- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Higher structures on boundary conformal manifolds: Higher Berry phase and boundary conformal field theory

Yichul Choi1, Hyunsoo Ha2, Dongyeob Kim2, Yuya Kusuki3,4,5, Shuhei Ohyama6, and Shinsei Ryu2

PDF

Phys. Rev. D 113, 106005 – Published 7 May, 2026

DOI: https://doi.org/10.1103/z8pw-8ckx

Abstract

We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in (1+1)d conformal field theories, related to each other by exactly marginal boundary deformations, forming a “boundary conformal manifold.” Our definition builds upon previous works on tensor networks, such as matrix product states, where the triple inner product or multiwavefunction overlap plays a key geometric role. On the one hand, our boundary conformal field theory formulation of higher Berry phase provides a new analytic tool to study families of invertible phases in condensed matter systems. On the other hand, it uncovers a new geometric structure on the moduli space of conformal boundary conditions, beyond the usual Riemannian structure defined through the Zamolodchikov metric. When the boundary conformal manifold has an interpretation as the position moduli space of a D-brane, our higher Berry connection coincides with the Neveu-Schwarz-Neveu-Schwarz (NS-NS) B-field in string theory. The general definition does not require such an interpretation and is formulated purely field theoretically, in terms of correlation functions of boundary-condition-changing operators. We also explore a connection between higher Berry connections and functional Berry connections in the loop spaces of boundary conformal manifolds.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (159)

  1. A. B. Zamolodchikov, “Irreversibility” of the flux of the renormalization group in a 2D field theory, JETP Lett. 43, 730 (1986).
  2. N. Seiberg, Observations on the moduli space of superconformal field theories, Nucl. Phys. B303, 286 (1988).
  3. D. Kutasov, Geometry on the space of conformal field theories and contact terms, Phys. Lett. B 220, 153 (1989).
  4. K. Ranganathan, Nearby CFTs in the operator formalism: The role of a connection, Nucl. Phys. B408, 180 (1993).
  5. K. Ranganathan, H. Sonoda, and B. Zwiebach, Connections on the state space over conformal field theories, Nucl. Phys. B414, 405 (1994).
  6. D. Friedan and A. Konechny, Curvature formula for the space of 2-d conformal field theories, J. High Energy Phys. 09 (2012) 113.
  7. J. de Boer, J. Manschot, K. Papadodimas, and E. Verlinde, The Chiral ring of AdS(3)/CFT(2) and the attractor mechanism, J. High Energy Phys. 03 (2009) 030.
  8. B. Balthazar and C. Cordova, Geometry of conformal manifolds and the inversion formula, J. High Energy Phys. 07 (2023) 205.
  9. P. H. Ginsparg, Curiosities at c=1, Nucl. Phys. B295, 153 (1988).
  10. 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, Nucl. Phys. B359, 21 (1991).
  11. C. Behan, Conformal manifolds: ODEs from OPEs, J. High Energy Phys. 03 (2018) 127.
  12. S. Hollands, Action principle for OPE, Nucl. Phys. B926, 614 (2018).
  13. V. Bashmakov, M. Bertolini, and H. Raj, On non-supersymmetric conformal manifolds: field theory and holography, J. High Energy Phys. 11 (2017) 167.
  14. D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa, and B. Wecht, Exactly marginal deformations and global symmetries, J. High Energy Phys. 06 (2010) 106.
  15. J. Gomis, P.-S. Hsin, Z. Komargodski, A. Schwimmer, N. Seiberg, and S. Theisen, Anomalies, conformal manifolds, and spheres, J. High Energy Phys. 03 (2016) 022.
  16. J. Gomis, Z. Komargodski, H. Ooguri, N. Seiberg, and Y. Wang, Shortening anomalies in supersymmetric theories, J. High Energy Phys. 01 (2017) 067.
  17. N. Seiberg, Y. Tachikawa, and K. Yonekura, Anomalies of duality groups and extended conformal manifolds, Prog. Theor. Exp. Phys. 2018, 073B04 (2018).
