  69. Distler, J.: Resurrecting (2,0) compactifications. Phys. Lett. B188, 431–436 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  70. Distler, J., Greene, B.R.: Aspects of (2,0) string compactifications. Nucl. Phys. B304, 1 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  71. Distler, J., Sharpe, E.: Heterotic compactifications with principal bundles for general groups and general levels. Adv. Theor. Math. Phys. 14, 335–398 (2010). http://arxiv.org/abs/hep-th/0701244

    Article  MathSciNet  MATH  Google Scholar 

  72. 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 1987

    Google Scholar 

  73. 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)

    Article  ADS  MathSciNet  Google Scholar 

  74. Donaldson, S.K., Kronheimer, P.B.: The Geometry Of Four-Manifolds. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1990)

    MATH  Google Scholar 

  75. Eguchi, T., Gilkey, P.B., Hanson, A.J.: Gravitation, gauge theories and differential geometry. Phys. Rep. 66, 213 (1980). http://dx.doi.org/10.1016/0370-1573(80)90130-1

    Article  ADS  MathSciNet  Google Scholar 

  76. Fei, T., Huang, Z., Picard, S.: A construction of infinitely many solutions to the Strominger system. http://arxiv.org/abs/1703.10067

  77. Ferrara, S., Lust, D., Theisen, S.: World sheet versus spectrum symmetries in heterotic and type II superstrings. Nucl. Phys. B325, 501 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  78. Florakis, I., Garcia-Etxebarria, I., Lust, D., Regalado, D.: 2d orbifolds with exotic supersymmetry. http://arxiv.org/abs/1712.04318

  79. Freed, D.: Determinants, torsion, and strings. Commun. Math. Phys. 107, 483–513 (1986). http://dx.doi.org/10.1007/BF01221001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. Freed, D.S.: Special Kaehler manifolds. Commun. Math. Phys. 203, 31–52 (1999). http://arxiv.org/abs/hep-th/9712042

    Article  ADS  MATH  Google Scholar 

  81. Freed, D., Harvey, J.A.: Instantons and the spectrum of Bloch electrons in a magnetic field. Phys. Rev. B41, 11328 (1990). http://dx.doi.org/10.1103/PhysRevB.41.11328

    Article  ADS  MathSciNet  Google Scholar 

  82. Friedan, D.H.: Nonlinear models in two + epsilon dimensions. Ann. Phys. 163, 318 (1985). Ph.D. Thesis. http://dx.doi.org/10.1016/0003-4916(85)90384-7

  83. Friedan, D., Martinec, E.J., Shenker, S.H.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B271, 93 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  84. 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). http://arxiv.org/abs/hep-th/0604063

    Article  MATH  Google Scholar 

  85. 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)

    Article  MathSciNet  MATH  Google Scholar 

  86. Garcia-Fernandez, M.: Lectures on the Strominger system. http://arxiv.org/abs/1609.02615

  87. Gauntlett, J.P., Martelli, D., Waldram, D.: Superstrings with intrinsic torsion. Phys. Rev. D69, 086002 (2004). http://arxiv.org/abs/hep-th/0302158

    ADS  MathSciNet  Google Scholar 

  88. Gepner, D.: Exactly solvable string compactifications on manifolds of SU(N) holonomy. Phys. Lett. B199, 380–388 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  89. Gepner, D.: Space-time supersymmetry in compactified string theory and superconformal models. Nucl. Phys. B296, 757 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  90. Gepner, D.: Lectures on N=2 string theory. Lectures at Spring School on Superstrings, Trieste, 3–14 Apr 1989

    Google Scholar 

  91. Goldstein, E., Prokushkin, S.: Geometric model for complex non-Kaehler manifolds with SU(3) structure. Commun. Math. Phys. 251, 65–78 (2004). http://arxiv.org/abs/hep-th/0212307

