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A classification of modular functors via factorization homology

Adrien Brochier and Lukas Woike

Geometry & Topology 30 (2026) 2157–2216
Bibliography
1 L Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996) 569 MR1414088
2 Y Ai, L Kong, H Zheng, Topological orders and factorization homology, Adv. Theor. Math. Phys. 21 (2017) 1845 MR3783835
3 R Allen, S Lentner, C Schweigert, S Wood, Duality structures for representation categories of vertex operator algebras and the Feigin–Fuchs boson, Selecta Math. 31 (2025) 36 MR4883790
4 J E Andersen, Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups, Ann. of Math. 163 (2006) 347 MR2195137
5 J E Andersen, K Ueno, Modular functors are determined by their genus zero data, Quantum Topol. 3 (2012) 255 MR2928086
6 R Andrade, From manifolds to invariants of En-algebras, PhD thesis, Massachusetts Institute of Technology (2010) MR2898647
7 M Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988) 175 MR1001453
8 M Atiyah, On framings of 3-manifolds, Topology 29 (1990) 1 MR1046621
9 D Ayala, J Francis, Factorization homology of topological manifolds, J. Topol. 8 (2015) 1045 MR3431668
10 J C Baez, J Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995) 6073 MR1355899
11 B Bakalov, A Kirillov Jr., On the Lego-Teichmüller game, Transform. Groups 5 (2000) 207 MR1780935
12 B Bakalov, A Kirillov Jr., Lectures on tensor categories and modular functors, 21, Amer. Math. Soc. (2001) MR1797619
13 B Balsam, Turaev–Viro invariants as an extended TQFT, III, preprint (2010) arXiv:1012.0560
14 B Bartlett, C L Douglas, C J Schommer-Pries, J Vicary, Modular categories as representations of the 3-dimensional bordism 2-category, preprint (2015) arXiv:1509.06811
15 A Beilinson, V Drinfeld, Chiral algebras, 51, Amer. Math. Soc. (2004) MR2058353
16 D Ben-Zvi, E Frenkel, Vertex algebras and algebraic curves, 88, Amer. Math. Soc. (2004) MR2082709
17 D Ben-Zvi, A Brochier, D Jordan, Integrating quantum groups over surfaces, J. Topol. 11 (2018) 874 MR3847209
18 D Ben-Zvi, A Brochier, D Jordan, Quantum character varieties and braided module categories, Selecta Math. 24 (2018) 4711 MR3874702
19 V Blanco, M Bullejos, E Faro, Categorical non-abelian cohomology and the Schreier theory of groupoids, Math. Z. 251 (2005) 41 MR2176463
20 J M Boardman, R M Vogt, Homotopy-everything H-spaces, Bull. Amer. Math. Soc. 74 (1968) 1117 MR236922
21 J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, 347, Springer (1973) MR420609
22 M Boyarchenko, V Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier, Quantum Topol. 4 (2013) 447 MR3134025
23 A Brochier, D Jordan, P Safronov, N Snyder, Invertible braided tensor categories, Algebr. Geom. Topol. 21 (2021) 2107 MR4302495
24 A Brochier, D Jordan, N Snyder, On dualizability of braided tensor categories, Compos. Math. 157 (2021) 435 MR4228258
25 J L Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nuclear Phys. B 324 (1989) 581 MR1019724
26 P Carrasco, A M Cegarra, A R Garzón, Nerves and classifying spaces for bicategories, Algebr. Geom. Topol. 10 (2010) 219 MR2602835
27 A M Cegarra, Homotopy fiber sequences induced by 2-functors, J. Pure Appl. Algebra 215 (2011) 310 MR2738356
28 J Cooke, Excision of skein categories and factorisation homology, Adv. Math. 414 (2023) 108848 MR4536120
29 K Costello, The A-infinity operad and the moduli space of curves, preprint (2004) arXiv:math/0402015
30 K Costello, O Gwilliam, Factorization algebras in quantum field theory, I, 31, Cambridge Univ. Press (2017) MR3586504
31 C Damiolini, A Gibney, N Tarasca, Conformal blocks from vertex algebras and their connections on g,n, Geom. Topol. 25 (2021) 2235 MR4310890
32 C Damiolini, A Gibney, N Tarasca, On factorization and vector bundles of conformal blocks from vertex algebras, Ann. Sci. Éc. Norm. Supér. 57 (2024) 241 MR4732678
33 M De Renzi, A M Gainutdinov, N Geer, B Patureau-Mirand, I Runkel, 3-dimensional TQFTs from non-semisimple modular categories, Selecta Math. 28 (2022) 42 MR4370375
34 M De Renzi, A M Gainutdinov, N Geer, B Patureau-Mirand, I Runkel, Mapping class group representations from non-semisimple TQFTs, Commun. Contemp. Math. 25 (2023) 2150091 MR4523145
35 P Dedecker, Les foncteurs ExtΠ, HΠ2 et HΠ2 non abéliens, C. R. Acad. Sci. Paris 258 (1964) 4891 MR168613
36 C L Douglas, C Schommer-Pries, N Snyder, Dualizable tensor categories, 1308, Amer. Math. Soc. (2020) MR4254952
