Criteria for conformal invariance of (0,2) heterotic worldsheet theories beyond linear sigma model realizations

Silverstein and Witten (1995) argued that many two-dimensional (0,2) worldsheet theories that admit gauged linear sigma model ultraviolet completions flow to infrared-conformal heterotic string vacua whose gauge-singlet space-time superpotential vanishes. The open problem is whether this protection extends beyond the linear-sigma-model-realizable class: does a general Calabi-Yau threefold X with a stable holomorphic vector bundle V define a cut-off independent quantum field theory preserving the right-moving R symmetry and flowing to a conformal fixed point, or do worldsheet-instanton effects generically obstruct this? The problem includes classifying which bundles can degenerate on genus-zero curves, the geometric locus where instanton superpotentials can develop poles, beyond the quintic examples, together with two subsidiary gaps acknowledged in the source (linear-model vertex-operator representatives for reduced-gauge-group amplitudes, and the (0,2)-moduli dependence of the one-loop gauge coupling function near singular loci). Later work mapped protected subclasses (residue-theorem cancellation in linear and half-linear models) and unprotected ones (torsion curve classes; monad and extension bundles with non-compact instanton moduli spaces), and settled the perturbative side of conformal invariance for general bundles, but no general criterion for generic geometric data exists.

Background

Compactifications of the heterotic string to four-dimensional theories with N=1 supersymmetry require worldsheet theories with (0,2) supersymmetry: two supercharges acting on one chirality (the right-movers) of the two-dimensional worldsheet fields, and none on the left-movers; the more familiar (2,2) models have supersymmetry of both chiralities. A geometric way to specify such a theory is a pair $(X,V)$, where $X$ is a Calabi-Yau threefold (a compact complex three-dimensional Kaehler manifold with trivial canonical bundle) and $V$ is a stable holomorphic vector bundle over $X$ (stability being the algebro-geometric condition that gives the bundle a well-behaved moduli space) satisfying the anomaly-cancellation condition $c_2(V)=c_2(TX)$, where $c_2$ denotes the second Chern class and $TX$ the holomorphic tangent bundle of $X$; the nonlinear sigma model with target $X$ whose left-moving fermions couple to $V$ reproduces, at leading order in $\alpha'$, a string vacuum with gauge group containing $E_6\times E_8$. Whether the interacting two-dimensional theory is actually conformally invariant, and hence defines an exact string vacuum, is the central question. The obstruction is the space-time superpotential $W$: a holomorphic function of the chiral moduli (bundle, complex-structure, and Kaehler parameters) whose non-vanishing lifts the would-be moduli and destabilizes the vacuum. Worldsheet instantons, i.e. Euclidean worldsheets wrapping holomorphic curves $C\subset X$, can generate exponentially small contributions to $W$ with no symmetry reason to cancel in a generic $(0,2)$ model, unlike the protected $(2,2)$ case. Silverstein and Witten gave an argument for vanishing $W$ for the gauge-singlet moduli in models admitting a gauged linear sigma model (GLSM) ultraviolet completion: a two-dimensional abelian gauge theory whose Higgs phase reduces at large radius to the nonlinear sigma model on $(X,V)$, providing a definite, cut-off independent quantum field theory in the ultraviolet. Their strategy was that the covariant derivatives of $W$ are holomorphic functions on a compact parameter space of the GLSM, so they must vanish if they have no possible pole; they recovered the expected simple pole in the charged ${\bf\bar{27}}^3$ Yukawa coupling near the Kaehler-moduli singularity but showed no analogous singlet pole arises, and gave a localization argument that instanton sums do not see bundle singularities. They also derived a partial geometric criterion, valid beyond linear models, for failure of conformal invariance: a pole, and hence a nonzero $W$, can arise only if the gauge bundle degenerates on a curve of genus zero (a rational curve, i.e. the image of a holomorphic map from $\mathbb{CP}^1$), which they verified is avoided for the quintic hypersurface in $\mathbb{CP}^4$ but did not analyze for other classes. In this criterion, $V$ degenerates on a curve $C\subset X$ when, in a generic one-parameter family of gauge bundles, $V$ develops a singularity whose locus is the curve $C$, a subvariety of complex codimension two in $X$. The essential limitation, stated in the paper's Section 2, is that the arguments rely on the GLSM ultraviolet completion, while the $(X,V)$ realizable as linear sigma models, though vast in number, are a tiny fraction of abstract $(X,V)$ pairs. The computations also use the half-twisted model, a topological twist of the (0,2) theory in which the physical chiral primary vertex operators are represented by classes in the cohomology of one of the right-moving supercharges $\overline{Q}_+$, so that certain correlators can be computed away from the conformal point; within that framework the authors could not find a natural GLSM representative for the ${\bf 1}_2$ component of the ${\bf\bar{27}}$ of $E_6$ (in a decomposition with respect to an $SO(10)$ subgroup, relevant to models with reduced space-time gauge group), leaving part of the $E_6$ amplitude structure unreconciled in the linear model.