    Article  ADS  MATH  Google Scholar 

  92. Gomis, J., Komargodski, Z., Ooguri, H., Seiberg, N., Wang, Y.: Shortening anomalies in supersymmetric theories. J. High Energy Phys. 01, 067 (2017). http://dx.doi.org/10.1007/JHEP01(2017)067; http://arxiv.org/abs/1611.03101

  93. Grana, M., Minasian, R., Petrini, M., Waldram, D.: T-duality, generalized geometry and non-geometric backgrounds. J. High Energy Phys. 04, 075 (2009). http://dx.doi.org/10.1088/1126-6708/2009/04/075; http://arxiv.org/abs/0807.4527

    Article  MathSciNet  Google Scholar 

  94. Green, M.B., Seiberg, N.: Contact interactions in superstring theory. Nucl. Phys. B299, 559 (1988). http://dx.doi.org/10.1016/0550-3213(88)90549-4

    Article  ADS  MathSciNet  Google Scholar 

  95. Green, M.B., Schwarz, J.H., West, P.C.: Anomaly free chiral theories in six-dimensions. Nucl. Phys. B254, 327–348 (1985). http://dx.doi.org/10.1016/0550-3213(85)90222-6

    Article  ADS  MathSciNet  Google Scholar 

  96. Green, M., Schwarz, J., Witten, E.: Superstring Theory, Volume 1. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  97. Green, M., Schwarz, J., Witten, E.: Superstring Theory, Volume 2. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  98. Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys. B338, 15–37 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  99. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)

    MATH  Google Scholar 

  100. 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). http://dx.doi.org/10.1016/0550-3213(86)90448-7

    Article  ADS  Google Scholar 

  101. Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  102. Gross, D.J., Harvey, J.A., Martinec, E.J., Rohm, R.: Heterotic string theory. 1. The free heterotic string. Nucl. Phys. B256, 253 (1985). http://dx.doi.org/10.1016/0550-3213(85)90394-3

    Article  ADS  MathSciNet  Google Scholar 

  103. Harlow, D., Maltz, J., Witten, E.: Analytic continuation of Liouville theory. J. High Energy Phys. 12, 071 (2011). http://dx.doi.org/10.1007/JHEP12(2011)071; http://arxiv.org/abs/1108.4417

  104. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  105. Hatcher, A.: Vector Bundles and K-Theory. Online, 2.1 edn. (2009)

    Google Scholar 

  106. 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. http://arxiv.org/abs/math/9907034

    MATH  Google Scholar 

  107. Hohenberg, P.C.: Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967). http://dx.doi.org/10.1103/PhysRev.158.383

    Article  ADS  Google Scholar 

  108. Honecker, G.: Massive U(1)s and heterotic five-branes on K3. Nucl. Phys. B748, 126–148 (2006). http://arxiv.org/abs/hep-th/0602101

    Article  ADS  MathSciNet  MATH  Google Scholar 

  109. 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 Vafa

    Google Scholar 

  110. Howe, P.S., Papadopoulos, G.: Anomalies in two-dimensional supersymmetric nonlinear sigma models. Class. Quantum Gravity 4, 1749–1766 (1987)

    Article  ADS  MATH  Google Scholar 

  111. Howe, P.S., Papadopoulos, G.: Further remarks on the geometry of two-dimensional nonlinear sigma models. Class. Quantum Gravity 5, 1647–1661 (1988)

    Article  ADS  MATH  Google Scholar 

  112. Hull, C.: Compactifications of the Heterotic Superstring. Phys. Lett. B178, 357 (1986). http://dx.doi.org/10.1016/0370-2693(86)91393-6

    Article  ADS  MathSciNet  Google Scholar 

  113. Hull, C.M., Townsend, P.K.: Finiteness and conformal invariance in nonlinear σ models. Nucl. Phys. B274, 349–362 (1986). http://dx.doi.org/10.1016/0550-3213(86)90289-0