37 J W Duskin, Simplicial matrices and the nerves of weak n-categories, I : Nerves of bicategories, Theory Appl. Categ. 9 (2001/02) 198 MR1897816
38 S Eilenberg, S Mac Lane, On the groups H,n), I, Ann. of Math. 58 (1953) 55 MR56295
39 P Etingof, V Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004) 627, 782 MR2119143
40 P Etingof, D Nikshych, V Ostrik, An analogue of Radford’s S4 formula for finite tensor categories, Int. Math. Res. Not. 2004 (2004) 2915 MR2097289
41 P Etingof, S Gelaki, D Nikshych, V Ostrik, Tensor categories, 205, Amer. Math. Soc. (2015) MR3242743
42 B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012) MR2850125
43 D S Freed, 3-dimensional TQFTs through the lens of the cobordism hypothesis, lecture slides (2012)
44 D S Freed, 4-3-2-8-7-6, lecture slides (2012)
45 D S Freed, C Teleman, Relative quantum field theory, Comm. Math. Phys. 326 (2014) 459 MR3165462
46 M H Freedman, K Walker, Z Wang, Quantum SU(2) faithfully detects mapping class groups modulo center, Geom. Topol. 6 (2002) 523 MR1943758
47 E Frenkel, Vertex algebras and algebraic curves, from: "Séminaire Bourbaki, 1999/2000", Soc. Math. de France (2002) MR1886764
48 B Fresse, Homotopy of operads and Grothendieck–Teichmüller groups, I : The algebraic theory and its topological background, 217, Amer. Math. Soc. (2017)
49 J Fuchs, C Schweigert, Consistent systems of correlators in non-semisimple conformal field theory, Adv. Math. 307 (2017) 598 MR3590526
50 J Fuchs, I Runkel, C Schweigert, TFT construction of RCFT correlators, I : Partition functions, Nuclear Phys. B 646 (2002) 353 MR1940282
51 J Fuchs, I Runkel, C Schweigert, TFT construction of RCFT correlators, II : Unoriented world sheets, Nuclear Phys. B 678 (2004) 511 MR2026879
52 J Fuchs, I Runkel, C Schweigert, TFT construction of RCFT correlators, III : Simple currents, Nuclear Phys. B 694 (2004) 277 MR2076134
53 J Fuchs, I Runkel, C Schweigert, TFT construction of RCFT correlators, IV : Structure constants and correlation functions, Nuclear Phys. B 715 (2005) 539 MR2137114
54 J Fuchs, I Runkel, C Schweigert, Twenty five years of two-dimensional rational conformal field theory, J. Math. Phys. 51 (2010) 015210 MR2605843
55 J Fuchs, G Schaumann, C Schweigert, A modular functor from state sums for finite tensor categories and their bimodules, Theory Appl. Categ. 38 (2022) 436 MR4403279
56 J Fuchs, C Schweigert, Y Yang, String-net construction of RCFT correlators, 45, Springer (2022) MR4696498
57 E Getzler, M M Kapranov, Cyclic operads and cyclic homology, from: "Geometry, topology, & physics" (editor S T Yau), Conf. Proc. Lecture Notes Geom. Topology IV, Int. (1995) 167 MR1358617
58 E Getzler, M M Kapranov, Modular operads, Compositio Math. 110 (1998) 65 MR1601666
59 J Giansiracusa, The framed little 2-discs operad and diffeomorphisms of handlebodies, J. Topol. 4 (2011) 919 MR2860346
60 P M Gilmer, G Masbaum, Maslov index, lagrangians, mapping class groups and TQFT, Forum Math. 25 (2013) 1067 MR3100961
61 G Ginot, Notes on factorization algebras, factorization homology and applications, from: "Mathematical aspects of quantum field theories" (editors D Calaque, T Strobl), Springer (2015) 429 MR3330249
62 S Gunningham, D Jordan, P Safronov, The finiteness conjecture for skein modules, Invent. Math. 232 (2023) 301 MR4557403
63 P Hackney, M Robertson, D Yau, A graphical category for higher modular operads, Adv. Math. 365 (2020) 107044, 61 MR4064770
64 B Haïoun, Unit inclusion in a (nonsemisimple) braided tensor category and (noncompact) relative TQFTs, Geom. Topol. 29 (2025) 2175 MR4929475
65 A Hatcher, Spaces of incompressible surfaces, preprint (1999)
66 R Haugseng, The higher Morita category of 𝔼n-algebras, Geom. Topol. 21 (2017) 1631 MR3650080
67 S Hollander, A homotopy theory for stacks, Israel J. Math. 163 (2008) 93 MR2391126
68 Y Z Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008) 871 MR2468370
69 C Kassel, Quantum groups, 155, Springer (1995) MR1321145
70 T Kerler, V V Lyubashenko, Non-semisimple topological quantum field theories for 3-manifolds with corners, 1765, Springer (2001) MR1862634
71 A Kirillov, String-net model of Turaev–Viro invariants, preprint (2011) arXiv:1106.6033
72 J Kock, Frobenius algebras and 2D topological quantum field theories, 59, Cambridge Univ. Press (2004) MR2037238
73 S Lentner, S N Mierach, C Schweigert, Y Sommerhäuser, Hochschild cohomology, modular tensor categories, and mapping class groups, I, 44, Springer (2023) MR4841457
74 I López Franco, Tensor products of finitely cocomplete and abelian categories, J. Algebra 396 (2013) 207 MR3108080
