Associative algebras for (logarithmic) twisted modules for a vertex operator algebra

Huang, Yi-Zhi; Yang, Jinwei

doi:10.1090/tran/7490

Associative algebras for (logarithmic) twisted modules for a vertex operator algebra
HTML articles powered by AMS MathViewer

by Yi-Zhi Huang and Jinwei Yang PDF

Trans. Amer. Math. Soc. 371 (2019), 3747-3786 Request permission

Abstract:

We construct two associative algebras from a vertex operator algebra $V$ and a general automorphism $g$ of $V$. The first, called a $g$-twisted zero-mode algebra, is a subquotient of what we call a $g$-twisted universal enveloping algebra of $V$. These algebras are generalizations of the corresponding algebras introduced and studied by Frenkel-Zhu and Nagatomo-Tsuchiya in the (untwisted) case that $g$ is the identity. The other is a generalization of the $g$-twisted version of Zhu’s algebra for suitable $g$-twisted modules constructed by Dong-Li-Mason when the order of $g$ is finite. We are mainly interested in $g$-twisted $V$-modules introduced by the first author in the case that $g$ is of infinite order and does not act on $V$ semisimply. In this case, twisted vertex operators in general involve the logarithm of the variable. We construct functors between categories of suitable modules for these associative algebras and categories of suitable (logarithmic) $g$-twisted $V$-modules. Using these functors, we prove that the $g$-twisted zero-mode algebra and the $g$-twisted generalization of Zhu’s algebra are in fact isomorphic.

References

Dražen Adamović and Antun Milas, Lattice construction of logarithmic modules for certain vertex algebras, Selecta Math. (N.S.) 15 (2009), no. 4, 535–561. MR 2565050, DOI 10.1007/s00029-009-0009-z
Bojko Bakalov, Twisted logarithmic modules of vertex algebras, Comm. Math. Phys. 345 (2016), no. 1, 355–383. MR 3509017, DOI 10.1007/s00220-015-2503-9
Bojko Bakalov and McKay Sullivan, Twisted logarithmic modules of free field algebras, J. Math. Phys. 57 (2016), no. 6, 061701, 18. MR 3510308, DOI 10.1063/1.4953249
David Brungs and Werner Nahm, The associative algebras of conformal field theory, Lett. Math. Phys. 47 (1999), no. 4, 379–383. MR 1693759, DOI 10.1023/A:1007525300192
Chongying Dong, Haisheng Li, and Geoffrey Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), no. 3, 571–600. MR 1615132, DOI 10.1007/s002080050161
Chongying Dong, Haisheng Li, and Geoffrey Mason, Vertex operator algebras and associative algebras, J. Algebra 206 (1998), no. 1, 67–96. MR 1637252, DOI 10.1006/jabr.1998.7425
Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
Igor B. Frenkel and Yongchang Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), no. 1, 123–168. MR 1159433, DOI 10.1215/S0012-7094-92-06604-X
Yi-Zhi Huang, Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N.S.) 6 (2000), no. 3, 225–267. MR 1817614, DOI 10.1007/PL00001389
Yi-Zhi Huang, Differential equations, duality and modular invariance, Commun. Contemp. Math. 7 (2005), no. 5, 649–706. MR 2175093, DOI 10.1142/S021919970500191X
Yi-Zhi Huang, Generalized twisted modules associated to general automorphisms of a vertex operator algebra, Comm. Math. Phys. 298 (2010), no. 1, 265–292. MR 2657819, DOI 10.1007/s00220-010-0999-6
Y.-Z. Huang, J. Lepowsky, and L. Zhang, Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition, to appear; arXiv:1012.4199.
Yi-Zhi Huang and Jinwei Yang, On functors between module categories for associative algebras and for $\Bbb {N}$-graded vertex algebras, J. Algebra 409 (2014), 344–361. MR 3198845, DOI 10.1016/j.jalgebra.2014.04.004
Kiyokazu Nagatomo and Akihiro Tsuchiya, Conformal field theories associated to regular chiral vertex operator algebras. I. Theories over the projective line, Duke Math. J. 128 (2005), no. 3, 393–471. MR 2145740, DOI 10.1215/S0012-7094-04-12831-3
V. Turaev, Homotopy field theory in dimension 3 and crossed group-categories, arxiv:math.GT/0005291. 2000.
Jinwei Yang, Twisted representations of vertex operator algebras associated to affine Lie algebras, J. Algebra 484 (2017), 88–108. MR 3656714, DOI 10.1016/j.jalgebra.2017.03.041
Yongchang Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237–302. MR 1317233, DOI 10.1090/S0894-0347-96-00182-8