    Article  ADS  MathSciNet  Google Scholar 

  114. Hull, C.M., Townsend, P.K.: World sheet supersymmetry and anomaly cancellation in the heterotic string. Phys. Lett. B178, 187 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  115. Hull, C.M., Witten, E.: Supersymmetric sigma models and the heterotic string. Phys. Lett. B160, 398–402 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  116. Israel, D., Sarkis, M.: New supersymmetric index of heterotic compactifications with torsion. J. High Energy Phys. 12, 069 (2015). http://dx.doi.org/10.1007/JHEP12(2015)069; http://arxiv.org/abs/1509.05704

  117. Israel, D., Sarkis, M.: Dressed elliptic genus of heterotic compactifications with torsion and general bundles. J. High Energy Phys. 08, 176 (2016). http://dx.doi.org/10.1007/JHEP08(2016)176; http://arxiv.org/abs/1606.08982

  118. Ivanov, S., Ugarte, L.: On the Strominger system and holomorphic deformations. http://arxiv.org/abs/1705.02792

  119. Jardine, I.T., Quigley, C.: Conformal invariance of (0, 2) sigma models on Calabi-Yau manifolds. J. High Energy Phys. 03, 090 (2018). http://dx.doi.org/10.1007/JHEP03(2018)090; http://arxiv.org/abs/1801.04336

  120. Jockers, H., Kumar, V., Lapan, J.M., Morrison, D.R., Romo, M.: Two-sphere partition functions and Gromov-Witten invariants. http://arxiv.org/abs/1208.6244

  121. Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  122. Joyce, D.D.: Riemannian Holonomy Groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics, vol. 12. Oxford University Press, Oxford (2007)

    Google Scholar 

  123. Kachru, S., Vafa, C.: Exact results for N=2 compactifications of heterotic strings. Nucl. Phys. B450, 69–89 (1995). http://dx.doi.org/10.1016/0550-3213(95)00307-E; http://arxiv.org/abs/hep-th/9505105

    Article  ADS  MathSciNet  MATH  Google Scholar 

  124. Kapustin, A.: Chiral de Rham complex and the half-twisted sigma-model. http://arxiv.org/abs/hep-th/0504074

  125. Ketov, S.: Quantum Non-linear Sigma Models. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  126. Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  127. Kumar, V., Taylor, W.: Freedom and constraints in the K3 landscape. J. High Energy Phys. 0905, 066 (2009). http://dx.doi.org/10.1088/1126-6708/2009/05/066; http://arxiv.org/abs/0903.0386

    Article  ADS  MathSciNet  Google Scholar 

  128. Lawson, H.B., Jr., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)

    Google Scholar 

  129. Lerche, W., Vafa, C., Warner, N.P.: Chiral rings in N=2 superconformal theories. Nucl. Phys. B324, 427 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  130. 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). http://dx.doi.org/10.4310/SDG.2014.v19.n1.a6

    Article  MathSciNet  MATH  Google Scholar 

  131. 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. Lutken, C., Ross, G.G.: Taxonomy of heterotic superconformal field theories. Phys. Lett. B213, 152 (1988). http://dx.doi.org/10.1016/0370-2693(88)91016-7

    Article  ADS  MathSciNet  Google Scholar 

  133. McOrist, J., Melnikov, I.V.: Old issues and linear sigma models. Adv. Theor. Math. Phys. 16, 251–288 (2012). http://arxiv.org/abs/1103.1322

    Article  MathSciNet  MATH  Google Scholar 

  134. Melnikov, I.V., Minasian, R.: Heterotic sigma models with N=2 space-time supersymmetry. J. High Energy Phys. 1109, 065 (2011). http://dx.doi.org/10.1007/JHEP09(2011)065; http://arxiv.org/abs/1010.5365

  135. Melnikov, I.V., Sharpe, E.: On marginal deformations of (0,2) non-linear sigma models. Phys. Lett. B705, 529–534 (2011). http://dx.doi.org/10.1016/j.physletb.2011.10.055; http://arxiv.org/abs/1110.1886

  136. Melnikov, I.V., Quigley, C., Sethi, S., Stern, M.: Target spaces from chiral gauge theories. J. High Energy Phys. 1302, 111 (2013). http://dx.doi.org/10.1007/JHEP02(2013)111; http://arxiv.org/abs/1212.1212

