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LATTICE VERTEX ALGEBRAS AND LOOP GRASSMANNIANS

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For an integral symmetric matrix κ we construct a new “nonabelian homology localization” of the lattice vertex algebra Lκ on the corresponding loop Grassmannian space 𝒢κ. We attempt to motivate our construction by presenting related topics in the language of “interaction of particles over algebraic curves”.

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References

  1. E. Artin, J. Tate, Class Field Theory, Addison-Wesley, Redwood City, Ca., 1990.

  2. L. Bandklyder, A proof of the Dold-Thom theorem via factorization homology, ar**v:math/1703.09170 (2017).

  3. A. Beilinson, J. Bernstein, Localisation de 𝔤-modules C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18.

  4. A. Beilinson, S. Bloch, H. Esnault, ε-factors for Gauss-Manin determinants, ar**v:math/0111277v3 (2003).

  5. A. Beilinson, V. Drinfeld, Chiral Algebras, Colloquium Publications, Vol. 51, American Mathematical Society, Providence, RI, 2004.

  6. T. Barnett-Lamb, The Dold-Thom theorem, available at https://people.brandeis.edu/~tbl/dold-thom.pdf.

  7. Bezrukavnikov, R., Mirković, I., Rumynin, D.: Localization of modules for a semi-simple Lie algebra in prime characteristic. Ann. of Math. 167(3), 945–991 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves, 2nd edn. American Mathematical Society, Providence, RI (2004)

    Book  MATH  Google Scholar 

  9. Contou-Carrère, C.E.: Corps de classes local geometrique relatif. C. R. Acad. Sci. Paris Sér. I Math. 292(9), 481–484 (1981)

    MathSciNet  MATH  Google Scholar 

  10. Contou-Carrère, C.E.: Jacobienne locale d’une courbe formelle relative. Rend. Sem. Mat. Univ. Padova. 130, 1–106 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. V. Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, ar**v: math/0309155v4 (2004).

  12. M. Finkelberg, I. Mirković, Semiinfinite flags. I. Case of the global curve1, arxiv.org/abs/alg-geom/9707010v2 (1998).

  13. Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62, 23–66 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. E. M. Friedlander, Motivic complexes of Suslin and Voevodsky, Astérisque 245 (1997), Séminaire Bourbaki, exp. no. 833, 355–378.

  15. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  16. J. Hilburn, S. Raskin, Tate’s thesis in the de Rham setting, ar**v:2107.11325 (2021).

  17. V. Kac, Vertex Algebras for Beginners, 2nd Edn., American Mathematical Society, Providence, RI, 1998.

  18. G. Laumon, Transformation de Fourier generalisee, ar**v:alg-geom/9603004 (1996).

  19. Mazur, B.: Notes on étale cohomology of number fields. Ann. Sci. École Norm. Sup. 6(4), 521–552 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Mirković, Loop Grassmannians in the framework of local spaces over a curve, in: Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, Contemp. Math. 623, American Mathematical Society, Providence, RI, 2014, pp. 215–226.

  21. I. Mirković, Some extensions of the notion of loop Grassmannians, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić issue 532 (2017), 53–74.

  22. I. Mirković, K. Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. 166 (2007), no. 1, 95–143.

  23. I. Mirković, Y. Yang, G. Zhao, Loop Grassmannians of quivers and affine quantum groups, accepted for the conference volume for Alexander Beilinson and Victor Ginzburg.

  24. Serre, J.-P.: Groupes Algébriques et Corps de Classes. Hermann, Paris (1959)

    MATH  Google Scholar 

  25. J. Tao, Y. Zhao, Extensions by K2 and factorization line bundles, ar**v:1901. 08760 (2019).

  26. V. Voevodsky, The 𝔸1-homotopy theory, Proc. ICM, Vol. 1, 1998, pp. 579–604.

  27. Zhu, X.: Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian. Adv. Math. 221(2), 570–600 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Ivan Mirković.

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Mirković, I. LATTICE VERTEX ALGEBRAS AND LOOP GRASSMANNIANS. Transformation Groups 28, 1221–1243 (2023). https://doi.org/10.1007/s00031-023-09821-4

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