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arxiv: 2605.30418 · v1 · pith:GZ3JW5KYnew · submitted 2026-05-28 · ✦ hep-th

Hodge Loci and Complex Multiplication via Generalized Symmetries in Calabi-Yau sigma models

Roberta Angius , Roberto Volpato This is my paper

Pith reviewed 2026-06-29 06:02 UTC · model grok-4.3

classification ✦ hep-th
keywords Hodge lociCalabi-Yau sigma modelstopological defectscomplex multiplicationN=(2,2) superconformal algebraBPS boundary statesK3 surfaceselliptic curves
0
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The pith

Hodge loci in Calabi-Yau sigma models correspond to points with non-trivial topological defect categories preserving the superconformal algebra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes identifying Hodge loci in Calabi-Yau sigma models by the presence of non-trivial categories of topological defects. These defects preserve the N=(2,2) superconformal algebra and act invertibly on spectral-flow generators. This characterization uses the CFT description where cohomology and Hodge structure come from Ramond-Ramond states and BPS boundary states. If correct, it connects geometric Hodge theory to generalized symmetries in string theory and reveals arithmetic structures at special points. The analysis focuses on elliptic curves and K3 surfaces to illustrate stronger constraints on boundary states.

Core claim

In the CFT description of Calabi-Yau sigma models, the complex cohomology is spanned by Ramond-Ramond ground states, the Hodge decomposition is determined by the U(1)×U(1) R-charges, and the rational structure is provided by BPS boundary states, with polarization induced by the open string Witten index. Hodge loci are identified by the existence of a non-trivial category TDL of topological defects preserving the N=(2,2) superconformal algebra and acting invertibly on the spectral-flow generators. At special points on these loci, the category TDL exhibits additional arithmetic structure and admits embeddings of finite products of number fields with Complex Multiplication, leading to stronger

What carries the argument

The category TDL of topological defects preserving the N=(2,2) superconformal algebra and acting invertibly on the spectral-flow generators, which identifies the Hodge loci and carries additional arithmetic structure at special points.

If this is right

  • Special points admit embeddings of number fields with complex multiplication in the TDL category.
  • Stronger constraints on boundary states arise at these points.
  • The construction applies generally to Calabi-Yau sigma models with detailed checks for elliptic curves and K3 surfaces.
  • Non-trivial rational Hodge endomorphisms appear at the identified loci.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could be used to locate Hodge loci by searching for non-trivial defect categories in known CFTs.
  • The complex multiplication embeddings might connect to known arithmetic properties of Calabi-Yau varieties.
  • Further study of TDL in other dimensions could uncover additional examples of such arithmetic structures.

Load-bearing premise

The rational structure on the complex cohomology is provided by BPS boundary states, with polarization induced by the open string Witten index.

What would settle it

A point in the moduli space that has a non-trivial TDL category but lacks non-trivial rational Hodge endomorphisms would contradict the proposed identification.

Figures

Figures reproduced from arXiv: 2605.30418 by Roberta Angius, Roberto Volpato.

Figure 1
Figure 1. Figure 1: Hodge diamonds of elliptic curves (a), K3 surfaces (b) and Calabi-Yau three-folds (c). Equipped C with a rational Hodge structure (H• (C, Q), H• (C, C)), the next step consists in identifying the objects in the theory that play the role of Hodge endomorphisms (Hodge tensors), namely endomorphisms of the Q-vector space H• (C, Q) that extend by C-linearity to transformations of H• (C, C) preserving the Hodge… view at source ↗
read the original abstract

We propose a sigma-model analogue of Hodge loci in the moduli space of geometric Calabi-Yau compactifications, characterized by the emergence of non-trivial rational Hodge endomorphisms, using generalized symmetries. In the CFT description, the complex cohomology is spanned by Ramond-Ramond ground states, the Hodge decomposition is determined by the $U(1)\times U(1)$ R-charges, and the rational structure is provided by BPS boundary states, with polarization induced by the open string Witten index. Hodge loci are identified by the existence of a non-trivial category $TDL$ of topological defects preserving the $N=(2,2)$ superconformal algebra and acting invertibly on the spectral-flow generators. At special points on these loci, the category $TDL$ exhibits additional arithmetic structure and admits embeddings of finite products of number fields with Complex Multiplication, leading to stronger constraints on the boundary states of the theory. Although the construction is general, we analyze in detail the cases of elliptic curves and $K3$ surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes a CFT analogue of Hodge loci in Calabi-Yau moduli spaces, identified by the existence of a non-trivial category TDL of topological defects that preserve the N=(2,2) superconformal algebra and act invertibly on spectral-flow generators. The rational structure on RR ground states is supplied by BPS boundary states with polarization from the open-string Witten index; at special points the TDL admits embeddings of CM number fields, with explicit analysis for elliptic curves and K3 surfaces.

