Varying-dilaton extension of the heterotic $A=\bar\omega$ equivalence at quartic order in the string effective action

The heterotic string effective action computed at tree level by Gross and Sloan (1987) - quartic in the massless fields, containing the contact term $t_8(\mathrm{Tr}F^2-\mathrm{Tr}\bar R^2/g_{10}^2)^2$ built from the Yang-Mills field strength and the generalized (torsion- and dilaton-corrected) curvature, together with $H^4$, $H^2(\nabla D)^2$, $(\nabla D)^4$ and $\zeta(3)\bar R^4$ terms - was shown to reduce, when the ten-dimensional gauge connection is identified with the generalized spin connection ($A=\bar\omega$), to the N=1 supergravity sector action of the type II superstring, up to an explicit local field redefinition that was verified only for constant dilaton. The open question is the varying-dilaton extension of the $A=\bar\omega$ equivalence: no published work constructs the dilaton-dependent field redefinition mapping the full non-constant-dilaton quartic heterotic action to the type-II N=1-sector action, or demonstrates an obstruction.

Background

The heterotic string is a ten-dimensional closed-string theory whose right-moving modes are those of the superstring while its left-moving modes are bosonic, leaving room for an internal gauge symmetry ($SO(32)$ or $E_8\times E_8$) carried by sixteen extra left-moving degrees of freedom; its massless spectrum contains the metric $g_{ab}$, an antisymmetric two-form $B_{ab}$ with field strength $H_{abc}$ (the torsion), a scalar dilaton $D$ (whose vacuum value sets the string coupling), and ten-dimensional Yang-Mills gauge fields with field strength $F_{ab}$. The low-energy dynamics is encoded in a spacetime effective action - an action whose tree-level scattering amplitudes for the massless states reproduce the string S-matrix - organized as a derivative expansion in $\alpha'$, the Regge slope (the squared string length, which sets the scale of higher-derivative corrections).

Gross and Sloan (1987) computed this action at tree level to quartic order in the massless fields. Beyond the two-derivative supergravity action and the standard order-$\alpha'$ corrections, the quartic action consists of order-$\alpha'^2$ terms - most notably the contact term $t_8(\mathrm{Tr}F^2-\mathrm{Tr}\bar R^2/g_{10}^2)^2$, where $t_8$ is a standard rank-eight tensor projecting four antisymmetric two-forms onto a scalar and $g_{10}$ is the ten-dimensional gauge coupling - together with $H^4$, $H^2(\nabla D)^2$ and $(\nabla D)^4$ terms, and the order-$\alpha'^3$ term $\zeta(3)\bar R^4$ ($\zeta$ is the Riemann zeta function). Here $\bar R$ is the generalized curvature of the connection $\bar\omega_{m}{}^{ab}=\omega_{m}{}^{ab}+\kappa e^{-\kappa D/\sqrt{2}}H_{m}{}^{ab}+\tilde\omega_{m}{}^{ab},$ which combines the spin connection $\omega$, the torsion $H$, and a supercovariantizing dilaton piece $\tilde\omega$ ($\kappa$ is the gravitational coupling); the torsion is itself modified by Yang-Mills and Lorentz Chern-Simons three-form terms, a structure forced by anomaly cancellation.

A structural result of that paper concerns the standard embedding, the identification of the gauge connection with the generalized spin connection, $A_{m}{}^{ab}=\bar\omega_{m}{}^{ab}$. Comparing amplitudes, Gross and Sloan found that under this identification the heterotic four-point S-matrix for states polarized along the embedding coincides with that of the N=1 supergravity sector of the type II superstring (the subset of the type II effective action carrying one of its two supersymmetry multiplets), so the two quartic effective actions must agree up to a local field redefinition - a transformation of the fields that leaves the S-matrix unchanged while reshuffling the action's off-shell form. They verified this explicitly only for a constant dilaton, giving the redefinition of $g_{ab}$ and $B_{ab}$ (their equation (6.4)) and showing that it maps the constant-dilaton quartic heterotic action to the type-II N=1-sector action; they stated that extending the analysis to nontrivial, varying dilaton configurations remains to be done. Cai and Nunez (1987) supplied contemporary amplitude-level support, showing that heterotic-specific quartic interactions vanish under the torsionful spin connection. On the worldsheet side - the two-dimensional sigma model whose conformal, Weyl-anomaly-free consistency defines the allowed string backgrounds - the embedding is known to enhance the heterotic theory's (0,1) supersymmetry (right-moving superfields only) to (1,1), making the embedded heterotic worldsheet coincide with a type-II worldsheet; how this fact relates to the spacetime-action statement at non-constant dilaton is part of what the source left unsettled.

Problem Statement

Determine whether the standard-embedding equivalence of Gross and Sloan extends from constant to non-constant dilaton at the quartic order they computed.

The quartic (in massless fields) tree-level heterotic effective action $S_{\mathrm{het}}^{(4)}[g,B,D,A]$ contains the dilaton-dependent structures $H^4$, $H^2(\nabla D)^2$, $(\nabla D)^4$, the contact term $t_8(\mathrm{Tr}F^2-\mathrm{Tr}\bar R^2/g_{10}^2)^2$ whose generalized curvature $\bar R$ includes $\nabla\nabla D$ pieces, and $\zeta(3)\bar R^4$, with $\bar R$ built from $\bar\omega=\omega+\kappa e^{-\kappa D/\sqrt{2}}H+\tilde\omega$. Gross and Sloan showed that at the standard embedding $A=\bar\omega$ and for constant dilaton ($\nabla D=0$), $S_{\mathrm{het}}^{(4)}$ reduces, under the explicit local field redefinition (their equation (6.4)) $g_{ab}\ \to\ \left(1-\frac{1}{16\kappa^2}R+\frac{13\kappa^2}{96g_{10}^2}H^2\right)g_{ab}+\frac{\kappa^2}{g_{10}^2}H_{acd}H_{b}{}^{cd}-\frac{1}{\kappa^2}R_{ab},$ $B_{ab}\ \to\ B_{ab}+\frac{\kappa^2}{2g_{10}^2}\nabla^{c}H_{abc},$ to the N=1 supergravity sector effective action of the type II superstring.