  137. Melnikov, I.V., Minasian, R., Theisen, S.: Heterotic flux backgrounds and their IIA duals. J. High Energy Phys. 07, 023 (2014). http://dx.doi.org/10.1007/JHEP07(2014)023; http://arxiv.org/abs/1206.1417

  138. Melnikov, I.V., Minasian, R., Sethi, S.: Heterotic fluxes and supersymmetry. J. High Energy Phys. 06, 174 (2014). http://dx.doi.org/10.1007/JHEP06(2014)174; http://arxiv.org/abs/1403.4298

  139. Melnikov, I.V., Minasian, R., Sethi, S.: Spacetime supersymmetry in low-dimensional perturbative heterotic compactifications. http://arxiv.org/abs/1707.04613

  140. 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). http://dx.doi.org/10.1103/PhysRevLett.17.1133

    Article  ADS  Google Scholar 

  141. Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  142. Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton (1974)

    Google Scholar 

  143. Moore, G.W., Nelson, P.C.: The etiology of sigma model anomalies. Commun. Math. Phys. 100, 83 (1985). http://dx.doi.org/10.1007/BF01212688

    Article  ADS  MATH  Google Scholar 

  144. Nekrasov, N.A.: Lectures on curved beta-gamma systems, pure spinors, and anomalies. http://arxiv.org/abs/hep-th/0511008

  145. Nemeschansky, D., Sen, A.: Conformal invariance of supersymmetric sigma models on Calabi-Yau manifolds. Phys. Lett. B178, 365 (1986). http://dx.doi.org/10.1016/0370-2693(86)91394-8

    Article  ADS  Google Scholar 

  146. Nibbelink, S.G.: Heterotic orbifold resolutions as (2,0) gauged linear sigma models. Fortschr. Phys. 59, 454–493 (2011). http://dx.doi.org/10.1002/prop.201100002; http://arxiv.org/abs/1012.3350

    Article  ADS  MathSciNet  MATH  Google Scholar 

  147. Nibbelink, S.G., Horstmeyer, L.: Super Weyl invariance: BPS equations from heterotic worldsheets. http://arxiv.org/abs/1203.6827

  148. Polchinski, J.: Scale and conformal invariance in quantum field theory. Nucl. Phys. B303, 226 (1988). http://dx.doi.org/10.1016/0550-3213(88)90179-4

    Article  ADS  MathSciNet  Google Scholar 

  149. Polchinski, J.: String Theory, Volume 2. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  150. Polchinski, J.: String Theory. Volume 1: An Introduction to the Bosonic String. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  151. Rohm, R., Witten, E.: The antisymmetric tensor field in superstring theory. Ann. Phys. 170, 454 (1986). http://dx.doi.org/10.1016/0003-4916(86)90099-0

    Article  ADS  MathSciNet  Google Scholar 

  152. Sagnotti, A.: A note on the Green-Schwarz mechanism in open string theories. Phys. Lett. B294, 196–203 (1992). http://dx.doi.org/10.1016/0370-2693(92)90682-T; http://arxiv.org/abs/hep-th/9210127

    Article  ADS  Google Scholar 

  153. 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. Seiberg, N., Tachikawa, Y., Yonekura, K.: Anomalies of duality groups and extended conformal manifolds. http://arxiv.org/abs/1803.07366

  155. Sen, A.: (2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory. Nucl. Phys. B278, 289 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  156. Sen, A.: Supersymmetry restoration in superstring perturbation theory. J. High Energy Phys. 12, 075 (2015). http://dx.doi.org/10.1007/JHEP12(2015)075; http://arxiv.org/abs/1508.02481

  157. Shatashvili, S.L., Vafa, C.: Superstrings and manifold of exceptional holonomy. Sel. Math. 1, 347 (1995). http://dx.doi.org/10.1007/BF01671569; http://arxiv.org/abs/hep-th/9407025