Significance. If the proposed dictionary holds, the construction supplies a symmetry-based criterion for detecting rational Hodge endomorphisms and CM points directly in the SCFT, which could complement geometric methods and constrain boundary states in string compactifications. The low-dimensional cases provide concrete tests of the arithmetic structure.

major comments (2)
  1. [CFT description] CFT description (abstract and opening paragraphs): the identification of Hodge loci via non-trivial TDL rests on the assertion that BPS boundary states furnish a rational lattice inside the RR ground-state space whose endomorphisms are automatically Hodge; no explicit dictionary or verification is given showing why TDL invertibility on spectral-flow generators implies these endomorphisms lie in the rational structure and preserve the Hodge decomposition defined geometrically.
  2. [Elliptic curves and K3 surfaces] Elliptic curve and K3 analysis sections: while explicit TDL categories are constructed, the manuscript does not compare the resulting arithmetic embeddings against the known geometric Hodge loci or CM points (e.g., via periods or Picard lattices) to confirm the correspondence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the two major comments point by point below, providing clarifications from the manuscript and indicating revisions that will strengthen the explicitness of the proposed dictionary.

read point-by-point responses

  1. Referee: [CFT description] CFT description (abstract and opening paragraphs): the identification of Hodge loci via non-trivial TDL rests on the assertion that BPS boundary states furnish a rational lattice inside the RR ground-state space whose endomorphisms are automatically Hodge; no explicit dictionary or verification is given showing why TDL invertibility on spectral-flow generators implies these endomorphisms lie in the rational structure and preserve the Hodge decomposition defined geometrically.

    Authors: Section 2 defines the rational structure on RR ground states via BPS boundary states and the open-string Witten index polarization. The TDL category is required to preserve the N=(2,2) SCA and act invertibly on spectral-flow generators; this action commutes with the R-charge operators that define the Hodge decomposition, ensuring endomorphisms preserve both the rational lattice and the geometric Hodge filtration. We agree an explicit step-by-step dictionary would improve clarity and will insert a new paragraph in the introduction plus a short subsection in Section 3 mapping TDL actions to Hodge endomorphisms. revision: yes

  2. Referee: [Elliptic curves and K3 surfaces] Elliptic curve and K3 analysis sections: while explicit TDL categories are constructed, the manuscript does not compare the resulting arithmetic embeddings against the known geometric Hodge loci or CM points (e.g., via periods or Picard lattices) to confirm the correspondence.

    Authors: For elliptic curves the TDL-derived CM embeddings reproduce the known j-invariants and period ratios of CM elliptic curves; for K3 the enhanced Picard lattices from the TDL match the geometric CM loci via the Shioda-Inose correspondence. To make the match fully explicit we will add a comparison table and brief discussion in the elliptic-curve and K3 sections relating the CFT arithmetic data to periods and Picard numbers. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected.

full rationale

The paper proposes a CFT-based analogue of Hodge loci identified by non-trivial TDL categories preserving the N=(2,2) SCA and acting invertibly on spectral-flow generators, with the rational structure on RR ground states stated as furnished by BPS boundary states polarized by the open-string Witten index. This is presented as a definitional correspondence within the standard SCFT framework rather than a derivation in which a central quantity is shown equal to an input by construction, a fitted parameter renamed as a prediction, or a load-bearing premise justified solely by overlapping self-citation. The explicit checks on elliptic curves and K3 surfaces supply independent content, and no equations or self-citations in the provided text reduce the identification to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proposal rests on standard CFT axioms for N=(2,2) theories and introduces the TDL category as the central new object without independent evidence outside the construction.

axioms (2)
  • domain assumption Complex cohomology is spanned by Ramond-Ramond ground states and Hodge decomposition is determined by U(1)×U(1) R-charges
    Stated directly in the abstract as the CFT description of the geometric data.
  • domain assumption Rational structure is provided by BPS boundary states with polarization from the open string Witten index
    Invoked in the abstract to supply the rational Hodge structure.
invented entities (1)
  • category TDL of topological defects no independent evidence
    purpose: To characterize Hodge loci by preserving N=(2,2) SCA and acting invertibly on spectral-flow generators
    Newly defined object whose non-triviality identifies the loci; no independent falsifiable handle supplied in the abstract.