  18. H. Ooguri and C. Vafa, On the geometry of the string landscape and the Swampland, Nucl. Phys. B766, 21 (2007).
  19. E. Perlmutter, L. Rastelli, C. Vafa, and I. Valenzuela, A CFT distance conjecture, J. High Energy Phys. 10 (2021) 070.
  20. H. Ooguri and Y. Wang, Universal bounds on CFT distance conjecture, J. High Energy Phys. 12 (2024) 154.
  21. A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of D-branes, Nucl. Phys. B545, 233 (1999).
  22. C. G. Callan, I. R. Klebanov, A. W. W. Ludwig, and J. M. Maldacena, Exact solution of a boundary conformal field theory, Nucl. Phys. B422, 417 (1994).
  23. M. R. Gaberdiel, A. Konechny, and C. Schmidt-Colinet, Conformal perturbation theory beyond the leading order, J. Phys. A 42, 105402 (2009).
  24. A. Karch and Y. Sato, Conformal manifolds with boundaries or defects, J. High Energy Phys. 07 (2018) 156.
  25. C. P. Herzog and V. Schaub, Tilting space of boundary conformal field theories, Phys. Rev. D 109, L061701 (2024).
  26. S. Bartlett-Tisdall, C. P. Herzog, and V. Schaub, Bootstrapping boundary QED. Part I, J. High Energy Phys. 05 (2024) 235.
  27. N. Drukker, Z. Kong, and G. Sakkas, Broken global symmetries and defect conformal manifolds, Phys. Rev. Lett. 129, 201603 (2022).
  28. A. Antinucci, C. Copetti, G. Galati, and G. Rizi, Defect conformal manifolds from phantom (non-invertible) symmetries, Phys. Rev. Lett. 135, 211602 (2025).
  29. J. A. Damia, R. Argurio, and S. Chaudhuri, When the moduli space is an orbifold: Spontaneous breaking of continuous non-invertible symmetries, J. High Energy Phys. 03 (2024) 042.
  30. M. R. Gaberdiel and A. Recknagel, Conformal boundary states for free bosons and fermions, J. High Energy Phys. 11 (2001) 016.
  31. A. Y. Kitaev, Differential forms on the space of statistical mechanical lattice models, talk at Between Topology and Quantum Field Theory: A conference in celebration of Dan Freed’s 60th birthday [https://web.ma.utexas.edu/topqft/talkslides/kitaev.pdf].
  32. A. Kapustin and L. Spodyneiko, Higher-dimensional generalizations of Berry curvature, Phys. Rev. B 101, 235130 (2020).
  33. A. Kapustin and L. Spodyneiko, Higher-dimensional generalizations of the Thouless charge pump, arXiv:2003.09519.
  34. P.-S. Hsin, A. Kapustin, and R. Thorngren, Berry phase in quantum field theory: Diabolical points and boundary phenomena, Phys. Rev. B 102, 245113 (2020).
  35. C. Córdova, D. S. Freed, H. T. Lam, and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications I, SciPost Phys. 8, 001 (2020).
  36. C. Córdova, D. S. Freed, H. T. Lam, and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications II, SciPost Phys. 8, 002 (2020).
  37. A. Kapustin and N. Sopenko, Local Noether theorem for quantum lattice systems and topological invariants of gapped states, J. Math. Phys. (N.Y.) 63, 091903 (2022).
  38. Y. Choi and K. Ohmori, Higher Berry phase of fermions and index theorem, J. High Energy Phys. 09 (2022) 022.
  39. X. Wen, M. Qi, A. Beaudry, J. Moreno, M. J. Pflaum, D. Spiegel, A. Vishwanath, and M. Hermele, Flow of higher Berry curvature and bulk-boundary correspondence in parametrized quantum systems, Phys. Rev. B 108, 125147 (2023).
  40. S. Ohyama, Y. Terashima, and K. Shiozaki, Discrete higher Berry phases and matrix product states, Phys. Rev. B 110, 035114 (2024).
  41. A. Beaudry, M. Hermele, J. Moreno, M. J. Pflaum, M. Qi, and D. D. Spiegel, Homotopical foundations of parametrized quantum spin systems, Rev. Math. Phys. 36, 2460003 (2024).
