Closed string field theory is constructed from a reference world-sheet conformal field theory — the 'background' — which fixes the BRST operator, the state space and, through the correlation functions of that conformal field theory, the interaction vertices of a non-polynomial action satisfying the Batalin–Vilkovisky master equation. Background independence — the statement that the closed string field theories formulated on two different backgrounds are the same physical theory, related by a field redefinition together with a shift of the vacuum — has been proven only when the two backgrounds are connected by (infinitesimal, or finitely integrated) exactly marginal deformations of the world-sheet theory. Muenster and Sachs (2014), after proving uniqueness of closed string field theory on a given background and interpreting background shifts as gauge transformations of the underlying loop homotopy algebra, noted that generic string backgrounds are not related by exactly marginal deformations, so that a complete proof of background independence remains open. The literature lineage through 2026 bears this out: the Sen–Zwiebach classical and quantum proofs and Sen's superstring proof remain local in the space of backgrounds; Hull's new action and the 2024 Sen–Zwiebach review both explicitly limit established background independence to closely related backgrounds; no general proof, manifest formulation, or no-go result exists. The open problem is to extend background independence beyond marginally connected backgrounds, including backgrounds with different BRST cohomology and different perturbative state spaces.

Background

Perturbative string theory is normally formulated around a fixed solution ('background') of the string equations of motion, encoded mathematically by a two-dimensional conformal field theory (CFT) living on the string world sheet. Closed string field theory (SFT) promotes this to a second-quantized spacetime theory: the dynamical variable is a closed string field, a vector in the state space AA of the world-sheet CFT equipped with its BRST operator QQ (the world-sheet symmetry generator whose cohomology H(A,Q)H(A,Q) is the space of physical string states), and Zwiebach's non-polynomial action — whose interaction vertices are built from moduli spaces of punctured Riemann surfaces — satisfies the Batalin–Vilkovisky (BV) master equation, the consistency condition guaranteeing gauge invariance and unitarity order by order in the string coupling. The reference CFT enters this construction irreducibly: it fixes QQ, the state space, and all correlation functions from which the vertices are built. Background independence is the expectation that the closed SFTs constructed on two different backgrounds are nevertheless physically the same theory — related by a field redefinition together with a shift of the vacuum — so that no background is privileged; this is the property that would make SFT a background-independent definition of string theory rather than a family of background-indexed theories. In the homotopy-algebraic language used by Muenster and Sachs, closed SFT on a given background is a loop homotopy algebra (an involutive Lie bialgebra-type L∞L_\infty structure) on the BRST complex, solutions of the equations of motion are Maurer–Cartan elements (degree-zero fields satisfying the non-linear field equation), and shifting the background corresponds to conjugating the algebra by such an element, i.e. to a gauge transformation; the minimal model of the algebra encodes the S-matrix. What is proven: Sen and Zwiebach showed that closed SFTs on backgrounds related by infinitesimal exactly marginal deformations — deformations by world-sheet operators that remain marginal to all orders — are related by field redefinition, classically and, including the path-integral measure, at the quantum level, and argued that this integrates to finite distance along marginal directions; a world-sheet operator is exactly marginal precisely when the corresponding deformation stays conformal, and this is exactly the class of background changes realizable as solutions of the SFT equations of motion near the reference background. Sen extended the proof to closed superstring field theory within marginally connected families. What is not proven: generic string backgrounds are not connected by exactly marginal deformations — they can lie in different components of the conformal manifold (the space of exactly marginal couplings of the world-sheet theory, which need not be connected), have different BRST cohomology (different perturbative spectra), or otherwise require changes that no shift inside a fixed off-shell state space can represent — and the finite-distance integration argument leaves possible divergences whose physical status is unresolved. Muenster and Sachs therefore state that 'the proof of background independence in general is still not complete', and the 2024 Sen–Zwiebach review confirms that background independence is established only 'for backgrounds that are nearby' and that a significant new idea appears needed for a manifestly background-independent formulation.

