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Conformal blocks from vertex algebras and their connections on $\overline{\mathcal{M}}_{g,n}$

Chiara Damiolini, Angela Gibney and Nicola Tarasca

Geometry & Topology 25 (2021) 2235–2286
Bibliography
1 T Abe, Rationality of the vertex operator algebra V L+ for a positive definite even lattice L, Math. Z. 249 (2005) 455 MR2115454
2 T Abe, C2–cofiniteness of the 2–cycle permutation orbifold models of minimal Virasoro vertex operator algebras, Comm. Math. Phys. 303 (2011) 825 MR2786218
3 T Abe, C2–cofiniteness of 2–cyclic permutation orbifold models, Comm. Math. Phys. 317 (2013) 425 MR3010190
4 T Abe, G Buhl, C Dong, Rationality, regularity, and C2–cofiniteness, Trans. Amer. Math. Soc. 356 (2004) 3391 MR2052955
5 T Abe, K Nagatomo, Finiteness of conformal blocks over compact Riemann surfaces, Osaka J. Math. 40 (2003) 375 MR1988696
6 D Adamović, A Milas, On the triplet vertex algebra 𝒲(p), Adv. Math. 217 (2008) 2664 MR2397463
7 C Ai, C2–cofiniteness of cyclic-orbifold vertex operator superalgebras, Algebra Colloq. 24 (2017) 315 MR3639037
8 V Alexeev, A Gibney, D Swinarski, Higher-level 𝔰𝔩2 conformal blocks divisors on 0,n, Proc. Edinb. Math. Soc. 57 (2014) 7 MR3165010
9 T Arakawa, A remark on the C2–cofiniteness condition on vertex algebras, Math. Z. 270 (2012) 559 MR2875849
10 T Arakawa, Associated varieties of modules over Kac–Moody algebras and C2–cofiniteness of W–algebras, Int. Math. Res. Not. 2015 (2015) 11605 MR3456698
11 T Arakawa, Rationality of W–algebras : principal nilpotent cases, Ann. of Math. 182 (2015) 565 MR3418525
12 M Arap, A Gibney, J Stankewicz, D Swinarski, 𝔰𝔩n level 1 conformal blocks divisors on 0,n, Int. Math. Res. Not. 2012 (2012) 1634 MR2913186
13 E Arbarello, C De Concini, V G Kac, C Procesi, Moduli spaces of curves and representation theory, Comm. Math. Phys. 117 (1988) 1 MR946992
14 M F Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957) 181 MR86359
15 B Bakalov, A Kirillov Jr., Lectures on tensor categories and modular functors, 21, Amer. Math. Soc. (2001) MR1797619
16 A Beauville, Y Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994) 385 MR1289330
17 A Beauville, Y Laszlo, C Sorger, The Picard group of the moduli of G–bundles on a curve, Compos. Math. 112 (1998) 183 MR1626025
18 A Beilinson, J Bernstein, A proof of Jantzen conjectures, from: "I M Gelfand seminar, I" (editors S Gelfand, S Gindikin), Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 1 MR1237825
19 A Beilinson, V Drinfeld, Chiral algebras, 51, Amer. Math. Soc. (2004) MR2058353
20 A Beilinson, B Feigin, B Mazur, Introduction to algebraic field theory on curves, unpublished manuscript (1991)
21 A Beilinson, D Kazhdan, Flat projective connections, preprint (1991)
22 A A Beilinson, V V Schechtman, Determinant bundles and Virasoro algebras, Comm. Math. Phys. 118 (1988) 651 MR962493
23 P Belkale, Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 for arbitrary Lie algebras, J. Math. Pures Appl. 98 (2012) 367 MR2968161
24 R E Borcherds, Vertex algebras, Kac–Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986) 3068 MR843307
25 G Codogni, Vertex algebras and Teichmüller modular forms, preprint (2019) arXiv:1901.03079
26 C Damiolini, A Gibney, On global generation of vector bundles on the moduli space of curves from representations of vertex operator algebras, preprint (2021) arXiv:2107.06923