  42. M. Qi, D. T. Stephen, X. Wen, D. Spiegel, M. J. Pflaum, A. Beaudry, and M. Hermele, Charting the space of ground states with tensor networks, SciPost Phys. 18, 168 (2025).
  43. D. D. Spiegel, A C*-algebraic approach to parametrized quantum spin systems and their phases in one spatial dimension, arXiv:2305.07951.
  44. K. Shiozaki, N. Heinsdorf, and S. Ohyama, Higher Berry curvature from matrix product states, Phys. Rev. B 112, 035154 (2025).
  45. O. E. Sommer, X. Wen, and A. Vishwanath, Higher Berry curvature from the wave function. I. Schmidt decomposition and matrix product states, Phys. Rev. Lett. 134, 146601 (2025).
  46. O. E. Sommer, A. Vishwanath, and X. Wen, Higher Berry curvature from the wave function. II. Locally parametrized states beyond one dimension, Phys. Rev. B 111, 155110 (2025).
  47. A. Artymowicz, A. Kapustin, and N. Sopenko, Quantization of the higher Berry curvature and the higher Thouless pump, Commun. Math. Phys. 405, 191 (2024).
  48. R. Geiko, Parametrized topological phases in 1d and T-duality, arXiv:2412.20905.
  49. N. Manjunath and D. V. Else, Anomalous continuous symmetries and quantum topology of Goldstone modes, Phys. Rev. B 111, 125151 (2025).
  50. S. Ohyama and S. Ryu, Higher structures in matrix product states, Phys. Rev. B 109, 115152 (2024).
  51. J. L. Cardy, Boundary conformal field theory, arXiv:hep-th/0411189.
  52. X.-L. Qi, H. Katsura, and A. W. W. Ludwig, General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states, Phys. Rev. Lett. 108, 196402 (2012).
  53. M. Miyaji, S. Ryu, T. Takayanagi, and X. Wen, Boundary states as holographic duals of trivial spacetimes, J. High Energy Phys. 05 (2015) 152.
  54. G. Y. Cho, K. Shiozaki, S. Ryu, and A. W. W. Ludwig, Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum, J. Phys. A 50, 304002 (2017).
  55. J. Cardy, Bulk renormalization group flows and boundary states in conformal field theories, SciPost Phys. 3, 011 (2017).
  56. E. Witten, Noncommutative geometry and string field theory, Nucl. Phys. B268, 253 (1986).
  57. V. Schomerus, Lectures on branes in curved backgrounds, Classical Quantum Gravity 19, 5781 (2002).
  58. D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401 (2007).
  59. D. Pérez-García, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, String order and symmetries in quantum spin lattices, Phys. Rev. Lett. 100, 167202 (2008).
  60. J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics Vol. 107 (Birkhäuser, Boston, 1993).
  61. A. Beaudry, M. Hermele, M. J. Pflaum, M. Qi, D. D. Spiegel, and D. T. Stephen, A classifying space for phases of matrix product states, Commun. Math. Phys. 407, 17 (2026).
  62. T. T. Wu and C. N. Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12, 3845 (1975).
  63. T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005).
  64. J. Cardy, Boundary conditions in conformal field theory, Adv. Stud. Pure Math. 19, 127 (1989).
  65. J. L. Cardy, Conformal invariance and statistical mechanics, in Fields, Strings and Critical Phenomena, edited by E. Brézin and J. Zinn-Justin (North-Holland, Amsterdam, 1990), pp. 169–245.
  66. E. Date, M. Jimbo, T. Miwa, and M. Okado, Automorphic properties of local height probabilities for integrable solid-on-solid models, Phys. Rev. B 35, 2105 (1987).
  67. H. Saleur and M. Bauer, On some relations between local height probabilities and conformal invariance, Nucl. Phys. B320, 591 (1989).
  68. P. Calabrese, J. Cardy, and I. Peschel, Corrections to scaling for block entanglement in massive spin chains, J. Stat. Mech. (2010) P09003.
  69. T. Erler, Four lectures on analytic solutions in open string field theory, Phys. Rep. 980, 1 (2022).
  70. J. Fuchs, I. Runkel, and C. Schweigert, TFT construction of RCFT correlators I: Partition functions, Nucl. Phys. B646, 353 (2002).