Problem Statement

Determine whether closed string field theory is background independent in general. Given any two world-sheet (super)conformal field theories that are not related by finite exactly marginal deformations (and that may therefore have different state spaces and, in general, different BRST cohomology), resolve the question in either direction: either establish the equivalence of the two closed string field theories built on them, extending (not reproducing) the known results for infinitesimal and marginally connected backgrounds: a field redefinition, together with a shift of the vacuum, that relates the two closed string field theories — their one-particle-irreducible effective actions and their underlying loop homotopy algebras — by an equivalence compatible with the quantum Batalin–Vilkovisky structure and the path-integration measure, from which agreement of the S-matrices about the corresponding vacua follows. Where the two theories share a common off-shell state space on which such a field redefinition is defined, this BV- and path-integration-measure-compatible equivalence is the criterion, and neither a morphism of loop homotopy algebras nor agreement of the S-matrices by itself resolves the task; where no such common off-shell space exists — as when the two backgrounds have different state spaces (different off-shell BRST complexes) — the equivalence to be established is instead the on-shell equivalence of the two theories, whose criterion is the agreement of their on-shell S-matrices induced by a map between the two physical state spaces; whether those two physical state spaces (the BRST cohomologies) are isomorphic is itself part of what the equivalence must establish, rather than an object presupposed to be distinct, and a bare agreement of the S-matrices, without such a map between the physical state spaces, does not by itself resolve the task; and in the superstring case, where Ramond-sector field doubling prevents a strict field-redefinition equivalence of actions even within marginal families, the equivalence may likewise be established at the S-matrix level, as in Sen's 2018 result; or prove that no such equivalence and no manifestly background-independent formulation exists (a no-go result, i.e. that closed string field theory is not background independent in general); such a negative result likewise resolves the question.

