Hull's new two-field action for superstring field theory is the first construction in which the closed superstring field theory action itself is background independent, but only locally: the statement is proven for backgrounds that arise as classical solutions of the single defining superconformal field theory (SCFT), while invariance of the action under marginal deformations of the SCFT remains conjectural, and backgrounds not continuously connected to the defining SCFT are out of reach. The open problem is the global extension: a superstring field theory that can be used on any string background, with the theories constructed about arbitrarily different SCFTs related by field redefinition, together with the source-identified milestones of proving the action's invariance under marginal deformations and constructing the two-SCFT generalization of the action, or else a precise obstruction showing why no such formulation exists. This is the modern, source-anchored form of the background-independence problem of closed string field theory, open since the Sen-Zwiebach local proofs of the 1990s.

Background

Closed string field theory (SFT) reformulates perturbative string theory as a field theory on the space of all string configurations: the string field Ψ\Psi collects all modes of the string at once, its free dynamics and gauge symmetry are governed by the BRST charge QQ (the nilpotent operator arising from the world-sheet ghosts), and interactions are encoded in geometric multi-string vertices organized into an L∞L_\infty algebra, a hierarchy of multilinear brackets whose generalized Jacobi identities are equivalent to gauge invariance of the action. Any perturbative SFT is built by choosing a background world-sheet conformal field theory (CFT; for the superstring, a superconformal field theory, SCFT, for example the ten free bosons and ten free fermions describing flat ten-dimensional Minkowski space): the BRST charge, the brackets, and the vertices are all constructed from that SCFT. The formalism therefore appears tied to one spacetime background, in tension with the expectation that a fundamental theory should not depend on a split into background plus fluctuations.

The accepted criterion for background independence of closed SFT is: the string field theories constructed about two different CFTs C\mathcal{C} and C′\mathcal{C}' are the same theory if they are related by a field redefinition. Sen and Zwiebach proved this for the bosonic closed string when C′\mathcal{C}' is near C\mathcal{C}, i.e. reachable by an infinitesimal or finite marginal deformation (a continuous deformation by an exactly-marginal world-sheet operator, such as varying a compactification modulus), both for the classical action and for the quantum theory in Batalin-Vilkovisky (BV) form, the gauge-fixed functional-integral formalism of closed SFT. This is background independence only in a local sense, along families of nearby conformal field theories, using a connection on the space of CFTs to identify their state spaces.

For superstrings the situation was until recently worse. Sen's superstring field theory for the type II and heterotic superstrings (the two standard classes of ten-dimensional superstrings), defined by a quantum BV master action, uses two string fields, a physical field Ψ\Psi and an auxiliary field Ψ~\widetilde{\Psi}; the doubling makes an action possible and yields the physical interacting superstring plus a second, so-called shadow superstring that is free and completely decoupled. Sen proved that under marginal NS-NS deformations of the defining SCFT the field equation for Ψ\Psi alone is background independent, but the action is not; moreover the gauge symmetry of the resulting effective theory acts like a spacetime diffeomorphism on some fields while leaving others inert (exotic diffeomorphisms), obscuring how general relativity emerges.

Hull's new action repairs this in the solution sector: adding self-interactions for the shadow superstring (coupling κ^\hat{\kappa}, with Sen's theory recovered as κ^→0\hat{\kappa}\to 0) produces an action that depends only on the total fields, so that expanding about a classical solution Ψˉ\bar{\Psi} of the theory defined by one SCFT gives S(Ψˉ+ψ,ψ~)=SΨˉ(ψ,ψ~)S(\bar{\Psi}+\psi,\widetilde{\psi})=S_{\bar{\Psi}}(\psi,\widetilde{\psi}), with a shift symmetry δΨˉ=A\delta\bar{\Psi}=A, δψ=−A\delta\psi=-A compensating any change of background solution. The two-graviton low-energy action becomes the difference of two Einstein-Hilbert terms for two metrics g=gˉ+κhg=\bar{g}+\kappa h and g^=gˉ+κ^h^\hat{g}=\bar{g}+\hat{\kappa}\hat{h}, with ordinary diffeomorphisms arising from the diagonal combination of two spin-two gauge symmetries. However, invariance of the action under marginal deformations of the defining SCFT itself remains a conjecture, and the generalization in which the two sectors are expanded about two different SCFTs was deferred to work in preparation. All background-independence results for closed SFT, bosonic and superstring alike, are thus restricted to close backgrounds: classical solutions of one SFT, or conformal field theories connected by marginal deformations. Whether a formulation exists that can be used on any background is the open problem, which Hull's paper states as a big challenge that his construction may be a first step toward.