Problem Statement

Determine criteria for conformal invariance of (0,2) heterotic worldsheet theories that cover geometric data $(X,V)$ not realizable as linear sigma models, where $X$ is a Calabi-Yau threefold and $V$ a stable holomorphic vector bundle satisfying the heterotic anomaly condition. The common objective is a usable decision procedure for when a (0,2) construction is a genuine string vacuum; the requested results jointly serve it as follows. (1) Core question: does a general geometric $(X,V)$ define a cut-off independent quantum field theory in the ultraviolet that preserves the right-moving R symmetry (equivalently, whose chiral vertex operators are organized in $\overline{Q}_+$ cohomology) and flows to an infrared conformal fixed point with vanishing gauge-singlet space-time superpotential, or is such protection special to $(X,V)$ realizable as linear sigma models or protected by analogous mechanisms? The core question concerns classes of geometric data specified by conditions on $(X,V)$ rather than by named examples; a result restricted to a single family or a finite set of examples concerns only that special case. (2) Geometric component: classify which $(X,V)$ can degenerate on genus-zero curves (in the sense defined in Background), the locus on which the source paper showed worldsheet instantons can generate poles in the superpotential, extending the partial criterion verified for the quintic to classes of Calabi-Yau manifolds and bundles specified by conditions on $(X,V)$ rather than by named examples. (3) Subsidiary gaps acknowledged in the source, to the extent they bear on the same objective: construct natural linear-sigma-model representatives for the charged vertex operators (notably the ${\bf 1}_2$ component of the ${\bf\bar{27}}$ under an $SO(10)\times U(1)$ decomposition) needed to reconcile reduced-gauge-group amplitudes in the half-twisted framework, and determine the moduli-dependence, including $(0,2)$ moduli, of the one-loop gauge coupling function near the singular loci of the linear model parameter space. The two gaps in (3) are subsidiary to the core question (1) and the geometric classification (2) and do not bear on their resolution.