References

  1. Korbinian Muenster and Ivo Sachs, 'Homotopy Classification of Bosonic String Field Theory', Commun. Math. Phys. 330 (2014) 1227–1262, arXiv:1208.5626, doi:10.1007/s00220-014-2027-8, https://arxiv.org/abs/1208.5626
  2. Barton Zwiebach, 'Closed string field theory: Quantum action and the B-V master equation', Nucl. Phys. B 390 (1993) 33–152, arXiv:hep-th/9206084, doi:10.1016/0550-3213(93)90388-6, https://arxiv.org/abs/hep-th/9206084
  3. Ashoke Sen and Barton Zwiebach, 'A Proof of local background independence of classical closed string field theory', Nucl. Phys. B 414 (1994) 649–680, arXiv:hep-th/9307088, doi:10.1016/0550-3213(94)90258-5, https://arxiv.org/abs/hep-th/9307088
  4. Ashoke Sen and Barton Zwiebach, 'Quantum background independence of closed string field theory', Nucl. Phys. B 423 (1994) 580–590, arXiv:hep-th/9311009, doi:10.1016/0550-3213(94)90145-7, https://arxiv.org/abs/hep-th/9311009
  5. Ashoke Sen, 'Background Independence of Closed Superstring Field Theory', JHEP 02 (2018) 155, arXiv:1711.08468, doi:10.1007/JHEP02(2018)155, https://arxiv.org/abs/1711.08468
  6. Ashoke Sen and Barton Zwiebach, 'String Field Theory: A Review' (2024), arXiv:2405.19421, https://arxiv.org/abs/2405.19421
  7. Chris Hull, 'A New Action for Superstring Field Theory' (2025), arXiv:2508.03902, https://arxiv.org/abs/2508.03902
  8. Matěj Krátký, Bhanu Narra and James Read, 'Is String Field Theory Background Independent?', Foundations of Physics 56 (2026) 26, arXiv:2509.21159, doi:10.1007/s10701-026-00925-x, https://arxiv.org/abs/2509.21159
  9. Branislav Jurčo and Korbinian Münster, 'Type II Superstring Field Theory: Geometric Approach and Operadic Description', JHEP 04 (2013) 126, arXiv:1303.2323, doi:10.1007/JHEP04(2013)126, https://arxiv.org/abs/1303.2323
  10. Martin Doubek, Branislav Jurčo and Korbinian Münster, 'Modular operads and the quantum open-closed homotopy algebra', JHEP 12 (2015) 158, arXiv:1308.3223, doi:10.1007/JHEP12(2015)158, https://arxiv.org/abs/1308.3223
  11. Ashoke Sen, 'Gauge Invariant 1PI Effective Action for Superstring Field Theory', JHEP 06 (2015) 022, arXiv:1411.7478, doi:10.1007/JHEP06(2015)022, https://arxiv.org/abs/1411.7478
  12. Ashoke Sen, 'BV Master Action for Heterotic and Type II String Field Theories', JHEP 02 (2016) 087, arXiv:1508.05387, doi:10.1007/JHEP02(2016)087, https://arxiv.org/abs/1508.05387
  13. Hiroshi Kunitomo and Tatsuya Sugimoto, 'Heterotic string field theory with cyclic L∞L_\infty structure', PTEP 2019 (2019) 063B02, arXiv:1902.02991, doi:10.1093/ptep/ptz051, https://arxiv.org/abs/1902.02991
  14. Hiroshi Kunitomo and Tatsuya Sugimoto, 'Type II superstring field theory with cyclic L∞L_\infty structure', PTEP 2020 (2020) 033B06, arXiv:1911.04103, doi:10.1093/ptep/ptaa013, https://arxiv.org/abs/1911.04103
  15. Hiroshi Kunitomo, 'Tree-level S-matrix of superstring field theory with homotopy algebra structure', JHEP 03 (2021) 193, arXiv:2011.11975, doi:10.1007/JHEP03(2021)193, https://arxiv.org/abs/2011.11975
  16. Hiroshi Kunitomo, 'Open-closed homotopy algebra in superstring field theory', PTEP 2022 (2022) 093B07, arXiv:2204.01249, doi:10.1093/ptep/ptac108, https://arxiv.org/abs/2204.01249
  17. Ranveer Kumar Singh, 'Algebraic Structures In Closed Superstring Field Theory, Homotopy Transfer And Effective Actions', Phys. Rev. D 110 (2024) 126007, arXiv:2405.08063, doi:10.1103/PhysRevD.110.126007, https://arxiv.org/abs/2405.08063
  18. Seyed Faroogh Moosavian and Yehao Zhou, 'On the existence of heterotic-string and type-II-superstring field theory vertices', J. Geom. Phys. 205 (2024) 105307, arXiv:1911.04343, doi:10.1016/j.geomphys.2024.105307, https://arxiv.org/abs/1911.04343
  19. Nils Carqueville and Michael M. Kay, 'Bulk deformations of open topological string theory', Commun. Math. Phys. 315 (2012) 739–769, arXiv:1104.5438, doi:10.1007/s00220-012-1513-0, https://arxiv.org/abs/1104.5438
  20. Martin Doubek, Branislav Jurčo and Ján Pulmann, 'Quantum L∞L_\infty Algebras and the Homological Perturbation Lemma', Commun. Math. Phys. 367 (2019) 215–240, arXiv:1712.02696, doi:10.1007/s00220-019-03375-x, https://arxiv.org/abs/1712.02696
  21. Christopher Braun and James Maunder, 'Minimal models of quantum homotopy Lie algebras via the BV-formalism', J. Math. Phys. 59 (2018) 063512, arXiv:1703.00082, doi:10.1063/1.5022890, https://arxiv.org/abs/1703.00082
  22. Corinne de Lacroix, Harold Erbin, Sitender Pratap Kashyap, Ashoke Sen and Mritunjay Verma, 'Closed Superstring Field Theory and its Applications', Int. J. Mod. Phys. A 32 (2017) 1730021, arXiv:1703.06410, doi:10.1142/S0217751X17300216, https://arxiv.org/abs/1703.06410
  23. Michael Kiermaier, Yuji Okawa, Leonardo Rastelli and Barton Zwiebach, 'Analytic solutions for marginal deformations in open string field theory', JHEP 01 (2008) 028, arXiv:hep-th/0701249, doi:10.1088/1126-6708/2008/01/028, https://arxiv.org/abs/hep-th/0701249
  24. Michael Kiermaier and Yuji Okawa, 'Exact marginality in open string field theory: A General framework', JHEP 11 (2009) 041, arXiv:0707.4472, doi:10.1088/1126-6708/2009/11/041, https://arxiv.org/abs/0707.4472
  25. Amr Ahmadain, Alexander Frenkel and Aron C. Wall, 'A Background-Independent Closed String Action at Tree Level' (2024), arXiv:2410.11938, https://arxiv.org/abs/2410.11938