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Works this paper leans on

68 extracted references · 59 canonical work pages · 19 internal anchors

  1. [1]

    On the geometry of moduli space of vacua in n = 2 supersymmetric yang-mills theory,

    A. Ceresole, R. D’Auria, and S. Ferrara, “On the geometry of moduli space of vacua in n = 2 supersymmetric yang-mills theory,”Physics Letters B, vol. 339, no. 1-2, pp. 71–76, Nov. 1994,issn: 0370-2693.doi:10.1016/0370-2693(94) 91134- 7[Online]. Available:http://dx.doi.org/10.1016/0370- 2693(94) 91134-7

    work page doi:10.1016/0370-2693(94 1994

  2. [2]

    What is special k¨ ahler geometry?

    B. Craps, F. Roose, W. Troost, and A. Van Proeyen, “What is special k¨ ahler geometry?”Nuclear Physics B, vol. 503, no. 3, pp. 565–613, Oct. 1997,issn: 0550-3213.doi:10.1016/s0550-3213(97)00408-2[Online]. Available:http: //dx.doi.org/10.1016/S0550-3213(97)00408-2

    work page doi:10.1016/s0550-3213(97)00408-2 1997

  3. [3]

    The string landscape and the swampland,

    C. Vafa, “The string landscape and the swampland,” 2005. arXiv:hep - th / 0509212 [hep-th]. [Online]. Available:https : / / arxiv . org / abs / hep - th / 0509212

    2005

  4. [4]

    T. D. Brennan, F. Carta, and C. Vafa,The string landscape, the swampland, and the missing corner, 2018. arXiv:1711.00864 [hep-th]. [Online]. Available: https://arxiv.org/abs/1711.00864 43

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [5]

    The swampland: Introduction and review,

    E. Palti, “The swampland: Introduction and review,”Fortschritte der Physik, vol. 67, no. 6, Apr. 2019,issn: 1521-3978.doi:10 . 1002 / prop . 201900037 [Online]. Available:http://dx.doi.org/10.1002/prop.201900037

    work page doi:10.1002/prop.201900037 2019

  6. [6]

    Lec- tures on the swampland program in string compactifications,

    M. van Beest, J. Calder´ on-Infante, D. Mirfendereski, and I. Valenzuela, “Lec- tures on the swampland program in string compactifications,”Physics Reports, vol. 989, pp. 1–50, Nov. 2022,issn: 0370-1573.doi:10.1016/j.physrep.2022. 09.002[Online]. Available:http://dx.doi.org/10.1016/j.physrep.2022. 09.002

    work page doi:10.1016/j.physrep.2022 2022

  7. [7]

    Mirror symmetry, mirror map and applications to calabi-yau hypersurfaces,

    S. Hosono, A. Klemm, S. Thiesen, and S.-T. Yau, “Mirror symmetry, mirror map and applications to calabi-yau hypersurfaces,”Communications in Mathematical Physics, vol. 167, no. 2, pp. 301–350, Feb. 1995,issn: 1432-0916.doi:10.1007/ bf02100589[Online]. Available:http://dx.doi.org/10.1007/BF02100589

    work page doi:10.1007/bf02100589 1995

  8. [9]

    A pair of calabi- yau manifolds as an exactly soluble superconformal theory,

    P. Candelas, X. C. De La Ossa, P. S. Green, and L. Parkes, “A pair of calabi- yau manifolds as an exactly soluble superconformal theory,”Nuclear Physics B, vol. 359, no. 1, pp. 21–74, 1991,issn: 0550-3213.doi:https : / / doi . org / 10 . 1016 / 0550 - 3213(91 ) 90292 - 6[Online]. Available:https : / / www . sciencedirect.com/science/article/pii/0550321391902926

    work page arXiv 1991

  9. [10]

    Special geometry and the swampland,

    S. Cecotti, “Special geometry and the swampland,”Journal of High Energy Physics, vol. 2020, no. 9, 2020,issn: 1029-8479.doi:10.1007/jhep09(2020)147 [Online]. Available:http://dx.doi.org/10.1007/JHEP09(2020)147

    work page doi:10.1007/jhep09(2020)147 2020

  10. [11]

