Skip to main content
SpringerLink
Log in

On the Extensions of the Left Modules for a Meromorphic Open-String Vertex Algebra, I

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the extensions of two left modules \(W_1, W_2\) for a meromorphic open-string vertex algebra V. We show that the extensions satisfying some technical but natural convergence conditions are in bijective correspondence to the first cohomology classes associated to the V-bimodule \({{\mathcal {H}}}_N(W_1, W_2)\) constructed in Huang and Qi (The first cohomology, derivations and the reductivity of a meromorphic open-string vertex algebra, Transactions of American Mathematical Society, 373 (2020)). When V is grading-restricted and contains a nice vertex subalgebra \(V_0\), those convergence conditions hold automatically. In addition, we show that the dimension of \(\text {Ext} ^1(W_1, W_2)\) is bounded above by the fusion rule \(N\left( {\begin{array}{c}W_2\\ VW_1\end{array}}\right) \) in the category of \(V_0\)-modules. In particular, if the fusion rule is finite, then \(\text {Ext} ^1(W_1, W_2)\) is finite-dimensional. We also give an example of an abelian category consisting of certain modules of the Virasoro VOA that does not contain any nice subalgebras, while the convergence conditions hold for every object.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Astashkevich, A.: On the structure of Verma modules over Virasoro and Neveu-Schwarz algebras. Commun. Math. Phys. 186, 531–562 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  2. Boe, B.D., Nakano, D.K., Wiesner, E.: Category \({\mathscr {O}}\) for the Virasoro algebra: cohomology and Koszulity. Pac. J. Math. 234(1), 1–21 (2008)

    Article  MathSciNet  Google Scholar 

  3. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  4. Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  5. Creutzig, T., Yang, J.: Tensor categories of affine Lie algebras beyond admissible level. Math. Ann. 380, 1991–2040 (2021)

    Article  MathSciNet  Google Scholar 

  6. Feign, B., Fuchs, D.: Representations of the Virasoro algebra, Representation of Lie groups and related topics, Advanced Study Contemp. Math., vol. 7, pp. 465–554. Gordon and Breach, New York (1990)

  7. Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebra and the monster. Pure and Appl. Math., vol. 134. Academic Press, New York (1988)

  8. Fiordalisi, F., Qi, F.: Fermionic constructions of the \({\mathbb{Z}}/{2}\)-graded meromorphic open-string vertex algebra and its \(\mathbb{Z}_2\)-twisted module, I. arXiv:2309.04620

  9. Huang, Y.-Z.: A cohomology theory of grading-restricted vertex algebras. Commun. Math. Phys. 327, 279–307 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  10. Huang, Y.-Z.: First and second cohomologies of grading-restricted vertex algebras. Commun. Math. Phys. 327, 261–278 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  11. Huang, Y.-Z., Kirillov, A., Jr., Lepowsky, J.: Braided tensor categories and extensions of vertex operator algebras. Commun. Math. Phys. 337, 1143–1159 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  12. Huang, Y.-Z., Lepowsky, J., Zhang, L.: Logarithmic tensor category theory, II: logarithmic formal calculus and properties of logarithmic intertwining operators. arXiv:1012.4196

  13. Huang, Y.-Z.: Meromorphic open string vertex algebras. J. Math. Phys. 54, 051702 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  14. Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and their Representations, Progress in Mathematics, vol. 227. Birkhäuser, Boston (2004)

    Google Scholar 

  15. Huang, Y.-Z., Qi, F.: The first cohomology, derivations and the reductivity of a (meromorphic open-string) vertex algebra. Transact. Amer. Math. Soc. 373 (2020)

  16. Iohara, K., Koga, Y.: Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics, Springer, London (2011)

    Book  Google Scholar 

  17. Moody, R.V., Pianzola, A.: Lie Algebras with Triangular Decompositions. Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, Hoboken (1995)

    Google Scholar 

  18. Qi, F.: On modules for meromorphic open-string vertex algebras. J. Math. Phys. 60, 031701 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  19. Qi, F.: On the cohomology of meromorphic open-string vertex algebras. N. Y. J. Math. 25, 467–517 (2019)

    MathSciNet  Google Scholar 

  20. Qi, F.: Representation theory and cohomology theory of meromorphic open-string vertex algebras. Ph.D. Thesis, Rutgers University (2018)

  21. Qi, F.: Meromorphic open-string vertex algebras and modules over two-dimensional orientable space forms. Lett. Math. Phys. 111, 1–54 (2021)

    Article  MathSciNet  Google Scholar 

  22. Qi, F.: First cohomologies of affine, Virasoro and lattice vertex operator algebras. Lett. Math. Phys. 112, 1–53 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  23. Qi, F.: On the extensions of the left modules for meromorphic open-string vertex algebra, II (in preparation)

  24. Qi, F.: Fermionic construction of the \(\frac{\mathbb{Z}}{2}\)-graded meromorphic open-string vertex algebra and its \({\mathbb{Z}}_2\)-twisted module, II. arXiv:2309.05673

Download references

Acknowledgements

I would like to thank Yi-Zhi Huang for his long-term constant support, especially for his encouragement in writing down the convergence results for the Virasoro VOA. I would also like to thank Kenji Iohara for answering my questions about the representation theory of Virasoro algebra and providing the idea for Theorem 5.14. Thanks also to Florencia Hunziker, Shashank Kanade, Andrew Linshaw, Jiayin Pan, and Eric Schippers for discussing various aspects of the current work. Finally, my deepest gratitude to the anonymous Reviewer 1, whose meticulous review and constructive suggestions greatly improved the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Denver, 2390 S York St Rm 216, Denver, 80210, USA

    Fei Qi

Authors
  1. Fei Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Qi.

Ethics declarations

Conflict of interest

The author states that there is no conflict of interest.

Additional information

Communicated by C. Schweigert.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, F. On the Extensions of the Left Modules for a Meromorphic Open-String Vertex Algebra, I. Commun. Math. Phys. 405, 65 (2024). https://doi.org/10.1007/s00220-023-04930-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00220-023-04930-3

Search

Navigation