  71. J. L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B324, 581 (1989).
  72. D. C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B372, 654 (1992).
  73. Y. Choi, B. C. Rayhaun, Y. Sanghavi, and S.-H. Shao, Remarks on boundaries, anomalies, and noninvertible symmetries, Phys. Rev. D 108, 125005 (2023).
  74. R. A. Janik, Exceptional boundary states at c=1, Nucl. Phys. B618, 675 (2001).
  75. I. Affleck and A. W. W. Ludwig, Universal noninteger “ground state degeneracy” in critical quantum systems, Phys. Rev. Lett. 67, 161 (1991).
  76. D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93, 030402 (2004).
  77. H. Casini, I. Salazar Landea, and G. Torroba, The g-theorem and quantum information theory, J. High Energy Phys. 10 (2016) 140.
  78. G. Cuomo, Z. Komargodski, and A. Raviv-Moshe, Renormalization Group flows on line defects, Phys. Rev. Lett. 128, 021603 (2022).
  79. M. R. Gaberdiel, A. Recknagel, and G. M. T. Watts, The conformal boundary states for SU(2) at level 1, Nucl. Phys. B626, 344 (2002).
  80. J. L. Cardy, Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories, Nucl. Phys. B275, 200 (1986).
  81. E. M. Brehm and I. Runkel, Lattice models from CFT on surfaces with holes: I. Torus partition function via two lattice cells, J. Phys. A 55, 235001 (2022).
  82. E. M. Brehm and I. Runkel, Lattice models from CFT on surfaces with holes II: Cloaking boundary conditions and loop models, arXiv:2410.19938.
  83. G. Cheng, L. Chen, Z.-C. Gu, and L.-Y. Hung, Precision reconstruction of rational CFT from exact fixed point tensor network, Phys. Rev. X 15, 011073 (2025).
  84. C. Bachas and M. Gaberdiel, Loop operators and the kondo problem, J. High Energy Phys. 11 (2004) 065.
  85. I. Affleck and A. W. W. Ludwig, The Fermi edge singularity and boundary condition changing operators, J. Phys. A 27, 5375 (1994).
  86. M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional ising model with a defect line, Nucl. Phys. B495, 533 (1997).
  87. I. Affleck and A. W. W. Ludwig, The fermi edge singularity and boundary condition changing operators, J. Phys. A 27, 5375 (1994).
  88. A. Giveon, M. Porrati, and E. Rabinovici, Target space duality in string theory, Phys. Rep. 244, 77 (1994).
  89. I. Affleck, Quantum impurity problems in condensed matter physics, arXiv:0809.3474.
  90. C. G. Callan, Jr., C. Lovelace, C. R. Nappi, and S. A. Yost, Loop corrections to superstring equations of motion, Nucl. Phys. B308, 221 (1988).
  91. A. Debray, S. K. Devalapurkar, C. Krulewski, Y. L. Liu, N. Pacheco-Tallaj, and R. Thorngren, A long exact sequence in symmetry breaking: Order parameter constraints, defect anomaly-matching, and higher Berry phases, J. High Energy Phys. 07 (2025) 007.
  92. M. B. Green and M. Gutperle, Symmetry breaking at enhanced symmetry points, Nucl. Phys. B460, 77 (1996).
  93. A. Recknagel and V. Schomerus, D-branes in Gepner models, Nucl. Phys. B531, 185 (1998).
  94. A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of d-branes, Nucl. Phys. B545, 233 (1999).
  95. A. Kapustin and N. Saulina, Surface operators in 3d topological field theory and 2d rational conformal field theory, arXiv:1012.0911.
  96. S. Elitzur, G. W. Moore, A. Schwimmer, and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B326, 108 (1989).
  97. O. Alvarez, Topological quantization and cohomology, Commun. Math. Phys. 100, 279 (1985).
  98. K. Gawędzki, Topological actions in two-dimensional quantum field theories, Nonperturbative Quantum Field Theory (Springer, New York, 1988), p. 101.