    Infinite Distances in Field Space and Massless Towers of States,

    T. W. Grimm, E. Palti, and I. Valenzuela, “Infinite Distances in Field Space and Massless Towers of States,”JHEP, vol. 08, p. 143, 2018.doi:10.1007/ JHEP08(2018)143arXiv:1802.08264 [hep-th]

    work page arXiv 2018

  11. [12]

    Infinite Distance Networks in Field Space and Charge Orbits

    T. W. Grimm, C. Li, and E. Palti, “Infinite Distance Networks in Field Space and Charge Orbits,”JHEP, vol. 03, p. 016, 2019.doi:10.1007/JHEP03(2019) 016arXiv:1811.02571 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2019 2019

  12. [13]

    Cedzich, J

    T. W. Grimm, F. Ruehle, and D. van de Heisteeg, “Classifying Calabi–Yau Threefolds Using Infinite Distance Limits,”Commun. Math. Phys., vol. 382, no. 1, pp. 239–275, 2021.doi:10.1007/s00220- 021- 03972- 9arXiv:1910. 02963 [hep-th]

    work page doi:10.1007/s00220- 2021

  13. [14]

    Weak gravity bounds in asymptotic string compactifications,

    B. Bastian, T. W. Grimm, and D. van de Heisteeg, “Weak gravity bounds in asymptotic string compactifications,”JHEP, vol. 06, p. 162, 2021.doi:10 . 1007/JHEP06(2021)162arXiv:2011.08854 [hep-th]. 44

    work page arXiv 2021

  14. [15]

    Bastian, T

    B. Bastian, T. W. Grimm, and D. van de Heisteeg, “Modeling General Asymp- totic Calabi–Yau Periods,”Fortsch. Phys., vol. 73, no. 7, e70010, 2025.doi: 10.1002/prop.70010arXiv:2105.02232 [hep-th]

    work page doi:10.1002/prop.70010arxiv:2105.02232 2025

  15. [16]

    Elliptic k3 surfaces at infinite complex structure and their refined kulikov models,

    S.-J. Lee and T. Weigand, “Elliptic k3 surfaces at infinite complex structure and their refined kulikov models,”Journal of High Energy Physics, vol. 2022, no. 9, 2022,issn: 1029-8479.doi:10 . 1007 / jhep09(2022 ) 143[Online]. Available: http://dx.doi.org/10.1007/JHEP09(2022)143

    work page doi:10.1007/jhep09(2022)143 2022

  16. [17]

    Non-minimal elliptic three- folds at infinite distance. part i. log calabi-yau resolutions,

    R. ´Alvarez-Garc´ ıa, S.-J. Lee, and T. Weigand, “Non-minimal elliptic three- folds at infinite distance. part i. log calabi-yau resolutions,”Journal of High Energy Physics, vol. 2024, no. 8, Aug. 2024,issn: 1029-8479.doi:10.1007/ jhep08(2024)240[Online]. Available:http://dx.doi.org/10.1007/JHEP08(2024) 240

    work page doi:10.1007/jhep08(2024 2024

  17. [18]

    Non-minimal elliptic threefolds at infinite distance ii: Asymptotic physics,

    R. ´Alvarez-Garc´ ıa, S.-J. Lee, and T. Weigand, “Non-minimal elliptic threefolds at infinite distance ii: Asymptotic physics,”Journal of High Energy Physics, vol. 2025, no. 1, Jan. 2025,issn: 1029-8479.doi:10.1007/jhep01(2025)058 [Online]. Available:http://dx.doi.org/10.1007/JHEP01(2025)058

    work page doi:10.1007/jhep01(2025)058 2025

  18. [19]

    Emergent strings in Type IIB Calabi-Yau compactifications,

    B. Hassfeld, J. Monnee, T. Weigand, and M. Wiesner, “Emergent strings in Type IIB Calabi-Yau compactifications,”JHEP, vol. 01, p. 140, 2026.doi:10. 1007/JHEP01(2026)140arXiv:2504.01066 [hep-th]

    work page arXiv 2026

  19. [20]

    Monnee, T

    J. Monnee, T. Weigand, and M. Wiesner,Physics and geometry of complex structure limits in type iib calabi-yau compactifications, 2025. arXiv:2509.07056 [hep-th]. [Online]. Available:https://arxiv.org/abs/2509.07056

    work page arXiv 2025

  20. [21]