27 C Damiolini, A Gibney, N Tarasca, On factorization and vector bundles of conformal blocks from vertex algebras, preprint (2019) arXiv:1909.04683
28 C Damiolini, A Gibney, N Tarasca, Vertex algebras of CohFT-type, preprint (2019) arXiv:1910.01658
29 C Dong, J Lepowsky, Generalized vertex algebras and relative vertex operators, 112, Birkhäuser (1993) MR1233387
30 C Dong, H Li, G Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997) 148 MR1488241
31 C Dong, H Li, G Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998) 571 MR1615132
32 C Dong, H Li, G Mason, Modular-invariance of trace functions in orbifold theory and generalized moonshine, Comm. Math. Phys. 214 (2000) 1 MR1794264
33 N Fakhruddin, Chern classes of conformal blocks, from: "Compact moduli spaces and vector bundles" (editors V Alexeev, A Gibney, E Izadi, J Kollár, E Looijenga), Contemp. Math. 564, Amer. Math. Soc. (2012) 145 MR2894632
34 F Falceto, K Gawędzki, A Kupiainen, Scalar product of current blocks in WZW theory, Phys. Lett. B 260 (1991) 101 MR1107911
35 G Faltings, Stable G–bundles and projective connections, J. Algebraic Geom. 2 (1993) 507 MR1211997
36 G Faltings, A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994) 347 MR1257326
37 G Felder, The KZB equations on Riemann surfaces, from: "Symétries quantiques" (editors A Connes, K Gawedzki, J Zinn-Justin), North-Holland (1998) 687 MR1616332
38 E Frenkel, D Ben-Zvi, Vertex algebras and algebraic curves, 88, Amer. Math. Soc. (2004) MR2082709
39 E Frenkel, V Kac, M Wakimoto, Characters and fusion rules for W–algebras via quantized Drinfeld–Sokolov reduction, Comm. Math. Phys. 147 (1992) 295 MR1174415
40 I B Frenkel, Y Z Huang, J Lepowsky, On axiomatic approaches to vertex operator algebras and modules, 494, Amer. Math. Soc. (1993) MR1142494
41 I B Frenkel, J Lepowsky, A Meurman, A natural representation of the Fischer–Griess Monster with the modular function J as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984) 3256 MR747596
42 I Frenkel, J Lepowsky, A Meurman, Vertex operator algebras and the Monster, 134, Academic (1988) MR996026
43 M R Gaberdiel, A Neitzke, Rationality, quasirationality and finite W–algebras, Comm. Math. Phys. 238 (2003) 305 MR1990879
44 K Gawędzki, Lectures on conformal field theory, from: "Quantum fields and strings: a course for mathematicians, II" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 727 MR1701610
45 K Gawędzki, A Kupiainen, SU(2) Chern–Simons theory at genus zero, Comm. Math. Phys. 135 (1991) 531 MR1091577
46 N Giansiracusa, A Gibney, The cone of type A, level 1, conformal blocks divisors, Adv. Math. 231 (2012) 798 MR2955192
47 A Gibney, D Jensen, H B Moon, D Swinarski, Veronese quotient models of M0,n and conformal blocks, Michigan Math. J. 62 (2013) 721 MR3160539
48 N J Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990) 347 MR1065677
49 G Höhn, Conformal designs based on vertex operator algebras, Adv. Math. 217 (2008) 2301 MR2388095
50 Y Z Huang, Two-dimensional conformal geometry and vertex operator algebras, 148, Birkhäuser (1997) MR1448404
51 Y Z Huang, Vertex operator algebras, the Verlinde conjecture, and modular tensor categories, Proc. Nat. Acad. Sci. U.S.A. 102 (2005) 5352 MR2140309
52 P Jitjankarn, G Yamskulna, C2–cofiniteness of the vertex algebra V L+ when L is a nondegenerate even lattice, Comm. Algebra 38 (2010) 4404 MR2764827