  99. A. Kapustin, D-branes in a topologically nontrivial B field, Adv. Theor. Math. Phys. 4, 127 (2000).
  100. D. S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math. 3, 819 (1999).
  101. A. L. Carey, S. Johnson, and M. K. Murray, Holonomy on D-branes, arXiv:hep-th/0204199.
  102. A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories, Commun. Math. Phys. 259, 577 (2005).
  103. U. Schreiber, C. Schweigert, and K. Waldorf, Unoriented WZW models and holonomy of bundle gerbes, Commun. Math. Phys. 274, 31 (2007).
  104. K. Waldorf, Gerbes in unoriented WZW models, SFIN A 1, 423 (2006).
  105. I. Runkel and R. R. Suszek, Gerbe-holonomy for surfaces with defect networks, Adv. Theor. Math. Phys. 13, 1137 (2009).
  106. J. M. Maldacena, G. W. Moore, and N. Seiberg, Geometrical interpretation of D-branes in gauged WZW models, J. High Energy Phys. 07 (2001) 046.
  107. D. Blakeley and A. Recknagel, Symmetry-breaking boundary states for WZW models, Nucl. Phys. B806, 636 (2009).
  108. M. Kudrna, Boundary states in the SU(2)k WZW model from open string field theory, J. High Energy Phys. 03 (2023) 228.
  109. Y. Lee, K. Ohmori, and Y. Tachikawa, Revisiting Wess-Zumino-Witten terms, SciPost Phys. 10, 061 (2021).
  110. C. G. Callan, Jr. and I. R. Klebanov, Exact C=1 boundary conformal field theories, Phys. Rev. Lett. 72, 1968 (1994).
  111. S. Collier, D. Mazac, and Y. Wang, Bootstrapping boundaries and branes, J. High Energy Phys. 02 (2023) 019.
  112. E. Lupercio and B. Uribe, An introduction to gerbes on orbifolds, Ann. Math. Blaise Pascal 11, 155 (2004).
  113. E. Lupercio and B. Uribe, Deligne cohomology for orbifolds, discrete torsion and B fields, in Geometric and Topological Methods for Quantum Field Theory, edited by A. Cardona, H. Ocampo, and S. Paycha (World Scientific, Singapore, 2003), pp. 468–482.
  114. A. Adem, J. Leida, and Y. Ruan, Orbifolds and Stringy Topology, Vol. 171 of Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, England, 2009).
  115. T. Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann. 206, 243 (1973).
  116. B. Liu, J. Zhang, S. Ohyama, Y. Kusuki, and S. Ryu, Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions, Phys. Rev. B 112, 125160 (2025).
  117. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013).
  118. Y. Choi, B. C. Rayhaun, and Y. Zheng, Generalized tube algebras, symmetry-resolved partition functions, and twisted boundary states, Commun. Math. Phys. 407, 62 (2026).
  119. V. Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8, 177 (2003).
  120. L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, J. High Energy Phys. 03 (2018) 189.
  121. C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, Topological defect lines and renormalization group flows in two dimensions, J. High Energy Phys. 01 (2019) 026.
  122. R. Thorngren and Y. Wang, Fusion category symmetry. Part I. Anomaly in-flow and gapped phases, J. High Energy Phys. 04 (2024) 132.
  123. K. Inamura and S. Ohyama, 1+1d spt phases with fusion category symmetry: Interface modes and non-abelian thouless pump, arXiv:2408.15960.
  124. J. Mickelsson, Kac-Moody Groups, topology of the Dirac determinant bundle and fermionization, Commun. Math. Phys. 110, 173 (1987).
  125. M. Stone, Coherent state path integrals for loop groups and nonabelian bosonization, Nucl. Phys. B327, 399 (1989).
  126. S. Iso, C. Itoi, and H. Mukaida, Necessity of a finite size term in the WZW model, Phys. Lett. B 244, 241 (1990).
  127. K. Waldorf, Transgressive loop group extensions, Math. Z. 286, 325 (2016).
  128. T. Kohno, Conformal Field Theory and Topology, Vol. 210 of Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs (American Mathematical Society, Providence, 2002).
  129. N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys. 09 (1999) 032.
  130. G. Arcioni, M. Carfora, C. Dappiaggi, and A. Marzuoli, The WZW model on random Regge triangulations, J. Geom. Phys. 52, 137 (2004).