    Exact flux vacua, symmetries, and the structure of the landscape,

    T. W. Grimm and D. van de Heisteeg, “Exact flux vacua, symmetries, and the structure of the landscape,” 2024. arXiv:2404.12422 [hep-th]. [Online]. Available:https://arxiv.org/abs/2404.12422

    work page arXiv 2024

  21. [22]

    Fusion Rules and Modular Transformations in 2D Conformal Field Theory,

    E. Verlinde, “Fusion Rules and Modular Transformations in 2D Conformal Field Theory,”Nucl. Phys. B, vol. 300, pp. 360–376, 1988.doi:10 . 1016 / 0550 - 3213(88)90603-7

    1988

  22. [23]

    Generalised twisted partition functions

    V. Petkova and J.-B. Zuber, “Generalized twisted partition functions,”Phys. Lett. B, vol. 504, pp. 157–164, 2001.doi:10.1016/S0370- 2693(01)00276- 3 arXiv:hep-th/0011021 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370- 2001

  23. [24]

    Kramers-Wannier duality from conformal defects

    J. Froehlich, J. Fuchs, I. Runkel, and C. Schweigert, “Kramers-Wannier duality from conformal defects,”Phys. Rev. Lett., vol. 93,070601, 2004.doi:10.1103/ PhysRevLett.93.070601arXiv:cond-mat/0404051 [cond-mat]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  24. [25]

    Defect lines, dualities, and generalised orbifolds

    J. Froehlich, J. Fuchs, I. Runkel, and C. Schweigert, “Defect lines, dualities, and generalised orbifolds,” in16th International Congress on Mathematical Physics, Sep. 2009.doi:10.1142/9789814304634_0056arXiv:0909.5013 [math-ph]. 45

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/9789814304634_0056arxiv:0909.5013 2009

  25. [26]

    Duality and defects in rational conformal field theory

    J. Froehlich, J. Fuchs, I. Runkel, and C. C. Schweigert, “Duality and defects in rational conformal field theory,”Nucl. Phys. B, vol. 763, pp. 354–430, 2007. doi:10.1016/j.nuclphysb.2006.11.017arXiv:hep-th/0607247 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2006.11.017arxiv:hep-th/0607247 2007

  26. [27]

    Topological Defect Lines and Renormalization Group Flows in Two Dimensions

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Topological defect lines and renormalization group flows in two dimensions,”JHEP, vol. 01, 2019. doi:10.1007/JHEP01(2019)026arXiv:1802.04445 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2019)026arxiv:1802.04445 2019

  27. [28]

    Topological defects,

    N. Carqueville, M. Del Zotto, and I. Runkel, “Topological defects,” Nov. 2023. doi:10.1016/B978-0-323-95703-8.00098-7arXiv:2311.02449 [math-ph]

    work page doi:10.1016/b978-0-323-95703-8.00098-7arxiv:2311.02449 2023

  28. [30]

    Topological defects for the free boson CFT

    J. Fuchs, M. Gaberdiel, I. Runkel, and C. Schweigert, “Topological defects for the free boson CFT,”J. Phys. A, vol. 40, 11403, 2007.doi:10.1088/1751- 8113/40/37/016arXiv:0705.3129 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1751- 2007

  29. [31]

    A worldsheet extension of O(d,d;Z)

    C. Bachas, I. Brunner, and D. Roggenkamp, “A worldsheet extension of O(d,d:Z),” JHEP, vol. 10, p. 039, 2012.doi:10.1007/JHEP10(2012)039arXiv:1205.4647 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep10(2012)039arxiv:1205.4647 2012

  30. [32]

    Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases

    R. Thorngren and Y. Wang, “Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases,” 2019. arXiv:1912.02817 [hep-th]. [Online]. Available: https://arxiv.org/abs/1912.02817

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  31. [33]

    Thorngren and Y

    R. Thorngren and Y. Wang,Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond, 2021. arXiv:2106 . 12577 [hep-th]. [Online]. Available: https://arxiv.org/abs/2106.12577

    work page arXiv 2021

  32. [34]

    Topological defects in K3 sigma mod- els,

    R. Angius, S. Giaccari, and R. Volpato, “Topological defects in K3 sigma mod- els,”JHEP, vol. 07, p. 111, 2024.doi:10 . 1007 / JHEP07(2024 ) 111arXiv: 2402.08719 [hep-th]

    work page arXiv 2024

  33. [35]

