The hexagon ('hexagonalization') bootstrap — computing planar three-point functions by cutting the string worldsheet into two-particle form factors fixed by symmetry, crossing and unitarity — was extended from AdS5/CFT4 to superstrings on AdS3×S3×T4 with mixed Ramond–Ramond and Neveu–Schwarz fluxes by Eden, le Plat and Sfondrini (JHEP 08 (2021) 049), where it was verified on protected correlators and left with explicit gaps: unknown mixed-flux dressing factors, uncontrolled finite-size (wrapping) corrections from massless modes, the pure-NSNS long-string limit, and higher-point and non-planar functions. Later literature resolved the first gap — the mixed-flux worldsheet S-matrix dressing factors were constructed and the S-matrix bootstrap completed in 2023–2026 — and computed mirror corrections in the pure-RR limit, where they cancel for protected structure constants. The surviving open core is the completion of the programme itself: construct mixed-flux hexagon form factors using the now-known dressing factors, incorporate wrapping corrections from massless mirror modes for non-protected operators at generic flux, recover the long-string (continuous-representation) correlators in the h→0 limit where the hexagon is no longer fixed by symmetry, and extend to four-point and higher correlators as well as to non-planar correlators. The joint objective is an exact, finite-size determination of the CFT2 data of the AdS3/CFT2 correspondence at generic flux.

Background

Type-IIB string theory on AdS3×S3×T4AdS_3\times S^3\times T^4, supported by a mixture of Ramond–Ramond (RR) and Neveu–Schwarz (NSNS) three-form flux, is a principal laboratory of holography with sixteen supercharges (the AdS3/CFT2AdS_3/CFT_2 correspondence). The fluxes are parametrised by an integer kk (the NSNS flux; with NSNS flux alone the worldsheet theory is a supersymmetric Wess–Zumino–Witten model) and a continuous RR coupling h≥0h\ge 0 analogous to the 't Hooft coupling of AdS5/CFT4AdS_5/CFT_4. In light-cone gauge the worldsheet theory is integrable: the two-particle worldsheet SS matrix is fixed, up to scalar 'dressing factors', by the residual symmetry algebra, and the dressing factors are pinned by crossing symmetry, unitarity and matching to perturbative worldsheet computations (Lloyd, Ohlsson Sax, Sfondrini and Stefański, Nucl. Phys. B 891 (2015) 570). Before 2021 they were known only in the pure-RR limit (k=0k=0; Borsato et al., J. Phys. A 50 (2017) 024004) and the pure-NSNS limit (h=0h=0; Baggio–Sfondrini, Phys. Rev. D 98 (2018) 021902, and Dei–Sfondrini, JHEP 07 (2018) 109).

A distinctive feature of this background, with no analogue in AdS5×S5AdS_5\times S^5, is a family of exactly massless worldsheet excitations (originating from the T4T^4 directions) with no mass gap, which split into independently propagating left- and right-moving (chiral) sectors. These massless modes make both the spectrum and finite-size corrections much harder: massless Lüscher terms already signal the breakdown of the naive asymptotic Bethe ansatz (Abbott–Aniceto, Phys. Rev. D 93 (2016) 106006).

The hexagon or hexagonalization approach, introduced for AdS5/CFT4AdS_5/CFT_4 by Basso, Komatsu and Vieira (arXiv:1505.06745), computes planar three-point functions — structure constants — by cutting the string worldsheet into hexagon form factors: two-particle patches bootstrapped from the Watson equation (a form-factor analogue of crossing) together with unitarity, with the worldsheet SS matrix, including its dressing factors, as the only dynamical input. Eden, le Plat and Sfondrini (JHEP 08 (2021) 049) extended this construction to AdS3×S3×T4AdS_3\times S^3\times T^4 with arbitrary mixed flux, adapting the bootstrap principle to a diagonal su(1∣1)⊕su(1∣1)\mathfrak{su}(1|1)\oplus\mathfrak{su}(1|1) subalgebra, checking consistency conditions, and verifying it against protected (non-renormalised) correlators of the chiral ring (the operators in short supersymmetry multiplets whose correlators receive no quantum corrections). Their construction is 'asymptotic' — exact for operators separated by an asymptotically large bridge length (the number of elementary worldsheet excitations between two operator insertions) — and its scalar prefactors inherit the dressing factors of the spectral problem.

Finite-size ('wrapping') corrections arise when virtual particles of the mirror theory — the analytically continued theory obtained by exchanging space and time on the worldsheet cylinder — wind around the short direction; they are suppressed only at large operator length and must be added order by order (Lüscher-type corrections) or resummed. In AdS3AdS_3 the massless modes make these corrections non-standard, and no general resummation is available.