    Article  MathSciNet  MATH  Google Scholar 

  158. Silverstein, E., Witten, E.: Criteria for conformal invariance of (0,2) models. Nucl. Phys. B444, 161–190 (1995). http://arxiv.org/abs/hep-th/9503212

    Article  ADS  MathSciNet  MATH  Google Scholar 

  159. Strominger, A.: Superstrings with torsion. Nucl. Phys. B274, 253 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  160. Strominger, A.: Special geometry. Commun. Math. Phys. 133, 163–180 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  161. Tachikawa, Y.: N=2 Supersymmetric Dynamics for Pedestrians, vol. 890. Springer, Berlin (2014). http://dx.doi.org/10.1007/978-3-319-08822-8; http://arxiv.org/abs/1312.2684

  162. Tan, M.-C.: Two-dimensional twisted sigma models and the theory of chiral differential operators. Adv. Theor. Math. Phys. 10, 759–851 (2006). http://arxiv.org/abs/hep-th/0604179

    Article  MathSciNet  MATH  Google Scholar 

  163. Tan, M.-C., Yagi, J.: Chiral algebras of (0,2) sigma models: beyond perturbation theory. Lett. Math. Phys. 84, 257–273 (2008). http://arxiv.org/abs/0801.4782

    Article  ADS  MathSciNet  MATH  Google Scholar 

  164. Taubes, C.H.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differ. Geom. 17(1), 139–170 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  165. Taylor, W.: TASI lectures on supergravity and string vacua in various dimensions. http://arxiv.org/abs/1104.2051

  166. 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. Todorov, A.: Weil-Petersson volumes of the moduli spaces of CY manifolds. Commun. Anal. Geom. 15(2), 407–434 (2007). http://dx.doi.org/10.4310/CAG.2007.v15.n2.a8

    Article  MathSciNet  MATH  Google Scholar 

  168. Tseytlin, A.A.: σ model Weyl invariance conditions and string equations of motion. Nucl. Phys. B294, 383–411 (1987). http://dx.doi.org/10.1016/0550-3213(87)90588-8

  169. Uhlenbeck, K., Yau, S.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39, S257–S293 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  170. Vafa, C.: String vacua and orbifoldized L-G models. Mod. Phys. Lett. A4, 1169 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  171. Weinberg, S.: The Quantum Theory of Fields, vol. 2. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  172. West, P.: Introduction to Supersymmetry and Supergravity. World Scientific, Singapore (1990)

    Book  MATH  Google Scholar 

  173. 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. Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  175. Witten, E.: Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  176. Witten, E.: Global gravitational anomalies. Commun. Math. Phys. 100, 197 (1985). http://dx.doi.org/10.1007/BF01212448

    Article  ADS  MathSciNet  MATH  Google Scholar 

  177. 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. Witten, E.: New issues in manifolds of SU(3) holonomy. Nucl. Phys. B268, 79 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  179. Witten, E.: Topological sigma models. Commun. Math. Phys. 118, 411 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  180. Witten, E.: Mirror manifolds and topological field theory. http://arxiv.org/abs/hep-th/9112056

  181. Witten, E.: Two-dimensional models with (0,2) supersymmetry: perturbative aspects. http://arxiv.org/abs/hep-th/0504078

  182. Yagi, J.: Chiral algebras of (0,2) models. http://arxiv.org/abs/1001.0118

  183. 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). https://doi.org/10.1002/cpa.3160310304

    Article  MATH  Google Scholar 

  184. 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). http://dx.doi.org/10.4310/SDG.2008.v13.n1.a9

    Article  MathSciNet  MATH  Google Scholar 

  185. Zumino, B.: Supersymmetry and Kahler manifolds. Phys. Lett. 87B, 203 (1979). http://dx.doi.org/10.1016/0370-2693(79)90964-X

    Article  ADS  Google Scholar 

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Melnikov, I.V. (2019). Heterotic Non-linear Sigma Models. In: An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry. Lecture Notes in Physics, vol 951. Springer, Cham. https://doi.org/10.1007/978-3-030-05085-6_4

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