53 S Kobayashi, Differential geometry of complex vector bundles, 15, Princeton Univ. Press (1987) MR909698
54 M L Kontsevich, The Virasoro algebra and Teichmüller spaces, Funktsional. Anal. i Prilozhen. 21 (1987) 78 MR902301
55 S Kumar, M S Narasimhan, A Ramanathan, Infinite Grassmannians and moduli spaces of G–bundles, Math. Ann. 300 (1994) 41 MR1289830
56 Y Laszlo, Hitchin’s and WZW connections are the same, J. Differential Geom. 49 (1998) 547 MR1669720
57 J Lepowsky, H Li, Introduction to vertex operator algebras and their representations, 227, Birkhäuser (2004) MR2023933
58 H S Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996) 143 MR1387738
59 E Looijenga, From WZW models to modular functors, from: "Handbook of moduli, II" (editors G Farkas, I Morrison), Adv. Lect. Math. 25, International (2013) 427 MR3184182
60 A Marian, D Oprea, R Pandharipande, The first Chern class of the Verlinde bundles, from: "String–Math 2012" (editors R Donagi, S Katz, A Klemm, D R Morrison), Proc. Sympos. Pure Math. 90, Amer. Math. Soc. (2015) 87 MR3409789
61 A Marian, D Oprea, R Pandharipande, A Pixton, D Zvonkine, The Chern character of the Verlinde bundle over g,n, J. Reine Angew. Math. 732 (2017) 147 MR3717090
62 M Miyamoto, Modular invariance of vertex operator algebras satisfying C2–cofiniteness, Duke Math. J. 122 (2004) 51 MR2046807
63 M Miyamoto, A 3–orbifold theory of lattice vertex operator algebra and 3–orbifold constructions, from: "Symmetries, integrable systems and representations" (editors K Iohara, S Morier-Genoud, B Rémy), Springer Proc. Math. Stat. 40, Springer (2013) 319 MR3077690
64 K Nagatomo, A Tsuchiya, Conformal field theories associated to regular chiral vertex operator algebras, I : Theories over the projective line, Duke Math. J. 128 (2005) 393 MR2145740
65 C Pauly, Espaces de modules de fibrés paraboliques et blocs conformes, Duke Math. J. 84 (1996) 217 MR1394754
66 T R Ramadas, The “Harder–Narasimhan trace” and unitarity of the KZ/Hitchin connection : genus 0, Ann. of Math. 169 (2009) 1 MR2480600
67 G Segal, The definition of conformal field theory, from: "Topology, geometry and quantum field theory" (editor U Tillmann), Lond. Math. Soc. Lect. Note Ser. 308, Cambridge Univ. Press (2004) 421 MR2079383
68 C Sorger, La formule de Verlinde, from: "Séminaire Bourbaki, 1994/95", Astérisque 237, Soc. Math. France (1996) 87 MR1423621
69 Y Tsuchimoto, On the coordinate-free description of the conformal blocks, J. Math. Kyoto Univ. 33 (1993) 29 MR1203889
70 A Tsuchiya, Y Kanie, Vertex operators in the conformal field theory on 1 and monodromy representations of the braid group, Lett. Math. Phys. 13 (1987) 303 MR895293
71 A Tsuchiya, K Ueno, Y Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, from: "Integrable systems in quantum field theory and statistical mechanics" (editors M Jimbo, T Miwa, A Tsuchiya), Adv. Stud. Pure Math. 19, Academic (1989) 459 MR1048605
72 K Ueno, On conformal field theory, from: "Vector bundles in algebraic geometry" (editors N J Hitchin, P E Newstead, W M Oxbury), Lond. Math. Soc. Lect. Note Ser. 208, Cambridge Univ. Press (1995) 283 MR1338420
73 G Yamskulna, C2–cofiniteness of the vertex operator algebra V L+ when L is a rank one lattice, Comm. Algebra 32 (2004) 927 MR2063790
74 Y Zhu, Global vertex operators on Riemann surfaces, Comm. Math. Phys. 165 (1994) 485 MR1301621
75 Y Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996) 237 MR1317233