The Lorentz- and gauge-invariant one-particle-irreducible (1PI) superstring field theory containing both open and closed strings constructed by Moosavian, Sen and Verma is not a single theory but a family: its interaction vertices depend on auxiliary choices (local coordinates at the punctures and locations of picture-changing operators) and on the world-sheet background around which it is expanded. Two equivalence questions, both posed explicitly in the source paper and both left unproven for this open-closed theory, ask whether this arbitrariness is only apparent: (1) are two formulations differing only in these auxiliary choices related by a field redefinition, and (2) are the theories formulated around world-sheet backgrounds related by marginal deformation — including the required simultaneous deformation of the D-brane system — also related by a field redefinition? Partial analogs are known for purely closed string field theory: vertex uniqueness was proved at the infinitesimal level by Hata and Zwiebach for closed bosonic string field theory (the 2024 Sen–Zwiebach review asserts the argument generalizes to the superstring case, but prints no complete proof), and background independence was proved by Sen for closed superstring field theory with the important caveat that, because of the doubled free fields needed for the Ramond sector, strict action-level equivalence fails and only the interacting equations of motion and S-matrix agree. No proof of either statement for the open-closed superstring theory has appeared in the literature as of 2026.

Background

String field theory (SFT) is the second-quantized formulation of string theory: a spacetime action for a string field whose components are the infinitely many string modes, such that the classical equations of motion reproduce interacting string dynamics and gauge symmetries are encoded in the Batalin–Vilkovisky (BV) formalism, an extension of the path-integral formalism that handles gauge symmetry through fields paired with antifields and an odd antibracket. The construction of the interaction vertices proceeds by integrating suitable differential forms over sections of fiber bundles whose base is the moduli space of punctured Riemann surfaces (with boundaries, when open strings are present) and whose fiber records the auxiliary data: the choice of local coordinates at the punctures, needed to insert off-shell (non on-shell) vertex operators, and the locations of the picture-changing operators. Picture-changing operators (PCOs) are insertions of the operator X(z)={Q,ξ(z)}\mathcal{X}(z) = \{Q, \xi(z)\} — where QQ is the BRST operator generating world-sheet gauge symmetry and ξ(z)\xi(z) is a field of the bosonized superghost system — required in the Neveu–Schwarz–Ramond (NSR) world-sheet formalism to reconcile picture-number bookkeeping (the superghost system carries a conserved charge, the picture number, that correlators must respect) with supermoduli-space geometry; PCO positions must avoid spurious poles (singularities of correlators produced by the PCO insertions themselves, not by the amplitude) by a finite margin. Sen and Witten showed that the correct off-shell amplitudes are obtained by integrating over 'generalized sections' that may include vertical segments across which PCO locations jump discontinuously; this technology is essential on surfaces with boundaries, where holomorphic and anti-holomorphic picture numbers are only jointly conserved, so the fiber has several branches between which one must jump. The open-closed superstring field theory of Moosavian, Sen and Verma (2020) realizes this program completely at the classical and quantum level, including orientifolds (quotients of the world-sheet theory by a reflection symmetry, which produce unoriented strings and image branes); in the Ramond sector (the sector with periodic world-sheet fermions, describing spacetime fermions) the construction requires doubling the string fields, with one linear combination remaining a free, decoupled field. Because the interaction vertices depend on the auxiliary choices and on the background world-sheet conformal field theory, two questions of well-definedness arise. First, uniqueness: two SFTs are regarded as equivalent if a field redefinition (a canonical transformation preserving the BV antibracket and, at the quantum level, the functional measure) relates their actions, which implies identical S-matrices; Hata and Zwiebach proved at the infinitesimal level that closed bosonic SFTs built from different vertices are so related, and the 2024 review by Sen and Zwiebach asserts the argument generalizes to open and to superstring cases, but a complete proof for the open-closed superstring theory has not been written. Second, background independence: Sen and Zwiebach proved local background independence (classical and quantum) for closed bosonic SFT — theories formulated around backgrounds related by marginal deformations (deformations by exactly-marginal world-sheet operators, which preserve conformal invariance) are related by field redefinition — and Sen proved the superstring version for closed strings, finding that the doubled free field prevents strict action-level equivalence while the interacting equations of motion and the S-matrix still agree. For open-closed bosonic SFT the generalization is considered straightforward, but for the open-closed superstring theory an additional complication arises that has no closed-string counterpart: a marginal deformation of the bulk world-sheet theory forces a simultaneous deformation of the D-brane system (D-branes are the dynamical hypersurfaces on which open strings end; their collective description is the boundary conformal field theory), so disk tadpoles (one-point functions of closed-string fields on the disk) and open-closed factorization enter the analysis. The source paper poses both questions for its theory and explicitly defers them ('We hope to return to these problems in the future'); neither has been answered since.

Problem Statement

For the Lorentz- and gauge-invariant one-particle-irreducible (1PI) open-closed superstring field theory of Moosavian, Sen and Verma (JHEP 01 (2020) 183), establish in what precise sense the theory is canonical, by proving or refuting each of the following two equivalence-up-to-field-redefinition statements, at the classical level and — since the construction also yields a quantum BV master action — at the quantum level, including the BV (Batalin–Vilkovisky) functional measure:

(1) Uniqueness of the formulation: any two superstring field theories of this type that are formulated around the same world-sheet background but differ in the choice of local coordinates at the punctures and in the choice of PCO (picture-changing operator) locations (i.e., in the generalized sections and section segments defining the interaction vertices, including choices that jump between the holomorphic/anti-holomorphic PCO branches on surfaces with boundaries) are related by a field redefinition preserving the BV structure.

(2) Background independence: given two world-sheet backgrounds related by marginal deformation (deformation by an exactly-marginal world-sheet operator, preserving conformal invariance), together with the required simultaneous deformation of the D-brane system (the boundary conformal field theory), the superstring field theories formulated around the two backgrounds are related by a field redefinition. In view of the closed-superstring precedent, where the doubled free field was found to obstruct strict action-level equivalence while the interacting equations of motion and S-matrix still agree, the question is in what precise sense the two formulations are related: whether the field redefinition can be exhibited at the level of the full action and measure, or only at the level of the interacting equations of motion and S-matrix, and what role the doubled free open and closed fields play.

The common objective of both parts is to show that the open-closed superstring field theory is a well-defined object independent of the auxiliary geometric data used in its construction and of the choice of background within a marginal-deformation family, so that physical predictions (the S-matrix) do not depend on these choices.

References

  1. Seyed Faroogh Moosavian, Ashoke Sen and Mritunjay Verma, 'Superstring Field Theory with Open and Closed Strings', JHEP 01 (2020) 183, doi:10.1007/JHEP01(2020)183, arXiv:1907.10632, https://arxiv.org/abs/1907.10632
  2. Ashoke 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
  3. Ashoke Sen and Barton Zwiebach, 'String Field Theory: A Review', arXiv:2405.19421 (2024), https://arxiv.org/abs/2405.19421
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