Finally, the AdS3AdS_3 string spectrum contains 'short strings' (discrete-series representations of the worldsheet affine sl(2,R)\mathfrak{sl}(2,\mathbb{R}) algebra), which are the excitations captured by the light-cone-gauge integrability construction, and 'long strings' (continuous-series representations, produced by spectral flow — a winding automorphism of the worldsheet affine algebra whose integer parameter equals the string winding number), which dominate the pure-NSNS WZW spectrum and are described exactly by the level-kk worldsheet conformal field theory, in particular at k=1k=1 where the dual CFT is the symmetric-product orbifold SymN(T4)\mathrm{Sym}_N(T^4). How the long-string sector emerges from the integrability description was argued at the level of the spectrum (Sfondrini, arXiv:2010.02782), but not for correlation functions, where exact worldsheet-CFT results with arbitrary spectral flow now exist as targets (Dei–Eberhardt, JHEP 08 (2021) 025).

The source paper's outlook enumerated the obstacles to completing this programme; the most concrete of them — the unknown mixed-flux dressing factors — has since been removed by the construction of those factors and the completion of the mixed-flux SS-matrix bootstrap (Ohlsson Sax–Riabchenko–Stefański, JHEP 09 (2024) 132; Frolov–Polvara–Sfondrini, Phys. Rev. D 111 (2025) L081901, JHEP 07 (2025) 171, J. Phys. A 58 (2025) 415402, and JHEP 02 (2026) 237), which also conjectures the mirror thermodynamic Bethe ansatz for the mixed-flux spectrum. What remains is to carry these inputs through to correlation functions.

Problem Statement

Building on the now-complete exact worldsheet S matrix of the mixed-flux AdS3×S3×T4AdS_3\times S^3\times T^4 superstring — including the dressing factors constructed in 2023–2026 — complete the integrable (hexagon) bootstrap programme for AdS3/CFT2AdS_3/CFT_2 correlation functions initiated by Eden–le Plat–Sfondrini, beyond the asymptotic, protected level at which it was verified. Concretely:

  1. Mixed-flux hexagons at finite size. Construct the hexagon form factors at generic mixed RR–NSNS flux with their dressing-factor prefactors, and incorporate finite-size (wrapping) corrections — including the contributions of massless mirror modes — into structure constants of non-protected operators, extending the pure-RR mirror-correction computation of Fabri and characterising when such corrections vanish.

  2. The pure-NSNS limit and long strings. Determine how the hexagon construction, built on short-string (discrete-series) excitations, behaves as h→0h\to 0, where all excitations become massless and the hexagon symmetry algebra's central charge vanishes so that the form factors are no longer fixed by symmetry, and recover from it the long-string (continuous-representation, spectrally flowed) correlation functions, matching the exact level-kk worldsheet-CFT three-point functions with arbitrary spectral flow at generic level kk.

  3. Higher-point and non-planar functions. Extend the tessellation (the cutting of the worldsheet into hexagon patches) to four-point and higher correlators, as well as to non-planar correlators.

These deliverables are independent, individually refutable sub-claims of one scientific objective: an exact, finite-size determination of the CFT2 data (structure constants and higher-point and non-planar correlators) of the AdS3/CFT2AdS_3/CFT_2 correspondence at generic flux by hexagonalization, with the pure-RR and pure-NSNS limits as consistency anchors. In particular, deliverable 1 asks for the mixed-flux hexagon prefactors together with the massless-mode wrapping correction to the structure constants of non-protected operators at generic flux, or a proof that this correction vanishes. The extension of the hexagon method to other integrable backgrounds such as AdS4×CP3AdS_4\times\mathbb{CP}^3 or AdS3×S3×S3×S1AdS_3\times S^3\times S^3\times S^1, which lack an obvious factorised symmetry algebra, is a related but separate problem.