  131. L.-Y. Hung and Y. Jiang, Building up quantum spacetimes with bcft legos, arXiv:2404.00877.
  132. N. Sopenko, Chiral topologically ordered states on a lattice from vertex operator algebras, Adv. Theor. Math. Phys. 28, 1 (2024).
  133. I. Runkel, Boundary structure constants for the a-series Virasoro minimal models, Nucl. Phys. B549, 563 (1999).
  134. G. W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123, 177 (1989).
  135. S. Iino, S. Morita, and N. Kawashima, Boundary tensor renormalization group, Phys. Rev. B 100, 035449 (2019).
  136. S. Iino, S. Morita, and N. Kawashima, Boundary conformal spectrum and surface critical behavior of classical spin systems: A tensor network renormalization study, Phys. Rev. B 101, 155418 (2020).
  137. S. Iino, Boundary CFT and tensor network approach to surface critical phenomena of the tricritical 3-state Potts model, J. Stat. Phys. 182, 56 (2021).
  138. A. Ueda and M. Yamazaki, Fixed-point tensor is a four-point function, arXiv:2307.02523.
  139. L. Rastelli and B. Zwiebach, Tachyon potentials, star products and universality, J. High Energy Phys. 09 (2001) 038.
  140. M. Schnabl, Analytic solution for tachyon condensation in open string field theory, Adv. Theor. Math. Phys. 10, 433 (2006).
  141. M. Kiermaier, Y. Okawa, and P. Soler, Solutions from boundary condition changing operators in open string field theory, J. High Energy Phys. 03 (2011) 122.
  142. A. Perez-Lona, E. Sharpe, and X. Yu, Categorified structures over moduli spaces: Anomalies, non-invertible symmetries, and exceptional holonomy, arXiv:2506.19909.
  143. E. Witten and J. Bagger, Quantization of Newton’s constant in certain supergravity theories, Phys. Lett. 115B, 202 (1982).
  144. J. Distler, Notes on N=2 sigma models, arXiv:hep-th/9212062.
  145. V. Periwal and A. Strominger, Kahler geometry of the space of N=2 superconformal field theories, Phys. Lett. B 235, 261 (1990).
  146. E. Sharpe, An overview of Bagger-Witten line bundles, arXiv:2412.09198.
  147. M. Baggio, V. Niarchos, and K. Papadodimas, Aspects of Berry phase in QFT, J. High Energy Phys. 04 (2017) 062.
  148. C. Bachas, J. de Boer, R. Dijkgraaf, and H. Ooguri, Permeable conformal walls and holography, J. High Energy Phys. 06 (2002) 027.
  149. S. Ohyama and S. Ryu, Higher Berry phase from projected entangled pair states in (2+1) dimensions, Phys. Rev. B 111, 045112 (2025).
  150. T. Quella, I. Runkel, and G. M. T. Watts, Reflection and transmission for conformal defects, J. High Energy Phys. 04 (2007) 095.
  151. K. Sakai and Y. Satoh, Entanglement through conformal interfaces, J. High Energy Phys. 12 (2008) 001.
  152. P.-S. Hsin and Z. Wang, On topology of the moduli space of gapped Hamiltonians for topological phases, J. Math. Phys. (N.Y.) 64, 041901 (2023).
  153. S. Ohyama and S. Ryu, Higher Berry connection for matrix product states, Phys. Rev. B 111, 035121 (2025).
  154. A. Henriques, What Chern-Simons theory assigns to a point, arXiv:1503.06254.
  155. X. Wen, Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs, arXiv:2507.12546.
  156. D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27, 6083 (1983).
  157. D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, Theta, Time reversal, and temperature, J. High Energy Phys. 05 (2017) 091.
  158. M. Oshikawa, C. Chamon, and I. Affleck, Junctions of three quantum wires, J. Stat. Mech. (2006) P02008.
  159. V. G. Kac, Infinite dimensional Lie algebras, 3rd ed. (Cambridge University Press, Cambridge, England, 1990).

Outline

Information

Phys. Rev. D 113, 106005– Published 7 May, 2026
  • Received 7 December 2025
  • Accepted 10 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation