Abstract
We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.
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A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, arXiv:1309.4721 [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
V. B. Petkova and J. B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
C. Bachas and M. Gaberdiel, Loop operators and the Kondo problem, JHEP 11 (2004) 065 [hep-th/0411067] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
A. Davydov, L. Kong and I. Runkel, Invertible Defects and Isomorphisms of Rational CFTs, Adv. Theor. Math. Phys. 15 (2011) 43 [arXiv:1004.4725] [INSPIRE].
L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].
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 01 (2019) 026 [arXiv:1802.04445] [INSPIRE].
R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases, arXiv:1912.02817 [INSPIRE].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].
R. Thorngren and Y. Wang, Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond, arXiv:2106.12577 [INSPIRE].
K. Kikuchi, Symmetry enhancement in RCFT, arXiv:2109.02672 [INSPIRE].
A. Feiguin et al., Interacting anyons in topological quantum liquids: The golden chain, Phys. Rev. Lett. 98 (2007) 160409 [cond-mat/0612341] [INSPIRE].
D. Aasen, R. S. K. Mong and P. Fendley, Topological Defects on the Lattice I: The Ising model, J. Phys. A 49 (2016) 354001 [arXiv:1601.07185] [INSPIRE].
D. Aasen, P. Fendley and R. S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies, arXiv:2008.08598 [INSPIRE].
R. H. Dijkgraaf, A geometrical approach to two-dimensional conformal field theory, Ph.D. Thesis, Utrecht University, The Netherlands (1989).
M. Fukuma, S. Hosono and H. Kawai, Lattice topological field theory in two-dimensions, Commun. Math. Phys. 161 (1994) 157 [hep-th/9212154] [INSPIRE].
B. Durhuus and T. Jonsson, Classification and construction of unitary topological field theories in two-dimensions, J. Math. Phys. 35 (1994) 5306 [hep-th/9308043] [INSPIRE].
S. Sawin, Direct sum decompositions and indecomposable TQFTs, J. Math. Phys. 36 (1995) 6673 [q-alg/9505026] [INSPIRE].
L. S. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theor. Ramifications 5 (1996) 569.
J. Kock, Frobenius algebras and 2-d topological quantum field theories, vol. 59, Cambridge University Press (2004).
C. I. Lazaroiu, On the structure of open-closed topological field theory in two-dimensions, Nucl. Phys. B 603 (2001) 497 [hep-th/0010269] [INSPIRE].
A. Alexeevski and S. Natanzon, Noncommutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves, math/0202164 [INSPIRE].
A. D. Lauda and H. Pfeiffer, Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras, math/0510664 [INSPIRE].
A. D. Lauda and H. Pfeiffer, State sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theor. Ramifications 16 (2007) 1121 [math/0602047] [INSPIRE].
G. W. Moore and G. Segal, D-branes and k-theory in 2D topological field theory, hep-th/0609042 [INSPIRE].
I. Runkel and R. R. Suszek, Gerbe-holonomy for surfaces with defect networks, Adv. Theor. Math. Phys. 13 (2009) 1137 [arXiv:0808.1419] [INSPIRE].
A. Davydov, L. Kong and I. Runkel, Field theories with defects and the centre functor, arXiv:1107.0495 [INSPIRE].
K. Inamura, Topological field theories and symmetry protected topological phases with fusion category symmetries, JHEP 05 (2021) 204 [arXiv:2103.15588] [INSPIRE].
H. Sonoda, Sewing conformal field theories, Nucl. Phys. B 311 (1988) 401 [INSPIRE].
H. Sonoda, Sewing conformal field theories. 2, Nucl. Phys. B 311 (1988) 417 [INSPIRE].
G. W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
G. W. Moore and N. Seiberg, LECTURES ON RCFT, in 1989 Banff NATO ASI: Physics, Geometry and Topology, (1989), pp. 1–129.
B. Bakalov and A. Kirillov, On the lego-teichmüller game, Transform. Groups 5 (2000) 207.
V. G. Turaev and O. Y. Viro, State sum invariants of 3 manifolds and quantum 6j symbols, Topology 31 (1992) 865 [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions, Phys. Rev. X 8 (2018) 031048 [arXiv:1705.06728] [INSPIRE].
Y. Tachikawa, On gauging finite subgroups, SciPost Phys. 8 (2020) 015 [arXiv:1712.09542] [INSPIRE].
T.-C. Huang and Y.-H. Lin, Topological Field Theory with Haagerup Symmetry, arXiv:2102.05664 [INSPIRE].
U. Haagerup, Principal graphs of subfactors in the index range 4 < [M : N] < 3 + \( \sqrt{2} \), in Subfactors (Kyuzeso, 1993), World. Sci. Publ., River Edge, NJ, U.S.A. (1994), pg. 1–38.