References

1. Eva Silverstein and Edward Witten, Criteria for conformal invariance of (0,2) models, Nucl. Phys. B444 (1995) 161-190, DOI 10.1016/0550-3213(95)00186-V, https://doi.org/10.1016/0550-3213(95)00186-V; preprint arXiv:hep-th/9503212, https://arxiv.org/abs/hep-th/9503212
2. Jacques Distler and Shamit Kachru, (0,2) Landau-Ginzburg theory, Nucl. Phys. B413 (1994) 213-243, DOI 10.1016/0550-3213(94)90619-X, https://doi.org/10.1016/0550-3213(94)90619-X; preprint arXiv:hep-th/9309110, https://arxiv.org/abs/hep-th/9309110
3. Eva Silverstein, Miracle at the Gepner point, Phys. Lett. B352 (1995) 69-74, DOI 10.1016/0370-2693(95)00478-4, https://doi.org/10.1016/0370-2693(95)00478-4; preprint arXiv:hep-th/9503150, https://arxiv.org/abs/hep-th/9503150
4. Chris Beasley and Edward Witten, Residues and world-sheet instantons, JHEP 0310 (2003) 065, DOI 10.1088/1126-6708/2003/10/065, https://doi.org/10.1088/1126-6708/2003/10/065; preprint arXiv:hep-th/0304115, https://arxiv.org/abs/hep-th/0304115
5. Anirban Basu and Savdeep Sethi, World-sheet stability of (0,2) linear sigma models, Phys. Rev. D68 (2003) 025003, DOI 10.1103/PhysRevD.68.025003, https://doi.org/10.1103/PhysRevD.68.025003; preprint arXiv:hep-th/0303066, https://arxiv.org/abs/hep-th/0303066
6. Volker Braun, Maximilian Kreuzer, Burt A. Ovrut and Emanuel Scheidegger, Worldsheet instantons, torsion curves and non-perturbative superpotentials, Phys. Lett. B649 (2007) 334-341, DOI 10.1016/j.physletb.2007.03.066, https://doi.org/10.1016/j.physletb.2007.03.066; preprint arXiv:hep-th/0703134, https://arxiv.org/abs/hep-th/0703134
7. Paul S. Aspinwall and Benjamin Gaines, Rational Curves and (0,2)-Deformations, J. Geom. Phys. 88 (2014) 1-15, DOI 10.1016/j.geomphys.2014.09.012, https://doi.org/10.1016/j.geomphys.2014.09.012; preprint arXiv:1404.7802, https://arxiv.org/abs/1404.7802
8. Marco Bertolini and M. Ronen Plesser, Worldsheet instantons and (0,2) linear models, JHEP 1508 (2015) 081, DOI 10.1007/JHEP08(2015)081, https://doi.org/10.1007/JHEP08(2015)081; preprint arXiv:1410.4541, https://arxiv.org/abs/1410.4541
9. Evgeny I. Buchbinder and Burt A. Ovrut, Non-vanishing Superpotentials in Heterotic String Theory and Discrete Torsion, JHEP 1701 (2017) 038, DOI 10.1007/JHEP01(2017)038, https://doi.org/10.1007/JHEP01(2017)038; preprint arXiv:1611.01922, https://arxiv.org/abs/1611.01922
10. Evgeny I. Buchbinder, Ling Lin and Burt A. Ovrut, Non-vanishing heterotic superpotentials on elliptic fibrations, JHEP 1809 (2018) 111, DOI 10.1007/JHEP09(2018)111, https://doi.org/10.1007/JHEP09(2018)111; preprint arXiv:1806.04669, https://arxiv.org/abs/1806.04669
11. Evgeny I. Buchbinder, Andre Lukas, Burt A. Ovrut and Fabian Ruehle, Heterotic Instantons for Monad and Extension Bundles, JHEP 2002 (2020) 081, DOI 10.1007/JHEP02(2020)081, https://doi.org/10.1007/JHEP02(2020)081; preprint arXiv:1912.07222, https://arxiv.org/abs/1912.07222
12. Evgeny I. Buchbinder, Andre Lukas, Burt A. Ovrut and Fabian Ruehle, Instantons and Hilbert Functions, Phys. Rev. D102 (2020) 026019, DOI 10.1103/PhysRevD.102.026019, https://doi.org/10.1103/PhysRevD.102.026019; preprint arXiv:1912.08358, https://arxiv.org/abs/1912.08358
13. Ian T. Jardine and Callum Quigley, Conformal invariance of (0,2) sigma models on Calabi-Yau manifolds, JHEP 1803 (2018) 090, DOI 10.1007/JHEP03(2018)090, https://doi.org/10.1007/JHEP03(2018)090; preprint arXiv:1801.04336, https://arxiv.org/abs/1801.04336
14. Jock McOrist and Ilarion V. Melnikov, Half-Twisted Correlators from the Coulomb Branch, JHEP 0804 (2008) 071, DOI 10.1088/1126-6708/2008/04/071, https://doi.org/10.1088/1126-6708/2008/04/071; preprint arXiv:0712.3272, https://arxiv.org/abs/0712.3272
15. Jock McOrist and Ilarion V. Melnikov, Summing the Instantons in Half-Twisted Linear Sigma Models, JHEP 0902 (2009) 026, DOI 10.1088/1126-6708/2009/02/026, https://doi.org/10.1088/1126-6708/2009/02/026; preprint arXiv:0810.0012, https://arxiv.org/abs/0810.0012
16. Matthias Klein and Jan Louis, Singular Yukawa and gauge couplings in d=4 heterotic string vacua, Nucl. Phys. B515 (1998) 146-186, DOI 10.1016/S0550-3213(97)00729-3, https://doi.org/10.1016/S0550-3213(97)00729-3; preprint arXiv:hep-th/9707212, https://arxiv.org/abs/hep-th/9707212
17. Jock McOrist, The Revival of (0,2) Sigma Models, Int. J. Mod. Phys. A26 (2011) 1-41, DOI 10.1142/S0217751X11051366, https://doi.org/10.1142/S0217751X11051366; preprint arXiv:1010.4667 (preprint title: The Revival of (0,2) Linear Sigma Models), https://arxiv.org/abs/1010.4667
18. Ilarion V. Melnikov, An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry, Springer (2019), DOI 10.1007/978-3-030-05085-6, https://doi.org/10.1007/978-3-030-05085-6
19. Marco Bertolini, Ilarion V. Melnikov and M. Ronen Plesser, Accidents in (0,2) Landau-Ginzburg theories, JHEP 1412 (2014) 157, DOI 10.1007/JHEP12(2014)157, https://doi.org/10.1007/JHEP12(2014)157; preprint arXiv:1405.4266, https://arxiv.org/abs/1405.4266