×
Damiolini, Chiara; Gibney, Angela; Tarasca, Nicola

Conformal blocks from vertex algebras and their connections on \(\overline{\mathcal{M}}_{g,n}\). (English) Zbl 1484.14011

Geom. Topol. 25, No. 5, 2235-2286 (2021).
Summary: We show that coinvariants of modules over vertex operator algebras give rise to quasicoherent sheaves on moduli of stable pointed curves. These generalize Verlinde bundles or vector bundles of conformal blocks defined using affine Lie algebras studied first by A. Tsuchiya and Y. Kanie [Lett. Math. Phys. 13, 303–312 (1987; Zbl 0631.17010)], K. Ueno [Lond. Math. Soc. Lect. Note Ser. 208, 283–345 (1995; Zbl 0846.17027)], and extend work of others. The sheaves carry a twisted logarithmic \(\mathcal{D}\)-module structure, and hence support a projectively flat connection. We identify the logarithmic Atiyah algebra acting on them, generalizing work of Tsuchimoto for affine Lie algebras.

Cited in 2 Reviews
Cited in 6 Documents

MSC:

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14H10 Families, moduli of curves (algebraic)
17B69 Vertex operators; vertex operator algebras and related structures
16D90 Module categories in associative algebras
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Keywords:

vertex algebras; conformal blocks and coinvariants; connections and Atiyah algebras; sheaves on moduli of curves; Chern classes of vector bundles on moduli of curves

Citations:

Zbl 0631.17010; Zbl 0846.17027
Cite Review PDF
Full Text: DOI arXiv

References:

[1] 10.1007/s00209-004-0709-1 · Zbl 1136.17021 · doi:10.1007/s00209-004-0709-1
[2] 10.1007/s00220-011-1209-x · Zbl 1261.17026 · doi:10.1007/s00220-011-1209-x
[3] 10.1007/s00220-012-1618-5 · Zbl 1290.17024 · doi:10.1007/s00220-012-1618-5
[4] 10.1090/S0002-9947-03-03413-5 · Zbl 1070.17011 · doi:10.1090/S0002-9947-03-03413-5
[5] ; Abe, Osaka J. Math., 40, 375 (2003) · Zbl 1052.17013
[6] 10.1016/j.aim.2007.11.012 · Zbl 1177.17017 · doi:10.1016/j.aim.2007.11.012
[7] 10.1142/S1005386717000190 · Zbl 1416.17016 · doi:10.1142/S1005386717000190
[8] 10.1017/S0013091513000941 · Zbl 1285.14012 · doi:10.1017/S0013091513000941
[9] 10.1007/s00209-010-0812-4 · Zbl 1351.17031 · doi:10.1007/s00209-010-0812-4
[10] 10.1093/imrn/rnu277 · Zbl 1335.17011 · doi:10.1093/imrn/rnu277
[11] 10.4007/annals.2015.182.2.4 · Zbl 1394.17058 · doi:10.4007/annals.2015.182.2.4
[12] 10.1093/imrn/rnr064 · Zbl 1271.14034 · doi:10.1093/imrn/rnr064
[13] 10.1007/BF01228409 · Zbl 0647.17010 · doi:10.1007/BF01228409
[14] 10.2307/1992969 · Zbl 0078.16002 · doi:10.2307/1992969
[15] 10.1090/ulect/021 · doi:10.1090/ulect/021
[16] 10.1007/BF02101707 · Zbl 0815.14015 · doi:10.1007/BF02101707
[17] 10.1023/A:1000477122220 · Zbl 0976.14024 · doi:10.1023/A:1000477122220
[18] ; Beilinson, I M Gelfand seminar, I. Adv. Soviet Math., 16, 1 (1993) · Zbl 0790.22007
[19] 10.1090/coll/051 · doi:10.1090/coll/051
[20] 10.1007/BF01221114 · Zbl 0665.17010 · doi:10.1007/BF01221114
[21] 10.1016/j.matpur.2012.02.008 · Zbl 1277.14035 · doi:10.1016/j.matpur.2012.02.008
[22] 10.1073/pnas.83.10.3068 · Zbl 0613.17012 · doi:10.1073/pnas.83.10.3068
[23] 10.1007/978-1-4612-0353-7 · doi:10.1007/978-1-4612-0353-7
[24] 10.1006/aima.1997.1681 · Zbl 0902.17014 · doi:10.1006/aima.1997.1681
[25] 10.1007/s002080050161 · Zbl 0890.17029 · doi:10.1007/s002080050161
[26] 10.1007/s002200000242 · Zbl 1061.17025 · doi:10.1007/s002200000242
[27] 10.1090/conm/564/11148 · doi:10.1090/conm/564/11148
[28] 10.1016/0370-2693(91)90975-V · doi:10.1016/0370-2693(91)90975-V
[29] ; Faltings, J. Algebraic Geom., 2, 507 (1993) · Zbl 0790.14019
[30] ; Faltings, J. Algebraic Geom., 3, 347 (1994) · Zbl 0809.14009
[31] ; Felder, Symétries quantiques, 687 (1998)
[32] 10.1090/surv/088 · doi:10.1090/surv/088
[33] 10.1007/BF02096589 · Zbl 0768.17008 · doi:10.1007/BF02096589
[34] 10.1090/memo/0494 · Zbl 0789.17022 · doi:10.1090/memo/0494
[35] 10.1073/pnas.81.10.3256 · Zbl 0543.20016 · doi:10.1073/pnas.81.10.3256
[36] ; Frenkel, Vertex operator algebras and the Monster. Pure Appl. Math., 134 (1988) · Zbl 0674.17001
[37] 10.1007/s00220-003-0845-1 · Zbl 1042.17025 · doi:10.1007/s00220-003-0845-1
[38] 10.1007/s002200050573 · Zbl 0938.37048 · doi:10.1007/s002200050573
[39] 10.1007/BF02104120 · Zbl 0722.53084 · doi:10.1007/BF02104120
[40] 10.1016/j.aim.2012.05.017 · Zbl 1316.14051 · doi:10.1016/j.aim.2012.05.017
[41] 10.1307/mmj/1387226162 · Zbl 1287.14011 · doi:10.1307/mmj/1387226162
[42] 10.1007/BF02161419 · Zbl 0718.53021 · doi:10.1007/BF02161419
[43] 10.1016/j.aim.2007.11.003 · Zbl 1157.17008 · doi:10.1016/j.aim.2007.11.003
[44] 10.1007/978-1-4612-4276-5 · doi:10.1007/978-1-4612-4276-5
[45] 10.1073/pnas.0409901102 · Zbl 1112.17029 · doi:10.1073/pnas.0409901102
[46] 10.1080/00927870903386478 · Zbl 1269.17012 · doi:10.1080/00927870903386478
[47] 10.1515/9781400858682 · doi:10.1515/9781400858682
[48] ; Kontsevich, Funktsional. Anal. i Prilozhen., 21, 78 (1987) · Zbl 0647.58012
[49] 10.1007/BF01450475 · Zbl 0803.14012 · doi:10.1007/BF01450475
[50] 10.4310/jdg/1214461110 · Zbl 0987.14027 · doi:10.4310/jdg/1214461110
[51] 10.1007/978-0-8176-8186-9 · doi:10.1007/978-0-8176-8186-9
[52] 10.1016/0022-4049(95)00079-8 · Zbl 0854.17035 · doi:10.1016/0022-4049(95)00079-8
[53] ; Looijenga, Handbook of moduli, II. Adv. Lect. Math., 25, 427 (2013)
[54] 10.1090/pspum/090/01521 · Zbl 1356.14022 · doi:10.1090/pspum/090/01521
[55] 10.1515/crelle-2015-0003 · Zbl 1453.14138 · doi:10.1515/crelle-2015-0003
[56] 10.1215/S0012-7094-04-12212-2 · Zbl 1165.17311 · doi:10.1215/S0012-7094-04-12212-2
[57] 10.1007/978-1-4471-4863-0_13 · Zbl 1287.17049 · doi:10.1007/978-1-4471-4863-0_13
[58] 10.1215/S0012-7094-04-12831-3 · Zbl 1074.81065 · doi:10.1215/S0012-7094-04-12831-3
[59] 10.1215/S0012-7094-96-08408-2 · Zbl 0877.14031 · doi:10.1215/S0012-7094-96-08408-2
[60] 10.4007/annals.2009.169.1 · Zbl 1167.32011 · doi:10.4007/annals.2009.169.1
[61] ; Segal, Topology, geometry and quantum field theory. Lond. Math. Soc. Lect. Note Ser., 308, 421 (2004)
[62] ; Sorger, Séminaire Bourbaki, 1994/95. Astérisque, 237, 87 (1996) · Zbl 0878.17024
[63] 10.1215/kjm/1250519338 · Zbl 0790.14022 · doi:10.1215/kjm/1250519338
[64] 10.1007/BF00401159 · Zbl 0631.17010 · doi:10.1007/BF00401159
[65] 10.2969/aspm/01910459 · doi:10.2969/aspm/01910459
[66] 10.1017/CBO9780511569319.011 · doi:10.1017/CBO9780511569319.011
[67] 10.1081/AGB-120027959 · Zbl 1093.17008 · doi:10.1081/AGB-120027959
[68] 10.1007/BF02099421 · Zbl 0819.17019 · doi:10.1007/BF02099421
[69] 10.1090/S0894-0347-96-00182-8 · Zbl 0854.17034 · doi:10.1090/S0894-0347-96-00182-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.