75 J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) MR2522659
76 J Lurie, On the classification of topological field theories, from: "Current developments in mathematics, 2008", Int. (2009) 129 MR2555928
77 J Lurie, Higher algebra, preprint (2017)
78 V V Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172 (1995) 467 MR1354257
79 V Lyubashenko, Modular transformations for tensor categories, J. Pure Appl. Algebra 98 (1995) 279 MR1324034
80 V Lyubashenko, Ribbon abelian categories as modular categories, J. Knot Theory Ramifications 5 (1996) 311 MR1405715
81 S Mac Lane, I Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory, Springer (1994) MR1300636
82 G Masbaum, J D Roberts, On central extensions of mapping class groups, Math. Ann. 302 (1995) 131 MR1329450
83 J P May, The geometry of iterated loop spaces, 271, Springer (1972) MR420610
84 G Moore, N Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212 (1988) 451 MR962600
85 G Moore, N Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989) 177 MR1002038
86 G Moore, N Seiberg, Lectures on RCFT, from: "Physics, geometry, and topology" (editor H C Lee), NATO Adv. Sci. Inst. Ser. B: Phys. 238, Plenum (1990) 263 MR1153682
87 L Müller, L Woike, Cyclic framed little disks algebras, Grothendieck–Verdier duality and handlebody group representations, Q. J. Math. 74 (2023) 163 MR4571627
88 L Müller, L Woike, The diffeomorphism group of the solid closed torus and Hochschild homology, Proc. Amer. Math. Soc. 151 (2023) 2311 MR4576300
89 L Müller, L Woike, Classification of consistent systems of handlebody group representations, Int. Math. Res. Not. 2024 (2024) 4767 MR4721656
90 L Müller, L Woike, The distinguished invertible object as ribbon dualizing object in the Drinfeld center, Selecta Math. 30 (2024) 98 MR4814800
91 N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1 MR1036112
92 N Reshetikhin, V G Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547 MR1091619
93 E Riehl, Categorical homotopy theory, 24, Cambridge Univ. Press (2014) MR3221774
94 J Roberts, Skeins and mapping class groups, Math. Proc. Cambridge Philos. Soc. 115 (1994) 53 MR1253282
95 I Runkel, String-net models for nonspherical pivotal fusion categories, J. Knot Theory Ramifications 29 (2020) 2050035 MR4125172
96 P Salvatore, N Wahl, Framed discs operads and Batalin–Vilkovisky algebras, Q. J. Math. 54 (2003) 213 MR1989873
97 C J Schommer-Pries, The classification of two-dimensional extended topological field theories, PhD thesis, University of California, Berkeley (2009) MR2713992
98 C Schommer-Pries, Invertible topological field theories, J. Topol. 17 (2024) MR4821355
99 O Schreier, Über die Erweiterung von Gruppen I, Monatsh. Math. Phys. 34 (1926) 165 MR1549403
100 C Schweigert, L Woike, Homotopy coherent mapping class group actions and excision for Hochschild complexes of modular categories, Adv. Math. 386 (2021) 107814 MR4270050
101 G Segal, Two-dimensional conformal field theories and modular functors, from: "IXth International Congress on Mathematical Physics", Hilger (1989) 22 MR1033753
102 K Shimizu, Non-degeneracy conditions for braided finite tensor categories, Adv. Math. 355 (2019) 106778 MR3996323
103 R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91 MR510404
104 U Tillmann, 𝒮-structures for k-linear categories and the definition of a modular functor, J. London Math. Soc. 58 (1998) 208 MR1670122
105 P Tingley, A minus sign that used to annoy me but now I know why it is there (two constructions of the Jones polynomial), from: "Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F R Jones’ 60th birthday" (editors S Morrison, D Penneys), Proc. Centre Math. Appl. Austral. Nat. Univ. 46, Austral. Nat. Univ. (2017) 415 MR3635680
106 V G Turaev, Modular categories and 3-manifold invariants, Internat. J. Modern Phys. B 6 (1992) 1807 MR1186845
107 V G Turaev, Quantum invariants of knots and 3-manifolds, 18, de Gruyter (1994) MR1292673
108 V Turaev, A Virelizier, On two approaches to 3-dimensional TQFTs, preprint (2010) arXiv:1006.3501
109 E Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nuclear Phys. B 300 (1988) 360 MR954762
110 N Wahl, Ribbon braids and related operads, PhD thesis, University of Oxford (2001)
111 K Walker, TQFTs, book draft (2006)
112 E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351 MR990772
113 S J Zetzsche, Generalised duality theory for monoidal categories and applications, master’s thesis, University of Hamburg (2018) arXiv:2301.10039