References

  1. B. Eden, D. le Plat and A. Sfondrini, "Integrable bootstrap for AdS3/CFT2 correlation functions", J. High Energ. Phys. 2021, 049 (2021), doi:10.1007/JHEP08(2021)049, https://arxiv.org/abs/2102.08365
  2. B. Basso, S. Komatsu and P. Vieira, "Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory", arXiv:1505.06745, https://arxiv.org/abs/1505.06745
  3. T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefański jr., "The complete worldsheet S matrix of superstrings on AdS3 x S3 x T4 with mixed three-form flux", Nucl. Phys. B 891 (2015) 570, doi:10.1016/j.nuclphysb.2014.12.019, https://arxiv.org/abs/1410.0866
  4. R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański jr. and A. Torrielli, "On the dressing factors, Bethe equations and Yangian symmetry of strings on AdS3 x S3 x T4", J. Phys. A 50 (2017) 024004, doi:10.1088/1751-8121/50/2/024004, https://arxiv.org/abs/1607.00914
  5. M. Baggio and A. Sfondrini, "Strings on NS-NS Backgrounds as Integrable Deformations", Phys. Rev. D 98 (2018) 021902, doi:10.1103/PhysRevD.98.021902, https://arxiv.org/abs/1804.01998
  6. A. Dei and A. Sfondrini, "Integrable spin chain for stringy Wess-Zumino-Witten models", J. High Energ. Phys. 2018, 109 (2018), doi:10.1007/JHEP07(2018)109, https://arxiv.org/abs/1806.00422
  7. M. C. Abbott and I. Aniceto, "Massless Lüscher terms and the limitations of the AdS3 asymptotic Bethe ansatz", Phys. Rev. D 93 (2016) 106006, doi:10.1103/PhysRevD.93.106006, https://arxiv.org/abs/1512.08761
  8. A. Sfondrini, "Long strings and symmetric product orbifold from the AdS3 Bethe equations", EPL 133 (2021) 10004, doi:10.1209/0295-5075/133/10004, https://arxiv.org/abs/2010.02782
  9. S. Frolov and A. Sfondrini, "New dressing factors for AdS3/CFT2", J. High Energ. Phys. 2022, 162 (2022), doi:10.1007/JHEP04(2022)162, https://arxiv.org/abs/2112.08896
  10. M. Fabri, "Hexagonalization in AdS3 x S3 x T4: Mirror Corrections", Phys. Rev. D 106 (2022) 126008, doi:10.1103/PhysRevD.106.126008, https://arxiv.org/abs/2209.01959
  11. O. Ohlsson Sax, D. Riabchenko and B. Stefański jr., "Worldsheet kinematics, dressing factors and odd crossing in mixed-flux AdS3 backgrounds", J. High Energ. Phys. 2024, 132 (2024), doi:10.1007/JHEP09(2024)132, https://arxiv.org/abs/2312.09288
  12. S. Frolov, D. Polvara and A. Sfondrini, "On mixed-flux worldsheet scattering in AdS3/CFT2", J. High Energ. Phys. 2023, 055 (2023), doi:10.1007/JHEP11(2023)055, https://arxiv.org/abs/2306.17553
  13. N. Baglioni, D. Polvara, A. Pone and A. Sfondrini, "On the worldsheet S matrix of the AdS3/CFT2 mixed-flux mirror model", J. High Energ. Phys. 2024, 237 (2024), doi:10.1007/JHEP05(2024)237, https://arxiv.org/abs/2308.15927
  14. A. Torrielli, "A study of form factors in relativistic mixed-flux AdS3", J. High Energ. Phys. 2024, 082 (2024), doi:10.1007/JHEP03(2024)082, https://arxiv.org/abs/2312.17557
  15. S. Frolov, D. Polvara and A. Sfondrini, "Dressing Factors for Mixed-Flux AdS3 x S3 x T4 Superstrings", Phys. Rev. D 111 (2025) L081901, doi:10.1103/PhysRevD.111.L081901, https://arxiv.org/abs/2402.11732
  16. S. Frolov, D. Polvara and A. Sfondrini, "Massive dressing factors for mixed-flux AdS3/CFT2", J. High Energ. Phys. 2025, 171 (2025), doi:10.1007/JHEP07(2025)171, https://arxiv.org/abs/2501.05995
  17. S. Frolov, D. Polvara and A. Sfondrini, "Exchange relations and crossing", J. Phys. A 58 (2025) 415402, doi:10.1088/1751-8121/ae0edb, https://arxiv.org/abs/2506.04096
  18. S. Frolov, D. Polvara and A. Sfondrini, "Dressing Factors and Mirror Thermodynamic Bethe Ansatz for mixed-flux AdS3/CFT2", J. High Energ. Phys. 2026, 237 (2026), doi:10.1007/JHEP02(2026)237, https://arxiv.org/abs/2507.12191
  19. A. Dei and L. Eberhardt, "String correlators on AdS3: three-point functions", J. High Energ. Phys. 2021, 025 (2021), doi:10.1007/JHEP08(2021)025, https://arxiv.org/abs/2105.12130
  20. D. le Plat and T. Skrzypek, "Three-point functions from integrability in N=2 orbifold theories", J. High Energ. Phys. 2025, 172 (2025), doi:10.1007/JHEP12(2025)172, https://arxiv.org/abs/2506.21323
  21. S. Frolov and A. Sfondrini, "On the AdS3 x S3 x S3 x S1 dressing factors", arXiv:2512.07721, https://arxiv.org/abs/2512.07721