Problem Statement

Extend background independence of closed superstring field theory beyond the local notion to a global formulation. Under the standard criterion: construct a superstring field theory that can be used on any string background, meaning that for any two admissible superconformal field theories C\mathcal{C} and C′\mathcal{C}' of the same string theory (both type II, or both heterotic) — where admissible means a critical world-sheet SCFT of the closed superstring, namely an N=(1,1)\mathcal{N}=(1,1) superconformal field theory with central charge c=15c=15 for the type II superstring and an N=(1,0)\mathcal{N}=(1,0) superconformal field theory with left- and right-moving central charges cL=15c_L=15, cR=26c_R=26 for the heterotic superstring, in both cases including backgrounds not related by marginal deformations or by classical solutions of a single string field theory — the theories constructed about C\mathcal{C} and C′\mathcal{C}' are related by a field redefinition, or a formulation in which no background SCFT enters the definition at all; or else prove that no such formulation exists within the class of perturbative string field theories or a well-specified broader class, identifying precisely what fails. Within Hull's framework the source leaves open the linked questions: (i) prove or refute that the new action is invariant, up to field redefinition, under marginal deformations of the defining SCFT, currently a conjecture with only the field equation of the physical string field known to be invariant; (ii) construct the two-SCFT generalization of the action, in which the two string sectors are expanded about different background SCFTs with distinct BRST charges and brackets, and establish its consistency and symmetries; and (iii) go beyond closely related backgrounds altogether, to arbitrarily different SCFTs, or identify and characterize the obstruction to doing so. A resolution establishes (iii), by either constructive route (field redefinition or a background-free formulation) or by obstruction; (i) and (ii) are linked sub-questions whose resolution alone does not establish (iii). Any construction must preserve the physical content of superstring theory, reproducing the perturbative superstring amplitudes about each background; within Hull's framework this includes reducing to Sen's theory in the appropriate limit κ^→0\hat{\kappa}\to 0.

References

  1. C. Hull, 'A New Action for Superstring Field Theory', arXiv:2508.03902, doi:10.48550/arXiv.2508.03902, https://arxiv.org/abs/2508.03902
  2. C. Hull, 'Self-dual gauge fields from superstring field theory', JHEP 06 (2026) 232, doi:10.1007/JHEP06(2026)232, arXiv:2602.22420, https://arxiv.org/abs/2602.22420
  3. A. Sen, 'Background Independence of Closed Superstring Field Theory', JHEP 02 (2018) 155, doi:10.1007/JHEP02(2018)155, arXiv:1711.08468, https://arxiv.org/abs/1711.08468
  4. A. Sen and B. Zwiebach, 'A Proof of Local Background Independence of Classical Closed String Field Theory', Nucl. Phys. B 414 (1994) 649-711, doi:10.1016/0550-3213(94)90258-5, arXiv:hep-th/9307088, https://arxiv.org/abs/hep-th/9307088
  5. A. Sen and B. Zwiebach, 'Quantum Background Independence of Closed String Field Theory', Nucl. Phys. B 423 (1994) 580-630, doi:10.1016/0550-3213(94)90145-7, arXiv:hep-th/9311009, https://arxiv.org/abs/hep-th/9311009
  6. A. Sen, 'BV Master Action for Heterotic and Type II String Field Theories', JHEP 02 (2016) 087, doi:10.1007/JHEP02(2016)087, arXiv:1508.05387, https://arxiv.org/abs/1508.05387
  7. A. Ahmadain, A. Frenkel and X. Yin, 'Closed String Field Theory in 25.99 Dimensions', arXiv:2605.20324, https://arxiv.org/abs/2605.20324
  8. M. Krátký, B. Narra and J. Read, 'Is String Field Theory Background Independent?', Found. Phys. 56 (2026) 26, doi:10.1007/s10701-026-00925-x, arXiv:2509.21159, https://arxiv.org/abs/2509.21159
  9. R. A. Mamade and B. Zwiebach, 'Gauge algebra and diffeomorphisms in string field theory', JHEP 02 (2026) 182, doi:10.1007/JHEP02(2026)182, arXiv:2505.23924, https://arxiv.org/abs/2505.23924
  10. R. A. Mamade and B. Zwiebach, 'Type II RR string fields and exotic diffeomorphisms', JHEP 09 (2025) 063, doi:10.1007/JHEP09(2025)063, arXiv:2506.00120, https://arxiv.org/abs/2506.00120
  11. B. Mazel, C. Wang and X. Yin, 'Diffeomorphism in Closed String Field Theory', arXiv:2504.12290, https://arxiv.org/abs/2504.12290
  12. T. Erler and C. Maccaferri, 'String Field Theory Solution for Any Open String Background', JHEP 10 (2014) 029, doi:10.1007/JHEP10(2014)029, arXiv:1406.3021, https://arxiv.org/abs/1406.3021
  13. N. Lambert and Y. Zhou, 'Compactifying the Sen Action: Six Dimensions', arXiv:2604.07621, https://arxiv.org/abs/2604.07621
  14. B. Mazel and X. Yin, 'String Field Theory for Nontrivial Backgrounds', PhD thesis, Harvard University, 2026, https://dash.harvard.edu/handle/1/42740414