M. Asaeda and U. Haagerup, Exotic subfactors of finite depth with jones indices \( \left(5+\sqrt{13}\right)/2 \) and \( \left(5+\sqrt{17}\right)/2 \), Commun. Math. Phys. 202 (1999) 1.
P. Grossman and N. Snyder, Quantum subgroups of the haagerup fusion categories, Commun. Math. Phys. 311 (2012) 617.
J. C. Baez and J. Dolan, Higher dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995) 6073 [q-alg/9503002] [INSPIRE].
A. Kapustin, Topological Field Theory, Higher Categories, and Their Applications, in International Congress of Mathematicians, (2010) [arXiv:1004.2307] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
F. Benini, C. Córdova and P.-S. Hsin, On 2-Group Global Symmetries and their Anomalies, JHEP 03 (2019) 118 [arXiv:1803.09336] [INSPIRE].
P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, math/0203060.
P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, vol. 205, American Mathematical Soc., (2016).
S. MacLane, Natural Associativity and Commutativity, Rice Univ. Studies 49 (1963) 28.
K. Fredenhagen, K.-H. Rehren and B. Schroer, Superselection sectors with braid group statistics and exchange algebras ii: Geometric aspects and conformal covariance, Rev. Math. Phys. 4 (1992) 113.
S.-H. Ng and P. Schauenburg, Higher frobenius-schur indicators for pivotal categories, Hopf algebras and generalizations 441 (2007) 63.
E. P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
J. L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, Boundary conditions in rational conformal field theories, Nucl. Phys. B 570 (2000) 525 [hep-th/9908036] [INSPIRE].
T. Gannon, Boundary conformal field theory and fusion ring representations, Nucl. Phys. B 627 (2002) 506 [hep-th/0106105] [INSPIRE].
M. R. Gaberdiel and T. Gannon, Boundary states for WZW models, Nucl. Phys. B 639 (2002) 471 [hep-th/0202067] [INSPIRE].
G. Schaumann, Traces on module categories over fusion categories, J. Algebra 379 (2013) 382.
K. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007) 165 [math/0412149] [INSPIRE].
J. Fjelstad, J. Fuchs, I. Runkel and C. Schweigert, Uniqueness of open / closed rational CFT with given algebra of open states, Adv. Theor. Math. Phys. 12 (2008) 1283 [hep-th/0612306] [INSPIRE].
T.-C. Huang and Y.-H. Lin, The F -Symbols for Transparent Haagerup-Izumi Categories with G = ℤ2n+1, arXiv:2007.00670 [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Defect lines, dualities, and generalised orbifolds, in 16th International Congress on Mathematical Physics, (2009), DOI [arXiv:0909.5013] [INSPIRE].
N. Carqueville and I. Runkel, Orbifold completion of defect bicategories, Quantum Topol. 7 (2016) 203 [arXiv:1210.6363] [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, A quick guide to defect orbifolds, Proc. Symp. Pure Math. 88 (2014) 231 [arXiv:1310.0062] [INSPIRE].
C. Vafa, Modular Invariance and Discrete Torsion on Orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].
A. Kirillov Jr and V. Ostrik, On a q-analogue of the mckay correspondence and the ade classification of sl2 conformal field theories, Adv. Math. 171 (2002) 183.
M. Müger, From subfactors to categories and topology I: Frobenius algebras in and morita equivalence of tensor categories, J. Pure Appl. Algebra 180 (2003) 81.
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators 1. Partition functions, Nucl. Phys. B 646 (2002) 353 [hep-th/0204148] [INSPIRE].
J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri and S.-H. Shao, Higher central charges and topological boundaries in 2+1-dimensional TQFTs, arXiv:2107.13091 [INSPIRE].
D. Gaiotto and T. Johnson-Freyd, Condensations in higher categories, arXiv:1905.09566 [INSPIRE].
C. Vafa, Quantum Symmetries of String Vacua, Mod. Phys. Lett. A 4 (1989) 1615 [INSPIRE].
V. Ostrik, Module categories, weak hopf algebras and modular invariants, Transform. Groups 8 (2003) 177.
P. M. Cohn, Basic algebra: groups, rings and fields, Springer Science & Business Media, (2012).
C.-M. Chang and Y.-H. Lin, On exotic consistent anomalies in (1+1)d: A ghost story, SciPost Phys. 10 (2021) 119 [arXiv:2009.07273] [INSPIRE].
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Huang, TC., Lin, YH. & Seifnashri, S. Construction of two-dimensional topological field theories with non-invertible symmetries. J. High Energ. Phys. 2021, 28 (2021). https://doi.org/10.1007/JHEP12(2021)028
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DOI: https://doi.org/10.1007/JHEP12(2021)028