    Vertex algebras, topological defects, and Moonshine,

    R. Volpato, “Vertex algebras, topological defects, and Moonshine,” Dec. 2024. arXiv:2412.21141 [hep-th]

    work page arXiv 2024

  34. [36]

    Non-invertible defects from the Conway SCFT to K3 sigma models. Part I. General results,

    R. Angius, S. Giaccari, S. M. Harrison, and R. Volpato, “Non-invertible defects from the Conway SCFT to K3 sigma models. Part I. General results,”JHEP, vol. 10, p. 046, 2025.doi:10 . 1007 / JHEP10(2025 ) 046arXiv:2504 . 18619 [hep-th]

    2025

  35. [37]

    Towards a classification of topological defects in K3 sigma models,

    R. Angius and S. Giaccari, “Towards a classification of topological defects in K3 sigma models,” Aug. 2025. arXiv:2508.03612 [hep-th]

    work page arXiv 2025

  36. [38]

    Non-invertible defects from the Conway SCFT to K3 sigma models II: duality and Fibonacci defects,

    R. Angius, S. Giaccari, S. M. Harrison, and R. Volpato, “Non-invertible defects from the Conway SCFT to K3 sigma models II: duality and Fibonacci defects,” Dec. 2025. arXiv:2512.19640 [hep-th]. 46

    work page arXiv 2025

  37. [39]

    Hodge Structures of Complex Mul- tiplication Type from Rational Conformal Field Theories,

    H. Jockers, P. Kuusela, and M. Sarve, “Hodge Structures of Complex Mul- tiplication Type from Rational Conformal Field Theories,” Oct. 2025. arXiv: 2510.25708 [hep-th]

    work page arXiv 2025

  38. [40]

    Topological de- fect lines and renormalization group flows in two dimensions,

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Topological de- fect lines and renormalization group flows in two dimensions,”Journal of High Energy Physics, vol. 2019, no. 1, Jan. 2019,issn: 1029-8479.doi:10 . 1007 / jhep01(2019)026[Online]. Available:http://dx.doi.org/10.1007/JHEP01(2019) 026

    work page doi:10.1007/jhep01(2019 2019

  39. [41]

    On finite symmetries and their gauging in two dimensions,

    L. Bhardwaj and Y. Tachikawa, “On finite symmetries and their gauging in two dimensions,”Journal of High Energy Physics, vol. 2018, no. 3, Mar. 2018, issn: 1029-8479.doi:10.1007/jhep03(2018)189[Online]. Available:http: //dx.doi.org/10.1007/JHEP03(2018)189

    work page doi:10.1007/jhep03(2018)189 2018

  40. [42]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik,Tensor categories(Mathemat- ical Surveys and Monographs). American Mathematical Society, Providence, RI, 2015, vol. 205, pp. xvi+343,isbn: 978-1-4704-2024-6.doi:10.1090/surv/205 [Online]. Available:https://doi.org/10.1090/surv/205

    work page doi:10.1090/surv/205 2015

  41. [43]

    Generalized calabi-yau manifolds,

    N. Hitchin, “Generalized calabi-yau manifolds,”The Quarterly Journal of Math- ematics, vol. 54, no. 3, pp. 281–308, 2003,issn: 1464-3847.doi:10.1093/qmath/ hag025[Online]. Available:http://dx.doi.org/10.1093/qmath/hag025

    work page doi:10.1093/qmath/ 2003

  42. [44]

    Generalized calabi-yau structures, k3 surfaces, and b-fields,

    D. Huybrechts, “Generalized calabi-yau structures, k3 surfaces, and b-fields,”

  43. [45]

    Generalized Calabi-Yau structures, K3 surfaces, and B-fields

    arXiv:math/0306162 [math.AG]. [Online]. Available:https://arxiv. org/abs/math/0306162

    work page internal anchor Pith review Pith/arXiv arXiv

  44. [46]

    ’t Hooft’s Consistency Condition as a Consequence of Analyticity and Unitarity,

    H. Ooguri, Y. Oz, and Z. Yin, “D-branes on calabi-yau spaces and their mirrors,” Nuclear Physics B, vol. 477, no. 2, pp. 407–430, Oct. 1996,issn: 0550-3213.doi: 10.1016/0550- 3213(96)00379- 3[Online]. Available:http://dx.doi.org/ 10.1016/0550-3213(96)00379-3

    work page doi:10.1016/0550- 1996

  45. [47]

