Abstract
Attached to a vertex algebra \({\mathcal V}\) are two geometric objects. The associated scheme of \({\mathcal V}\) is the spectrum of Zhu’s Poisson algebra \(R_{{\mathcal V}}\). The singular support of \({\mathcal V}\) is the spectrum of the associated graded algebra \(\text{gr}({\mathcal V})\) with respect to Li’s canonical decreasing filtration. There is a closed embedding from the singular support to the arc space of the associated scheme, which is an isomorphism in many interesting cases. In this note we give an example of a non-quasi-lisse vertex algebra whose associated scheme is reduced, for which the isomorphism is not true as schemes but true as varieties.
Dedicated to Professor Anthony Joseph on his seventy-fifth birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Provided that \({\mathcal V}\) is finitely strongly generated.
- 2.
Unless \({\mathcal V}\) is finite-dimensional.
References
D. Adamovic, Classification of irreducible modules of certain subalgebras of free boson vertex algebra, Journal of Algebra 270 (2003) 115–132.
D. Adamovic and A. Milas, On the triplet vertex algebra \({\mathcal W}(p)\), Adv. Math. 217 (2008), no. 6, 2664–2699.
T. Arakawa, A remark on the C 2 cofiniteness condition on vertex algebras, Math. Z. 270 (2012), no. 1–2, 559–575.
T. Arakawa, Associated Varieties and Higgs Branches (A Survey), Contemp. Math. 711 (2018), 37–44.
T. Arakawa, Representation theory of W-algebras and Higgs branch conjecture, Proc. Int. Cong. of Math 2018 Rio de Janeiro, Vol. 1 (1261–1278).
T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, In: V. G. Kac, V. L. Popov (eds.), Lie Groups, Geometry, and Representation Theory, A Tribute to the Life and Work of Bertram Kostant, Progr. Math., 326, Birkhauser, 2018.
T. Arakawa and A. Moreau, Lectures on \({\mathcal W}\) -algebras, preprint.
T. Arakawa and A. Moreau, Arc spaces and chiral symplectic cores, to appear in the special issue of Publ. Res. Inst. Math. in honor of Professor Masaki Kashiwara’s 70th birthday.
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, Infinite chiral symmetry in four dimensions, Comm. Math. Phys., 336(3):1359–1433, 2015.
C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, J. High Energ. Phys. (2018) 2018:114. https://doi.org/10.1007/JHEP08(2018)114
R. Borcherds, Vertex operator algebras, Kac-Moody algebras and the monster, Proc. Nat. Acad. Sci. USA 83 (1986) 3068–3071.
D. Bourqui and J. Sebag, The radical of the differential ideal generated by XY in the ring of two variable differential polynomials is not differentially finitely generated, to appear Journal of Comm. Algebra (2017).
A. de Sole and V. Kac, Freely generated vertex algebras and non-linear Lie conformal algebras, Comm. Math. Phys. 254 (2005), no. 3, 659–694.
C. Dong and K. Nagatomo, Classification of irreducible modules for the vertex operator algebra M(1)+, J. Algebra 216 (1999), no. 1, 384–404.
L. Ein and M. Mustata, Jet schemes and singularities, Algebraic geometry—Seattle 2005. Part 2, 505–546, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
J. van Ekeren and R. Heluani, Chiral homology of elliptic curves and Zhu’s algebra, arXiv:1804.00017 [math.QA].
E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Math. Surveys and Monographs, Vol. 88, American Math. Soc., 2001.
I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, New York, 1988.
V. Kac, Vertex Algebras for Beginners, University Lecture Series, Vol. 10. American Math. Soc., 1998
K. Kpognon, J. Sebag, Nilpotency in arc schemes of plane curves, Comm. in Algebra, Vol. 45 no 5 (2017), 2195–2221
E. Kolchin, Differential algebra and algebraic groups, Academic Press, New York 1973.
H. Li, Vertex algebras and vertex Poisson algebras, Commun. Contemp. Math. 6 (2004) 61–110.
M. Miyamoto, Modular invariance of vertex operator algebras satisfying C2-cofiniteness. Duke Math. J. 122(1), 51–91 (2004).
J. Sebag, Arcs schemes, derivations and Lipman’s theorem, J. Algebra 347 (2011) 173–183.
J. Sebag, A remark on Berger’s conjecture, Kolchin’s theorem and arc schemes, Archiv der Math., Vol. 108 no (2017), 145–150
W. Wang, \({\mathcal W}_{1+\infty }\) algebra, \({\mathcal W}_3\) algebra, and Friedan-Martinec-Shenker bosonization, Comm. Math. Phys. 195 (1998), no. 1, 95–111.
W. Wang, Classification of irreducible modules of \({\mathcal W}_3\) algebra with c = −2, Comm. Math. Phys. 195 (1998), no. 1, 113–128.
A.B. Zamolodchikov, Infinite extra symmetries in two-dimensional conformal quantum field theory (Russian), Teoret. Mat. Fiz. 65 (1985), 347–359. English translation, Theoret. and Math. Phys. 65 (1985), 1205–1213.
Y. Zhu, Modular invariants of characters of vertex operators, J. Amer. Soc. 9 (1996) 237–302.
Acknowledgements
T. A. is supported by JSPS KAKENHI Grants #17H01086 and #17K18724. A. L. is supported by Simons Foundation Grant #318755. We thank Julien Sebag for helpful comments on an earlier draft of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Arakawa, T., Linshaw, A.R. (2019). Singular Support of a Vertex Algebra and the Arc Space of Its Associated Scheme. In: Gorelik, M., Hinich, V., Melnikov, A. (eds) Representations and Nilpotent Orbits of Lie Algebraic Systems. Progress in Mathematics, vol 330. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23531-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-23531-4_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-23530-7
Online ISBN: 978-3-030-23531-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)