The question: does there exist a local field redefinition - now possibly including a redefinition of the dilaton itself - under which the full dilaton-dependent quartic heterotic action evaluated at $A=\bar\omega$ equals the type-II N=1-sector quartic action, up to terms that vanish on shell or are themselves removable by further local redefinitions? Equivalently, is there an obstruction: a quantity built from the dilaton-coupled quartic structures, invariant under all admissible local redefinitions, that fails to match the type-II side? The required scope is the quartic order computed in the source; an all-orders statement subsumes it.

References

1. David J. Gross and John H. Sloan, "The quartic effective action for the heterotic string", Nuclear Physics B 291 (1987) 41-89. DOI: 10.1016/0550-3213(87)90465-2. URL: https://doi.org/10.1016/0550-3213(87)90465-2
2. Yunhai Cai and C. A. Nunez, "Heterotic string covariant amplitudes and low-energy effective action", Nuclear Physics B 287 (1987) 279-303. DOI: 10.1016/0550-3213(87)90106-4. URL: https://doi.org/10.1016/0550-3213(87)90106-4
3. R. R. Metsaev and A. A. Tseytlin, "Order alpha' (two-loop) equivalence of the string equations of motion and the sigma-model Weyl invariance conditions", Nuclear Physics B 293 (1987) 385-419. DOI: 10.1016/0550-3213(87)90077-0. URL: https://doi.org/10.1016/0550-3213(87)90077-0
4. M. Daniel, D. Hochberg and N. E. Mavromatos, "The dilaton and quartic curvature terms in the heterotic string effective action", Physics Letters B 187 (1987) 79-84. DOI: 10.1016/0370-2693(87)90075-x. URL: https://doi.org/10.1016/0370-2693(87)90075-x
5. U. Ellwanger, J. Fuchs and M. G. Schmidt, "The heterotic sigma model in the presence of background gauge fields. Three-loop results", Physics Letters B 203 (1988) 244-250. DOI: 10.1016/0370-2693(88)90546-1. URL: https://doi.org/10.1016/0370-2693(88)90546-1
6. U. Ellwanger, J. Fuchs and M. G. Schmidt, "The heterotic sigma-model with background gauge fields", Nuclear Physics B 314 (1989) 175-208. DOI: 10.1016/0550-3213(89)90117-X. URL: https://doi.org/10.1016/0550-3213(89)90117-X
7. A. P. Foakes, N. Mohammedi and D. A. Ross, "Effective action and beta-functions for the heterotic string", Physics Letters B 206 (1988) 57-61. DOI: 10.1016/0370-2693(88)91262-2. URL: https://doi.org/10.1016/0370-2693(88)91262-2
8. A. P. Foakes, N. Mohammedi and D. A. Ross, "Three-loop beta-functions for the superstring and heterotic string", Nuclear Physics B 310 (1988) 335-354. DOI: 10.1016/0550-3213(88)90152-6. URL: https://doi.org/10.1016/0550-3213(88)90152-6
9. E. A. Bergshoeff and M. de Roo, "The quartic effective action of the heterotic string and supersymmetry", Nuclear Physics B 328 (1989) 439-468. DOI: 10.1016/0550-3213(89)90336-2. URL: https://doi.org/10.1016/0550-3213(89)90336-2
10. Wissam A. Chemissany, Mees de Roo and Sudhakar Panda, "alpha'-corrections to heterotic superstring effective action revisited", Journal of High Energy Physics 2007 (2007) 037. DOI: 10.1088/1126-6708/2007/08/037. URL: https://doi.org/10.1088/1126-6708/2007/08/037
11. Nathan Berkovits, Yuji Okawa and Barton Zwiebach, "WZW-like action for heterotic string field theory", Journal of High Energy Physics 2004 (2004) 038. DOI: 10.1088/1126-6708/2004/11/038. URL: https://doi.org/10.1088/1126-6708/2004/11/038
12. Hiroshi Kunitomo, "The Ramond sector of heterotic string field theory", Progress of Theoretical and Experimental Physics 2014 (2014) 043B07. DOI: 10.1093/ptep/ptu032. URL: https://doi.org/10.1093/ptep/ptu032
13. Mohammad R. Garousi, "Duality constraints on effective actions", Physics Reports 702 (2017) 1-30. DOI: 10.1016/j.physrep.2017.07.009. URL: https://doi.org/10.1016/j.physrep.2017.07.009
14. Eric Lescano, Carmen A. Nunez and Jesus A. Rodriguez, "Supersymmetry, T-duality and heterotic alpha'-corrections", Journal of High Energy Physics 2021 (2021) 092. DOI: 10.1007/JHEP07(2021)092. URL: https://doi.org/10.1007/JHEP07(2021)092
15. Mohammad R. Garousi, "Effective action of heterotic string theory at order alpha'2", Journal of High Energy Physics 2023 (2023) 020. DOI: 10.1007/JHEP09(2023)020. URL: https://doi.org/10.1007/JHEP09(2023)020
16. Georgios Papadopoulos, "Scale and Conformal Invariance in Heterotic sigma-Models", arXiv:2409.01818 [hep-th] (v1 2024, v2 2025). DOI: 10.48550/arXiv.2409.01818. URL: https://arxiv.org/abs/2409.01818