    Noninvertible sym- metries in the B model TFT,

    A. C˘ ald˘ araru, T. Pantev, E. Sharpe, B. Sung, and X. Yu, “Noninvertible sym- metries in the B model TFT,”J. Geom. Phys., vol. 218, p. 105 653, 2025.doi: 10.1016/j.geomphys.2025.105653arXiv:2504.02023 [hep-th]

    work page doi:10.1016/j.geomphys.2025.105653arxiv:2504.02023 2025

  46. [48]

    Non-invertible defects on the worldsheet,

    S. Bharadwaj, P. Niro, and K. Roumpedakis, “Non-invertible defects on the worldsheet,”JHEP, vol. 03, p. 164, 2025.doi:10 . 1007 / JHEP03(2025 ) 164 arXiv:2408.14556 [hep-th]

    work page arXiv 2025

  47. [49]

    Rational Conformal Field Theories and Complex Multiplication

    S. Gukov and C. Vafa, “Rational conformal field theories and complex multi- plication,”Commun. Math. Phys., vol. 246, pp. 181–210, 2004.doi:10.1007/ s00220-003-1032-0arXiv:hep-th/0203213

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  48. [50]

    Complex and real multiplication for k3 surfaces,

    D. Huybrechts, “Complex and real multiplication for k3 surfaces,” 2009. [On- line]. Available:https://api.semanticscholar.org/CorpusID:220726890 47

    2009

  49. [51]

    Mirror symmetry for lattice polarizedK3 surfaces,

    I. V. Dolgachev, “Mirror symmetry for lattice polarizedK3 surfaces,” in 3, vol. 81, Algebraic geometry, 4, 1996, pp. 2599–2630.doi:10.1007/BF02362332 [Online]. Available:https://doi.org/10.1007/BF02362332

    work page doi:10.1007/bf02362332 1996

  50. [52]

    K3 Surfaces and String Duality

    P. S. Aspinwall, “K3 surfaces and string duality,” inTheoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, Nov. 1996, pp. 421–540. arXiv:hep-th/9611137

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  51. [53]

    Mirror Symmetry on Kummer Type K3 Surfaces

    W. Nahm and K. Wendland, “Mirror symmetry on Kummer type K3 surfaces,” Commun. Math. Phys., vol. 243, pp. 557–582, 2003.doi:10 . 1007 / s00220 - 003-0985-3arXiv:hep-th/0106104

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  52. [54]

    Mirror symmetry and rigid structures of generalized k3 surfaces,

    A. Kanazawa, “Mirror symmetry and rigid structures of generalized k3 surfaces,” Advances in Mathematics, vol. 460, p. 110 050, Jan. 2025,issn: 0001-8708.doi: 10.1016/j.aim.2024.110050[Online]. Available:http://dx.doi.org/10. 1016/j.aim.2024.110050

    work page doi:10.1016/j.aim.2024.110050 2025

  53. [55]

    Boundary states as modules over categories of defects,

    R. Angius, A. Miccich` e, and R. Volpato, “Boundary states as modules over categories of defects,” to appear

  54. [56]

    Non-invertible symmetry in calabi-yau conformal field theories,

    C. C´ ordova and G. Rizi, “Non-invertible symmetry in calabi-yau conformal field theories,”Journal of High Energy Physics, vol. 2025, no. 1, Jan. 2025,issn: 1029- 8479.doi:10.1007/jhep01(2025)045[Online]. Available:http://dx.doi. org/10.1007/JHEP01(2025)045

    work page doi:10.1007/jhep01(2025)045 2025

  55. [57]

    B-type defects in Landau-Ginzburg models

    I. Brunner and D. Roggenkamp, “B-type defects in Landau-Ginzburg models,” JHEP, vol. 08, p. 093, 2007.doi:10.1088/1126- 6708/2007/08/093arXiv: 0707.0922 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126- 2007

  56. [58]

    Orbifolds and topological defects

    I. Brunner, N. Carqueville, and D. Plencner, “Orbifolds and topological defects,” Commun. Math. Phys., vol. 332, pp. 669–712, 2014.doi:10.1007/s00220-014- 2056-3arXiv:1307.3141 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00220-014- 2014

  57. [59]

    Non-invertible symmetries of two-dimensional non-linear sigma models,

    G. Arias-Tamargo, C. Hull, and M. L. Vel´ asquez Cotini Hutt, “Non-invertible symmetries of two-dimensional non-linear sigma models,”SciPost Phys., vol. 19, no. 5, p. 126, 2025.doi:10.21468/SciPostPhys.19.5.126arXiv:2503.20865 [hep-th]

    work page doi:10.21468/scipostphys.19.5.126arxiv:2503.20865 2025

  58. [60]

    Symmetries, anomalies, and dualities of two-dimensional Non-Linear Sigma Models,

    G. Arias-Tamargo and M. L. Vel´ asquez Cotini Hutt, “Symmetries, anomalies, and dualities of two-dimensional Non-Linear Sigma Models,”JHEP, vol. 02, p. 013, 2026.doi:10.1007/JHEP02(2026)013arXiv:2508.16721 [hep-th]

    work page doi:10.1007/jhep02(2026)013arxiv:2508.16721 2026

  59. [61]

    Lectures on D-branes and sheaves,

    E. Sharpe, “Lectures on D-branes and sheaves,” Jul. 2003. arXiv:hep - th / 0307245. 48

    2003

  60. [62]

    D-Branes on Calabi-Yau Manifolds

    P. S. Aspinwall, “D-branes on Calabi-Yau manifolds,” inTheoretical Advanced Study Institute in Elementary Particle Physics (TASI 2003): Recent Trends in String Theory, Mar. 2004, pp. 1–152.doi:10.1142/9789812775108_0001arXiv: hep-th/0403166

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/9789812775108_0001arxiv: 2003

  61. [63]

    Homological Algebra of Mirror Symmetry

    M. Kontsevich, “Homological Algebra of Mirror Symmetry,” Nov. 1994. arXiv: alg-geom/9411018

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  62. [64]

    Complex multiplication, rationality and mirror symmetry for abelian varieties

    M. Chen, “Complex multiplication, rationality and mirror symmetry for Abelian varieties,”J. Geom. Phys., vol. 58, pp. 633–653, 2008.doi:10 . 1016 / j . geomphys.2008.01.001arXiv:math/0512470

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  63. [65]

    Towards Hodge Theoretic Characterizations of 2d Rational SCFTs: II,

    M. Okada and T. Watari, “Towards Hodge Theoretic Characterizations of 2d Rational SCFTs: II,” Dec. 2022. arXiv:2212.13028 [hep-th]

    work page arXiv 2022

  64. [66]

    Towards Hodge Theoretic Characteri- zations of 2d Rational SCFTs,

    A. Kidambi, M. Okada, and T. Watari, “Towards Hodge Theoretic Characteri- zations of 2d Rational SCFTs,” May 2022. arXiv:2205.10299 [hep-th]

    work page arXiv 2022

  65. [67]

    Notes on Characterizations of 2d Ra- tional SCFTs: Algebraicity, Mirror Symmetry, and Complex Multiplication,

    A. Kidambi, M. Okada, and T. Watari, “Notes on Characterizations of 2d Ra- tional SCFTs: Algebraicity, Mirror Symmetry, and Complex Multiplication,” Fortsch. Phys., vol. 73, no. 1-2, p. 2 400 161, 2025.doi:10.1002/prop.202400161 arXiv:2408.00861 [hep-th]

    work page doi:10.1002/prop.202400161 2025

  66. [68]

    Variation of hodge structure: The singularities of the period map- ping.,

    W. Schmid, “Variation of hodge structure: The singularities of the period map- ping.,”Inventiones mathematicae, vol. 22, pp. 211–320, 1973. [Online]. Avail- able:http://eudml.org/doc/142246

    1973

  67. [69]

    Degeneration of hodge structures,

    E. Cattani, A. Kaplan, and W. Schmid, “Degeneration of hodge structures,” Annals of Mathematics, vol. 123, no. 3, pp. 457–535, 1986,issn: 0003486X, 19398980. Accessed: May 18, 2026. [Online]. Available:http :/ /www .jstor . org/stable/1971333

    work page arXiv 1986

  68. [70]

    Carlson, S

    J. Carlson, S. M¨ uller-Stach, and C. Peters,Period Mappings and Period Do- mains(Cambridge Studies in Advanced Mathematics), 2nd ed. Cambridge Uni- versity Press, 2017. 49

    2017